On Instanton Contributions to Anomalous Dimensions in N = 4 Supersymmetric Yang–Mills Theory

Nuclear Physics B 684 (2004) 3–74 www.elsevier.com/locate/npe

On instanton contributions to anomalous dimensions in N = 4 supersymmetric Yang–Mills theory

Stefano Kovacs

Max-Planck Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg, 1-14476 Golm, Germany Received 12 January 2004; accepted 11 February 2004

Abstract Instanton contributions to the anomalous dimensions of gauge-invariant composite operators in the N = 4 supersymmetric SU(N) Yang–Mills theory are studied in the one-instanton sector. Independent sets of scalar operators of bare dimension ∆0 5 are constructed in all the allowed representations of the SU(4) R-symmetry group and their two-point functions are computed in the semiclassical approximation. Analysing the moduli space integrals the sectors in which the scaling dimensions receive non-perturbative contributions are identiﬁed. The requirement that the integrations over the fermionic collective coordinates which arise in the instanton background are saturated leads to non-renormalisation properties for a large class of operators. Instanton-induced corrections to the scaling dimensions are found only for dimension 4 SU(4) singlets and for dimension 5 operators in the 6 of SU(4). In many cases the non-renormalisation results are argued to be speciﬁc to operators of small dimension, but for some special sectors it is shown that they are valid for arbitrary dimension. Comments are also made on the implications of the results on the form of the instanton contributions to the dilation operator of the theory and on the possibility of realising its action on the instanton moduli space.  2004 Elsevier B.V. All rights reserved.

PACS: 11.15.Tk; 11.30.Pb; 12.60.Jv

1. Introduction and summary of results

Recent progress in the study of the interconnections between string and gauge theories has revived the interest for the computation of anomalous dimensions in the N = 4 supersymmetric Yang–Mills (SYM) theory. In a conformal ﬁeld theory the spectrum of

E-mail address: [email protected] (S. Kovacs).

0550-3213/$ – see front matter  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2004.02.014 4 S. Kovacs / Nuclear Physics B 684 (2004) 3–74 scaling dimensions, together with the set of OPE coefficients for a complete basis of operators, constitutes the fundamental physical information. In the original formulation of the AdS/CFT correspondence [1–3], which in its best understood form relates the N = 4 5 SYM theory with SU(N) gauge group to type IIB string theory in AdS5 ×S , the possibility of comparing this physical information with the dual string theory quantities is limited by the present inability of quantising strings in the relevant background. In the correspondence scaling dimensions of gauge-invariant composite operators are related to the energy of the dual string/supergravity states. The parameters of the string and gauge theories are related by L2 2 4πg = g2 , = g2 N ≡ λ, (1) s YM α YM which imply the strong/weak coupling nature of the duality: in the limit in which classical supergravity is a good approximation, which is the only limit in which string theory in 5 AdS5 × S is under control, the gauge theory is strongly coupled. This makes a direct comparison of quantities calculated at weak coupling in field theory with quantities computed in supergravity impossible. This is true in particular for the comparison of field theory scaling dimensions and energies of supergravity states. In the early days of the correspondence anomalous dimensions first emerged in connection with logarithmic singularities in the short distance limit of four-point functions [4]. The singularities in the so-called conformal partial wave expansion of four-point amplitudes in supergravity [5] are interpreted as due to quantum corrections to the mass of states exchanged in intermediate channels. A similar analysis of the operator product expansion (OPE) of the dual four-point correlation functions in field theory displays the same type of singularities [6], which in turn are attributed to anomalous dimensions of operators entering the OPE [4]. Although the physical mechanisms which are responsible for the observed divergences are correctly identified, a quantitative comparison is impossible because the two calculations have different regimes of validity. This is a general feature and until recently, in spite of what appear to be non-trivial tests of the duality, relatively little insight into truly dynamical aspects had been obtained. An important step forward was made with the proposal of [7] where a limit of the AdS/CFT correspondence was identified in which the comparison between string theory and gauge theory can be carried on avoiding the strong/weak coupling problem. On the string side one considers a maximally supersymmetric plane-wave background obtained 5 as Penrose limit of AdS5 × S [8]. Although there is a non-trivial R ⊗ R condensate, the quantisation of string theory in this background is feasible because the world sheet theory in the light-cone gauge reduces to a free massive model [9]. The dual of string theory in this background was identified in [7] as a particular sector of N = 4 SYM made of operators of large scaling dimension, ∆, and with large value of the charge, J , with respect to one of the R-symmetry generators. The holographic nature of the duality in this limit, referred to as BMN limit, is not well understood, but a prescription for comparing the mass spectrum of the string with the spectrum of dimensions of gauge theory operators has been given. Using this prescription a remarkable agreement between string and field theory results has been obtained in perturbation theory [10–12]. What is particularly interesting in the BMN correspondence is that the comparison can be extended beyond supergravity S. Kovacs / Nuclear Physics B 684 (2004) 3–74 5 to include massive string modes and moreover it can be carried on in a quantitative way. This is because the duality arises in a double scaling limit in which the rank of the gauge group is sent to infinity and at the same time J →∞,insuchawaythatJ 2/N is kept fixed. In this limit and in the BMN sector of the gauge theory the relevant parameters are = 2 2 = 2 λ gYMN/J and g2 J /N [13,14]. The dual string theory can also be formulated in terms of the same parameters and in the BMN limit on both sides one can consider an expansion for small values of λ and g2. Another new idea, which was suggested in [15] extending considerations in [16], has lead to further exciting developments. What was noticed in these papers is that in certain limits, in which some quantum numbers take large values, the semiclassical 5 description of the AdS5 × S string sigma model can represent a good approximation. These concepts have been employed in several recent papers [17–25] in which solitonic 5 solutions of the sigma-model, corresponding to strings rotating in AdS5 × S , have been put in correspondence with gauge theory operators. The comparison of the energy of such solitonic string configurations with the anomalous dimensions of the dual operators has opened the possibility of new quantitative tests of the AdS/CFT correspondence, which appear to have a dynamical nature. Very accurate comparisons of the spectra on the two sides of the correspondence have been performed. An important rôle in these calculations in the gauge theory is played by the observation that the problem of computing anomalous dimensions can be rephrased as an eigenvalue problem for the dilation operator of the theory [26–29]. When the problem is recast in these terms the structure of an integrable system emerges in the planar limit [27,30]. This allows to apply the techniques of the Bethe ansatz to the computation of anomalous dimensions. The integrability emerging in the planar limit of the N = 4 theory appears to play a crucial rôle in these recent tests of the AdS/CFT duality and is believed to have a counterpart in the integrability of the free 5 string theory in AdS5 × S [31–34]. All the recent developments mentioned here concern the perturbative sector of the gauge theory and very little has been done in the study of quantum corrections to anomalous dimensions beyond perturbation theory. One of the purposes of this paper is to initiate a systematic analysis of the non-perturbative instanton-induced corrections to the spectrum of anomalous dimensions in N = 4 SYM. A fundamental motivation for this comes from the observation that instantons are expected to play a special rôle in N = 4 SYM due to the S-duality of the theory. The study of instanton effects in the AdS/CFT correspondence provides an example of a situation in which the agreement between string and gauge theory results appears highly non-trivial already at the level of supergravity. Here the duality relates effects of instantons in N = 4 SYM and D-instantons in type IIB string theory [35]. Instanton corrections to correlation functions have a counterpart in contributions to string/supergravity amplitudes induced by the presence of D-instantons. In the large-N limit multi-instanton effects in N = 4 SYM can be explicitly computed and compared with AdS amplitudes involving certain D-instanton induced vertices in the type IIB effective action. Such vertices appear in the α expansion, i.e., the low energy derivative expansion, of type IIB supergravity and their structure is determined by supersymmetry and S-duality constraints. For a large class of processes a striking agreement between supergravity multi-particle amplitudes involving the leading D-instanton vertices and multi-instanton contributions to the dual 6 S. Kovacs / Nuclear Physics B 684 (2004) 3–74

Yang–Mills correlation functions was found in [36], generalising the results of [35,37]. The correspondence has been further extended to include higher-order D-instanton terms in [38]. What makes the result of the comparison remarkable is that perfect agreement is found in the functional form of correlation functions which depend non-trivially on the coupling. The non-perturbative results just described refer to a class of four- and higher-point correlation functions of protected operators. All the cases analysed in the past involve 1/2 BPS operators, which are dual to supergravity states and their Kaluza–Klein excited modes. In view of the success in the comparison with D-instanton effects at the level of supergravity on the one hand and of the recent progress in matching the gauge and string theory spectra at the level of massive string excitations on the other, it is interesting to study instanton contributions to correlation functions of operators dual to massive string modes. In the present paper we shall consider two-point functions of gauge-invariant scalar operators of bare dimension ∆0 5. These include BPS operators dual to supergravity fields, multi-trace operators dual to multi-particle bound states as well as single-trace operators in long multiplets dual to massive excitations of type IIB string theory. As is well known, two-point functions in a conformal field theory like N = 4 supersymmetric Yang–Mills encode the information about scaling dimensions. For primary operators O(x) conformal invariance determines the form of two-point functions to be c O¯ (x)O(y) = , (2) (x − y)2∆ where c is a constant and the scaling dimension is ∆ = ∆0 + γ , with γ the anomalous part which has an expansion at weak coupling. By computing (2) at the instanton level one can extract the corresponding contribution to γ . In general the anomalous dimensions have an expansion of the form ∞ ∞ = pert 2n + (K) 2m 2πiτK + γ(gYM,N) γn (N)gYM γm (N)gYMe c.c. , (3) n=1 K>0 m=0 = + 2 where τ θ/(2π) i4π/gYM and anti-instantons give the complex conjugate of the instanton contributions, since the anomalous dimensions are real quantities. The N = 4 theory is believed to possess a SL(2, Z) S-duality symmetry under which the complexified coupling τ transforms projectively, τ → (aτ + b)/(cτ + d) (where ad − bc = 1 with integer a, b, c, d). In the dual type IIB string theory D-instantons are instrumental in implementing the constraints imposed by S-duality, for instance, on the form of the low-energy effective action. On the field theory side S-duality requires the invariance of the spectrum of scaling dimensions and instantons are expected to play a crucial rôle in the implementation of this symmetry. The invariance of the spectrum under SL(2, Z) transformations suggests that single anomalous dimensions should be functions of τ and τ¯,

∆ = ∆(τ, τ¯; N)≡ ∆0 + γ(τ,τ¯; N), (4) so that in general a contribution from the non-perturbative part in (3) is expected. In order to compute instanton corrections to anomalous dimensions in the following we shall study two-point functions of gauge-invariant scalar operators of bare dimension S. Kovacs / Nuclear Physics B 684 (2004) 3–74 7

∆0 = 2, 3, 4 and 5. The calculations will be performed in the semiclassical approximation and in the one-instanton sector of the SU(N) N = 4 supersymmetric Yang–Mills theory. This means that expectation values reduce to finite dimensional integrals over the moduli space parametrised by the bosonic and fermionic collective coordinates which arise in the instanton background. The general aspects of these calculations are reviewed in Section 2 and the case of two-point correlators is discussed in Section 4.1. The methods developed in this paper can be applied to non-Lorentz-scalar operators as well. In this case additional quantum numbers need to be taken into account in the construction of independent sets of operators, but the calculation of instanton corrections to two-point correlation functions does not present any new complications. The quantum numbers that characterise scalar operators are, besides the dimension ∆0, the Dynkin labels, [a,b,c], determining their transformation under the SU(4) R-symmetry. The parameters (∆0;[a,b,c]) identify sectors in the theory and as discussed in Section 4.2 in order to extract the anomalous dimensions one has to compute all the two-point functions to resolve the mixing among operators in any given sector. One of the outcomes of our analysis is that the problem of operator mixing is in general more complicated at the instanton level than in perturbation theory: in sectors where instantons contribute we find that mixing occurs among all the operators at the same leading order in the coupling. We shall not compute all the entries in the mixing matrices for the sectors we shall examine and thus it will not be possible to pin down the actual values of the instanton induced anomalous dimensions. However, it will be possible, analysing the moduli space integrations, to unambiguously identify the sectors in which the scaling dimensions receive instanton corrections. Rather surprisingly the calculations of Section 5 show that the majority of the scalar operators of dimension ∆0 5 do not get instanton corrections. This is an unexpected result in view of the above considerations on S-duality: many of the operators that we examine are corrected in perturbation theory, but not at the instanton level. A similar behaviour had been observed in [4,39] for operators in the Konishi multiplet. The results presented here might suggest that this is a more general feature of the theory, however, in the concluding discussion we shall argue that this is probably not the case and that the non-renormalisation properties we find are likely to be specific to operators of small dimension. At ∆0 = 2 there are two operators and all their two-point functions vanish in the one-instanton sector. This was a known result since one of the operators belongs to a protected 1/2 BPS multiplet and the other is a component of the Konishi multiplet, whose anomalous dimension was shown not to receive instanton corrections in [4,39]. Among the operators of dimension 3 there is only one whose structure allows instanton i contributions, it is the unique operator O3,6 in the 6 of SU(4). However, although this is hard to see from the two-point functions, we shall give another argument, based on the i OPE analysis of the four-point function computed in Appendix C, showing that O3,6 has no instanton induced anomalous dimension. In all the other sectors at ∆0 = 3, corresponding to the representations 10, 10 and 50, there are no instanton corrections. At ∆0 = 4we find the first non-vanishing instanton contributions. For operators in the SU(4) singlet we find non-vanishing two-point functions. Even without computing the actual values of the anomalous dimensions, the results of Section 5.3.1 show that operators in this sector have scaling dimensions which are corrected by instantons. Operators in all the remaining 8 S. Kovacs / Nuclear Physics B 684 (2004) 3–74

sectors at ∆0 = 4, i.e., in the SU(4) representations 15, 20 , 45, 45, 84 and 105, are not renormalised by instantons. Similarly among the operators of bare dimension ∆0 = 5 only those transforming in the 6 have non-vanishing two-point functions in the one-instanton background. These are more complicated to compute because their evaluation at leading order in the coupling requires the inclusion of the leading quantum fluctuations around the instanton configuration. Explicit examples of calculations of such effects will be given in Section 5.4.1. Again, in all the other SU(4) sectors at ∆0 = 5(10, 10, 50, 64, 70, 70, 126, 126, 196 and 300) no instanton corrections to two-point functions are found. For both the singlet at ∆0 = 4andthe6 at ∆0 = 5 the OPE analysis of the four-point function computed in Appendix C confirms the presence of non-perturbative corrections to the scaling dimensions. In the calculation of anomalous dimensions important simplifications arise from the use of the constraints imposed by the PSU(2, 2|4) global symmetry. First of all supersymmetry implies that all the operators in a multiplet must have the same anomalous dimension. Therefore the above results for operators with ∆0 5 imply the absence of instanton corrections to a much larger set of operators: all the components of the multiplets for which a representative is found to be non-renormalised share the same (vanishing) anomalous dimension. Taking into account this fact and the structure of the PSU(2, 2|4) multiplets, which can be determined systematically using the method of [40], the problem of extracting the values of the anomalous dimension for operators that receive corrections can be significantly simplified. When a direct analysis is complicated it is usually possible to identify superconformal descendants which can be more easily studied. Then superconformal symmetry implies that the result for the anomalous dimension extends to the original operators. An explicit example of this is represented by the case of the Konishi multiplet: the absence of instanton corrections to the superconformal primary (as well as to the whole multiplet) is obtained as consequence of the non-renormalisation of a suitably chosen descendant. The same idea can be used to simplify the resolution of the operator- mixing when instanton corrections are present. As already observed the above results for ∆0 5 probably do not reflect a general property of the theory, but rather are characteristic of small dimension sectors. However, some non-renormalisation properties can be generalised to arbitrary ∆0.Thisisin particular the case for the absence of instanton corrections in the SU(2) subsector identified in [29] consisting of scalar operators transforming in the representation [a,b,a] and with bare dimension ∆0 = 2a +b. In [29,42] it was shown that the action of the dilation operator on such operators is particularly simple. We shall argue that operators of this type do not receive instanton corrections. Superconformal symmetry implies that this result is valid for all the components of multiplets containing operators of this type. The paper is organised as follows. In Section 2 we review general aspects of instanton calculus in N = 4 SYM. Bosonic and fermionic collective coordinates in the one-instanton sector are described together with the integration measure on the moduli space used in the semiclassical calculations. The structure of the N = 4 instanton supermultiplet, i.e., of the instanton solution for the elementary fields, is described in Section 3. Generalities on the calculation of instanton induced two-point functions and their relation with corrections to the anomalous dimensions are discussed in Section 4. Section 5 presents the calculation of one-instanton contributions to two-point functions of operators with ∆0 5 and contains S. Kovacs / Nuclear Physics B 684 (2004) 3–74 9 the main results of the paper. The final section contains a discussion of the results and some comments on the possibility of realising the action of the dilation operator on the instanton moduli space. Appendix A summarises our notation and in Appendix B a brief summary of the ADHM description of the one-instanton sector of N = 4 SYM is provided. In Appendix C we present the calculation of a four-point function, which allows to resolve ambiguities left from the analysis of two-point functions in Sections 5.2.1 and 5.4.1.

2. One-instanton contributions to correlation functions in N = 4SYM

In this section we present general aspects of the computation of instanton effects in semiclassical approximation in the N = 4 supersymmetric Yang–Mills theory. The general instanton configuration can be described using the ADHM formalism [43]. The original construction of self-dual configurations in pure Yang–Mills theory with arbitrary classical gauge group has been generalised to the case of theories with different field content and in particular to supersymmetric theories. Comprehensive reviews of instanton calculus in supersymmetric gauge theories can be found in [44,45]. In particular [45] contains a detailed description of multi-instantons in the N = 4SYMtheory.Inthe following we shall focus on the one-instanton sector of the N = 4 theory with SU(N) gauge group. Many of the results of the present paper, however, remain valid in sectors with arbitrary instanton number, K. The extension to orthogonal and symplectic groups also does not present particular problems from the point of view of the instanton calculations. Appendix B contains a brief summary of the ADHM description of the one-instanton sector of the SU(N) N = 4SYMtheory. We begin by discussing instanton contributions to generic correlation functions of gauge invariant composite operators in semiclassical approximation. The specific case of two- point functions of such operators, which is relevant for the computation of instanton- induced anomalous dimensions will be considered in Section 4. At the semiclassical level expectation values are evaluated using a saddle point approximation around the instanton configuration. For a correlation function of generic local operators Oi the path integral reduces to a finite-dimensional integration over the moduli space of the instanton, i.e., an integration over the unfixed parameters (moduli or collective coordinates) of the generic instanton configuration as described by the ADHM construction O (x ) ···O (x ) 1 1 n n

−Sinst ˆ ˆ = dµinst(m b, m f) e O1(x1; m b, m f) ···On(xn; m b, m f). (5)

In (5) we have denoted the bosonic and fermionic collective coordinates by m b and m f, respectively, and by dµinst(m b, m f) the corresponding integration measure; Sinst is the ˆ classical action evaluated on the instanton solution and Or , r = 1,...,n, denotes the classical expression for the operator Or computed in the instanton background. In the one-instanton sector and with SU(N) gauge group there are 4N bosonic collective coordinates entering into the integration measure in (5). With a particular choice of parametrisation these bosonic moduli can be identiﬁed with the size, ρ, and position, x0,of 10 S. Kovacs / Nuclear Physics B 684 (2004) 3–74 the instanton as well as its global gauge orientation. The latter can in turn be described by three angles identifying an SU(2) instanton and 4N additional constrained variables, wuα˙ and w¯ αu˙ (where u is a colour index), in the coset SU(N)/(SU(N − 2) × U(1)) describing the embedding of the SU(2) conﬁguration into SU(N).IntheN = 4 theory there are 8N fermionic collective coordinates as well in the one-instanton sector. These correspond to zero modes of the Dirac operator in the background of an instanton. They comprise the 16 moduli associated with Poincaré and special supersymmetries broken by the instanton and A ¯ αA˙ denoted respectively by ηα and ξ (where A is an index in the fundamental of the SU(4) A ¯ Au R-symmetry group) and 8N additional parameters, νu and ν , which can be considered A as the fermionic superpartners of the gauge orientation parameters. The fermion modes νu and ν¯Au satisfy the constraints

¯ αu˙ A = ¯ Au = w νu 0, ν wuα˙ 0, (6) which effectively reduce the number of independent variables of this type to 8(N − 2). In the computation of correlation functions of gauge-invariant operators as the ones we shall be interested in the moduli space integration measure in (5) simpliﬁes. The only non-trivial bosonic integrals are over ρ and x0. Moreover in the case of the N = 4 A theory of the total set of 8N fermionic moduli dictated by the index theorem, only the ηα and ξ¯αA˙ moduli associated with broken supersymmetries correspond to true zero-modes when the interactions are taken into account. The instanton action acquires a non-trivial A ¯Au dependence on the other modes, νu and ν , which means that the integration measure has an explicit dependence on these modes. The moduli space integration measure for the N = 4 supersymmetric Yang–Mills theory in the generic K instanton sector was constructed in [36]. In the one-instanton sector the gauge-invariant measure takes the form −S dµphys e inst

π−4N g4N e2πiτ = YM (N − 1)!(N − 2)! 4 4 2 A 2 ¯ A N−2 A N−2 A 4N−13 −S4F × dρ d x0 d η d ξ d ν d ν¯ ρ e , (7) A=1 where the instanton action is [46]

π2 S =−2πiτ + S =−2πiτ + ε F ABF CD (8) inst 4F 2 2 ABCD 2gYMρ with

= 4πi + θ AB = √1 ¯ Au B −¯Bu A τ 2 , F ν νu ν νu . (9) gYM 2π 2 2

In (7) we have omitted an overall numerical constant, independent of gYM and N, that will be reinstated in the ﬁnal expression. Following [37] the integration measure can be written S. Kovacs / Nuclear Physics B 684 (2004) 3–74 11 in the form −4N 4N 2πiτ 4 π g e − − YM dρ d4x d6χ d2ηA d2ξ¯ A dN 2νA dN 2ν¯ A (N − 1)!(N − 2)! 0 A=1 4N−7 2 i i 4πi AB × ρ exp −2ρ χ χ + χAB F , (10) gYM where auxiliary bosonic variables, χi , i = 1,...,6, have been introduced to rewrite the exponential of the quartic fermionic action as a Gaussian integral. In (10)

1 i i χAB = √ Σ χ , (11) 8 AB i where the symbols ΣAB denote Clebsch–Gordan coefficients projecting the product of two 4sofSU(4) onto the 6 and are defined in Appendix A. Once the measure on the instanton moduli space is written in the form (10) the A ¯ Au integration over νu and ν is Gaussian and can be immediately performed. However, in general in correlation functions of gauge-invariant operators there is also a non-trivial dependence on these variables coming from the expressions for the operators in the instanton background. It is thus convenient to construct a generating function as in [38], which allows to deal easily with the otherwise complicated combinatorics associated with A ¯ Au ¯ u the fermionic integrations over νu and ν . We introduce sources, ϑA and ϑAu, coupled A ¯ Au to νu and ν and define π−4N g4N e2πiτ 4 Z[ϑ, ϑ¯ ]= YM ρ 4x 6χ 2ηA 2ξ¯A N−2ν¯A N−2νA − ! − ! d d 0 d d d d d (N 1) (N 2) = √ A 1 × 4N−7 − 2 i i + 8 πi ¯ Au B + ¯ u A + ¯ Au ρ exp 2ρ χ χ ν χABνu ϑAνu ϑAuν . gYM (12) Performing the Gaussian ν¯ and ν integrals and introducing polar coordinates, 6 χi → (r, Ω), χi 2 = r2, (13) i=1 Z[ϑ, ϑ¯ ] can be put in the form −29 −13 8 2πiτ 4 2 π g e − Z[ϑ, ϑ¯ ]= YM dρ d4x d5Ω d2ηA d2ξ¯A ρ4N 7 (N − 1)!(N − 2)! 0 A=1 ∞ − − 2 2 × drr4N 3e 2ρ r Z(ϑ, ϑ¯ ; Ω,r), (14) 0 where all the numerical coefficients have been reinstated. In (14) we have introduced the density

igYM Z(ϑ, ϑ¯ ; Ω,r)= exp − ϑ¯ uΩABϑ , (15) πr A Bu 12 S. Kovacs / Nuclear Physics B 684 (2004) 3–74 where the symplectic form ΩAB is given by 6 AB = √1 ¯ AB i i 2 = Ω Σi Ω , Ω 1 (16) 8 i=1 ¯ AB and the symbols Σi are deﬁned in Appendix A. In conclusion the semiclassical approximation to a correlation function (5) is computed as O1(x1) ···On(xn) −29 −13 8 2πiτ 4 2 π g e − = YM dρ d4x d5Ω d2ηA d2ξ¯ A ρ4N 7 (N − 1)!(N − 2)! 0 A=1 ∞ 4N−3 −2ρ2r2 ¯ ˆ ¯ × drr e Z(ϑ, ϑ; Ω,r)O1 x1; ρ,x0; η,ξ,ν(Ω),ν(Ω)¯ ×··· 0 × ˆ ; ; ¯ ¯ On xn ρ,x0 η,ξ,ν(Ω),ν(Ω) ϑ=ϑ¯ =0, (17) A ¯ Au ˆ where the νu and ν variables in each Or are understood to be rewritten in terms of ¯ u derivatives with respect to the sources, ϑA and ϑAu, before setting the latter to zero. As discussed in [38] the use of the generating function to express a correlator as in (17) allows in particular to easily determine the dependence on the parameters gYM and N. A ¯ Au The variables νu and ν enter in the expressions of gauge-invariant composite operators in the instanton background always in colour singlet bilinears. They arise in pairs either in a combination which is in the 6 of the SU(4) R-symmetry

¯ A B ≡¯u[A B] = ¯ Au B −¯Bu A ν ν 6 ν νu ν νu ν νu , (18) or in the 10 of SU(4)

¯ A B ≡¯u(A B) = ¯Au B +¯Bu A ν ν 10 ν νu ν νu ν νu . (19) Using the generating function defined in (14) we can determine the dependence of a generic correlator on the parameters gYM and N. In the one-instanton sector the N-dependence can be computed exactly. In particular in the large-N limit from (17) we get O1(x1) ···On(xn) 4 ∼ ; 8+p+q 2πiτ 4 5 2 A 2 ¯A p+q−5 α(p,q N)gYM e dρ d x0 d Ω d η d ξ ρ = A 1 ˆ ¯ ˆ ¯ × O1 x1; ρ,x0; η,ξ,νν(Ω)¯ ···On xn; ρ,x0; η,ξ,νν(Ω)¯ , (20) with 2−2N+(p+q)/2π−(p+q)Γ(2N − 1 − (p + q)/2) α(p,q; N)= − ! − ! (N 1) (N 2) + + − × Np q/2 + O Np q/2 1 + 1 ∼ N(p 1)/2 1 + O , (21) N S. Kovacs / Nuclear Physics B 684 (2004) 3–74 13 where p and q denote respectively the number of (νν)¯ 6 and (νν)¯ 10 bilinears entering the integrand in (20). As will be discussed explicitly in the case of scalar operators of bare dimension 5, in general the computation of two-point functions in the instanton background at the first non-trivial order in the coupling, gYM, requires to take into account the effect of the leading quantum fluctuations around the classical configuration. To compute leading order effects we need to include contributions in which, instead of replacing all the fields with their background value in the presence of an instanton, pairs of fields are contracted via a propagator. The general construction of the scalar propagator in the instanton background in terms of ADHM variables was given in [47], an explicit expression for the scalar propagator in the adjoint representation of the SU(N) gauge group was presented in [38]. In general the propagators for vectors and spinors are also needed. These propagators can be deduced from the scalar propagator as discussed in [48]. The Green function for the adjoint scalars which will be relevant for some of the calculations in Section 5.4 is given in Appendix B.

3. The N = 4 instanton supermultiplet As discussed in the previous section, in order to evaluate instanton induced correlation functions we need to integrate the classical profiles1 of the relevant composite operators over the instanton moduli space. In preparation for such computations in this section we shall discuss the structure of the instanton solution for the elementary fields in the N = 4 supermultiplet. We are interested in the dependence on the collective coordinates and of particular relevance will be the way the fermionic modes enter into the expressions for the various fields. = AB A The N 4 multiplet consists of six real scalars, ϕ , four Weyl spinors, λα ,anda vector, Aµ. Our notation is summarised in Appendix A. The field equations in the N = 4 SYM theory take the form µν αA ν ¯ α˙ 1 ν AB DµF + i λ σ ˙ , λ + ϕ¯AB, D ϕ = 0, αα A 2 √ 2 AB αA B 1 ABCD ¯ ¯ α˙ 1 AB CD D ϕ + 2 λ ,λ + √ ε λαC˙ , λ − ϕ¯CD, ϕ ,ϕ = 0, α D 2 √ 2 ¯ αA AB ¯ D/ ˙ λ + i 2 ϕ , λαB˙ = 0, αα √ ¯ α˙ − ¯ B = D/ αα˙ λA i 2 ϕAB,λα 0. (22) A solution to these equations is given by = I AB = A = ¯ α˙ = Aµ Aµ,ϕ λα λA 0, (23) I where Aµ is the standard instanton solution of SU(N) pure Yang–Mills theory. However, the Dirac operator has zero modes in the background of this solution, i.e., the equation ¯ αA = D/ αα˙ λ 0 has non-trivial solutions when the covariant derivative is evaluated in the

1 In the following we shall use the word “profile” in a slightly loose sense to indicate the expression for a composite operator computed in an instanton background including the dependence on fermion zero-modes. 14 S. Kovacs / Nuclear Physics B 684 (2004) 3–74 background of the instanton. This is the origin of the fermionic zero-mode integrations in the semiclassical expression for correlation functions (17). Because of these fermionic ˆ integrations using the solution (23) to compute the classical profiles of the operators Oi in (17) is not sufficient and we need to include the zero-mode dependence in the operators. We must thus solve Eq. (22) iteratively in order to determine the complete dependence on the fermion zero modes in the multiplet of elementary fields [49]. In the following we shall use the notation Φ(n) to denote a term in the solution for the field Φ containing n fermion zero modes. Notice that in the simple case of SU(2) gauge group there are only 16 fermionic modes in a one-instanton background, the ones associated with broken superconformal symmetries. In this case it is possible to determine completely the zero mode dependence in the N = 4 supermultiplet acting with the broken supersymmetries on (23). Substituting (0) ≡ I A (1)A Aµ Aµ in the supersymmetry transformation of λα gives a configuration, λα ,which A ¯αA˙ is linear in the fermion modes ηα and ξ and solves the corresponding field equation. Then plugging λ(1)A into the variation of ϕAB generates a solution for the scalar which is quadratic in the fermion modes, ϕ(2)AB. Iteration of this procedure gives rise to a ¯ (3) solution λαA˙ and then to a contribution to Aµ which corrects the original solution with (4) the addition of a term quartic in the fermion modes, Aµ . In this way one can construct the complete dependence on the fermion zero modes in the N = 4 supermultiplet in the case of SU(2) gauge group. The iteration continues until the number of zero modes in the fields exceeds 16, at which point further variations do not produce new independent terms in the solutions. This procedure can be implemented in an efficient way using a superspace formalism. A general discussion in the case of N = 1 supersymmetric Yang–Mills theory can be found in [50]. In the case of SU(N) gauge group the situation is more complicated and the above procedure cannot be utilised. As already discussed in this case there are the additional A ¯ Au fermion zero modes νu and ν which are not associated with symmetries broken by the bosonic instanton solution. The dependence on these modes, which is crucial in computing Green functions in semiclassical approximation, cannot be obtained using symmetry arguments. Therefore we need to explicitly construct the zero-mode dependence by solving the field equations. (0) Starting with the solution Aµ in (23), we solve Eq. (22) iteratively to generate solutions for all the fields in the multiplet with an increasing number of fermionic zero-modes. The first few steps in this construction have been carried out in [49]. (1)A The term linear in the fermion modes in the solution for the spinor, λα , is determined by the equation ˙ ¯ (0)αα (1)A = D/ λα 0, (24) ¯ (0)αα˙ (0) where the covariant derivative D/ contains Aµ . The subsequent steps give rise (2)AB ¯ (3) to ϕ and λαA˙ , which are obtained solving respectively √ (0)2 (2)AB + (1)αA (1)B = D/ ϕ 2 λ ,λα 0 (25) and S. Kovacs / Nuclear Physics B 684 (2004) 3–74 15 √ (0) ¯ (3)α˙ − ¯(2) (1)B = D/ αα˙ λA i 2 ϕAB,λα 0. (26) Further iteration gives rise to corrections to the above lowest order solution in which each { (0) (1)A (2)AB ¯ (3)α˙ } field has the minimal number of fermion modes, Aµ ,λα ,ϕ , λA . The following (4) (5)A (6)AB steps generate the terms Aµ , λα and ϕ which solve respectively (0)2 (4) − (0) (0) (4)ν + (0) (4)ν − ¯ (3) ¯ αα˙ (1)A = D/ Aµ D/ µ D/ ν A 2 Fµν ,A i λαA˙ σµ ,λα 0, (27) √ ¯ (0) (5)αA + ¯ (4) (1)αA + (2)AB ¯ (3) = D/ αα˙ λ D/ αα˙ λ i 2 ϕ , λαB˙ 0 (28) and √ √ (0)2 (6)AB + (4)2 (2)AB + (1)αA (5)B + (5)αA (1)B D ϕ D ϕ 2 λ ,λα 2 λ ,λα 1 ABCD ¯ (3) ¯ (3)α˙ 1 (2) (2)AB (2)CD + √ ε λ ˙ , λ − ϕ¯ , ϕ ,ϕ = 0. (29) 2 αC D 2 CD For generic operators in the N = 4 supersymmetric Yang–Mills theory these new terms contribute and are needed to compute correlation functions at leading order in gYM.Inthe special case of 1/2 BPS operators, as those in the supercurrent multiplet, the AdS/CFT correspondence suggests that only terms with the minimal number of superconformal modes are allowed [38]. Operators of this type are dual to the supergravity multiplet and 5 its Kaluza–Klein excitations in the type IIB string theory in AdS5 × S and the AdS/CFT correspondence combined with knowledge of the structure of the low-energy effective action for the type IIB “massless” fields puts restrictions on the set of correlation functions which can receive instanton contributions. Such constraints restrict the maximal number of superconformal modes that each operator can saturate. The same constraints do not apply to the unprotected operators we shall consider in the following and thus higher order terms will be needed as well. The procedure outlined here can in principle be employed to determine the exact zero- mode structure of the solution. The actual implementation of this construction becomes soon very involved as one gets to higher-order terms. The general solution takes the form = (4n) AB = (4n+2)AB Aµ Aµ ,ϕ ϕ , n=0 n=0 + 4n8N 4n2 8N A = (4n+1)A ¯ = ¯ (4n+3) λα λα , λαA˙ λαA˙ , (30) n=0 n=0 4n+18N 4n+38N where it is also understood that in each field the number of superconformal modes does A ¯ Au not exceed 16 and the remaining modes are of νu and ν type. In computing the expressions for gauge invariant composite operators we shall make use of the ADHM description in which the elementary fields are written as [N + 2]×[N + 2] matrices as discussed in Appendix B. In the same appendix the leading order terms in the A AB solution for Aµ, λα and ϕ , which will be used in some of the examples presented in Section 5, are given explicitly. Here we shall not discuss the details of the solution of the iterative equations, but instead we shall only analyse the SU(4) structure, which will suffice for the study of two-point 16 S. Kovacs / Nuclear Physics B 684 (2004) 3–74 functions to be carried out in later sections. The scalar fields in the N = 4 multiplet, ϕi ∼ ϕAB , transform in the representation 6 (with Dynkin labels [0, 1, 0])oftheSU(4) A ¯ α˙ [ ] R-symmetry group, the fermions, λα and λA, transform respectively in the 4 ( 1, 0, 0 )and 4 ([0, 0, 1]) and the vector, Aµ, (as well as its field strength and the covariant derivatives) is a singlet. The SU(4) structure of the combination of fermion zero modes in the various terms in the iterative solution discussed above can be determined without solving the A equations explicitly. All the fermion zero modes, both the superconformal ones, ηα and ¯ αA˙ A ¯Au ξ , and the modes of type νu and ν , transform in the 4 of SU(4). We shall denote A a generic fermion mode by mf . Inspecting the iterative equations that determine the instanton multiplet we can deduce in which SU(4) combinations the fermion modes enter (0) each term. The starting point is the classical instanton, Aµ , which has no fermions. The A first term in λα is linear in the fermion modes (1)A ∼ A λα mf . (31) For the term ϕ(2)AB in the scalar solution we find (2)AB ∼ [A B] ϕ mf mf , (32) i.e., the two fermion modes are antisymmetrised in order to form a combination in the 6. The schematic notation of (32) indicates that the [N + 2]×[N + 2] matrix ϕ(2)AB has entries which involve one mode of flavour A and one of flavour B in all the possible combinations [A B] [A ¯ αB˙ ] ¯α˙ [A ¯β˙B] [A ¯ B]v [A B] ¯α˙ [A ¯ B]v ηα ηβ ,ηα ξ , ξ ξ ,ηα ν ,νu ηα , ξ ν , [A ¯ β˙B] [A ¯ B]v νu ξ ,νu ν . (33) ¯ (3)α˙ The 4 spinor λA contains fermion modes in the combination ¯ (3)α˙ ∼ B C D λA εABCDmf mf mf , (34) (3) so that the component λA has three fermion modes, one of each of the flavours apart from A. Proceeding in the multiplet we find the quartic term in the solution for the vector, (4) Aµ , which contains one fermion mode of each flavour in a singlet combination (4) ∼ A B C D Aµ εABCDmf mf mf mf . (35) (5)A The following term is λα , which has flavour structure (5)A ∼ A A B C D λα εA B C D mf mf mf mf mf , (36) i.e., it involves a mode of flavour A plus one of each flavour. Then we find ϕ(6)AB that contains an antisymmetric combination of a mode of flavour A and one of flavour B plus one mode of each flavour [A B] (6)AB ∼ A B C D ϕ εA B C D mf mf mf mf mf mf . (37) As observed after equation (32) the previous expressions are symbolic and the products of A ¯ αA˙ A ¯Au mfs in (34)–(37) correspond to different combinations of the modes ηα , ξ , νu and ν in the various entries of the ADHM matrices for each field. S. Kovacs / Nuclear Physics B 684 (2004) 3–74 17

The SU(4) structure of the combinations of fermionic modes entering into the higher- order terms in the solution can be determined analogously. As already mentioned the knowledge of the ﬂavour structure of the solution described here will be enough to obtain some interesting results concerning the anomalous dimensions of composite operators without actually solving the ﬁeld equations.

4. Instanton induced two-point functions and anomalous dimensions

Before analysing speciﬁc cases of scalar operators in Section 5 we now describe the general strategy that will be followed in such calculations. As usual in instanton calculus in the evaluation of the moduli space integrals it is convenient to perform the fermionic integrals ﬁrst. In the case of two-point functions these are particularly simple and their analysis will allow us to show the absence of instanton corrections to a large class of operators.

4.1. Two-point functions in the one-instanton sector

In the special case of two-point functions the semiclassical approximation takes the form 2 ¯ O(x1)O(x2) −4N 4N 2πiτ 4 π g e − − − = YM dρ d4x d2ηA d2ξ¯ A dN 2νA dN 2ν¯A ρ4N 13 (N − 1)!(N − 2)! 0 A=1 π2 ¯ [A B] ¯ [C D] 2 2 εABCD (ν ν )(ν ν ) × e 16gYMρ ˆ¯ ¯ ˆ ¯ × O(x1; x0,ρ,η,ξ,ν,ν)¯ O(x2; x0,ρ,η,ξ,ν,ν),¯ (38) A ¯Au where we have not yet rewritten the fermionic variables νu and ν in terms of bosonic auxiliary variables. As already observed in the general discussion of Section 2 we have to A ¯ αA˙ distinguish between the superconformal fermionic modes, ηα and ξ , and the remaining A ¯ Au ones, νu and ν : the former do not appear in the measure and must be soaked up by the operator insertions in order for (38) to yield a non-vanishing result. The ‘non-exact’ modes A ¯Au of type νu and ν appear in the measure so that an explicit dependence on these variables in the integrand is not required, although in general the classical proﬁles of composite operators do depend on ν¯ AνB colour singlet bilinears. After translating the dependence A ¯ Au AB on νu and ν into a dependence on the angular variables Ω using the generating function as discussed in Section 2 the two-point function (38) becomes 8 2πiτ 4 c(gYM,N)g e − O¯ (x )O(x ) = YM dρ d4x d5Ω d2ηA d2ξ¯ A ρ4N 7 1 2 (N − 1)!(N − 2)! 0 A=1 ˆ¯ ¯ ˆ ¯ × O(x1; x0,ρ,η,ξ,Ω)O(x2; x0,ρ,η,ξ,Ω), (39)

2 In this general discussion we omit overall numerical constants in the integration measure. 18 S. Kovacs / Nuclear Physics B 684 (2004) 3–74 where the coefficient c(gYM,N) contains additional dependence on gYM and N arising from the integration over the radial variable r introduced in (13). As mentioned previously c(g√YM,N) contains a power of gYM for each νν¯ bilinear in the integrand and a factor of N plus 1/N corrections for each (νν)¯ 6. As will be shown explicitly in the examples presented in Section 5 the superconformal modes always appear in the expressions for gauge-invariant composite operators in the combination 1 ˙ ζ A(x) = √ ρηA − (x − x ) σ µ ξ¯αA . (40) α ρ α 0 µ αα˙ Since the two-component spinors ζ A(x) satisfy (ζ A(x))3 = 0, ∀A,x, it is clear that in a A ¯αA˙ two-point function the only way of saturating the 16 integrations over ηα and ξ in (39) is that each of the two operators provides the combination ζ 1(x) 2 ζ 2(x) 2 ζ 3(x) 2 ζ 4(x) 2, (41) i.e., each of the two operators must soak up two powers of the superconformal combination (40) for of each of the four flavours. Unless this can be achieved the two-point function vanishes and therefore the anomalous dimension of the operator does not get instanton correction. This is a rather strong condition and it will allow us, using the results of Section 3 to analyse the dependence on the superconformal modes, to show the absence of instanton corrections in many cases. If the operators can indeed soak up the right combination of superconformal modes the η and ξ¯ integrals are non-vanishing and can be easily evaluated. Using a simple Fierz rearrangement one finds 2 A 2 ¯ A A 2 A 2 2 d η d ξ ζ (x1) ζ (x2) =−(x1 − x2) , ∀A = 1,...,4. (42)

Once the integrations over the fermion superconformal modes have been performed we are left with an integration over the five-sphere parametrised by the angular variables AB Ω and the bosonic part of the moduli space integration over x0 and ρ. The five-sphere integration factorises and gives rise to further selection rules. It gives a non-vanishing result only if the SU(4) indices carried by the Ωs in the two operators can be combined to form a SU(4) singlet. This analysis of the fermionic moduli space integrations can be repeated in the case of contributions with contractions between pairs of fields in the operator insertions. The same selection rules apply in these cases. Before discussing specific examples we shall now briefly recall how the information on anomalous dimensions is encoded in the two-point functions and how it can be extracted from the final bosonic integration over the moduli space.

4.2. Anomalous dimensions

Gauge invariant composite operators in the N = 4 theory are classified in terms of their transformation under the PSU(2, 2|4) supergroup of global symmetries. Each operator is characterised by a set of quantum numbers identifying the irreducible representation of the maximal bosonic subgroup SO(2, 4) × SU(4) of PSU(2, 2|4) it transforms in. The S. Kovacs / Nuclear Physics B 684 (2004) 3–74 19 quantum numbers defining such irreducible representations are (∆, J1,J2;[a,b,c]),where the spins J1 and J2 characterise the Lorentz group transformation and together with the scaling dimension ∆ determine the transformation under the conformal group SO(2, 4) and the remaining three numbers, [a,b,c] are Dynkin labels of SU(4). Let us consider a sector in the N = 4 theory formed by operators belonging to the same SU(4) and SO(1, 3) representations (in the following we shall restrict our attention to Lorentz scalars) and with the same bare dimension, ∆0. In the quantum theory in general the operators acquire an anomalous dimension which corrects the bare value, ∆ = ∆0 + γ , where the anomalous term is a function of the parameters gYM and N. Let us consider in such sector a complete set of n primary operators forming an orthonormal basis with respect to the scalar product defined by the two-point functions. We shall suppress Lorentz r and SU(4) labels and denote the operators by O∆(x). We assume that the operators are the ones well defined, i.e., transforming properly under all the global symmetries, in the full quantum theory. As is well known the spatial dependence in two-point functions of primary operators is completely fixed by conformal invariance. For the sector we are considering we have δrs O¯ r (x )Os (x ) = . (43) ∆r 1 ∆s 2 2∆ (x1 − x2) r

In the full quantum theory ∆r = ∆r (gYM,N) = ∆0 + γr (gYM,N) and the two-point function is non-zero only if O¯ r (x ) and Os (x ) have the same dimension, ∆ .We ∆r 1 ∆s 2 r assume that in case there are operators with the same anomalous dimension they are made orthogonal, so that the set {Or } forms an orthonormal basis in the sector under investigation in this case as well. We want to compare (43) with the result of a small coupling calculation in order to extract from the latter the information about the anomalous dimensions. At small gYM we assume the anomalous dimension γ(g,N)to be small and hence expand (43) as ¯ r s O∆ (x1)O∆ (x2) r s δrs 1 x − x 2 = 1 − γ (g ,N)log 1 2 +··· , (44) 2∆ 2γ (g ,N) r YM (x1 − x2) 0 µ r YM µ where µ is an arbitrary length scale related to the renormalisation scale in the small gYM calculation. Clearly physical quantities, as the anomalous dimension γ , do not depend on µ. Eq. (44) is a small-γ expansion, further expanding γ(gYM,N)at small gYM we get ¯ r s O∆ (x1)O∆ (x2) r s rs − 2 = c(gYM,N)δ − 2 (1) x1 x2 +··· 2∆ 1 gYMγr (N) log (x1 − x2) 0 µ x − x 2 − e2πiτγ (inst) log 1 2 +··· , (45) r µ where only the leading perturbative and instanton-induced terms have been kept. At higher- orders in the double expansion (small γ and small gYM) the situation is more complicated and terms with different powers of logarithms appear at the same order in gYM.Theterms 20 S. Kovacs / Nuclear Physics B 684 (2004) 3–74 in (45) are sufficient for the purposes of the analysis to be carried on in the following sections. Eq. (45) is the small coupling expansion of the exact two-point function of primary operators in an orthonormal basis. When performing explicit calculations, in perturbation theory or semiclassically in an instanton background, one works with a set of independent operators of bare dimension ∆ , O˜ r , which are not in general orthonormal with respect to 0 ∆0 the scalar product defined by the two-point function. In extracting the physical information, i.e., the anomalous dimensions, one has thus to deal with the complications associated with operator mixing. The result for a two-point function of operators O˜ r including the first order ∆0 contributions in perturbation theory and the leading semiclassical instanton term is of the form ˜¯ r ˜ s O∆ (x1)O∆ (x2) 0 0 − 2 = 1 rs − 2 rs x1 x2 +··· T gYM L log (x − x )2∆0 µ 1 2 x − x 2 − e2πiτ Krs log 1 2 +··· , (46) µ where we have denoted by T , L and K the matrices arising at tree-level, one loop and in the one-instanton sector. To make contact with the expansion (45) of the exact function we rewrite (46) as 2 ¯ 1 x1 − x2 O˜ r (x )O˜ s (x ) = T rs 1s s − e2πiτ Γ s s log , (47) ∆0 1 ∆0 2 2∆ (x1 − x2) 0 µ where only the instanton part has been kept and we have defined Γ rs = (T −1K)rs.Inorder to bring (47) into the form (45) we need to perform a change of basis which takes from O˜ r to Or . This is achieved by acting with a matrix M ∗ ¯ O¯ r (x )Os(x ) = Mrr (M )s s O˜ r (x )O˜ s (x ) . (48) 1 2 ∆0 1 ∆0 2 Substituting (47) and (45) into this relation we find the matrix identity (inst) = † diag γr MΓM , (49) which implies that the instanton contributions to the anomalous dimensions in the sector under investigation can be identified with the eigenvalues of the matrix Γ rs = (T −1K)rs. As discussed in [26,28,29] the above steps define the dilation operator of the theory, Dˆ .In this language Γ is therefore identified with the leading instanton correction to the dilation operator.

5. Non-perturbative contributions to anomalous dimensions of scalar operators

Having described the general strategy for the computation of anomalous dimensions from two-point correlation functions we shall now consider explicit examples of Lorentz scalar operators of bare dimension ∆0 = 2, 3, 4, 5. S. Kovacs / Nuclear Physics B 684 (2004) 3–74 21

We shall work with the normalisation of the fields which is standard in instanton 2 calculus, in which the action is written with an overall factor of 1/gYM. The usual normalisation of perturbative calculations is recovered rescaling all the fields, Φ → gYMΦ, so that the kinetic terms become gYM-independent, the cubic couplings have a factor 2 of gYM and the quartic couplings are proportional to gYM. With our choice of normalisation 2 the propagators are proportional to gYM. We shall define composite operators in such a way that their correlation functions are independent of gYM at tree-level. Therefore for an operator made of elementary fields, which we denote generically by Φa,a= 1,...,, we adopt the normalisation N O() = Tr(Φ ···Φ ), (50) 2 /2 1 (gYMN) which is easily verified to agree with the standard convention used in the AdS/CFT correspondence in which two-point functions behave like N2 at large N and are independent of the coupling at tree-level. The same rescaling that leads to the ordinary perturbative normalisation makes the composite operator (50) independent of gYM. The operators we focus on are Lorentz scalars and for fixed bare dimension, ∆0,they are characterised by their SU(4) Dynkin labels. We use the notation O∆0,r to denote a scalar operator of bare dimension ∆0 transforming in the r-dimensional representation of SU(4). In each SU(4) sector we shall construct a complete set of independent operators and then study the instanton contributions to their two-point functions.

5.1. Dimension 2 scalar operators

At bare dimension 2 the situation is particularly simple. All operators are single trace and Lorentz scalars can only be obtained as bilinears in the elementary scalar ﬁelds,

ij ∼ i j O2,6⊗6 Tr ϕ ϕ . (51) The scalars transform in the 6 of SU(4) with Dynkin labels [0, 1, 0] and thus the possible sectors for composite bilinears are those in the decomposition

[0, 1, 0]⊗[0, 1, 0]=[0, 0, 0]s ⊕[1, 0, 1]a ⊕[0, 2, 0]s ⇔ ⊗ = ⊕ ⊕ 6 6 1s 15a 20s, (52) where the subscripts ‘s’ and ‘a’ indicate the symmetric and antisymmetric parts. There is actually no operator in the 15 because of cyclicity of the trace and the only two sectors of scalars with ∆0 = 2 are the singlet and 20 with one operator in each of the two SU(4) representations

1 i i [0, 0, 0]: O2,1 = Tr ϕ ϕ , (53) g2 YM ij [ ] {ij } = 1 i j − δ k k 0, 2, 0 : O2,20 2 Tr ϕ ϕ ϕ ϕ . (54) gYM 6 Here and in the following we indicate complete symmetrisation plus removal of traces by curly brackets, {i1i2 ···in}. We shall denote complete antisymmetrisation by square 22 S. Kovacs / Nuclear Physics B 684 (2004) 3–74 brackets, [i1i2 ···in] and symmetrisation without removal of traces by parenthesis (i1i2 ···in). The two operators in (53) and (54) are of course well known. The singlet is the ≡ ij ≡ ij lowest component of the Konishi multiplet, O2,1 K1,andO2,20 Q is the lowest component of the supercurrent multiplet containing the energy–momentum tensor and the supersymmetry and R-symmetry currents. The latter is a 1/2 BPS operator and thus its scaling dimension does not receive quantum corrections. Operators in the Konishi multiplet do have anomalous dimension in perturbation theory, but not at the instanton level. The one-loop contribution was ﬁrst computed in [51] and re-derived in [4] using OPE techniques in the context of the AdS/CFT correspondence. The perturbative result has been extended to two-loops in [52,53] and a three loop result has been obtained in [29] under certain assumptions studying the action of the dilation operator. The fact that the anomalous dimension of the Konishi multiplet does not receive instanton corrections has been shown through the OPE analysis of a four-point function of 1/2 BPS operators Qij in [4]. The derivation of this result directly from the study of the [AB] two-point function is rather subtle. Rewriting the scalar ﬁelds as ϕ , K1 takes the form

= 1 AB CD K1 2 εABCD Tr ϕ ϕ . (55) gYM As discussed in Section 4 in computing the instanton induced two-point function K1(x1)K1(x2) we integrate over the moduli space the classical expression of the operators in the background of an instanton and the superconformal fermionic integrations must be saturated. Each of the two operators must soak up eight fermion modes in the combination (41). In order to get the correct number of fermion modes for each of the two insertions we need to consider

(2)AB (6)CD (6)AB (2)CD εABCD Tr ϕ ϕ + ϕ ϕ . (56) This combination contains the minimal required number of fermionic modes. The other terms in the scalar solution, ϕ(10)AB and ϕ(14)AB, would give rise to contributions involving νν¯ bilinears as well and so would be of higher order in gYM and hence not relevant in semiclassical approximation. Using (32) and (37) the classical profile of K1 is seen to be proportional to [ ] [ ] A B C D A B C D εABCDεA B C D mf mf mf mf mf mf mf mf . (57) This expression indeed contains the combination (41). This means that the vanishing of the instanton contribution to the two-point function of the K1 operator and thus of its instanton induced anomalous dimension does not simply follow from the impossibility of saturating the fermionic integrals. The direct calculation of the (vanishing) anomalous dimension of K1 from the two-point function requires a cancellation among various terms for which we need the exact form of ϕ(6)AB, which we have not determined. We shall find in Section 5.3 that the absence of instanton correction for this operator can be shown analysing a superconformal descendant of K1 transforming in the representation 84. ij = {ij } For the 1/2 BPS operator Q O2,20 , on the other hand, we can easily verify the absence of instanton corrections to the two-point function. For this purpose it is convenient S. Kovacs / Nuclear Physics B 684 (2004) 3–74 23 to choose a specific component to evaluate the two-point function, e.g., 12 12 Q (x1)Q (x2) . (58) √ √ Recalling that ϕ1 = 2(ϕ14 + ϕ23) and ϕ2 = 2(ϕ24 + ϕ13) (see Appendix A) we find that in the instanton background the operator Q12 is proportional to

A B C D [1 4] [2 4] [2 3] [2 4] εABCD m m m m m m m m + m m m m f f f f f f f f f f f f + [1 4] [1 3] + [2 3] [1 3] mf mf mf mf mf mf mf mf , (59) and taking all the modes to be superconformal modes none of the four terms in this expression contains the combination (41). The above detailed discussion was clearly not needed for Qij which is known to be protected, as already remarked, but it is included in order to illustrate the strategy utilised in more complicated examples in the following sections. ij The operator O2,20 is the simplest example in a class of operators characterised by the fact that their bare dimension is ∆0 = and they transform in the representation [0,,0] of SU(4). Operators of this type which are superconformal primaries are 1/2 BPS [54,55]. As shown in [29,41] operators of this type form a ‘sector’ in the N = 4SYMtheory.They cannot mix with operators involving fermions, field strengths or covariant derivatives at any order in gYM. For operators of this type it is always possible to choose a component which in N = 1 notation can be written in terms of a single complex scalar. This makes proving the absence of instanton corrections very simple. For Qij we can consider the component Tr(φ1φ1) ∼ Tr(ϕ14ϕ14) (see Appendix A for the definition of the complex combinations φI ), so that it is immediately verified that it cannot saturate the eight fermion modes as in (41).

5.2. Dimension 3 scalar operators

Composite operators of bare dimension ∆0 = 3 are also necessarily single trace. They can be obtained either as products of three scalars or as fermion bilinears, which must involve spinors of the same chirality in order for the composite to be a Lorentz scalar. So we need to consider

ij k ∼ i j k AB ∼ A B O3,6⊗6⊗6 Tr ϕ ϕ ϕ , O3,4⊗4 Tr λ λ , ∼ ¯ ¯ O3,4⊗4;AB Tr(λAλB ). (60) Operators cubic in the elementary scalars belong to the sectors in the decomposition [0, 1, 0]⊗[0, 1, 0]⊗[0, 1, 0]=3[0, 1, 0]⊕[2, 0, 0]⊕[0, 0, 2]⊕[0, 3, 0] ⊕ 2[1, 1, 1] ⇔ 6 ⊗ 6 ⊗ 6 = 3 · 6 ⊕ 10 ⊕ 10 ⊕ 50 ⊕ 2 · 64, (61) whereas the fermionic bilinears contribute to the sectors

[1, 0, 0]⊗[1, 0, 0]=[0, 1, 0]a ⊕[2, 0, 0]s ⇔ 4 ⊗ 4 = 6a ⊕ 10s, (62)

[0, 0, 1]⊗[0, 0, 1]=[0, 1, 0]a ⊕[0, 0, 2]s ⇔ 4 ⊗ 4 = 6a ⊕ 10s. (63) 24 S. Kovacs / Nuclear Physics B 684 (2004) 3–74

5.2.1. ∆0 = 3, [0, 1, 0] In this sector there is only one operator coming from (61). The decomposition 6 ⊗ 6 ⊗ 6 contains the 6 with multiplicity 3, but taking into account the cyclicity of the trace there is only one independent operator, 1 Oi = Tr ϕi ϕj ϕj . (64) 3,6 3 1/2 gYMN The 6 is also contained in 4 ⊗4 and 4⊗4, but the corresponding fermionic bilinears vanish for the cyclicity of the trace,

i αA B = ¯ AB ¯ ¯ α˙ = ΣAB Tr λ λα Σi Tr λαA˙ λB 0, (65) i ¯ AB since ΣAB and Σi are antisymmetric in A,B. i Let us then consider the two-point function of operators O3,6. For definiteness we take = i j ij the component i 1, the correlator O3,6(x1)O3,6(x2) is clearly proportional to δ .The first step is to check whether each of the two operators can saturate the superconformal fermion integrations, i.e., whether their instanton profile contains the combination (41). We need to consider

Tr ϕ(2)1ϕ(2)j ϕ(6)j + ϕ(2)1ϕ(6)j ϕ(2)j + ϕ(6)1ϕ(2)j ϕ(2)j [1 4] [2 3] [A B] [C D] ∼ εABCDεABCD m m + m m m m m m f f f f f f f f × A B C D mf mf mf mf , (66) j j AB CD where we have used (32) and (37) and written the contraction ϕ ϕ as εABCDϕ ϕ . Inspecting (66) it is easily verified that indeed the superconformal modes can be saturated in the two-point function. A potentially non-vanishing contribution is obtained, for instance, from the first term in (66) replacing ϕ1 by νν¯ bilinears and using the remaining scalars to soak up the 16 ζ s in each of the two operators. The other terms give identical contributions up the overall coefficient. The resulting contribution to the two-point function is schematically of the form 1 1 O3,6(x1)O3,6(x2) 4 −Sinst [1 4] A 2 ∼ dµphys e f(x1; x0,ρ) ν¯ ν ζ (x1) A=1 4 [2 3] A 2 × f(x2; x0,ρ) ν¯ ν ζ (x2) , (67) A=1 (2) (6) where the function f(x; x0,ρ)depends on the details of the solutions ϕ and ϕ for the scalars. Writing the measure explicitly as in Section 2 to we get 1 1 O3,6(x1)O3,6(x2) 2 2πiτ 4 g e − ∼ YM dρ d4x d5Ω d2ηA d2ξ¯ A ρ4N 7 N(N − 1)!(N − 2)! 0 A=1 S. Kovacs / Nuclear Physics B 684 (2004) 3–74 25

∞ 4 4 4N−3 −2ρ2r2 A 2 A 2 × drr e f(x1; x0,ρ) ζ (x1) f(x2; x0,ρ) ζ (x2) , = = 0 A 1 A 1 4 δ ¯ Z(ϑ, ϑ; Ω,r) ¯ ¯ u ¯ v ϑ=ϑ=0 δϑ[ δϑ ]δϑ[ δϑ ] 1 u4 2 v3 g4 Γ(2N − 2) 2−2N e2πiτ dρ d4x ∼ YM (x − x )8 0 f(x ; x , ρ)f (x ; x ,ρ) − ! − ! 1 2 3 1 0 2 0 N(N 1) (N 2) ρ × d5Ω (N − 2)2Ω14Ω23 − (N − 2)Ω13Ω24 . (68)

The five-sphere integral is evaluated using 1 d5ΩΩABΩCD = εABCD (69) 4 and we finally find O1 (x )O1 (x ) 3,6 1 3,6 2 dρ ∼ g4 c(N)e2πiτ(x − x )8 d4x f(x ; x , ρ)f (x ; x ,ρ), (70) YM 1 2 ρ3 0 1 0 1 0 √ where c(N) = N[1 + O(1/N)] for large N. The complete result for the two-point function is a sum of terms of this form. A logarithmic divergence in the remaining bosonic i integrals would signal an instanton contribution to the anomalous dimension of O3,6.Since we have not determined the complete expression for ϕ(6) we cannot compute the exact coefficient in (70) and verify the presence of the divergence. Notice, however, that (70) contains, apart from the standard instanton weight, a factor 4 of gYM. Since in this sector there is only one operator there is no problem of mixing to take into account and the instanton induced anomalous dimension can be read directly from the two-point function (70) (after dividing by the tree-level coefficient). The result is that if i O3,6 does acquire an anomalous dimension, it is of order inst ∼ 4 2πiτ γ3,6 gYMe . (71) This is to be contrasted with the case of other operators that we shall examine in the following in which the leading instanton contribution to two-point functions behaves 2πiτ as e with no further factors of gYM. i We shall come back to the case of the operator O3,6 in Section 5.2.5.

5.2.2. ∆0 = 3, [2, 0, 0] and [0, 0, 2] There are two scalar operators with ∆0 = 3inthe[2, 0, 0], one is cubic in the scalars and the other is bilinear in the [1, 0, 0] fermions, 1 O(1)(AB) = t(AB) Tr ϕiϕj ϕk , (72) 3,10 3 1/2 [ij k] gYMN

(2)(AB) = 1 αA B O3,10 2 Tr λ λα , (73) gYM 26 S. Kovacs / Nuclear Physics B 684 (2004) 3–74

(AB) ⊗ ⊗ where the projector t[ij k] of the product 6 6 6 onto the 10 is deﬁned as (AB) = ¯ AC ¯ DB t10[ij k] Σ[i ΣjCDΣk] , (74) with the brackets indicating complete antisymmetrisation in i, j and k. A similar projector onto the 10 can be constructed as [ ] [ ] t¯ ij k = Σ i Σ¯ jCDΣk , (75) 10(AB) AC DB so that in the [0, 0, 2] we ﬁnd

1 [ ] O¯ (1) = t¯ ij k Tr ϕiϕj ϕk , (76) 3,10(AB) 3 1/2 (AB) gYMN

(2) 1 α˙ ¯ = ¯ ˙ ¯ O3,10(AB) 2 Tr λαAλB . (77) gYM We now consider the two-point functions in this sector. In order to saturate the fermionic integrals over the superconformal modes we need to consider the following terms

(1)(AB) ∼ (AB) (2)i (2)j (6)k + (2)i (6)j (2)k + (6)i (2)j (2)k O t[ij k] Tr ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ , (78) 3,10 (2)(AB) ∼ (5)αA (5)B + (1)αA (9)B + (9)αA (1)B O3,10 Tr λ λα λ λα λ λα (79) and similarly for the conjugate operators we need [ ] O¯ (1) ∼ t¯ ij k Tr ϕ(2)iϕ(2)j ϕ(6)k + ϕ(2)iϕ(6)j ϕ(2)k + ϕ(6)iϕ(2)j ϕ(2)k , (80) 3,10(AB) (AB) ¯ (2) ∼ ¯ (3) ¯ (7)α˙ + ¯ (7) ¯ (3)α˙ O3,10(AB) Tr λαA˙ λB λαA˙ λB . (81) (r) (11) × As usual it is convenient to pick a speciﬁc component, so, e.g., we consider O3,10 (x1) O¯ (s) (x ) , with r, s = 1, 2. The two operators O(r) (11) are 3,10(11) 2 3,10 √ 9 8 1 O(1)(11) = Tr ϕ12 ϕ13,ϕ14 , O(2)(11) = Tr λα1λ1 (82) 3,10 3 1/2 3,10 2 α gYMN gYM and their conjugates are √ ¯ (1) 9 8 23 24 34 ¯ (2) 1 ¯ ¯ α˙ O = Tr ϕ ϕ ,ϕ , O = Tr λ ˙ λ . (83) 3,10(11) 3 1/2 3,10(11) 2 α1 1 gYMN gYM (r) (11) Recalling the analysis of Section 3 we ﬁnd that the two operators O3,10 can soak up the fermionic modes in the combination required to get a non-zero contribution to the two-point function. Expanding (82) as in (78), (79) we obtain terms which contain the combination (41) and are proportional to ν¯ 1ν1 ζ 1(x) 2 ζ 2(x) 2 ζ 3(x) 2 ζ 4(x) 2. (84) None of the conjugate operators, however, can provide (41), they can at most contain one mode ζ 1. Hence the two-point functions are all zero in the instanton background and the scaling dimensions do not receive instanton corrections in this sector. S. Kovacs / Nuclear Physics B 684 (2004) 3–74 27

Notice that even before analysing the conjugate operators we could conclude that the two-point functions are not corrected by instantons. From (84) it is clear that the ﬁve-sphere integration would vanish as a consequence of (69). These results were to be expected. Two orthogonal operators of dimension 3 in the 10 can be deﬁned as √

(AB) 1 αA B 2 (AB) i j k E =− Tr λ λ + t[ ] Tr ϕ ϕ ϕ , (85) g2 α g2 ij k √YM YM 3 2 6N (AB) = t(AB) Tr ϕiϕj ϕk + Tr λαAλB . (86) K 2 [ij k] 2 α gYM 32π E (AB) is a component of the 1/2 BPS multiplet of the N = 4 supercurrents, it is found 2 {ij } (AB) at level δ starting from the lowest component O2,20 ,andK is a component at the same level of the Konishi multiplet. Therefore the former is protected and the latter does not receive instanton corrections. Notice that the relative coefficients in (85) can be fixed, for instance, by the requiring that the three-point function E (AB)E (CD)Qij does not receive perturbative corrections [56], whereas the coefficients in (86) are determined by the Konishi anomaly [39].

5.2.3. ∆0 = 3, [0, 3, 0] In this sector there is only one operator which is obtained from the fully symmetrised product of three scalars

{ } 1 { } O ij k = Tr ϕ i ϕj ϕk 3,50 g3 N1/2 YM 1 = Tr ϕiϕj ϕk + ϕiϕkϕj g3 N1/2 YM 1 − Tr δij ϕkϕlϕl + δikϕj ϕlϕl + δjkϕiϕlϕl . (87) 4 This sector is the next example of the type discussed at the end of Section 5.1. The operator in (87) is a protected 1/2 BPS operator and using the notation of Section 5.1 we define ij k ≡ {ij k} Q O3,50 .The1/2 BPS operator in the 20 is the lowest component of the supercurrent multiplet. It is dual in the AdS/CFT correspondence to a scalar in the type IIB supergravity multiplet which is a linear combination of the trace part of the metric and the R ⊗ R 4-form with indices in internal directions. The operator Qij k is dual to the first Kaluza–Klein excited mode of the same field.3 It is straightforward to verify the absence of instanton corrections to two point-functions of Qij k by choosing a suitable component to analyse the zero-mode structure. The simplest choice is a component written in terms of a single complex scalar, Tr(φ1φ1φ1). The terms in this operator with at least eight fermion modes in the instanton background are

3 i In this sense the operator O3,6 of Eq. (64) can be thought of as the “ﬁrst Kaluza–Klein excitation” of the Konishi operator, K1, considered in Section 5.1. 28 S. Kovacs / Nuclear Physics B 684 (2004) 3–74

(6)1 (2)1 (2)1 ∼ [1 4] [1 4] [1 4] A B C D Tr φ φ φ εABCD mf mf mf mf mf mf mf mf mf mf , (88) from which one immediately veriﬁes that only one mode of ﬂavours 2 and 3 can be saturated, so that the two-point functions of this operator cannot get instanton correction.

5.2.4. ∆0 = 3, [1, 1, 1] Operators with ∆0 = 3inthe64, corresponding to Dynkin labels [1, 1, 1],alsoinvolve only elementary scalars. The 64 occurs twice in the product 6 ⊗ 6 ⊗ 6, respectively, in 6 ⊗ 15 and 6 ⊗ 20. To obtain the operators in the 64 we should suitably project the two combinations

[ ] Tr ϕiϕ j ϕk , (89)

{ } Tr ϕiϕ j ϕk . (90) Taking into account the cyclicity of the trace one ﬁnds that (89) actually only contributes to the already examined 10 and 10, since the operator is automatically fully antisymmetric in i, j and k. Similarly (90) is found to only contribute to the 50 and 6 representations: it is not possible to make (90) orthogonal to both (64) and (87). Hence the 64, although allowed by the group theoretical analysis, is not realised in terms of gauge invariant operators.

5.2.5. ∆0 = 3, [0, 1, 0] revisited The non-renormalisation results for operators in the representations 10, 10 and 50 (and the absence of gauge-invariant operators in the 64) simplify the computation of the i anomalous dimension of O3,6 using an alternative approach, based on the OPE analysis of a four-point function. This method is more complicated, because it involves the computation of an instanton induced four-point function, but does not require to solve the field equation for ϕ(6). A four-point function of protected operators Q can be expanded in a double OPE as r s u v Q (x1)Q (x1)Q (x3)Q (x4) Crs(x ,∂ )Cuv(x ,∂ ) = k 12 1 k 34 3 k k + − + − O (x1)O (x3) , (91) (x )∆r ∆s ∆k (x )∆u ∆v ∆k k 12 34 where xij = xi − xj , ∂i = ∂/∂xi and the sum is over a generally infinite set of primary rs operators. The dependence on derivatives in the Wilson coefficients Ck implicitly represents the inclusion of descendants. The operators Q are chosen to be protected, but = (0) + in general some of the Ok’s may have anomalous dimension, so that ∆k ∆k γk. rs rs = (0)rs + rs Similarly for the Ck coefficients we have Ck Ck ωk . Expanding (91) for small rs γk and ω we get k r s u v Q (x1)Q (x1)Q (x3)Q (x4) k k = O (x1)O (x3) (0) ∆ +∆ −∆(0) ∆ +∆ −∆(0) (x ) r s k (x ) u v k k 12 34 γ x2 x2 × (0)rs (0)uv + (0)rs uv + rs (0)uv + k (0)rs (0)uv 12 34 Ck Ck Ck ωk ωk Ck Ck Ck log 4 , 2 x13 (92) S. Kovacs / Nuclear Physics B 684 (2004) 3–74 29 which shows how anomalous dimensions and corrections to the OPE coefficients can be extracted from the analysis of four-point functions. In order to have the exchange of the scalar in the 6 we can consider the instanton contribution to the correlation function G(x1,x2,x3,x4) = O3,50(x1)O2,20 (x2)O3,50(x3)O2,20 (x4) , (93) involving two ∆0 = 2andtwo∆0 = 31/2 BPS operators. In computing (93) at leading order we only need the scalar solution ϕ(2) which is given in Appendix B. The operators exchanged in the s-channel, corresponding to x12 → 0, x34 → 0, are in the decomposition 20 ⊗ 50 = 6 ⊕ 50 ⊕ 64 ⊕ 196 ⊕ 300 ⊕ 384. (94) According to the general formula (92) the presence of an instanton induced anomalous dimension for an operator of dimension 3 would give rise to a singularity of the type 1 (x )2(x )2 log 12 34 . (95) 2 2 4 (x12) (x34) (x13) i Since we have verified that O3,6 is the only operator of bare dimension 3 which can have contribution at the instanton level, the OPE analysis of a single four-point function unambiguously determines its anomalous dimension. It can be read off from the coefficient of the singular term (95) with no problem of mixing to take into account and no necessity to project onto the representation we are interested in. This is the same type of argument used in [4] to show that the anomalous dimension of the Konishi operator, K1, is not corrected non-perturbatively. In that case the argument took advantage of the fact that K1 is the only operator of bare dimension 2 which could possibly be corrected. The computation of the correlator (93) in the one-instanton sector is presented in Appendix C. As shown there in the limit x12 → 0, x34 → 0 this four-point function i does not have a singularity of the type (95), indicating that the operator O3,6 does not have anomalous dimension at the instanton level. This means that the coefficient of the logarithmically divergent term in the two-point function (70) actually vanishes.

5.3. Dimension 4 scalar operators

The analysis operators of bare dimension ∆0 = 4 is more involved, at this level derivatives and ﬁeld strengths also contribute to scalar operators and moreover we need to consider single- as well as double-trace operators. The combinations of elementary ﬁelds we need to consider here are

(a) µν (b) ˜ µν O ∼ Tr Fµν F , O ∼ Tr Fµν F , (96) 4,1 4,1 i i µ j O ⊗ ∼ Tr Dµϕ D ϕ , (97) 4,6 6 iAB i αA B i i ¯ α˙ O ∼ Tr ϕ λ λ , O ∼ Tr ϕ λ ˙ λ , (98) 4,6⊗4⊗4 α AB;4,6⊗4⊗4 αA B

(s)ijkl ∼ i j k l (d)ijkl ∼ i j k l O4,6⊗6⊗6⊗6 Tr ϕ ϕ ϕ ϕ , O4,6⊗6⊗6⊗6 Tr ϕ ϕ Tr ϕ ϕ , (99) 30 S. Kovacs / Nuclear Physics B 684 (2004) 3–74 where the notation is the same used in previous sections and superscripts (s) and (d) denote single- and double-trace operators. This shows that operators involving ﬁeld strengths can only contribute to the SU(4) singlet sector. Operators with covariant derivatives can be found in the sectors in the decomposition

[0, 1, 0]⊗[0, 1, 0]=[0, 0, 0]⊕[1, 0, 1]⊕[0, 2, 0] ⇔ 6 ⊗ 6 = 1 ⊕ 15 ⊕ 20 . (100) The operators of the two types in (98), made out of a scalar and a fermion bilinear, contribute respectively to

[0, 1, 0]⊗[1, 0, 0]⊗[1, 0, 0]=[0, 0, 0]⊕2[1, 0, 1]⊕[0, 2, 0]⊕[2, 1, 0] ⇔ 6 ⊗ 4 ⊗ 4 = 1 ⊕ 2 · 15 ⊕ 20 ⊕ 45 (101) and

[0, 1, 0]⊗[0, 0, 1]⊗[0, 0, 1]=[0, 0, 0]⊕2[1, 0, 1]⊕[0, 2, 0]⊕[0, 1, 2] ⇔ 6 ⊗ 4 ⊗ 4 = 1 ⊕ 2 · 15 ⊕ 20 ⊕ 45. (102) The single- and double-trace operators in (99) can contribute to the representations in

[0, 1, 0]⊗[0, 1, 0]⊗[0, 1, 0]⊗[0, 1, 0] = 3[0, 0, 0]⊕7[1, 0, 1]⊕6[0, 2, 0]⊕3([2, 1, 0]⊕[0, 1, 2]) ⊕ 2[2, 0, 2] ⊕[0, 4, 0]⊕[1, 2, 1] ⇔ 6 ⊗ 6 ⊗ 6 ⊗ 6 = 3 · 1 ⊕ 7 · 15 ⊕ 6 · 20 ⊕ 3 · (45 ⊕ 45) ⊕ 2 · 84 ⊕ 105 ⊕ 175. (103)

5.3.1. ∆0 = 4, [0, 0, 0] In the singlet sector at ∆0 = 4 one can consider the following basis of operators. There are the two operators (96) involving ﬁeld strengths, which including the normalisation read 1 1 (1) = Tr F F µν and (2) = Tr F F˜ µν . (104) O4,1 2 µν O4,1 2 µν gYM gYM With two derivatives there is the operator

(3) = 1 i µ i O4,1 2 Tr Dµϕ D ϕ . (105) gYM The two singlets in (98) are, respectively, 1 O(4) = Σi Tr ϕiλαAλB and 4,1 3 1/2 AB α gYMN

(5) 1 ¯ AB i ¯ ¯ α˙ O = Σ Tr ϕ λαA˙ λ . (106) 4,1 3 1/2 i B gYMN S. Kovacs / Nuclear Physics B 684 (2004) 3–74 31

Finally using cyclicity of the trace one ﬁnds four (out of the possible six) operators made out of four scalars, two single-trace and two double-trace,

(6) = 1 i j i j (7) = 1 i i j j O4,1 4 Tr ϕ ϕ ϕ ϕ , O4,1 4 Tr ϕ ϕ ϕ ϕ , (107) gYMN gYMN 1 1 (8) = Tr ϕiϕj Tr ϕiϕj , (9) = Tr ϕiϕi Tr ϕj ϕj . (108) O4,1 4 O4,1 4 gYMN gYMN Among the ∆0 = 4 operators in the SU(4) singlet two are known to be protected. They are the θ 4 and θ¯4 components of the N = 4 supercurrent multiplet, C− and C+.Theseare linear combinations of (104), (105), (106) and (107),

− = 1 − − µν +··· C 2 Tr Fµν F , gYM

+ 1 + + = Tr F F µν +···. (109) C 2 µν gYM These two operators are 1/2 BPS and dual to complex combinations of the dilaton and (0) −φ R ⊗ R scalar in the AdS/CFT correspondence, τ and τ¯,whereτ = τ1 + iτ2 = C + ie . The sum C− + C+ is proportional to the N = 4 on-shell Lagrangian. Another known operator in this sector is a component of the Konishi multiplet, K1, which is a different linear combination of the same operators, orthogonal to C− and C+. Since it belongs to the Konishi multiplet this operators is not corrected at the instanton level. The operators (107) and (108) have been studied in perturbation theory in [29,59] and their anomalous dimensions were computed at one loop, where there is no mixing with the remaining operators in the sector. At higher loops mixing is expected to occur. At the non-perturbative level there is mixing among all the operators (104)–(108) at leading order (r) = in gYM. As will be now shown all the operators O4,1,r 1,...,9, can soak up the correct combination of fermionic modes and hence all their two-point functions can get instanton corrections. For the two operators (104) the terms contributing to two-point functions in the one- instanton sector are

(1) → 1 (4) (4)µν O4,1 2 Tr Fµν F , (110) gYM

(2) → 1 (4) ˜ (4)µν O4,1 2 Tr Fµν F , (111) gYM which using (35) can be shown to be proportional to A B C D A B C D εABCDεA B C D mf mf mf mf mf mf mf mf , (112) so that they contain the combination (41). Similarly for the operator (105) one gets a potentially non-vanishing contribution considering

(3) → 1 (0) (6)i (0)µ (2)i + (4) (2)i (0)µ (2)i O4,1 2 Tr Dµ ϕ D ϕ Dµ ϕ D ϕ , (113) gYM 32 S. Kovacs / Nuclear Physics B 684 (2004) 3–74 which yields the combination [ ] [ ] A B C D A B C D εABCDεA B C D mf mf mf mf mf mf mf mf , (114) which can saturate the superconformal modes. In the expansion of the two operators (106) the relevant terms are 1 O(4) → Σi Tr ϕ(6)iλ(1)αAλ(1)B + ϕ(2)iλ(5)αAλ(1)B 4,1 g3 N1/2 AB α α YM + (2)i (1)αA (5)B ϕ λ λα , (115) 1 ˙ O(5) → Σ¯ AB Tr ϕ(2)iλ¯ (3) λ¯ (3)α , (116) 4,1 3 1/2 i αA˙ B gYMN which again can saturate the fermion integrations in a two-point function since these expressions contain, respectively, [ ] C D A B A B C D εABCDεA B C D mf mf mf mf mf mf mf mf (117) and [ ] A B C D E C D E εAC D E εBC D E mf mf mf mf mf mf mf mf . (118) Finally the operators (107) and (108), quartic in the scalars, can contribute to two-point correlators through the terms 1 (6) → Tr ϕ(2)iϕ(2)j ϕ(2)iϕ(2)j , (119) O4,1 4 gYMN

(7) → 1 (2)i (2)i (2)j (2)j O4,1 4 Tr ϕ ϕ ϕ ϕ , (120) gYMN 1 (8) → Tr ϕ(2)iϕ(2)j Tr ϕ(2)iϕ(2)j , (121) O4,1 4 gYMN

(9) → 1 (2)i (2)i (2)j (2)j O4,1 4 Tr ϕ ϕ Tr ϕ ϕ , (122) gYMN whose zero-mode structure is [ ] [ ] [ ] [ ] A B C D A B C D εABCDεA B C D mf mf mf mf mf mf mf mf . (123) From the above analysis it is clear that the classical expressions of all the fields in the singlet sector at ∆0 = 4 can contain the correct combination of fermion zero-modes to produce non-vanishing two-point functions. The expansion of all the combinations (112), (114), (118) and (123), in which all the modes are taken to be superconformal, involves exactly two ζ s for each flavour. Moreover at leading order in the coupling there is no dependence on the ν and ν¯ other than what comes from the measure. So the integration over the fermionic part of the moduli space is expected to be non-zero for all two- (r) ¯ (s) = point correlation functions O4,1(x1)O4,1(x2) , r, s 1,...,9. This is a general result, in sectors in which instanton contributions are present the non-perturbative mixing should be expected to be more complicated than at the first few orders in perturbation theory. In S. Kovacs / Nuclear Physics B 684 (2004) 3–74 33 this particular case one has to diagonalise the whole 9 × 9 problem to extract the instanton induced anomalous dimensions. As already remarked at least three of the eigenvalues of the anomalous dimensions matrix should vanish, since they correspond to the operators C−, + C and K1 discussed above. As discussed in the introduction further simplifications arise from taking into account the constraints imposed by the PSU(2, 2|4), which imply that all the operators in a multiplet have the same anomalous dimension. Therefore other operators can be eliminated from the mixing problem if they are identified as superconformal descendants of operators whose anomalous dimension is known. We shall now compute explicitly the two-point functions involving the quartic scalar operators in (107) and (108) in the one-instanton sector.4 This will prove that the situation in this sector is different from what was found in Sections 5.1 and 5.2.1 and indeed singlet operators at ∆0 = 4 do receive instanton corrections. In order to evaluate the semiclassical two-point functions of these operators we need their expression in the instanton background, i.e., the explicit form of (119)–(122). As usual it is convenient to work with the scalar fields written as ϕAB . The composite operators become 1 (6) = A1B1 A2B2 A3B3 A4B4 O 4 εA1B1A3B3 εA2B2A4B4 Tr ϕ ϕ ϕ ϕ , (124) gYMN 1 (7) = ε ε Tr ϕA1B1 ϕA2B2 ϕA3B3 ϕA4B4 , (125) O 4 A1B1A2B2 A3B3A4B4 gYMN 1 (8) = A1B1 A2B2 A3B3 A4B4 O 4 εA1B1A3B3 εA2B2A4B4 Tr ϕ ϕ Tr ϕ ϕ , (126) gYMN 1 (9) = A1B1 A2B2 A3B3 A4B4 O 4 εA1B1A2B2 εA3B3A4B4 Tr ϕ ϕ Tr ϕ ϕ . (127) gYMN The two combinations that are needed are thus Tr ϕA1B1 ϕA2B2 ϕA3B3 ϕA4B4 (x) 28ρ8 = B1 βA1 − A1 βB1 B2 γA2 − A2 γB2 2 2 8 ζα ζ ζα ζ ζβ ζ ζβ ζ [(x − x0) + ρ ] × B3 δA3 − A3 δB3 B4 αA4 − A4 αB4 +··· ζγ ζ ζγ ζ ζδ ζ ζδ ζ (x) (128) and Tr ϕA1B1 ϕA2B2 (x) 24ρ4 = B1 βA1 − A1 βB1 B2 αA2 − A2 αB2 +··· 2 2 4 ζα ζ ζα ζ ζβ ζ ζβ ζ (x) . [(x − x0) + ρ ] (129)

4 In the following discussion we omit the subscripts indicating the dimension and SU(4) representation. 34 S. Kovacs / Nuclear Physics B 684 (2004) 3–74

In both the previous expressions the ellipsis stands for terms involving the modes of type ν and ν¯ as well as terms with more than eight fermion zero modes which are not relevant in semiclassical approximation. Substituting into the expressions for the composite fields we find that only the operators O(6) and O(8) can soak up the superconformal modes in the combination (41). O(9) vanishes identically, whereas O(7) contains terms with eight superconformal modes, but not in the required combination. We thus find that the non-vanishing correlation functions in the one-instanton sector in semiclassical approximation are G(a)(x ,x ) = O(6)(x )O(6)(x ) , 1 2 1 2 G(b)(x ,x ) = O(8)(x )O(8)(x ) , 1 2 1 2 (c) (6) (8) G (x1,x2) = O (x1)O (x2) . (130) In computing these correlation functions at leading order in the coupling the ν and ν¯ modes appear only in the moduli space integration measure and we find (r) (s) O (x1)O (x2) 2πiτ 4 e − = dρ d4x d5Ω d2ηA d2ξ¯A ρ4N 7 N2(N − 1)!(N − 2)! 0 A=1 ∞ 4N−3 −2ρ2r2 ˆ (r) ˆ (s) × drr e O (x1; ρ,x0; ζ)O (x2; ρ,x0; ζ), (131) 0 where as usual the classical expressions for the operators in the presence of an instanton are denoted by a hat. The exact numerical coefficients will be reinstated in the final formulae. Since there is no dependence on ν and ν¯ in the integrand the integrations over the five- sphere and the radial variable r can be immediately performed and one obtains (r) (s) O (x1)O (x2) −2N − 2πiτ = 2 Γ(2N 1) e N2(N − 1)!(N − 2)! 4 4 dρ d x0 × d2ηA d2ξ¯AOˆ (r)(x ; ρ,x ; ζ)Oˆ (s)(x ; ρ,x ; ζ). (132) ρ5 1 0 2 0 A=1 Using (128) and (129) to compute (124) and (126) and integrating over the superconformal modes we then get 3452π−132−2N−15Γ(2N − 1)e2πiτ O(r)(x )O(s)(x ) = crs (x − x )8 1 2 N2(N − 1)!(N − 2)! 1 2 11 × 4 ρ dρ d x0 2 2 8 2 2 8 , (133) [(x1 − x0) + ρ ] [(x2 − x0) + ρ ] where the coefficients crs take into account the different prefactors in the term (41) in the expansion of the operators. By explicitly computing the contractions with the Levi-Civita S. Kovacs / Nuclear Physics B 684 (2004) 3–74 35 tensors in the definitions of the operators we find for these coefficients

c66 = 1,c88 = 24,c68 = c86 =−22. (134) The bosonic integrals which remain to be evaluated in (133) are logarithmically divergent as can be seen introducing Feynman parameters to rewrite ρ11 I = dρ d4x 0 2 2 8 2 2 8 [(x1 − x0) + ρ ] [(x2 − x0) + ρ ] 1 Γ(16) = dα dα δ(α + α − 1)α7α7 [Γ(8)]2 1 2 1 2 1 2 0 ρ11 × dρ d4x . (135) 0 [ − 2 + 2 + 2 ]16 (x0 i αi xi) ρ α1α2x12 After the standard manipulations the ρ integral can be performed and using dimensional regularisation for the x0 integration we get 1 + I = 2+ − Γ(14 ) 1 dα π c(4 ) 2 16− 1− , (136) [Γ(8)] (x1 − x2) [α(1 − α)] 0 where c(d) = 3840/[(d − 20)(d − 22)(d − 24)(d − 26)(d − 28)(d − 30)].Thefinal integration over α produces a 1/ pole which is the signal of a logarithmic divergence in dimensional regularisation. In conclusion we find non-zero entries in the matrix Krs defined in Section 4.2 for to the two-point functions (130), which can be read from (133), (134) and (136). As anticipated after Eq. (71), these matrix elements behave as e2πiτ with no additional powers of gYM. The above calculation shows that ∆0 = 4 operators in the singlet do acquire an instanton induced anomalous dimension. This can be confirmed analysing the OPE of a four-point function as in Section 5.2.5. In fact the t-channel, x13 → 0, x24 → 0, of the same four-point function (93), which is computed in Appendix C, displays a singularity corresponding to the exchange of operators of dimension 4. The results of the following subsections show that this contribution to the OPE must come from operators in the singlet. Although we have not computed all the entries in the matrix Krs and diagonalised Γ rs = (T −1K)rs, our results appear to disagree with those of [57]. There, on the basis of an OPE analysis, it was argued that only operators whose dimension remains finite in the large-N limit get an instanton correction. These are multi-trace operators dual to supergravity multi-particle bound states in the AdS/CFT correspondence. We have instead shown that single trace operators, which are dual to massive string modes and whose dimension diverges in the N →∞limit, have non-zero two-point functions in the one- instanton sector.

5.3.2. ∆0 = 4, [1, 0, 1] The representation [1, 0, 1]≡15 occurs in (100), in (101) and (102) with multiplicity 2 and in (103) with multiplicity 7, so there are in principle many operators to be considered. 36 S. Kovacs / Nuclear Physics B 684 (2004) 3–74

The actual number of independent gauge-invariant operators, however, reduces when the cyclicity of the trace is taken into account. In particular this implies that there are only single trace operators. No operators can be constructed from (100), i.e., with two scalars and two covariant derivatives, because of antisymmetry. A basis of independent operators can be obtained as follows. To realise the two operators in (101) we need to consider

[ ] 1 [ ] O(1) ij = Tr Σ i ϕj λαAλB (137) 4,15 3 1/2 AB α gYMN and

[ ] 1 [ ] O(2) ij kl = Tr t ij k ϕl λαAλB , (138) 4,15 3 1/2 AB α gYMN i [ij k] where the symbols ΣAB are deﬁned in (A.5) and tAB is the projector deﬁned in (74). Similarly from (102) we get

(3) 1 ¯ AB ¯ ¯ α˙ O = Tr Σ[ ϕj]λαA˙ λ (139) 4,15[ij ] 3 1/2 i B gYMN and

(4) 1 ¯AB ¯ ¯ α˙ O = Tr t[ ϕl]λαA˙ λ . (140) 4,15[ij kl] 3 1/2 ij k B gYMN Operators in the 15 made of four scalars can be constructed antisymmetrising a pair of indices and contracting the remaining two or completely antisymmetrising the four indices. We obtain however only one independent gauge-invariant operator,

[ ] 1 [ ] O(5) ij = Tr ϕ iϕj ϕ ϕk . (141) 4,15 g4N k The other combinations of four 6s forming a 15 are identically zero because of cyclicity of the trace. Antisymmetrisation in (141) implies that no double trace operators can be realised. We can now analyse the instanton contributions to two-point functions in this sector. An analysis on the lines of that in the previous subsection will show that none of the above operators can soak up the correct combination of superconformal modes to give rise to non-vanishing two-point correlators. As usual the strategy is to choose a speciﬁc component for the operators and study the zero-mode structure, the result is the same for all the components. For the operator (137) we consider, for instance, the component (1)[12] O4,15 which is proportional to

[ ] [ ] [ ] [ ] Tr ϕ13λ 1λ4 + ϕ13λ 2λ3 + ϕ24λ 1λ4 + ϕ24λ 2λ3

[ ] [ ] [ ] [ ] − ϕ14λ 1λ3 − ϕ14λ 2λ4 − ϕ23λ 1λ3 − ϕ23λ 2λ4 . (142) In order to saturate the eight superconformal modes required to give rise to a non-zero two-point functions in all these terms the contributions one needs to consider are

[ ] [ ] [ ] Tr ϕ(2)ABλ(1) Cλ(5)D + ϕ(2)ABλ(5) Cλ(1)D + ϕ(6)ABλ(1) Cλ(1)D . (143) S. Kovacs / Nuclear Physics B 684 (2004) 3–74 37

Expanding in this way the ﬁrst term in (142) yields a contribution proportional to 1 3 1 4 A B C D εABCDζ ζ ζ ζ ζ ζ ζ ζ , (144) which does not contain the required combination (41), having only one mode of ﬂavour 2. Similar considerations apply to the other terms in (142) as well as to the other components of the same operator. For the operator (139) the analysis is completely analogous and gives the same result. (2)[1234] For the operator (138) we can consider the component O4,15 . After contracting the indices the result one obtains is a sum of terms of the form

Tr ϕABλC λC + ϕAC λBλC , (145) which when expanded as in (143) in order to select the terms with eight fermion modes again give contributions of the type (144), where one flavour occurs thrice and one flavour only once. The result is analogous for other components and also for the operator (140), which can be analysed much in the same way. Similar results hold for the operator (141) constructed from only scalar fields. The [12] component of (141) is [ ] O(5) 12 ∼ Tr ϕ14 + ϕ23 −ϕ13 + ϕ24 4,15 × ϕ14 + ϕ23 2 + −ϕ13 + ϕ24 2 + ϕ12 + ϕ34 2 + −ϕ14 + ϕ23 2 + −ϕ13 − ϕ24 2 + ϕ12 − ϕ34 2 (146) and a combination of eight fermion modes is obtained selecting in each scalar ϕ the term with two fermion modes, ϕ(2)AB. It is then easy to verify that none of the terms in the resulting expansion can soak up two modes of each flavour. As in the case of the operators (137)–(140) in each term there is always a flavour occurring thrice and one occurring only once. Analogous considerations can be repeated for all the components. (r) (s) = In conclusion we find that none of the two-point functions O4,15(x1)O4,15(x2) ,r,s 1,...,7, in this sector receives instanton corrections, so that the corresponding non- perturbative contributions to all the anomalous dimensions vanish.

5.3.3. ∆0 = 4, [0, 2, 0] Scalar operators in the [0, 2, 0]≡20 were studied in [60], where the mixing at the per- 2 turbative level was resolved at order gYM. In the same paper it was also argued that anomalous dimensions in this sector do not receive instanton corrections. In this section we shall re-derive this result by directly analysing the instanton contributions to two-point functions. The basis of operators we consider is different from the starting point of [60]. Moreover as in previously considered sectors at the non-perturbative level there is mixing among all the operators at leading order in gYM. From (97) which admits the decomposition (100) there is one operator that can be constructed as

(1){ij } = 1 {i µ j} O4,20 2 Tr Dµϕ D ϕ gYM

≡ 1 i µ j − 1 ij k µ 2 Tr Dµϕ D ϕ 2 δ Tr Dµϕ D ϕk , (147) gYM 6gYM 38 S. Kovacs / Nuclear Physics B 684 (2004) 3–74 where as usual {ij } indicates symmetrisation and removal of traces in ﬂavour space, as explicitly indicated in (147). We then ﬁnd two operators involving fermionic bilinears, respectively, from (101) and (102). They read

(2){ij } 1 {i j} αA B O = Tr Σ ϕ λ λ (148) 4,20 3 1/2 AB α gYMN and

(3){ij } 1 ¯ AB{i j} ¯ ¯ α˙ O = Tr Σ ϕ λαA˙ λ . (149) 4,20 3 1/2 B gYMN Operators made out of four scalars in the 20 can be single or double trace. There is a total of four operators of this type and as a basis we consider the single trace

(4){ij } 1 {i j} k = Tr ϕ ϕ ϕ ϕ , (150) O4,20 4 k gYMN

(5){ij } 1 {i j} k = Tr ϕ ϕ ϕ ϕ (151) O4,20 4 k gYMN and the double trace combinations

(6){ij } 1 {i j} k = Tr ϕ ϕ Tr ϕ ϕ , (152) O4,20 4 k gYMN

(7){ij } 1 {i j} k = Tr ϕ ϕ Tr ϕ ϕ . (153) O4,20 4 k gYMN The analysis of the zero-mode structure of these operators follows closely what was done for the [1, 0, 1] sector. Notice that the antisymmetrisation was irrelevant in deriving the results of the previous subsection and only played a rôle in constructing the basis of operators. Therefore considering the {12} component of (148)–(153) we can immediately (r){12} (s){12} = conclude that all the two-point functions O4,20 (x1)O4,20 (x2) ,forr, s 2,...,7, vanish in the one-instanton sector and in semiclassical approximation. The zero-mode content of (148)–(153) is in fact exactly the same found in the analogous antisymmetric combinations belonging to the 15. If one chooses a component which is diagonal in ﬂavour {ii} space, O4,20 , the analysis is slightly more involved since the subtraction of the trace is then crucial in cancelling terms which could saturate the superconformal modes in a two-point function. A careful analysis of the instanton proﬁles of the operators shows that indeed for these components the two-point correlation functions vanish as well. The operator (147) must be analysed separately since it has no analogue in the [1, 0, 1] sector. Considering again the {12} component the terms in the expansion which soak up eight fermion modes are

(1){12} → 1 (0) (2){1 (4)µ (2)2} + (4) (2){1 (0)µ (2)2} O4,20 2 Tr Dµ ϕ D ϕ Dµ ϕ D ϕ gYM ∼ 1 (0) (2)14 + (2)23 (4)µ − (2)13 + (2)24 +··· 2 Tr Dµ ϕ ϕ D ϕ ϕ , (154) gYM S. Kovacs / Nuclear Physics B 684 (2004) 3–74 39 where the ellipsis stands for symmetrisation. Recalling that D(4) involves one fermion mode of each ﬂavour we ﬁnd that (154) contains the terms

A B C D 1 4 1 3 1 4 2 4 2 3 1 3 2 3 2 4 εABCDζ ζ ζ ζ ζ ζ ζ ζ + ζ ζ ζ ζ + ζ ζ ζ ζ + ζ ζ ζ ζ (155) none of which equals the required combination (41). Hence we ﬁnd that in the scalar [0, 2, 0] sector there are no operators which acquire an anomalous dimension at the instanton level.

5.3.4. ∆0 = 4, [2, 1, 0] and [0, 1, 2] Operators in the representations [2, 1, 0]≡45 and [0, 1, 2]≡45 arise respectively from (101) and (102), involving a scalar and a fermionic bilinear, and from (103), quartic in the scalars, which contains both 45 and 45 with multiplicity 3. Operators of the latter type can in principle be single- or double-trace. To project (98) onto the 45 we consider

(1)(AB) = 1 ¯ α(A B) − 1 A ¯ α(E B) − A ¯ α(E B) O4,45[CD] 3 1/2 Tr ϕCDλ λα δC ϕEDλ λα δD ϕCEλ λα g N 6

+ B ¯ α(A E) − B ¯ α(A E) δC ϕEDλ λα δD ϕCEλ λα . (156)

The operator in the 45 made of two fermions a scalar is obtained analogously as

(1)[CD] 1 CD ¯ ¯ α˙ 1 C ED ¯ ¯ α˙ D CE ¯ ¯ α˙ O = Tr ϕ λ ˙ λ − δ ϕ λ ˙ λ − δ ϕ λ ˙ λ 4,45(AB) 3 1/2 α(A B) A α(E B) A α(E B) g N 6

+ C ED ¯ ¯ α˙ − D CE ¯ ¯ α˙ δB ϕ λα(A˙ λE) δB ϕ λα(A˙ λE) . (157) Because of cyclicity of the trace from the product of four scalars we obtain only one operator in the 45 and one in the 45. In order to project (99) we can construct a projector onto the 45 as 45 (AB) = AE ¯ FB ¯ P[ij k]l [CD] Σ[i ΣjEFΣk] ΣlCD

− 1 A GE ¯ FB ¯ − A GE ¯ FB ¯ δC Σ[i ΣjEFΣk] ΣlGD δDΣ[i ΣjEFΣk] ΣlCG 6 + B AE ¯ FG ¯ − B AE ¯ FG ¯ δC Σ[i ΣjEFΣk] ΣlGD δDΣ[i ΣjEFΣk] ΣlCG (158) and similarly for the 45 we consider 45[CD] = ¯ EF ¯ CD P[ij k]l(AB) ΣAE[i Σj Σk]FBΣl

− 1 C ¯ EF ¯ GD − D ¯ EF ¯ CG δA ΣGE[i Σj Σk]FBΣl δA ΣGE[i Σj Σk]FBΣl 6 + C ¯ EF ¯ GD − D ¯ EF ¯ CG δB ΣAE[i Σj Σk]FGΣl δB ΣAE[i Σj Σk]FGΣl . (159) Then the two operators are, respectively,

(2)(AB) = 45 (AB) 1 i j k l O4,45[CD] P[ij k]l [CD] 4 Tr ϕ ϕ ϕ ϕ (160) gYMN 40 S. Kovacs / Nuclear Physics B 684 (2004) 3–74 and [ ] [ ] 1 O(2) CD = P 45 CD Tr ϕiϕj ϕkϕl . (161) 4,45(AB) [ij k]l(AB) 4 gYMN Notice that because of the symmetry properties of the projectors (158) and (159) there is no double trace operator in these sectors. To compute possible instanton corrections to the anomalous dimensions of the above operators we need to consider two-point functions O4,45(x1)O4,45(x2) . As usual it is convenient to pick a component and analyse the dependence on the superconformal zero- modes. A simple choice is to consider the set correlation functions [ ] G(r,s)(x ,x ) = O(r) (12) (x )O(s) 34 (x ) ,r,s= 1, 2, (162) 1 2 4,45[34] 1 4,45(12) 2 since for these components there is no trace to subtract in flavour space and (156), (157), (160) and (161) simplify. For the operators involving fermionic bilinears the possible non- vanishing contributions to (162) in semiclassical approximation come from the following terms 1 α(1 2) α(1 2) O(1)(12) → Tr ϕ¯(2) λ(1) λ(5) + λ(5) λ(1) , (163) 4,45[34] g3N1/2 [34] α α [ ] 1 [ ] α˙ O(1) 34 → Tr ϕ(2) 34 λ¯ (3) λ¯ (3) . (164) 4,45(12) g3N1/2 α(˙ 1 2) The combination of fermionic modes contained in (163) is (1)(12) ∼ 1 2 1 2 A B C D O4,45[34] εABCDmf mf mf mf mf mf mf mf (165) and the operator cannot provide two modes of each flavour as required. From (164) we find a contribution proportional to

(1)[34] 3 4 A B C A B C O ∼ ε ABCε m m m m m m m m , (166) 4,45(12) 1 2A B C f f f f f f f f so that it cannot give rise to a non-vanishing contribution when inserted in a two-point function since it contains only one mode of ﬂavours 1 and 2. For the two operators quartic in the scalars the terms in the solution which need to be considered for the semiclassical analysis are

(2)(12) → 45(12) 1 (2)i (2)j (2)k (2)l O4,45[34] P[ij k]l [34] 4 Tr ϕ ϕ ϕ ϕ , (167) gYMN [ ] [ ] 1 O(2) 34 → P 45 34 Tr ϕ(2)iϕ(2)j ϕ(2)kϕ(2)l . (168) 4,45(12) [ij k]l(12) 4 gYMN Using the solution (B.40) and performing the contraction to project onto the 45 and 45 one veriﬁes that none of these two operators can saturate the superconformal modes as required for producing a non-zero two-point correlation function. The same results can be shown for other components, in which case a cancellation involving the additional terms in the operators subtracting ﬂavour traces is required. We conclude that the scaling dimensions of operators in these sectors do not receive instanton corrections. S. Kovacs / Nuclear Physics B 684 (2004) 3–74 41

5.3.5. ∆0 = 4, [2, 0, 2] The representation [2, 0, 2]≡84 at ∆0 = 4 can only be obtained from (103), i.e., the operators in this sector are all quartic in the scalars. This sector is the simplest example of the type described in [29,41], which comprises operators transforming in representations with Dynkin labels [a,b,a] with bare dimension ∆0 = 2a + b. Operators of this type cannot involve covariant derivatives, field strengths or fermions and are only made out of elementary scalars. Among these operators those which are superconformal primaries belong to 1/4 BPS multiplets [54,58]. Moreover, as shown in [29], for fields in this class it is always possible to choose components that can be written in terms of only two complex scalars. The action of the dilation operator on composite fields belonging to these sectors is also particularly simple [29,41]. Notice that the [1, 1, 1] sector at ∆0 = 3 would also belong to this class, but, as observed in Section 5.2.4, it is not realised in terms of gauge invariant operators. The representation 84 occurs in (103) with multiplicity 2 allowing in principle to construct four independent operators, two single-trace and two double-trace operators. However, cyclicity of the trace implies that there exist only two independent gauge- invariant operators. In order to obtain a single-trace operator in this representation we consider (1)[ij ][kl] = 1 i j k l O4,84 4 Tr ϕ ,ϕ ϕ ,ϕ gYMN 1 − δik ϕm,ϕj ϕm,ϕl + δil ϕm,ϕj ϕk,ϕm 4 + δjk ϕi ,ϕm ϕm,ϕl + δjl ϕi,ϕm ϕk,ϕm 1 + δikδjl − δilδjk ϕm,ϕn ϕm,ϕn , (169) 20 i.e., we need to subtract the singlet, the 15 and the 20 from the symmetric part of 15 ⊗ 15. Similarly to project the double trace composite operator in (99) onto the 84 we consider

(2)[ij ][kl] = 1 i k j l − i l j k O4,84 4 Tr ϕ ϕ Tr ϕ ϕ Tr ϕ ϕ Tr ϕ ϕ gYMN 1 − δik Tr ϕmϕm Tr ϕj ϕl − Tr ϕj ϕm Tr ϕmϕl 4 + δil Tr ϕmϕk Tr ϕj ϕm − Tr ϕmϕm Tr ϕj ϕk

+ δjl Tr ϕiϕk Tr ϕmϕm − Tr ϕiϕm Tr ϕmϕk + δjk Tr ϕi ϕm Tr ϕmϕl − Tr ϕiϕl Tr ϕmϕm 1 + δikδjl − δilδjk Tr ϕmϕm Tr ϕnϕn 20 − Tr ϕmϕn Tr ϕmϕn . (170)

Operators in this sector were studied in perturbation theory in [4,29,52]. The single trace operator (169) belongs to the Konishi multiplet, being a superconformal descendant 42 S. Kovacs / Nuclear Physics B 684 (2004) 3–74

2 ¯2 of K1 at level δ δ . A linear combination of (170) and (169) is protected and indeed a superconformal primary and thus it is the lowest component of a 1/4 BPS multiplet. We should therefore expect to ﬁnd that the instanton contributions vanish in this sector. The computation of instanton contributions to the two-point functions (r)[ij ][kl] (s)[mn][pq] = O4,84 (x1)O4,84 (x2) ,r,s1, 2 (171) at leading order in gYM requires the expansion of the ﬁelds to the lowest order needed to saturate eight fermion modes at each point, so for the two above operators we need to compute (1)[ij ][kl] → 1 (2)i (2)j (2)k (2)l +··· O4,84 4 Tr ϕ ,ϕ ϕ ,ϕ , (172) gYMN [ ][ ] 1 O(2) ij kl → Tr ϕ(2)iϕ(2)k Tr ϕ(2)j ϕ(2)l 4,84 g4 N YM − Tr ϕ(2)iϕ(2)l Tr ϕ(2)j ϕ(2)k +··· . (173)

Using the expression for ϕ(2)i in (B.40) one can verify that the expansion of the two operators deﬁned in (169) and (170) in the instanton background does not contain the combination (41) needed to produce non-vanishing two-point functions. The simplest way of showing this is to exploit the observation of [29] and choose a component involving only two complex scalars, e.g.,

1 (1) → Tr φ1,φ2 φ1,φ2 , (174) O4,84 4 gYMN (2) → 1 1 2 1 2 − 1 1 2 2 O4,84 4 Tr φ φ Tr φ φ Tr φ φ Tr φ φ . (175) gYMN Notice that with this choice no subtraction of traces is required. For both of these operators the instanton proﬁle contains the combination

1 4 2 4 1 4 2 4 mf mf mf mf mf mf mf mf (176) of zero modes and so all their two-point functions vanish. Therefore the two operators in the 84 receive no instanton contribution to their scaling dimension as expected. The calculation just described provides another proof (and the most direct one) of the fact that the scaling dimension of the Konishi multiplet is not corrected by instantons. As observed in Section 5.1 a direct computation of two-point functions of the lowest component K1 is rather subtle. It is instead easy to analyse the superconformal descendant [ij ][kl] ≡ (1)[ij ][kl] K84 O4,84 , which has the same vanishing anomalous dimension because of supersymmetry. This situation resembles what is found in perturbation theory [29], where the action of the dilation operator is simpler on the descendant than on the primary ﬁeld. This example shows explicitly a general feature, i.e., how the superconformal symmetry of the theory can be used to simplify the computation of anomalous dimensions. S. Kovacs / Nuclear Physics B 684 (2004) 3–74 43

5.3.6. ∆0 = 4, [0, 4, 0] The sector of ∆0 = 4 operators transforming in the representation [0, 4, 0]≡105 is in the same class as the 20 at ∆0 = 2andthe50 at ∆0 = 3 examined in Sections 5.1 and 5.2.3, respectively. These are operators in representation with Dynkin labels [0,,0] and bare dimension ∆0 = and when primaries they belong to 1/2 BPS multiplets. The dimension 4 operators of this type are selected from (99) by taking the completely symmetric and traceless combination of SU(4) indices. There is a single- and a double-trace operator

(1){ij kl} = 1 {i j k l} O4,105 4 Tr ϕ ϕ ϕ ϕ , (177) gYMN

{ } 1 { } (2) ij kl = Tr ϕ i ϕj Tr ϕkϕl . (178) O4,105 4 gYMN In this case the projection onto the 105 requires the subtraction of the 20 and the singlet with coefﬁcients which are readily determined imposing tracelessness. In order to analyse the instanton contributions to these operators we consider a speciﬁc component and the simplest choice is to consider a component written in terms of a single complex scalar. In this way no trace subtraction is required and two representatives are, for instance,

(1) = 1 1 1 1 1 O4,105 4 Tr φ φ φ φ , (179) gYMN and

(2) = 1 1 1 1 1 O4,105 4 Tr φ φ Tr φ φ . (180) gYMN It is then clear that the corresponding two-point functions are not corrected at the instanton level since the operators (179) and (180) involve fermion modes in the combination 1 4 4 (mf mf ) .

5.3.7. ∆0 = 4, [1, 2, 1] The representation [1, 2, 1]≡175 is contained only in the decomposition (103), where it occurs with multiplicity 3. We can thus expect single- and double-trace operators made of four scalars. However cyclicity of the trace implies that actually no gauge-invariant operator exists in this sector. This can be seen analysing the origin of the three 175s in (103). They arise from the products (15 ⊗ 15)a, (20 ⊗ 20 )a and 15 ⊗ 20 .Theﬁrst two correspond to operators of the form [ ] [ ] Tr ϕ iϕj ,ϕ kϕl (181) and { } { } Tr ϕ iϕj ,ϕ kϕl (182) and thus vanish as traces of commutators. For the third combination which contains the 175 we should consider

[ ] { } Tr ϕ iϕj ϕ kϕl , (183) 44 S. Kovacs / Nuclear Physics B 684 (2004) 3–74 which again vanishes as can be shown by considering 5 [ ] { } [ ] { } [ ] { } Tr ϕ iϕj ϕ kϕl = Tr ϕ iϕj ϕ kϕl † =−Tr ϕ iϕj ϕ kϕl . (184) The representation 175 is thus not realised in terms of gauge-invariant operators. Notice that if operators in this sector could be constructed they would be in the same class as those in the 84 discussed in Section 5.3.5, i.e., operators transforming in a representation [a,b,a] with ∆0 = 2a + b:the175 corresponds to a = 1, b = 2.

5.4. Dimension 5 scalar operators

The analysis of gauge-invariant operators becomes increasingly complicated as their bare dimension grows and it is already very involved at dimension 5. The number of SU(4) representations which appear and their multiplicities increase rapidly, making the construction of a basis of independent operators in each sector rather non-trivial. The discussion in this section will therefore be less detailed than in previous sections. Instanton contributions to anomalous dimensions are expected only for operators in the 6 of SU(4).6 Explicit calculations of two-point functions in this sector will be presented in order to display a new feature which appears, i.e., the necessity of including the leading quantum fluctuations around the classical instanton configuration in order to compute two-point functions at leading order in the coupling. Although rather cumbersome, the analysis can, however, be carried on along the lines discussed in the previous cases without additional new conceptual difficulties for all the other sectors. The combinations of elementary fields which give rise to scalar composites of bare dimension 5 are

(a)i ∼ µν i (b)i ∼ ˜ µν i O5,6 Tr Fµν F ϕ , O5,6 Tr Fµν F ϕ , (185)

ij k ∼ i j µ k O5,6⊗6⊗6 Tr ϕ Dµϕ D ϕ , (186)

ij AB i j αA B ij i j ¯ ¯ α˙ O ∼ Tr ϕ ϕ λ λ , O ∼ Tr ϕ ϕ λ ˙ λ , (187) 5,6⊗6⊗4⊗4 α AB;5,6⊗6⊗4⊗4 αA B

ij A ∼ i αA µ ¯ α˙ O ; ⊗ ⊗ ⊗ Tr Dµϕ λ σαα˙ λB , (188) B 5,6 6 4 4 AB ∼ αA µν β B AB ∼ ˜ αA µν β B O5,4⊗4 Tr Fµν λ σα λβ , O5,4⊗4 Tr Fµν λ σα λβ , (189) ˙ ˙ ∼ ¯ ¯ µν α˙ ¯ β ∼ ˜ ¯ ¯ µν α˙ ¯ β OAB;5,4⊗4 Tr Fµν λαA˙ σ β˙λB , OAB;5,4⊗4 Tr Fµν λαA˙ σ β˙λB , (190) AB ∼ αA µ B ∼ ¯ µ ¯ α˙ O5,4⊗4 Tr Dµλ D λα , OAB;5,4⊗4 Tr DµλαA˙ D λB , (191)

ij klm ∼ i j k l m O5,6⊗6⊗6⊗6⊗6 Tr ϕ ϕ ϕ ϕ ϕ . (192) Moreover for the operators in (187) and in (192) we also need to consider the double-trace operators obtained splitting the product into two traces in all possible ways.

5 I thank Massimo Bianchi for a discussion on this point. 6 An argument supporting this claim will be presented in the next section. S. Kovacs / Nuclear Physics B 684 (2004) 3–74 45

Operators involving the field strength only contribute to the 6 of SU(4).Fromthe combination in (186) we obtain operators in the sectors appearing in the decomposition [0, 1, 0]⊗[0, 1, 0]⊗[0, 1, 0]=3[0, 1, 0]⊕[2, 0, 0]⊕[0, 0, 2]⊕[0, 3, 0] ⊕ 2[1, 1, 1] ⇔ 6 ⊗ 6 ⊗ 6 = 3 · 6 ⊕·10 ⊕ 10 ⊕ 50 ⊕ 2 · 64. (193) From (187) and the double-trace combinations involving the same fields we get operators, respectively, in [0, 1, 0]⊗[0, 1, 0]⊗[1, 0, 0]⊗[1, 0, 0] = 4[0, 1, 0]⊕3[2, 0, 0]⊕2[0, 0, 2]⊕[0, 3, 0]⊕4[1, 1, 1]⊕[3, 0, 1]⊕[2, 2, 0] ⇔ 6 ⊗ 6 ⊗ 4 ⊗ 4 = 4 · 6 ⊕ 3 · 10 ⊕ 2 · 10 ⊕ 50 ⊕ 4 · 64 ⊕ 70 ⊕ 126 (194) and [0, 1, 0]⊗[0, 1, 0]⊗[0, 0, 1]⊗[0, 0, 1] = 4[0, 1, 0]⊕2[2, 0, 0]⊕3[0, 0, 2]⊕[0, 3, 0]⊕4[1, 1, 1]⊕[1, 0, 3]⊕[0, 2, 2] ⇔ 6 ⊗ 6 ⊗ 4 ⊗ 4 = 4 · 6 ⊕ 2 · 10 ⊕ 3 · 10 ⊕ 50 ⊕ 4 · 64 ⊕ 70 ⊕ 126. (195) The sectors which can contain operators of the form (188) are [0, 1, 0]⊗[1, 0, 0]⊗[0, 0, 1]=2[0, 1, 0]⊕[2, 0, 0]⊕[0, 0, 2]⊕[1, 1, 1] ⇔ 6 ⊗ 4 ⊗ 4 = 2 · 6 ⊕ 10 ⊕ 10 ⊕ 64. (196) The two types of operators in (189) and the first in (191) contribute to [1, 0, 0]⊗[1, 0, 0]=[0, 1, 0]⊕[2, 0, 0]⇔4 ⊗ 4 = 6 ⊕ 10 (197) whereas those in (189) and the second in (191) contribute to [0, 0, 1]⊗[0, 0, 1]=[0, 1, 0]⊕[0, 0, 2]⇔4 ⊗ 4 = 6 ⊕ 10. (198) Finally operators of the form (192) (and the corresponding double-trace operators) are found in the decomposition [0, 1, 0]⊗[0, 1, 0]⊗[0, 1, 0]⊗[0, 1, 0]⊗[0, 1, 0] = 16[0, 1, 0]⊕10([2, 0, 0]⊕[0, 0, 2]) ⊕ 10[0, 3, 0]⊕24[1, 1, 1]⊕5([3, 0, 1]⊕[1, 0, 3]) ⊕ ([2, 2, 0]⊕[0, 2, 2]) ⊕[0, 5, 0]⊕5[2, 1, 2]⊕4[1, 3, 1] ⇔ 6 ⊗ 6 ⊗ 6 ⊗ 6 ⊗ 6 = 16 · 6 ⊕ 10 · (10 ⊕ 10) ⊕ 10 · 50 ⊕ 24 · 64 ⊕ 5 · (70 ⊕ 70) ⊕ 6 · (126 ⊕ 126) ⊕ 196 ⊕ 5 · 300 ⊕ 4 · 384. (199) From this preliminary analysis it is already apparent that many more representations appear at ∆0 = 5. Furthermore there are representations occurring with high multiplicity, which, as already observed, makes the construction of a basis of operators in the corresponding sectors rather tedious. 46 S. Kovacs / Nuclear Physics B 684 (2004) 3–74

5.4.1. ∆0 = 5, [0, 1, 0] Operators in the [0, 1, 0]≡6 can be obtained from the combinations (185), (187) (which both contain the 6 with multiplicity 4), (188) (where it occurs with multiplicity 2), (191) and (192) (where it appears with multiplicity 16). Clearly these multiplicities do not take into account the cyclicity of the trace which reduces the actual number of independent gauge invariant operators. On the other hand we must consider single- and double-trace operators of the types in (187) and (192). Two independent operators involving ﬁeld strengths are

(1)i 1 µν i (2)i 1 ˜ µν i O = Tr Fµν F ϕ , O = Tr Fµν F ϕ . (200) 5,6 3 1/2 5,6 3 1/2 gYMN gYMN With three scalars and two covariant derivatives we can construct the operators

(3)i 1 i j µ (4)i 1 j i µ O = Tr ϕ Dµϕ D ϕj , O = Tr ϕ Dµϕ D ϕj . 5,6 3 1/2 5,6 3 1/2 gYMN gYMN (201) Single trace operators with two scalars and two fermions are

(5)i = 1 i j j αA B (6)i = 1 j [i j] αA B O5,6 4 Tr ΣABϕ ϕ λ λα , O5,6 4 Tr ΣAB ϕ ϕ λ λα , gYMN gYMN

(7)i = 1 i {i j} αA B (8)i = 1 ¯[ij k] [j k] αA B O5,6 4 Tr ΣABϕ ϕ λ λα , O5,6 4 Tr t(AB)ϕ ϕ λ λα , gYMN gYMN

1 [ ] (9)i = Tr Σj λαAϕ iλB ϕj (202) O 5,6 4 AB α gYMN and similarly with fermions of the opposite chirality

(10)i 1 iAB j j α˙ = ¯ ¯ ˙ ¯ O5,6 4 Tr Σ ϕ ϕ λαAλB , gYMN

(11)i 1 jAB [i j] α˙ = ¯ ¯ ˙ ¯ O5,6 4 Tr Σ ϕ ϕ λαAλB , gYMN

(12)i 1 iAB {i j} α˙ = ¯ ¯ ˙ ¯ O5,6 4 Tr Σ ϕ ϕ λαAλB , gYMN

(13)i 1 (AB) [j k] α˙ = Tr t ϕ ϕ λ¯ ˙ λ¯ , O5,6 4 [ij k] αA B gYMN

(14)i 1 jAB [i α˙ j] = Tr Σ¯ λ¯ ˙ ϕ λ¯ ϕ . (203) O5,6 4 αA B gYMN Besides these operators there are double-trace operators obtained splitting the traces in (202) and (203) in all the ways allowed by the (anti)symmetry properties. The operators in (188) are

(15)i 1 i αA µ ¯ α˙ O = Tr Dµϕ λ σ λ , 5,6 3 1/2 αα˙ A gYMN S. Kovacs / Nuclear Physics B 684 (2004) 3–74 47

(16)i 1 αA i µ ¯ α˙ O = Tr λ Dµϕ σ ˙ λ , 5,6 g3 N1/2 αα A YM (17)i 1 ¯ iC[A αB] µ ¯ α˙ 1 i AB αC µ ¯ α˙ O = Tr Σ Dµϕiλ σ λ + Σ Dµϕ λ σ λ . 5,6 3 1/2 αα˙ C AB αα˙ C gYMN 4 (204) From (189) and (190) we get

(18)i 1 i αA µν β B O = Tr Σ Fµν λ σ λ , 5,6 3 1/2 AB α β gYMN

(19)i 1 i ˜ αA µν β B O = Tr Σ Fµν λ σ λ , 5,6 3 1/2 AB α β gYMN ˙ (20)i 1 ¯ iAB ¯ µν α˙ ¯ β O = Tr Σ F λ ˙ σ¯ ˙λ , 5,6 3 1/2 µν αA β B gYMN ˙ (21)i 1 ¯ iAB ˜ ¯ µν α˙ ¯ β O = Tr Σ Fµν λαA˙ σ¯ ˙λ . (205) 5,6 3 1/2 β B gYMN Operators of the form (191) actually do not contribute to the 6 because the trace of the antisymmetric combination vanishes. They are only found in the 10 and 10. Finally there are the operators made of only elementary scalars. As a set of independent gauge-invariant operators we can consider the single-trace combinations 1 1 O(22)i = Tr ϕiϕj ϕj ϕkϕk , O(23)i = Tr ϕiϕj ϕkϕj ϕk , 5,6 5 3/2 5,6 5 3/2 gYMN gYMN 1 O(24)i = Tr ϕiϕj ϕkϕkϕj , 5,6 5 3/2 gYMN 1 O(25)i = εij klmn Tr ϕj ϕkϕlϕmϕn (206) 5,6 5 3/2 gYMN and the double-trace combinations 1 O(26)i = Tr ϕiϕj Tr ϕj ϕkϕk , 5,6 5 3/2 gYMN 1 O(27)i = Tr ϕiϕj ϕj Tr ϕkϕk , 5,6 5 3/2 gYMN 1 O(28)i = Tr ϕiϕj ϕk Tr ϕj ϕk . (207) 5,6 5 3/2 gYMN There is therefore a large number of operators in this sector. Using the same techniques employed in the previous sections it is easy to verify that all the above operators can soak up the fermion superconformal modes in the required combination (41). The terms we need to consider in the expansion of all the operators in (200)–(207) at the semiclassical level involve 10 fermion modes. Analysing the structure of the single operators we ﬁnd that they all contain terms proportional to

ν¯ AνB ζ 1 2 ζ 2 2 ζ 3 2 ζ 4 2, (208) 48 S. Kovacs / Nuclear Physics B 684 (2004) 3–74 so that in the computation of two-point functions, (r,s) = (r)i (s)i = G (x1,x2) O5,6 (x1)O5,6 (x2) ,r,s1,...,28, (209) the integration over the superconformal modes is potentially non-zero. We should thus expect to find a non-vanishing result for all the two-point functions in this sector, so that the mixing problem in the one-instanton sector and at leading order in gYM is very involved. Since the total number of fermion modes entering the classical expressions of the operators (200)–(207) exceeds the minimum required of eight, there is also the possibility of non-vanishing contributions to two-point functions in which pairs of fields are contracted. Ordinary semiclassical contributions to the two-point functions (209), in which all the fields are replaced by their background expression in the presence of an instanton, involve a non-trivial five-sphere integration. The profiles of the operators contain a νν¯ bilinear each leading to an integral of the form (69). The insertion of two νν¯ bilinears 2 from the operators induces a factor of gYM in the expectation value. Similarly a factor 2 of gYM is produced by each propagator in our normalisations, so that the two types of contributions are of the same order and consistency requires that both are included. The counting of zero modes shows that the allowed contractions are between two ϕi s, between A ¯ a λ and a λA or between two vectors, Aµ. This is because in these cases the number of zero modes appearing in the correlation function is reduced by four, leaving a total of sixteen. The scalar contraction involves the propagator (B.41). The spinor and vector ¯ B propagators, λA(x1)λ (x2) and Aµ(x1)Aν(x2) , which have not been given explicitly, can be deduced from that for the scalar [48].7 In the two-point functions we are considering, however, we need the sixteen zero modes distributed in two groups of eight as in (41). This implies that fermion contractions cannot contribute because after the contraction the remaining sixteen modes are not evenly distributed. The same argument applies to vector contractions. A complete analysis and resolution of the mixing in this sector is beyond the scope of the present paper, we shall, however, compute the two types of contributions for a specific two-point function of operators in (206) in order to illustrate the features and difficulties of the calculation. Consider the correlation function = (23)1 (23)1 G(x1,x2) O5,6 (x1)O5,6 (x2) 1 = Tr ϕ1ϕiϕj ϕiϕj (x ) Tr ϕ1ϕkϕlϕkϕl (x ) . (210) 10 3 1 2 gYMN In order to soak up the sixteen superconformal modes in moduli space integration the relevant terms in the expansion of two composite operators are Tr(ϕ(2)1ϕ(2)iϕ(2)j ϕ(2)iϕ(2)j ). To compute the expression for the operators in (210) it is then convenient to rewrite them

7 It should be noted that the evaluation of contributions containing vector contractions presents subtleties related to infrared divergences. See [61], Chapter 4, and references therein. S. Kovacs / Nuclear Physics B 684 (2004) 3–74 49 in terms of ϕAB sas

(23)i 1 i AB AB AB CD CD O = √ Σ εABCD εABCD Tr ϕ ϕ ϕ ϕ ϕ 5,6 5 3/2 AB 2gYMN (211) and use Tr ϕA1B1 ϕA2B2 ϕA3B3 ϕA4B4 ϕA5B5 (x) 5 8 = 2 ρ [(x − x )2 + ρ2]9 0 [ ] × ν¯ A1 νA2 ζ B2 ζ B3 ζ A3 ζ B4 ζ A4 ζ B5 ζ A5 ζ B1

− (A4 ↔ B4) − (A5 ↔ B5) + (A4 ↔ B4,A5 ↔ B5) − (A3 ↔ B3) + (A ↔ B ,A ↔ B ) 3 3 4 4 + (A ↔ B ,A ↔ B ) − (A ↔ B ,A ↔ B ,A ↔ B ) 3 3 5 5 3 3 4 4 5 5 − (A ↔ B ) − (A ↔ B ) + (A ↔ B ,A ↔ B ) 1 1 2 2 1 1 2 2 + “cyclic permutations” [ ] + ν¯ A4 νB4 ζ A1 ζ B2 ζ A2 ζ B3 ζ A3 ζ B5 ζ A5 ζ B1

− (A1 ↔ B1) − (A2 ↔ B2) − (A3 ↔ B3) − (A5 ↔ B5)

+ (A1 ↔ B1,A2 ↔ B2)

+ (A1 ↔ B1,A3 ↔ B3) + (A1 ↔ B1,A5 ↔ B5) + (A2 ↔ B2,A3 ↔ B3)

+ (A2 ↔ B2,A5 ↔ B5) + (A3 ↔ B3,A5 ↔ B5)

− (A1 ↔ B1,A2 ↔ B2,A3 ↔ B3)

− (A1 ↔ B1,A2 ↔ B2,A5 ↔ B5) − (A1 ↔ B1,A3 ↔ B3,A5 ↔ B5) − (A ↔ B ,A ↔ B ,A ↔ B ) 2 2 3 3 5 5 + (A ↔ B ,A ↔ B ,A ↔ B ,A ↔ B ) 1 1 2 2 3 3 5 5 + “cyclic permutations” (212) which can be obtained after a lengthy but straightforward calculation utilising the solution (B.40) for the scalar ﬁeld. In (212) only the terms relevant to the computation of (210) have been kept. The ﬁrst contribution to the two-point function (210) in semiclassical approximation, obtained substituting the background values of the composite operators, is

(1) G (x 1,x2) − = Sinst ˆ (23)1 ; ; ¯ ¯ ˆ (23)1 ; ; ¯ ¯ dµphys e O5,6 (x1 ρ,x0 η,ξ,ν,ν)O5,6 (x2 ρ,x0 η,ξ,ν,ν) −4N 4N−10 2πiτ 4 π gYM e 4 2 A 2 ¯A N−2 A N−2 A 4N−13 = dρ d x0 d η d ξ d ν d ν¯ ρ N3(N − 1)!(N − 2)! A=1 50 S. Kovacs / Nuclear Physics B 684 (2004) 3–74

π2 ¯ [A B] ¯ [C D] 8 8 2 2 εABCD (ν ν )(ν ν ) ρ ρ × e 16gYMρ [(x − x )2 + ρ2]9 [(x − x )2 + ρ2]9 1 0 2 0 [1 4] [2 3] 2 1 2 2 2 3 2 4 2 × ν¯ ν + ν¯ ν ζ ζ ζ ζ (x1) 1 2 2 2 3 2 4 2 × ζ ζ ζ ζ (x2), (213) where only the terms giving a non-vanishing contribution have been included. An overall numerical coefficient has been omitted and will be reinstated in the final formula. The integrals over the ν and ν¯ modes can be performed with the aid of the generating function defined in Section 2, which allows to replace them with a five-sphere integral of the type (69). The integration over the 16 superconformal modes is also easily done using (42), so that we get

2−2N (N2 − 3N + 2)Γ (2N − 2) e2πiτ G(1)(x ,x ) = (x − x )8 1 2 N3(N − 1)!(N − 2)! 1 2 4 9 9 × d x0 dρ ρ ρ 5 2 2 9 2 2 9 . (214) ρ [(x1 − x0) + ρ ] [(x2 − x0) + ρ ] The second type of contribution involves one scalar contraction 1 G(2)(x ,x ) = Tr ϕ1ϕiϕj ϕiϕj (x ) Tr ϕ1ϕkϕlϕkϕl (x ) +···, 1 2 10 3 1 2 gYMN (215) where the ellipsis stands for the other contractions according to Wick’s theorem. The computation of this correlator is rather involved because of the complicated form of the scalar propagator in the instanton background. The insertion the propagator (B.41) eventually gives rise to two types of structures. Combining all the terms which arise from (215) we obtain (again up to an overall numerical coefﬁcient) 2−2N Γ(2N − 1)e2πiτ d4x dρ 4 G(2)(x ,x ) = 0 d5Ω d2ηA d2ξ¯ A 1 2 3 − ! − ! 5 N (N 1) (N 2) ρ = A 1 c ρ16 × 1 (x − x )2 [(x − x )2 + ρ2]8[(x − x )2 + ρ2]8 1 2 1 0 2 0 18 + c2ρ 2 2 9 2 2 9 [(x1 − x0) + ρ ] [(x2 − x0) + ρ ] 1 2 2 2 3 2 4 2 1 2 2 2 3 2 4 2 × ζ ζ ζ ζ (x1) ζ ζ ζ ζ (x2) −2N − 2πiτ 4 2 Γ(2N 1)e 8 d x0 dρ = (x1 − x2) N3(N − 1)!(N − 2)! ρ5

18 × c2ρ [(x − x )2 + ρ2]9[(x − x )2 + ρ2]9 1 0 2 0 c ρ16 + 1 2 2 2 8 2 2 8 . (216) (x1 − x2) [(x1 − x0) + ρ ] [(x2 − x0) + ρ ] S. Kovacs / Nuclear Physics B 684 (2004) 3–74 51

Summing (214) and (216) we obtain the complete result for the two-point function (210) in the one-instanton sector which reads 2 2 −15 −2N−16 2πiτ 3 5 π 2 e 8 4 G(x1,x2) = (x1 − x2) d x0 dρ N3(N − 1)!(N − 2)!

a (N)ρ13 1 a (N)ρ11 × 1 + 2 , 2 + 2 9 2 + 2 9 − 2 2 + 2 8 2 + 2 8 (y1 ρ ) (y2 ρ ) (x1 x2) (y1 ρ ) (y2 ρ ) (217) where yi = xi − x0 and all the numerical factors have been reinstated. The coefficients a1(N) and a2(N) are 2 a (N) = Γ(2N − 1), 1 3 2 a (N) = N2 − 3N + 2 Γ(2N − 2) + Γ(2N − 1). (218) 2 3 The final bosonic moduli space integrations in (217) are logarithmically divergent. The second term is exactly of the same form as that encountered in Section 5.3.1. The first term in the last line of (217) is evaluated in a completely analogous way and also leads to a simple pole singularity after dimensional regularisation of the x0 integral. As in the case of the singlets of dimension 4 we find non-zero entries in the matrix Krs for operators with ∆0 = 5 transforming in the 6 of SU(4). Therefore some of the operators in this sector acquire an anomalous dimension in the one-instanton sector. The calculation of the other two-point functions in this sector for operators made of only elementary scalars can be carried on in a similar fashion. The result is of the same form (21)i as (217) apart from the numerical coefficient. In particular the operator O5,6 vanishes in a one-instanton background, so that two-point functions involving this operator only receive contribution from terms in which pairs of scalars are contracted. The double-trace operators in (207) can be treated in a similar way. The two-point functions involving the remaining operators in this sector, (200)–(204), require higher order terms in the iterative solution of the field equations for the vector and the fermions. Without computing these two- point functions it is not possible to extract the values of the instanton induced anomalous dimensions of ∆0 = 5 operators in the 6 of SU(4). The fact that operators in this sector do receive instanton corrections is, however, also confirmed by the analysis of the four-point function in Appendix C. The singularity observed in the OPE studied there corresponds to the contribution of operators in the sector examined in this subsection.

5.4.2. ∆0 = 5, [2, 0, 0] and [0, 0, 2] In this and the following subsections we brieﬂy discuss the other SU(4) representations which appear at the level ∆0 = 5. The discussion will not be detailed. The analysis follows in a straightforward way what was done in previous sections. What makes the study of these sectors more involved is the large number of operators which makes the construction of a basis rather laborious. In particular we shall only consider single-trace operators. It is understood that for all operators involving at least four elementary ﬁelds there exist also double-trace operators. From the discussion in the previous sections it is evident that for a qualitative analysis of the zero-mode structure the number of traces is not relevant. 52 S. Kovacs / Nuclear Physics B 684 (2004) 3–74

Operators in the representation [2, 0, 0]≡10 are obtained from (186)–(192). The combination (186) can be projected onto the 10 by fully antisymmetrising the indices on the scalars

(1)[ij k] 1 µ [i j k] O = Tr D ϕ Dµϕ ϕ . (219) 5,10 3 1/2 gYMN Operators in the 10 containing fermions come from (187)–(191). We ﬁnd

(2)(AB) = 1 i i α(A B) (3)(AB) = 1 i α(A B) O5,10 4 Tr ϕ ϕ λ λα , O5,10 4 Tr ϕ λ ϕiλα , gYMN gYMN 1 (4)(AB) = t(AB) Tr Σi ϕj ϕkλαCλD , O5,10 4 [ij k] CD α gYMN (5)(AB) = 1 i j A α(B C) + B α(A C) O5,10 4 Tr ϕ ϕ Σij C λ λα Σij C λ λα , gYMN 1 (6)(AB) = Tr Σ Aϕi λα(Bϕj λC) + Σ Bϕiλα(Aϕj λC) , O5,10 4 ij C α ij C α gYMN

(7)(AB) = 1 α(A µ B) O5,10 2 Tr Dµλ D λα , gYM

(8) 1 (AB) ij D k αC µ ¯ α˙ O = t Tr Σ Dµϕ λ σ λ , 5,10 3 1/2 [ij k] C αα˙ D gYMN 1 O(9)(AB) = Tr F λα(Aσ µν β λB) , 5,10 3 1/2 µν α β gYMN

(10)(AB) 1 ˜ α(A µν β B) O = Tr Fµν λ σ λ , 5,10 3 1/2 α β gYMN

(11)(AB) 1 (AB) iCD j k α˙ = t Tr Σ¯ ϕ ϕ λ¯ ˙ λ¯ , O5,10 4 [ij k] αC D gYMN

(12)(AB) 1 (AB) iCD j k α˙ = ¯ ¯ ˙ ¯ O5,10 4 t[ij k] Tr Σ ϕ λαCϕ λD , gYMN

(13)(AB) 1 C(A B)D (i j) α˙ = ¯ ¯ ¯ ˙ ¯ O5,10 4 Σi Σj Tr ϕ ϕ λαCλD , gYMN

(14)(AB) 1 C(A B)D (i j) α˙ = Σ¯ Σ¯ Tr ϕ λ¯ ˙ ϕ λ¯ . (220) O5,10 4 i j αC D gYMN With only elementary scalars we can construct the following combinations in the 10 1 O(15)(AB) = t(AB) Tr ϕlϕlϕiϕj ϕk , 5,10 5 3/2 [ij k] gYMN 1 O(16)(AB) = t(AB) Tr ϕlϕiϕlϕj ϕk . (221) 5,10 5 3/2 [ij k] gYMN The operators in the representation [0, 0, 2]≡10 are built in a completely analogous way. Moreover for operators made either of two scalars and two fermions or of ﬁve S. Kovacs / Nuclear Physics B 684 (2004) 3–74 53 scalars there are also double-trace combinations. We should then consider the two-point functions O(r) (x )O¯ (s) (x ) to compute the instanton corrections to the matrix of 5,10 1 5,10 2 anomalous dimensions. By choosing speciﬁc components one can verify that for all the above operators and their conjugates the instanton contributions to two-point functions vanish. There are always 10 fermion modes involved and depending on the choice of components it may be possible to soak up the sixteen superconformal modes, but when this is the case the remaining integral over the ν and ν¯ fermion variables vanishes. There is no instanton correction to the scaling dimension of operators in these sectors.

5.4.3. ∆0 = 5, [0, 3, 0] The construction of operators in the representation [0, 3, 0]≡50 is very similar to what was done in the previous subsection for the 10: in the product of three 6sthe50 is obtained from the totally symmetric and traceless combination and the 10 arises as totally antisymmetric combination. We ﬁnd the following operators in this sector. From (186) we select the 50 taking

(1){ij k} 1 {i j µ k} O = Tr ϕ Dµϕ D ϕ . (222) 5,50 3 1/2 gYMN With two scalars and two fermions we get

(2){ij k} = 1 {i j k} αA B O5,50 4 Tr ΣAB ϕ ϕ λ λα , gYMN

(3){ij k} 1 AB{i j k} α˙ = Tr Σ¯ ϕ ϕ λ¯ ˙ λ¯ . (223) O5,50 4 αA B gYMN Finally there are the operators made of scalars only

(4){ij k} 1 l {i j k} O = Tr ϕ ϕlϕ ϕ ϕ , 5,50 5 3/2 gYMN

(5){ij k} 1 l {i j k} O = Tr ϕ ϕ ϕlϕ ϕ . (224) 5,50 5 3/2 gYMN As in the previous case one can easily verify that none of the above operators gives rise to non-vanishing two-point functions in the instanton background. Again depending on the component considered the vanishing of the two-point functions follows from the ﬁve- sphere integration after re-expressing the dependence on the ν and ν¯ modes in terms of angles ΩAB.

5.4.4. ∆0 = 5, [1, 1, 1], [3, 0, 1], [1, 0, 3], [2, 2, 0] and [0, 2, 2] The representations [1, 1, 1]≡64, [3, 0, 1]≡70, [1, 0, 3]≡70, [2, 2, 0]≡126 and [0, 2, 2]≡126 are rather complicated to analyse. They appear with high multiplicity and to disentangle them one needs projectors which are not straightforward to construct. We shall only brieﬂy sketch how one can proceed to deﬁne a basis of operators. In Section 6 an argument will be given that implies that there cannot be instanton contributions to two- point functions of operators in these sectors. 54 S. Kovacs / Nuclear Physics B 684 (2004) 3–74

The 64 appears in (186), (187), (188) and (192). The first type of operator, however, does not contribute for the same reason discussed in Section 5.2.4. The operators with fermions come from ⊗ → i [j k] αA B ⊗ → i {j k} αA B 15 6 Tr ΣAB ϕ ϕ λ λα 64, 20 6 Tr ΣAB ϕ ϕ λ λα 64, ⊗ → [i j] α(A B) ⊗ → {i j} α(A B) 15 10 Tr ϕ ϕ λ λα 64, 20 10 Tr ϕ ϕ λ λα 64, ⊗ → i αA µ ¯ α˙ − 1 A αC µ ¯ α˙ 15 6 Tr Dµϕ λ σαα˙ λB δ B λ σαα˙ λC , (225) 4 64 where the notation indicates that each combination has to be suitably projected onto the 64, which requires to make it orthogonal to the operators in the other representations appearing in the same tensor product. The operators involving spinors in the 4 can be obtained in a similar fashion. The construction of operators in the 64 made of five scalars is very involved. The 64 appears in 6 ⊗ 6 ⊗ 6 ⊗ 6 ⊗ 6 with multiplicity 24. This does not take into account the cyclicity of the trace. To incorporate this we can build the operators leaving one scalar fixed in the first position in the trace and combining the remaining four scalars respectively into the representations 15, 20, 45, 45, 84 and 175. The corresponding combinations have been discussed in the section on operators of bare dimension ∆0 = 4. We then need to project the tensor products 6 ⊗ 15, 6 ⊗ 20, 6 ⊗ 45, 6 ⊗ 45, 6 ⊗ 84 and 6 ⊗ 175, which all include the 64. The 70 appears in (187) and (192). The operator made of two scalars and two fermions is contained in the first combination in the second line of (225). The operators made of only scalars can be extracted with the procedure outlined in the previous paragraph from 6 ⊗ 45 and 6 ⊗ 84.Forthe70 one can proceed in a similar way. Operators in the 126 also arise from (187) and (192). One operator of the form (187) is in the decomposition of the second combination in the second line of (225). Then there are operators made of five scalars which can be obtained from 6 ⊗ 45 and 6 ⊗ 175. Again the conjugate operators in the 126 are treated in a completely analogous way. The same reasoning can be applied to the construction of double-trace operators in these sectors.

5.4.5. ∆0 = 5, [0, 5, 0] In the representation [0, 5, 0]≡196 we ﬁnd one single-trace and one double-trace operator. These only involve elementary scalars and belong to the same class as the operators discussed in Sections 5.1, 5.2.3 and 5.3.6, i.e., the class of scalar operators of dimension ∆0 = transforming in the representation [0,,0]. For such operators it is always possible to choose a component which can be written in terms of only one complex scalar, φI . For the single-trace operator in the 196 we can take Tr(φ1φ1φ1φ1φ1),which is immediately veriﬁed to have vanishing two-point functions in the instanton background. This is in fact a 1/2 BPS operator dual to a third Kaluza–Klein excited mode of a 5 supergravity scalar in AdS5 × S . A similar argument can be made for the double-trace operator. S. Kovacs / Nuclear Physics B 684 (2004) 3–74 55

5.4.6. ∆0 = 5, [2, 1, 2] and [1, 3, 1] Operators in the representations [2, 1, 2]≡300 and [1, 3, 1]≡384 can only be obtained from the product of five scalars, i.e., from (192) and the analogous double-trace combinations. Taking into account the cyclicity of the trace one finds that the representation 384 is not realised in terms of gauge-invariant operators. This is analogous to what was observed for the 64 at ∆0 = 3andthe175 at ∆0 = 4. Operators in the 300 belong to the class [29] of operators of dimension ∆0 = 2a + b transforming in the representation [a,b,a], for which it is always possible to single out components that only involve two complex scalars fields, φI and φJ . From Eq. (A.8) in Appendix A and the discussion in the previous sections it is then clear that operators of this type cannot contain the required combination (41) of superconformal fermion zero-modes and therefore all their two-point functions vanish in the instanton background. This applies to both single- and double-trace operators.

6. Discussion and conclusions

In the previous sections we have analysed instanton contributions to two-point correlation functions of scalar operators of bare dimension ∆0 = 2, 3, 4, 5intheN = 4 SYM theory in the semiclassical approximation. In this way it has been possible to identify the sectors in which operators get instanton corrections to their scaling dimensions. One of our motivations was to try to understand to what extent the S-duality of the theory manifests itself in the spectrum of anomalous dimensions at weak coupling. It is somewhat surprising that very few operators among those considered are corrected by instantons. Our results show that there is a large class of operators that display non-renormalisation properties in topologically non-trivial sectors. It was already known [39] that operators belonging to the Konishi multiplet, which acquire an anomalous dimension in perturbation theory, are not corrected by instanton effects. We have now shown that the same is true for the majority of the scalar operators of bare dimension ∆0 5. As already remarked these results imply the absence of instanton corrections to a much larger set of operators. The non-renormalisation properties extend to all the components of the multiplets for which a representative was found not to be corrected. Many of the operators that have been studied in this paper, and for which no instanton corrections arise, have also been studied in perturbation theory, 5 as well as at strong coupling via the dual supergravity amplitudes in AdS5 × S [29,41, 52,57,59,60]. At the perturbative level the scaling dimensions of those belonging to long multiplets do get an anomalous quantum correction. The non-renormalisation properties observed in this paper are therefore rather surprising, in particular in view of the S-duality of the theory. Although single operators might transform in a complicated way under S-duality the full spectrum of scaling dimensions must be invariant and instantons are expected to play a crucial rôle in implementing the duality as is the case in the dual type IIB string theory. A more careful analysis shows, however, that the observed non-renormalisation properties are speciﬁc to the cases of operators of relatively small bare dimension considered here. In order to explain this we need to recall the general discussion of instanton contributions to two-point functions in Section 4.1. In the one-instanton sector 56 S. Kovacs / Nuclear Physics B 684 (2004) 3–74

(and up to a non-zero overall numerical constant) a generic two-point function is given by ¯ r s O (x1)O (x2) 4 = rs ; 8+n+m 2πiτ 4 5 2 A 2 ¯A n+m−5 c α(m,n N)gYM e dρ d x0 d Ω d η d ξ ρ = A 1 ˆ r ¯ ˆ s ¯ × O x1; x0,ρ,η,ξ,νν(Ω)¯ O x2; x0,ρ,η,ξ,νν(Ω)¯ , (226) where n an m denote respectively the number of (νν)¯ 6 and (νν)¯ 10 bilinears entering the expressions for the operators in the instanton background. The notation in the second line indicates that these bilinears have been re-expressed in terms of the angular variables ΩAB . The N-dependence is contained in the function α(m,n; N) defined in Eq. (21). In evaluating (226) one first computes the integrations over the superconformal modes which yield a non-vanishing the result if and only if both operators contain the combination (ζ 1ζ 1)(ζ 2ζ 2)(ζ 3ζ 3)(ζ 4ζ 4). After performing the integrals over the ηsandξ¯s one is left with the integration over the five-sphere and over the original bosonic collective coordinates, x0 and ρ. The latter integrals are in general logarithmically divergent as follows from dimensional analysis, signalling a contribution to the matrix of anomalous dimensions. Now consider the cases of the operators studied in this paper. The profiles of operators AB of dimension ∆0 = 2 and 4 contain no dependence on Ω , those of operators of AB dimension ∆0 = 3 and 5 are linear in Ω . Let us assume that there are operators with ∆0 = 2, 4 in two different sectors, i.e., transforming in different representations r1 and r2, which both receive instanton corrections, so that the two-point functions ¯ = O∆0,ri (x1)O∆0,ri (x1) inst,i 1, 2, are different from zero. This would lead to a paradox. Since the five-sphere integral in this case is trivial we would find that the two-point function ¯ O∆0,r1 (x1)O∆0,r2 (x1) inst is also non-zero, but this is forbidden by the SU(4) symmetry. The same situation would arise if there were instanton corrections to two-point functions of both an operator of dimension 2 and one of dimension 4. In this case we would have an even worse situation: there would be a non-vanishing two-point function in which the two operators have different dimension and this would violate conformal invariance. A similar argument can be repeated in the cases ∆0 = 3 and 5. In these cases the five-sphere integrals are of the form 5 AB CD 1 d ΩΩ Ω = εABCD. (227) 4 Again we would find that if there were non-zero two-point functions in two different sectors, then the mixed two-point functions would also involve the same integral (227) and so would not vanish, thus violating the SU(4) symmetry. There is no similar problem associated with overlapping of different SU(4) representations for two-point functions of one operator with ∆0 = 3, 5 and one with ∆0 = 2, 4.Inthiscasewewouldgetafive-sphere integral of the form d5ΩΩAB, (228) which vanishes identically. S. Kovacs / Nuclear Physics B 684 (2004) 3–74 57

This argument shows that there can be instanton corrections to the anomalous dimensions in one and only one sector among those present at ∆0 = 2, 4 and in one and only one sector among those with ∆0 = 3, 5. This is consistent with our findings in the previous sections, where by direct inspection we have shown that instantons only correct operators in the sectors (∆0 = 4, [0, 0, 0]) and (∆0 = 5, [0, 1, 0]). What is special about the cases we have considered is that they always involve (almost) trivial five-sphere integrals. The general situation, expected for operators of larger dimension, is that a more complicated dependence on the angles ΩAB should enter in the operator profiles. In this way it is possible to have non-vanishing contributions in various sectors, but zero overlap between different sectors since the angular dependence makes them ‘orthogonal’ with respect to the five-sphere integration 5 ˜¯ ; ˜ ; = = d Ω Ori (x1 x0,ρ,Ω)Orj (x2 x0,ρ,Ω) 0iffri rj , (229) where we have indicated with a tilde the expressions of the operators after the integration over the superconformal zero-modes. It is therefore natural to conjecture that the operators analysed here are special and that generically for larger values of ∆0 most of the operators that receive corrections in perturbation theory are also corrected by instantons. Some of the non-renormalisation properties are not restricted to small values of the bare dimension. All the operators in the SU(2) subsector transforming in a representation [a,b,a] with ∆0 = 2a + b, as well as the other components of multiplets containing such operators, do not receive instanton corrections, irrespective of the value of ∆0. The above argument explains why so many operators among those considered appear to be ‘protected’ against instanton effects, whereas they get corrections in perturbation theory. On the other hand comparing the perturbative and non-perturbative sectors we notice that operator mixing is much more complicated at the instanton level. At small orders in perturbation theory it is possible to identify subsets of operators that do not mix with others in the same sector and thus compute the anomalous dimensions within the subsector. For instance, at one loop two-point functions involving one operator made of only scalars and one containing fermion bilinears in the same sector vanish. For this reason in [29,59] it was possible to compute the one-loop anomalous dimension of the SU(4) singlet scalar operators (107), (108) without having to consider the mixing with other operators in that sector. The study of instanton corrections to two-point functions shows that in general whenever a sector receives correction the mixing occurs among all the operators in that sector. This phenomenon was to be expected: in general the mixing begins to be relevant at some order in perturbation theory and the instanton effects we have considered are subleading with respect to contributions at any order in the perturbative expansion. In the present paper we have restricted our attention to the one-instanton sector. Part of the results remain valid in multi-instanton sectors. In particular the non-renormalisation results hold for arbitrary instanton number K, since they only rely on the analysis of the superconformal zero-modes. The calculation in sectors in which there are non-vanishing contributions is in general much more complicated for K>1. Some cases however can be treated for any K in the large-N limit. This is true for sectors in which at leading order in gYM no non-exact modes enter in the operators and contractions are also not 58 S. Kovacs / Nuclear Physics B 684 (2004) 3–74 allowed. This is the case in particular for the singlet at ∆0 = 4: the two-point functions for arbitrary K in the large-N limit coincide with the one-instanton results of Section 5.3.1 up to a calculable K-dependent coefficient. This is because a saddle-point approximation can be used for large N. When there is a non-trivial dependence on the non-exact modes their evaluation at the saddle-point is complicated and the generalisation much more difficult. The extension of the calculations presented here to the case of orthogonal or symplectic gauge groups is also straightforward. It should be noted, however, that the construction of bases of independent gauge-invariant operators is significantly different in these cases. We have considered only Lorentz scalars, but the same methods can be used to study non-scalar operators. In this case the full set of PSU(2, 2|4) quantum numbers, (∆0,J1,J2;[a,b,c]) is needed to identify the various sectors. Therefore the spectrum is richer and the construction of independent operators in each sector more involved, but once this is done the computation of anomalous dimensions proceeds on the same lines as in the cases considered in this paper. The non-renormalisation properties that we have derived in semiclassical approximation can be argued to remain valid at higher orders in gYM. The simplest contributions beyond the semiclassical result come from additional insertions of νν¯ bilinears, which modify the integrations over the angular variables ΩAB. The resulting five-sphere integrals are only non-vanishing if an equal number of modes of each flavour is contained in the combination of νν¯ s. This implies that including subleading terms with more fermion modes cannot produce non-zero results for two-point functions which vanish at leading order. A similar argument can be made for higher-order corrections involving contractions because the propagator (B.41) is proportional to εABCD. As previously observed non-protected operators of large dimension are expected generically to also receive instanton corrections. It is therefore natural to ask what the situation is for the operators which are relevant for the BMN limit. These are operators of large dimension and large charge with respect to one of the generators of the R-symmetry group and the techniques developed here can be applied to the study of such operators. Instanton effects in the BMN limit and the comparison with string theory in a plane wave background are currently under investigation [62]. As discussed in the introduction, recently there have been many interesting developments in connection with the computation of anomalous dimensions in the N = 4SYM theory, leading to new non-trivial tests of the AdS/CFT correspondence [2,7,10–12,18,19, 23,25]. Of particular interest is the emergence of an integrability structure in the theory [27,29,30,42], which appears to have a counterpart in the dual string theory. On the string side this integrability property appears at the level of the semiclassical analysis of solitonic string configurations [19,24] as well as through the emergence of an infinite set of nonlocal classically conserved charges [31–33]. In the gauge theory the integrability arises most naturally when the problem of computing scaling dimensions for gauge-invariant operators is reformulated in terms of an eigenvalue problem for the dilation operator, Dˆ , [29, 30,42]. Dˆ acts on gauge-invariant composite operators as

ˆ (r) (r) DO = ∆r O , (230) S. Kovacs / Nuclear Physics B 684 (2004) 3–74 59

(r) where ∆r is the scaling dimension of O . Therefore knowing the form of the dilation operator allows to compute anomalous dimensions in a very efficient way expanding (230). This is the approach followed in [29,41,42] at the perturbative level. The results of the present paper do not seem to be relevant for the issue of the integrability of N = 4 SYM. This is because if indeed an integrable structure survives in the full quantum theory it is expected to arise only in the planar limit. In this limit instanton effects are exponentially suppressed, since they produce contributions of order − π2/g2 − π2N/λ e 8 YM ∼ e 8 . On the other hand the possibility of computing instanton induced corrections to the dilation operator in N = 4 SYM is very interesting in itself and our results represent a first step in this direction. Although we have not addressed this issue in the previous sections, we can make some general considerations based on the results obtained for two-point functions. First of all in the closed SU(2) sector of operators of dimension ∆0 = 2a +b transforming in the [a,b,a] of SU(4) the dilation operator does not receive non-perturbative corrections at all. The same is true for the even simpler sector of ∆0 = operators in the [0,,0]. This has been verified explicitly in Sections 5.1 (∆0 = 2, [0, 2, 0]), 5.2.3 (∆0 = 3, [0, 3, 0]), 5.3.5 (∆0 = 4, [2, 0, 2]), 5.3.6 (∆0 = 4, [0, 4, 0]), 5.4.5 (∆0 = 5, [0, 5, 0]) and 5.4.6 (∆0 = 5, [2, 1, 2] and [1, 3, 1]). Using the fact that operators in these sectors have components that can be written in terms of only two complex scalars it is possible to generalise the result to arbitrary subsectors of the above type. In general, however, the dilation operator contains instanton corrections as well and it can be expanded as ∞ ∞ ˆ = n ˆ + m 2πiKτ ˆ D cn(N)gYMDn c(K,m)(N)gYMe D(K,m), (231) n=0 K>0 m=0 where the first sum denotes the perturbative contributions and the second double sum incorporates the instanton corrections including the perturbative fluctuations around the leading semiclassical term in each instanton sector. An analogous series of anti-instanton contributions (proportional to e2πiτK¯ ) has not been indicated explicitly. ˆ Focusing on the non-perturbative part, it is natural to construct the terms D(K,m) not as operators acting on the elementary fields of the N = 4 theory, as in perturbation theory, but as operators acting on the multi-instanton collective coordinates. In other words one can consider the action of the dilation operator as realised on the instanton moduli space. In [63] it was shown that the supersymmetry algebra (in the N = 2 case) can be realised in a simple and elegant way on the ADHM collective coordinates (see Appendix B) before imposing the ADHM constraints. The construction of the instanton supermultiplet acting with broken supersymmetries on the bosonic solution, which was outlined in Section 3 for the SU(2) case, is an example of application of this idea. As already observed, this approach can be implemented directly in superspace, at least for N = 1 supersymmetric theories. In [64] a superspace description of the SU(2) one-instanton moduli space in N = 4 SYM was presented and the corresponding superconformal transformations were used in the computation of instanton contributions to Wilson loops. The realisation of symmetries on the instanton moduli space can be generalised to the whole superconformal group and in particular the action of dilations can be analysed. The strategy of [29,41] was to write down the most general form of Dˆ compatible with the structure of the 60 S. Kovacs / Nuclear Physics B 684 (2004) 3–74 perturbative two-point functions and then fix the unknown coefficients using known values of anomalous dimensions. The same procedure can be used to determine the form of the ˜ dilation operator on the instanton moduli space. We shall denote the latter by D(K,m) to distinguish it from its counterpart acting on the space of fields. From our analysis of ˜ two-point functions it is clear that D(K,m) contains eight derivatives with respect to the variables ζ A. In the one-instanton sector it also involves derivatives with respect to νA and ν¯ A, which can be replaced by derivatives with respect to the angular variables ΩAB . ˜ We can argue that D(1,m) must be of the form ˜ ∼ 8 A1B1...AmBm D(1,m) gYMc(N)d(x0,ρ)tN δ8 δm × , (232) δ(ζ1)2δ(ζ2)2δ(ζ3)2δ(ζ4)2 δΩA1B1 ···δΩAmBm 8 where the factor of gYM comes from the measure and√ the dependence on N is contained A1B1···AmBm in the coefficient c(N), which for large N behaves as N, and in the tensor tN . The latter is a projector onto SU(4) singlets. It consists of various terms corresponding to the different ways of combining ν¯ [AνB] and ν¯(AνB) bilinears to form a singlet. The N-dependence is determined by the number of SU(4) indices paired in a 6 or in a 10. ˜ D(1,m) has also a dependence on the bosonic collective coordinates which is encoded in ˜ the function d(x0,ρ). Determining the exact form of D(1,m) appears to be feasible, but more data on instanton induced anomalous dimensions are needed. We hope to investigate further this issue in the future. In view of the previous discussion this requires the study of operators of larger dimension. The results obtained in this paper mostly show non-renormalisation properties for a large class of operators, which, however, are expected to be specific to small values of ∆0. For greater values of the bare dimension the situation is more involved because of the larger number of operators appearing. From the point of view of instanton calculations there is, however, a simplification since as ∆0 grows fewer terms in the iterative solution for the elementary fields are required in semiclassical approximation. Moreover as previously remarked further simplifications arise when imposing the constraints of PSU(2, 2|4), which imply that all operators in a same multiplet have equal anomalous dimension.

Acknowledgements

I would like to thank Gleb Arutyunov, Niklas Beisert, Massimo Bianchi, Michael Green, Jan Plefka, Yassen Stanev and Matthias Staudacher for useful and stimulating discussions. I am particularly grateful to Massimo Bianchi and Matthias Staudacher for their comments on the manuscript. This work was supported in part by the European Commission RTN Programme under Contracts HPRN-CT-2000-00122 and HPRN-CT-2000-00131.

Appendix A. Conventions and useful relations

In this appendix we summarise the notation used in the paper and collect some useful relations. S. Kovacs / Nuclear Physics B 684 (2004) 3–74 61

Lower case Latin letters, i,j,k,... are used for the 6 of SU(4) and capital letters, A,B,C,... for the 4.SO(4) Lorentz spinor and vector indices are indicated by Greek letters, respectively, from the beginning, α,β,...,α,˙ β,...˙ , and the mid, µ,ν,...,ofthe alphabet. SU(N) colour indices are denoted by Latin letters from the end of the alphabet, r,s,u,v,.... = i A The N 4 multiplet comprises six real scalars, ϕ , four Weyl fermions, λα ,anda vector, Aµ with ﬁeld strength Fµν , all transforming in the adjoint representation of the gauge group. It is often convenient to label the scalars by an antisymmetric pair of indices in the 4, ϕ[AB], subject to the reality condition

AB ∗ 1 CD ϕ¯AB ≡ ϕ = εABCDϕ . (A.1) 2 In some situations the N = 1 formulation proves very useful. The N = 1 decomposition of the N = 4 supermultiplet consists of three chiral multiplets and one vector multiplet and under this decomposition only a SU(3) × U(1) subgroup of the SU(4) R-symmetry group is manifest. The six scalars are combined into three complex ﬁelds, φI ,I= 1, 2, 3, according to

1 + 1 + φI = √ ϕI + iϕI 3 ,φ† = √ ϕI − iϕI 3 . (A.2) 2 I 2 I † × The complex scalars φ and φI transform, respectively, in the 31 and 3−1 of SU(3) U(1). The fermions in the chiral multiplets are

I = I ¯ α˙ = ¯ α˙ = ψα λα, ψI λI ,I1, 2, 3, (A.3) transforming in the 33/2 and 3−3/2. The fourth fermion and the vector form the N = 1 { = 4 } × vector multiplet, λα λα,Aµ ,andareSU(3) U(1) singlets. The two parametrisations of the N = 4 scalars, ϕi and ϕAB , are related by

i = √1 i AB AB = √1 ¯ AB i ϕ Σ ϕ ,ϕ Σi ϕ , (A.4) 2 AB 8 i ¯ AB where ΣAB (Σi ) are Clebsch–Gordan coefﬁcients projecting the product of two 4s(4s) onto the 6. These are in other words six-dimensional euclidean sigma matrices. They are deﬁned as

+ Σi = Σa ,Σa 3 = ηa ,iη¯a , AB AB AB AB AB ¯ AB = ¯ a ¯ a+3 = − AB ¯AB Σi ΣAB , ΣAB ηa ,iηa , (A.5) = a ¯a where a 1, 2, 3 and the ’t Hooft symbols ηAB and ηAB are a =¯a = = ηAB ηAB εaAB,A,B1, 2, 3, a =¯a = a ηA4 η4A δA, a =− a ¯a =−¯a ηAB ηBA, ηAB ηBA. (A.6) 62 S. Kovacs / Nuclear Physics B 684 (2004) 3–74

Using these deﬁnitions we ﬁnd the following relations among the scalars in the different formulations √ √ √ 2 2 2 ϕ1 = ϕ14 + ϕ23 ,ϕ2 = −ϕ13 + ϕ24 ,ϕ3 = ϕ12 + ϕ34 , √2 2 √ 2 √ i 2 i 2 i 2 ϕ4 = −ϕ14 + ϕ23 ,ϕ5 = −ϕ13 − ϕ24 ,ϕ6 = ϕ12 − ϕ34 2 2 2 (A.7) and φ1 = 2ϕ14,φ2 = 2ϕ24,φ3 = 2ϕ34, † = 23 † =− 13 † = 12 φ1 2ϕ ,φ2 2ϕ ,φ3 2ϕ . (A.8) The following properties of the six-dimensional sigma matrices are of use ABCD i =− ¯ iAB ¯ iCD =− i ε ΣCD 2Σ ,εABCDΣ 2ΣAB, (A.9) i ¯ jCB = ij B + ij B ΣAC Σ δ δA ΣA , (A.10)

ij B = iAC ¯ j − jAC ¯ i where ΣA Σ ΣCB Σ ΣCB /2,

¯ iAB i A B A B i i Σ Σ = 2 δ δ − δ δ ,ΣΣ = 2εABCD, (A.11) CD D C C D AB CD Σij B Σkl A =−4 δikδjl − δilδjk , (A.12) A B tr Σ¯ i Σj Σ¯ kΣl = 4δij δkl − 4δikδjl + 4δilδjk. (A.13)

Appendix B. One-instanton in N = 4SYM

In this appendix we briefly review the ADHM description of the one-instanton sector for the N = 4 SYM theory. A comprehensive review, including the treatment of multi- instanton configurations, can be found in [45]. Here we only recall a few elements useful for the calculations presented in this paper. The classical instanton solution is defined in terms of a [N +2]×[2]-dimensional matrix = µ ∆aα;˙α which is a linear function of the space–time coordinate xαα˙ σαα˙ xµ, β ∆uα;˙α = auα;˙α + buα; xβα˙ , (B.1) and its conjugate, ¯α˙;uα α˙;uα αβ˙ ¯ uα ∆ =¯a + x bβ; . (B.2) The ADHM gauge field is written in the form v ¯ rα v (Aµ)u; = Uu; ∂µUrα; , (B.3) where the complex [N]×[N + 2] matrix U(x) and its hermitian conjugate U(x)¯ satisfy ¯ rα v v Uu; Urα; = δu , ¯α˙;rβ v ¯ rα ∆ Urβ; = 0, Uu; ∆rα;˙α = 0. (B.4) S. Kovacs / Nuclear Physics B 684 (2004) 3–74 63

Eq. (B.3) gives a gauge configuration with self-dual field strength provided the matrices ∆ and ∆¯ satisfy ¯α˙;rβ = α˙ −1 ∆ ∆rβ;β˙ δ β˙f (x), (B.5) where f(x)is an arbitrary function. From this relation and the definitions (B.1) and (B.2) it follows that the coefficients a and b satisfy the bosonic ADHM constraints

α˙;uα 1 α˙ a¯ a ; ˙ = Tr(aa)δ¯ ˙ , (B.6) uα β 2 β ¯ α˙;uα β = βγ α˙ β˙ ¯ uα a buα; bγ ; auα;β˙, (B.7) ¯ uα γ 1 ¯ γ b ; b ; = Tr(bb)δ . (B.8) β uα 2 β A choice of special frame allows to put the bosonic parameters in the form ;β ;u β 0u ¯ uβ 0α b ; = , b ; = , (B.9) uα β α β δα δα

wu;˙α α˙;uα α˙ ;u ˙αα auα;˙α = , a¯ = w¯ , a¯ , (B.10) aαα˙ = µ where aαα˙ σαα˙ aµ and the components satisfy the matrix constraints c ¯ = ∗ = Tr2(τ aa) 0,(aµ) aµ. (B.11) These ADHM collective coordinates can be easily related to the usual variables describing the position, size and gauge orientation of the instanton. The second equation in (B.11) expressing the reality of a allows to identify it with the instanton position, =− aµ (x0)µ. (B.12) Using the ﬁrst equation in (B.11) the scale size of the instanton ρ is related to the ADHM variables by ¯ αu˙ = α˙ 2 w wuβ˙ δ β˙ρ . (B.13) α˙ ;u The bosonic coordinates wu;˙α and w¯ parametrise the embedding of an SU(2) instanton into SU(N). We start with the special embedding = 00 Aµ SU(2) , (B.14) 0 (Aµ) SU(2) where (Aµ) is the standard SU(2) instanton solution 2 a ν a β ρ η (x − x0) (τ )α (A )SU(2) = µν . (B.15) µ 2 2 2 (x − x0) [(x − x0) + ρ ] The general SU(N) conﬁguration is then given by = 00 † Aµ H SU(2) H , (B.16) 0 (Aµ) 64 S. Kovacs / Nuclear Physics B 684 (2004) 3–74 where 0[N−2]×[2] SU(N) wuα˙ = ρH , H ∈ . (B.17) 1[2]×[2] SU(N − 2) × U(1) The fermionic collective coordinates enter as [N + 2]×[1] Grassmann valued matrices, M and M¯ , that satisfy the ADHM constraints

¯ Auα =− ¯ β˙;uα A ¯ Auα β = βγ ¯ uα A M auα;˙α α˙β˙a Muα, M buα; bγ ; Muα. (B.18) The fermionic matrices can be parametrised by A αA˙ A ¯ αA˙ ν + wu;˙αµ ν + 4wu;˙αξ MA = u ≡ u , (B.19) uα M A 4ηA α α ¯ Auα = ¯ Au +¯A ¯ α˙ ;u ¯ Aα ≡ − ¯ A ¯ αu˙ +¯Au αA M ν µα˙ w , M 4ξα˙ w ν , 4η , (B.20) ¯ A ¯ Aα where the ADHM conditions (B.18) have been used to eliminate µα˙ and M in flavour A ¯ Au of the others and the variables νu and ν satisfy ¯ α˙ ;u A =¯Au = w νu ν wu;˙α 0. (B.21) A ¯A The sixteen fermionic collective coordinates ηα and ξα˙ are identified with the zero modes associated respectively with the Poincaré and special supersymmetries broken by the A ¯ Au − bosonic instanton solution. The coordinates νu and ν , whose total number is 8(N 2) because of the constraints (B.21), are the fermionic partners of the coset variables wu;˙α and w¯ α˙ ;u parametrising the gauge orientations. We now summarise the expressions for the leading order terms in the iterative solution of the field equations for the elementary fields in the N = 4 SYM multiplet as given in terms of the ADHM variables. All the fields in the multiplet are in the adjoint representation of the gauge group SU(N) and can be represented as [N]×[N] matrices. (0)− The (self-dual part of the) gauge field strength Fµν in the instanton background follows from the construction of the gauge field Aµ in the previous section. The result is

(0)µν v rα β µν γ sδ v = ¯ ; ; ¯ ; ; F u; Uu brα σ β f bγ Usδ . (B.22) (1)A The Weyl fermions λα solve the equation (0) (1)A = D/ λα 0 (B.23) and in terms of the ADHM variables can be written as

(1)A v rβ A sγ δ Asγ v = ¯ ; ¯ ; − ; ¯ ; λα u; Uu Mrβf bα εαδbrβ f M Usγ . (B.24) Analogously the term ϕ(2)AB in the solution for the scalar ﬁeld is determined by 1 D2ϕ(2)AB = √ λ(1)A,λ(1)B . (B.25) 2 S. Kovacs / Nuclear Physics B 684 (2004) 3–74 65

The solution was constructed in [46]. In the N = 4 SYM theory in the superconformal phase of interest here it can be written in the form

(2)AB = 1 ¯ rα B ¯ Asβ − A ¯ Bsβ v ϕ Uu; Mrαf M Mrαf M Usγ; 2 0 ;s 0 ;β 1 ¯ rα r r v + Uu; Usβ; , (B.26) s AB β 2 0α; A δα where AAB is deﬁned by

AB = √1 ¯ Arα B − ¯ Brα A A M Mrα M Mrα . (B.27) 2 2 ρ2 The gauge invariant composite operators we are interested in are traces over colour indices of products of elementary ﬁelds. In evaluating correlation functions of such operators in the semiclassical approximation in the instanton background one must then compute expressions of the form ¯ ˜ ¯ ˜ ¯ O = TrN (UFU···UλU ···UϕU˜ ···). (B.28) It is convenient to rewrite such expressions as traces over [N + 2]×[N + 2] matrices in the following way

O = Tr (U¯ FU˜ ···U¯ λU˜ ···U¯ ϕU˜ ···) N ˜ ˜ = TrN+2 (P F)···(P λ) ···(P ϕ)˜ ··· , (B.29) where we have deﬁned the projection operator vβ = r ¯ vβ = v β − γ ¯ vβ Puα; Uuα; Ur; δu; δα ∆uα; f ∆γ ; , (B.30) where ∆ and ∆¯ are the matrices of bosonic ADHM variables deﬁned in (B.1) and (B.2). In the one-instanton sector all the previous formulae can be made very explicit. The function f(x)entering the ADHM construction is

= ; = 1 = 1 f f(x x0,ρ) 2 2 2 2 (B.31) (x − x0) + ρ y + ρ and the projector P becomes ˙ ; ˙ w ;˙w¯ α v w ;˙yαβ vβ v β 1 u α u α Puα; = δu; δ − . (B.32) α 2 + 2 α˙ ;v 2 β y ρ yαα˙ w¯ y δα Notice in particular that it satisﬁes TrN+2 P (x) = N. (B.33) As observed above the gauge invariant composite operators can be expressed as traces over [N + 2]×[N + 2] matrices. In the following we give the formulae for the ‘projected’ [N + 2]-dimensional matrices corresponding to the solutions (B.22), (B.24) and (B.26) for the elementary ﬁelds. For this purpose we introduce the notation for a generic Yang–Mills 66 S. Kovacs / Nuclear Physics B 684 (2004) 3–74

ﬁeld Φ v ¯ rβ ˜ sγ v ˆ vβ rγ ˜ vβ Φu; = Uu; Φrβ; Usγ; , Φuα; = Puα; Φrγ; . (B.34) 8 For the ﬁeld strength Fµν we get ˆ (1) 0 (F ) ;β ˆ vβ µν u (Fµν )uα; = , (B.35) ˆ (2) β 0 (Fµν )α; where

ˆ (1) β 4 αγ˙ β F =− w ;˙y σ , mn u; (y2 + ρ2)2 u α mnγ 4 Fˆ (2) β = ρ2σ β . (B.36) mn α; (y2 + ρ2)2 mnα A The solution for the fermions λα is ˆ (1)A v ˆ (2)A γ (λα )u; (λα )u; λˆ A vγ = , (B.37) α uβ; ˆ (3)A v ˆ (4)A γ (λα )β; (λα )β; where

ˆ (1)A v 1 αδ˙ ¯ A β˙;v Av λ = ε w ;˙y −4ξ ˙ w¯ +¯ν , α u; (y2 + ρ2)2 αδ u α β ˆ (2)A γ 1 2 ¯ αA˙ γ αδ˙ A γ αδ˙ γA λ = 4 y w ;˙ξ δ − w ;˙y η δ + ε w ;˙y η α u; (y2 + ρ2)2 u α α u α δ α αδ u α + 2 + 2 A γ y ρ νu δα , 2 ˆ (3)A v ρ ¯ A α˙ ;v Av λ = ε 4ξ ˙ w¯ −¯ν , α β; (y2 + ρ2)2 αβ α 2 ˆ (4)A γ 4ρ ¯αA˙ γ A γ γA λ = −y ˙ ξ δ + η δ − ε η . (B.38) α β; (y2 + ρ2)2 βα α β α αβ Finally the scalar ﬁeld ϕAB reads (1)AB v (2)AB γ (ϕˆ )u; (ϕˆ )u; ϕˆAB vγ = , (B.39) uβ; (3)AB v (4)AB γ (ϕˆ )β; (ϕˆ )β; where (1)AB v 1 2 ¯αB˙ ¯ A ¯αA˙ ¯ B β˙;v ϕˆ = y −16 ξ ξ ˙ − ξ ξ ˙ w ;˙w¯ u; 4(y2 + ρ2)2 β β u α ¯αB˙ Av ¯αA˙ Bv + 4w ;˙ ξ ν¯ − ξ ν¯ u α 2 2 ¯ B A ¯ A B α˙ ;v B Av A Bv + y + ρ −4 ξ ˙ ν − ξ ˙ ν w¯ + ν ν¯ − ν ν¯ α u α u u u αδ˙ B A A B β˙;v B Av A Bv + y η ξ¯ − η ξ¯ w ;˙w¯ − w ;˙ η ν¯ − η ν¯ , 16 δ β˙ δ β˙ u α 4 u α δ δ

8 In the remainder of this appendix we omit the superscript denoting the number of fermion modes on the ﬁelds. Superscripts in parenthesis refer here to matrix elements. S. Kovacs / Nuclear Physics B 684 (2004) 3–74 67

(2)AB γ 1 2 ¯αB˙ γA ¯ αA˙ γB ϕˆ = 16y w ;˙ ξ η − ξ η u; 4(y2 + ρ2)2 u α

+ 4 y2 + ρ2 νBηγA − νAηγB u u

− αδ˙ B γA − A γB wu;˙α 16y ηδ η ηδ η 2 2 1 y + ρ ˙ + yαγ ν¯ AuνB −¯νBrνA , 2 ρ2 u r (3)AB v 1 2 ¯αB˙ ¯ A ¯ αA˙ ¯ B β˙;v ϕˆ = ρ 16yβα˙ ξ ξ ˙ − ξ ξ ˙ w¯ β; 4(y2 + ρ2)2 β β

¯αB˙ Av ¯ αA˙ Bv − 4yβα˙ ξ ν¯ − ξ ν¯ − B ¯ A − A ¯ B ¯ α˙ ;v + B ¯ Av − A ¯Bv 16 ηβ ξα˙ ηβ ξα˙ w 4 ηβ ν ηβ ν ,

2 (4)AB γ ρ ¯ αB˙ γA ¯αB˙ γA B γA B γA ϕˆ = −16yβα˙ ξ η − ξ η + 16 η η − η η β; 4(y2 + ρ2)2 β β

1 y2 + ρ2 + δγ ν¯ Ar νB −¯νBrνA . (B.40) 2 ρ2 β r r

In the calculation of correlation functions at leading order in gYM in non-trivial topological sectors we need the expression for the propagators in the instanton background. The propagator for the adjoint scalars in the one-instanton background takes the form ˜AB uγ ˜CDvδ ϕrα (x)ϕsβ (y) ≡ ˜ ABCD[ uγ,vδ ] G rα,sβ (x, y) g2 εABCD = YM P (x)P (y) vδ P (y)P (x) uγ 2π2(x − y)2 rα sβ − 1 uγ vδ + vδ uγ P (x) rα P (y)P (x)P (y) sβ P (y) sβ P (x)P (y)P (x) rα N 1 uγ vδ + P (x) [P (y)] Tr + P (x)P (y) N2 rα sβ N 2 g2 εABCD + YM P (x)bb¯P (x) uγ P (y)bb¯P (y) vδ 4π2ρ2 rα sβ − 1 uγ ¯ vδ ¯ P (x) rα P (y)bbP (y) sβ Tr2 bP (x)b N vδ ¯ uγ ¯ + P (y) P (x)bbP (x) Tr2 bP (y)b sβ rα 1 + P (x) uγ P (y) vδ Tr b¯P (x)b Tr b¯P (y)b . (B.41) N2 rα sβ 2 2 ¯ B The propagators for the fermions, λAα˙ λα , and for the vector, AµAν , can be deduced from the scalar Green function [48]. 68 S. Kovacs / Nuclear Physics B 684 (2004) 3–74

Appendix C. OPE analysis of a four-point function

In this appendix we present the calculation of the four-point function (93), i j k i j i j k i j G(x1,x2,x3,x4) = Q 1 1 1 (x1)Q 2 2 (x2)Q 3 3 3 (x3)Q 4 4 (x4) . (C.1)

As discussed in Section 5.2.5 the double pinching limit, x12 → 0, x34 → 0, allows to extract the anomalous dimension of the ∆0 = 3 operator in the 6 of SU(4) from its contribution to the OPE. The computation of the four-point function can be drastically simpliﬁed by a suitable choice of components in (C.1). It is convenient to work with complex ﬁelds in the N = 1 formulation. We use the SU(4) → SU(3) × U(1) branching rules

20 → 62 ⊕ 6−2 ⊕ 80, (C.2)

50 → 103 ⊕ 10−3 ⊕ 151 ⊕ 15−1, (C.3) where the subscript denotes the U(1) charge. Under this decomposition the operators in the 20 and 50, Qij and Qij k , decompose, respectively, into 1 IJ = Tr φI φJ ∈ 6 , (C.4) C 2 2 gYM

1 † † ¯ = ∈ − CIJ 2 Tr φI φJ 6 2, (C.5) gYM 1 1 J = Tr φ†φJ − δ J Tr φ† φK ∈ 8 (C.6) VI 2 I 2 I K 0 gYM 3gYM and 1 CIJK = Tr φI φJ φK + φI φK φJ ∈ 10 , (C.7) 3 1/2 3 gYMN

¯ 1 † † † † † † CIJK = Tr φ φ φ + φ φ φ ∈ 10− , (C.8) 3 1/2 I J K I K J 3 gYMN

JK 1 † J K † K J VI = Tr φ φ φ + φ φ φ 3 1/2 I I gYMN 1 J † L K † K L − δI Tr φ φ φ + φ φ φ 4g3 N1/2 L L YM + K † L J + † J L ∈ δI Tr φLφ φ φLφ φ 151, (C.9)

¯ I 1 I † † I † † V JK = Tr φ φ φ + φ φ φ 3 1/2 J K K J gYMN 1 − δIJ Tr φLφ† φ† + φLφ† φ† 4g3 N1/2 L K K L YM + IK L † † + L † † ∈ δ Tr φ φLφJ φ φJ φL 15−1. (C.10) S. Kovacs / Nuclear Physics B 684 (2004) 3–74 69

A simple choice of components, which leads to a contribution of the 6 in the x12 → 0 x34 → 0 channel, is then 113 ¯ 11 ¯ G(x1,x2,x3,x4) = C (x1)C11(x2)V3 (x3)C11(x4) . (C.11) Recalling the relation between the complex scalars φI and the ϕAB sweﬁnd

C113 ∼ Tr ϕ14ϕ14ϕ34 ,

¯ 23 23 11 12 14 14 C11 ∼ Tr ϕ ϕ , V3 ∼ Tr ϕ ϕ ϕ , (C.12) which shows that (C.11) can saturate the sixteen superconformal modes. In the one-instanton sector this correlation function receives two types of contributions. The operators in (C.11) contain at least twenty fermionic modes. Therefore one can either replace all the fields by their classical instanton solutions, in which case four of the fermion modes have to be of the ν and ν¯ type, or contract one pair of scalars. For the first type of contribution substituting the classical expressions for the composite fields and after some simple Fierz rearrangements we get

(1) G (x 1,x2,x3,x4)

−Sinst ˆ 113 ¯ˆ = dµphys e C (x1; x0,ρ; ζ,ν,ν)¯ C11(x2; x0,ρ; ζ,ν,ν)¯

ˆ 11 ¯ˆ × V3 (x3; x0,ρ; ζ,ν,ν)¯ C11(x4; x0,ρ; ζ,ν,ν)¯ −4N 4N−10 2πiτ 4 π g e − − − = YM d4x dρ d2ηA d2ξ¯ A dN 2νA dN 2ν¯ A e S4F N(N − 1)!(N − 2)! 0 A=1 4[ 1 1 ][ 4 4 ] ¯ [3 4] 4[ 2 2 ][ 3 3 ] × ρ ζ (x1)ζ (x1) ζ (x1)ζ (x1) (ν ν ) ρ ζ (x2)ζ (x2) ζ (x2)ζ (x2) 2 + 2 5 2 + 2 4 (y1 ρ ) (y2 ρ ) ρ4[ζ 1(x )ζ 1(x )][ζ 4(x )ζ 4(x )](ν¯ [1ν2]) ρ4[ζ 2(x )ζ 2(x )][ζ 3(x )ζ 3(x )] × 3 3 3 3 4 4 4 4 , 2 + 2 5 2 + 2 4 (y3 ρ ) (y4 ρ ) (C.13) where the spinor indices on pairs of ζ s in square brackets are understood to be contracted. In (C.13) as usual an overall numerical constant has been omitted and is to be restored in the ﬁnal result. The integrations over the superconformal fermion modes can then be performed using (42). The integrations over the ν and ν¯ modes can be treated similarly to what was done in previous cases and after simple manipulations one is left with ﬁve-sphere integrals of the form (69). After computing all the fermionic integrals we obtain

(1) G (x1,x2,x3,x4) 34π−152−2N−15(N2 − 3N + 2)Γ (2N − 2)e2πiτ = x4 x4 − ! − ! 13 24 N(N 1) (N 2) 4 13 2 2 5 2 2 4 × d x0 dρρ (x1 − x0) + ρ (x2 − x0) + ρ 2 2 5 2 2 4 −1 × (x3 − x0) + ρ (x4 − x0) + ρ , (C.14) 70 S. Kovacs / Nuclear Physics B 684 (2004) 3–74 where xpq = (xp − xq ). The ﬁnal integrations over ρ and x0 can be rewritten as 4 13 2 2 5 2 2 4 d x0 dρρ (x1 − x0) + ρ (x2 − x0) + ρ 2 2 5 2 2 4 −1 × (x3 − x0) + ρ (x4 − x0) + ρ ∂ ∂ = c B(x1,x2,x3,x4), (C.15) ∂x2 ∂x2 13 i

The four-point function (C.14) is ﬁnite and in the double limit x12 → 0, x34 → 0 its leading singularity is 2 2 (1) 1 x12x34 G (x1,x2,x3,x4) −→ log . (C.17) x →0 5 5 2 2 12 x13x24 x13x24 x34→0 The second type of contribution involves scalar propagators. Recalling that ϕAB ϕCD ∼ εABCD, from (C.11), (C.12) it follows that the allowed contractions are

G(2)(x ,x ,x ,x ) 1 2 3 4 14 14 34 23 23 ∼ Tr ϕ ϕ ϕ (x1) Tr ϕ ϕ (x2) × Tr ϕ12ϕ14ϕ14 (x ) Tr ϕ23ϕ23 (x ) +··· 3 4 14 14 34 23 23 12 14 14 + Tr ϕ ϕ ϕ (x1) Tr ϕ ϕ (x2) Tr ϕ ϕ ϕ (x3) 23 23 × Tr ϕ ϕ (x4) , (C.18) where the dots in (C.18) stand for the other contractions of the same type as those indicated as required by Wick’s theorem. The ﬁrst type of contraction, between a ϕ14 and a ϕ23, leads, however, to a vanishing contribution since after replacing the remaining ﬁelds with 1 their instanton solution one is forced to put three ζ modes at the same point, x3.Weare thus left with only one possible contraction, the one on the last line of (C.18). To evaluate the corresponding contribution to the four-point function we use the propagator (B.41). The calculation is rather involved and gives

(2) G (x1,x2,x3,x4) 34π−152−2N−13Γ(2N − 1) e2πiτ 4 = 4x ρ 5Ω 2ηA 2ξ¯A − ! − ! d 0 d d d d N(N 1) (N 2) = A 1 ρ8 × 4 − 2 2 + 2 4 2 + 2 4 (x1 x3) (y1 ρ ) (y3 ρ ) S. Kovacs / Nuclear Physics B 684 (2004) 3–74 71

2 + + 10 − (3N 10N 12) ρ 2 2 + 2 5 2 + 2 5 N (y1 ρ ) (y3 ρ ) 8 ρ 1 1 4 4 1 1 4 4 × ζ ζ ζ ζ (x1) ζ ζ ζ ζ (x3) (y2 + ρ2)4(y2 + ρ2)4 2 4 2 2 3 3 2 2 3 3 × ζ ζ ζ ζ (x2) ζ ζ ζ ζ (x4) , (C.19) where the ν and ν¯ integrals have been replaced by a ﬁve-sphere integral since in this contribution there is no explicit dependence on these variables in the integrand. The integrations over the superconformal modes can now be performed and we get

(2) G (x1,x2,x3,x4) 34π−152−2N−13Γ(2N − 1)e2πiτ = (x − x )4(x − x )4 N(N − 1)!(N − 2)! 1 3 2 4 4 11 2 2 4 2 2 4 × d x0 dρρ (x1 − x0) + ρ (x2 − x0) + ρ 2 2 4 2 2 4 −1 × (x3 − x0) + ρ (x4 − x0) + ρ

4 (3N2 + 10N + 12) ρ2 × − , (C.20) − 2 2 2 + 2 2 + 2 (x1 x3) N (y1 ρ )(y3 ρ ) which can be rewritten in terms of the box integral (C.16) similarly to the case of (1) G (x1,x2,x3,x4). In the limit x12 → 0 x34 → 0 the singularity is again of the type 2 2 (2) 1 x12x34 G (x1,x2,x3,x4) −→ log . (C.21) x →0 5 5 2 2 12 x13x24 x13x24 x34→0 In conclusion the OPE of the complete four-point function (C.11) does not present a singularity corresponding to the contribution of a scalar operator of bare dimension ∆0 = 3 in the 6 of SU(4) in the x12 → 0, x34 → 0 channel. Therefore the operator

1 Oi = Tr ϕi ϕj ϕj (C.22) 3,6 3 1/2 gYMN which according to the analysis of Section 5.2.1 could have an instanton induced anomalous dimension appears instead to be protected at the instanton level. The singularities observed in (C.17) and (C.21) on the other hand correspond to the contribution of operators of bare dimension 5. Since in Section 5.4 it was argued that at the level of dimension 5 operators only the ones in the 6 can have an instanton contribution, the result of the above OPE analysis conﬁrms that indeed at least one of the ∆0 = 5 operators in the 6 has scaling dimension corrected by instantons. The study of a single four- point function, however, does not allow to identify which operators in this sector receive corrections. 72 S. Kovacs / Nuclear Physics B 684 (2004) 3–74

References

[1] J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231; J.M. Maldacena, Int. J. Theor. Phys. 38 (1999) 1113, hep-th/9711200. [2] S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Gauge theory correlators from non-critical string theory, Phys. Lett. B 428 (1998) 105, hep-th/9802109. [3] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253, hep-th/9802150. [4] M. Bianchi, S. Kovacs, G.C. Rossi, Ya.S. Stanev, On the logarithmic behavior in N = 4SYMtheory, JHEP 9908 (1999) 020, hep-th/9906188. [5] H. Liu, Scattering in anti-de Sitter space and operator product expansion, Phys. Rev. D 60 (1999) 106005, hep-th/9811152; J.H. Brodie, M. Gutperle, String corrections to four point functions in the AdS/CFT correspondence, Phys. Lett. B 445 (1999) 296, hep-th/9809067; E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis, L. Rastelli, Graviton exchange and complete 4-point functions in the AdS/CFT correspondence, Nucl. Phys. B 562 (1999) 353, hep-th/9903196. [6] B. Eden, P.S. Howe, C. Schubert, E. Sokatchev, P.C. West, Four-point functions in N = 4 supersymmetric Yang–Mills theory at two loops, Nucl. Phys. B 557 (1999) 355, hep-th/9811172. [7] D. Berenstein, J.M. Maldacena, H. Nastase, Strings in flat space and pp waves from N = 4 super-Yang– Mills, JHEP 0204 (2002) 013, hep-th/0202021. [8] M. Blau, J. Figueroa-O’Farrill, C. Hull, G. Papadopoulos, A new maximally supersymmetric background of IIB superstring theory, JHEP 0201 (2002) 047, hep-th/0110242. [9] R.R. Metsaev, A.A. Tseytlin, Exactly solvable model of superstring in plane wave Ramond–Ramond background, Phys. Rev. D 65 (2002) 126004, hep-th/0202109. [10] A. Santambrogio, D. Zanon, Exact anomalous dimensions of N = 4 Yang–Mills operators with large R charge, Phys. Lett. B 545 (2002) 425, hep-th/0206079. [11] N. Beisert, C. Kristjansen, J. Plefka, G.W. Semenoff, M. Staudacher, BMN correlators and operator mixing in N = 4 super-Yang–Mills theory, Nucl. Phys. B 650 (2003) 125, hep-th/0208178. [12] N.R. Constable, D.Z. Freedman, M. Headrick, S. Minwalla, Operator mixing and the BMN correspondence, JHEP 0210 (2002) 068, hep-th/0209002. [13] C. Kristjansen, J. Plefka, G.W. Semenoff, M. Staudacher, A new double-scaling limit of N = 4 super-Yang– Mills theory and PP-wave strings, Nucl. Phys. B 643 (2002) 3, hep-th/0205033. [14] N.R. Constable, D.Z. Freedman, M. Headrick, S. Minwalla, L. Motl, A. Postnikov, W. Skiba, PP-wave string interactions from perturbative Yang–Mills theory, JHEP 0207 (2002) 017, hep-th/0205089. [15] S.S. Gubser, I.R. Klebanov, A.M. Polyakov, A semi-classical limit of the gauge/string correspondence, Nucl. Phys. B 636 (2002) 99, hep-th/0204051. [16] A.M. Polyakov, Gauge fields and space–time, Int. J. Mod. Phys. A 17S1 (2002) 119, hep-th/0110196. 5 [17] S. Frolov, A.A. Tseytlin, Semiclassical quantization of rotating superstring in AdS5 × S , JHEP 0206 (2002) 007, hep-th/0204226. 5 [18] J.G. Russo, Anomalous dimensions in gauge theories from rotating strings in AdS5 × S , JHEP 0206 (2002) 038, hep-th/0205244. [19] N. Beisert, J.A. Minahan, M. Staudacher, K. Zarembo, Stringing spins and spinning strings, JHEP 0309 (2003) 010, hep-th/0306139. 5 [20] A.A. Tseytlin, On semiclassical approximation and spinning string vertex operators in AdS5 × S ,Nucl. Phys. B 664 (2003) 247, hep-th/0304139. 5 [21] S. Frolov, A.A. Tseytlin, Multi-spin string solutions in AdS5 × S , Nucl. Phys. B 668 (2003) 77, hep- th/0304255. 5 [22] S. Frolov, A.A. Tseytlin, Quantizing three-spin string solution in AdS5 × S , JHEP 0307 (2003) 016, hep- th/0306130. [23] S. Frolov, A.A. Tseytlin, Rotating string solutions: AdS/CFT duality in non-supersymmetric sectors, Phys. Lett. B 570 (2003) 96, hep-th/0306143. 5 [24] G. Arutyunov, S. Frolov, J. Russo, A.A. Tseytlin, Spinning strings in AdS5 × S and integrable systems, hep-th/0307191. [25] N. Beisert, S. Frolov, M. Staudacher, A.A. Tseytlin, Precision spectroscopy of AdS/CFT, hep-th/0308117. S. Kovacs / Nuclear Physics B 684 (2004) 3–74 73

[26] R.A. Janik, BMN operators and string field theory, Phys. Lett. B 549 (2002) 237, hep-th/0209263. [27] J.A. Minahan, K. Zarembo, The Bethe-ansatz for N = 4 super-Yang–Mills, JHEP 0303 (2003) 013, hep- th/0212208. [28] N. Beisert, C. Kristjansen, J. Plefka, M. Staudacher, BMN gauge theory as a quantum mechanical system, Phys. Lett. B 558 (2003) 229, hep-th/0212269. [29] N. Beisert, C. Kristjansen, M. Staudacher, The dilatation operator of N = 4 super-Yang–Mills theory, Nucl. Phys. B 664 (2003) 131, hep-th/0303060. [30] N. Beisert, M. Staudacher, The N = 4 SYM integrable super-spin chain, Nucl. Phys. B 670 (2003) 439, hep-th/0307042. [31] G. Mandal, N.V. Suryanarayana, S.R. Wadia, Aspects of semiclassical strings in AdS5, Phys. Lett. B 543 (2002) 81, hep-th/0206103. 5 [32] I. Bena, J. Polchinski, R. Roiban, Hidden symmetries of the AdS5 × S superstring, hep-th/0305116. [33] L. Dolan, C.R. Nappi, E. Witten, A relation between approaches to integrability in superconformal Yang– Mills theory, hep-th/0308089. 5 [34] L.F. Alday, Nonlocal charges on AdS5 × S and pp-waves, JHEP 0312 (2003) 033, hep-th/0310146. [35] M. Bianchi, M.B. Green, S. Kovacs, G.C. Rossi, Instantons in supersymmetric Yang–Mills and D-instantons in IIB superstring theory, JHEP 9808 (1998) 013, hep-th/9807033. [36] N. Dorey, T.J. Hollowood, V.V. Khoze, M.P. Mattis, S. Vandoren, Multi-instanton calculus and the AdS/CFT correspondence in N = 4 superconformal field theory, Nucl. Phys. B 552 (1999) 88, hep-th/9901128. [37] N. Dorey, V.V. Khoze, M.P. Mattis, S. Vandoren, Yang–Mills instantons in the large-N limit and the AdS/CFT correspondence, Phys. Lett. B 442 (1998) 145, hep-th/9808157. 5 [38] M.B. Green, S. Kovacs, Instanton-induced Yang–Mills correlation functions at large N and their AdS5 × S duals, JHEP 0304 (2003) 058, hep-th/0212332. [39] M. Bianchi, S. Kovacs, G.C. Rossi, Ya.S. Stanev, Properties of the Konishi multiplet in N = 4SYMtheory, JHEP 0105 (2001) 042, hep-th/0104016. 5 [40] M. Bianchi, J.F. Morales, H. Samtleben, On stringy AdS5 × S and higher spin holography, JHEP 0307 (2003) 062, hep-th/0305052. [41] N. Beisert, The complete one-loop dilatation operator of N = 4 super-Yang–Mills theory, hep-th/0307015. [42] N. Beisert, Higher loops, integrability and the near BMN limit, JHEP 0309 (2003) 062, hep-th/0308074. [43] M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld, Yu.I. Manin, Construction of instantons, Phys. Lett. A 65 (1978) 185. [44] D. Amati, K. Konishi, Y. Meurice, G.C. Rossi, G. Veneziano, Nonperturbative aspects in supersymmetric gauge theories, Phys. Rep. 162 (1988) 169. [45] N. Dorey, T.J. Hollowood, V.V. Khoze, M.P. Mattis, The calculus of many instantons, Phys. Rep. 371 (2002) 231, hep-th/0206063. [46] N. Dorey, V.V. Khoze, M.P. Mattis, On mass-deformed N = 4 supersymmetric Yang–Mills theory, Phys. Lett. B 396 (1997) 141, hep-th/9612231. [47] E.F. Corrigan, P. Goddard, S. Templeton, Instanton Green’s functions and tensor products, Nucl. Phys. B 151 (1979) 93. [48] L.S. Brown, R.D. Carlitz, D.B. Creamer, C. Lee, Propagation functions in pseudoparticle fields, Phys. Rev. D 17 (1978) 1583. [49] A.V. Belitsky, S. Vandoren, P. van Nieuwenhuizen, Yang–Mills and D-instantons, Class. Quantum Grav. 17 (2000) 3521, hep-th/0004186. [50] M.A. Shifman, A.I. Vainshtein, Instantons versus supersymmetry: fifteen years later, hep-th/9902018. [51] D. Anselmi, M. Grisaru, A. Johansen, A critical behaviour of anomalous currents, electric–magnetic universality and CFT4, Nucl. Phys. B 491 (1997) 221, hep-th/9601023. [52] M. Bianchi, S. Kovacs, G.C. Rossi, Ya.S. Stanev, Anomalous dimensions in N = 4 SYM theory at order g4, Nucl. Phys. B 584 (2000) 216, hep-th/0003203. [53] G. Arutyunov, S. Frolov, A.C. Petkou, Operator product expansion of the lowest weight CPOs in N = 4 SYM4 at strong coupling, Nucl. Phys. B 586 (2000) 547, hep-th/0005182; G. Arutyunov, S. Frolov, A.C. Petkou, Nucl. Phys. B 609 (2001) 539, Erratum. [54] V.K. Dobrev, V.B. Petkova, All positive energy unitary irreducible representations of extended conformal supersymmetry, Phys. Lett. B 162 (1985) 127. 74 S. Kovacs / Nuclear Physics B 684 (2004) 3–74

[55] S. Ferrara, A. Zaffaroni, Superconformal ﬁeld theories, multiplet shortening, and the AdS5/CFT4 correspondence, hep-th/9908163. [56] E. D’Hoker, D.Z. Freedman, W. Skiba, Field theory tests for correlators in the AdS/CFT correspondence, Phys. Rev. D 59 (1999) 045008, hep-th/9807098. [57] G. Arutyunov, S. Frolov, A.C. Petkou, Perturbative and instanton corrections to the OPE of CPO’s in N = 4 SYM4, Nucl. Phys. B 602 (2001) 238, hep-th/0010137; G. Arutyunov, S. Frolov, A.C. Petkou, Nucl. Phys. B 609 (2001) 540, Erratum. [58] E. D’Hoker, P. Heslop, P. Howe, A.V. Ryzhov, Systematics of quarter BPS operators in N = 4SYM, JHEP 0304 (2003) 038, hep-th/0301104. [59] G. Arutyunov, S. Penati, A.C. Petkou, A. Santambrogio, E. Sokatchev, Non-protected operators in N = 4 SYM and multiparticle states of AdS5 SUGRA, Nucl. Phys. B 643 (2002) 49, hep-th/0206020. [60] M. Bianchi, B. Eden, G.C. Rossi, Ya.S. Stanev, On operator mixing in N = 4 SYM, Nucl. Phys. B 646 (2002) 69, hep-th/0205321. [61] M.A. Shifman (Ed.), Instantons in Gauge Theories, World Scientiﬁc, Singapore, 1994. [62] M.B. Green, S. Kovacs, A. Sinha, in preparation. [63] N. Dorey, V.V. Khoze, M.P. Mattis, Multi-instanton calculus in N = 2 supersymmetric gauge theory. II: coupling to matter, Phys. Rev. D 54 (1996) 7832, hep-th/9607202. [64] M. Bianchi, M.B. Green, S. Kovacs, Instanton corrections to circular Wilson loops in N = 4 supersymmetric Yang–Mills, JHEP 0204 (2002) 040, hep-th/0202003; M. Bianchi, M.B. Green, S. Kovacs, Instantons and BPS Wilson loops, hep-th/0107028. Nuclear Physics B 684 (2004) 75–99 www.elsevier.com/locate/npe

Quiver matrix mechanics for IIB string theory (I): wrapping membranes and emergent dimension

Jian Dai, Yong-Shi Wu

Department of Physics, University of Utah, Salt Lake City, UT 84112, USA Received 5 January 2004; accepted 11 February 2004

Abstract In this paper we present a discrete, nonperturbative formulation for type IIB string theory. Being a supersymmetric quiver matrix mechanics model in the framework of M(atrix) theory, it is a generalization of our previous proposal of compactification via orbifolding for deconstructed IIA strings. In the continuum limit, our matrix mechanics becomes a (2 + 1)-dimensional Yang–Mills theory with 16 supercharges. At the discrete level, we are able to construct explicitly the solitonic states that correspond to membranes wrapping on the compactified torus in target space. These states have a manifestly SL(2, Z)-invariant spectrum with correct membrane tension, and give rise to an emergent flat dimension when the compactified torus shrinks to vanishing size.  2004 Elsevier B.V. All rights reserved.

PACS: 11.25.-w; 11.10.Kk

1. Introduction

In the string/M-theory duality web, both the IIA/M and IIB/M duality conjectures involve the emergence of a new spatial dimension in a certain limit. M-theory, according to our current understanding, is the 11-dimensional dual of the strongly coupled 10- dimensional IIA string theory [1]. Here a new dimension emerges, because the D-particles, carrying RR charges, can be viewed as the Kaluza–Klein momentum modes along a circle in the “hidden eleventh” dimension. The size of the circle depends on the IIA string coupling in such a way that when the string coupling tends to inﬁnity, the size of the circle also becomes inﬁnite. In other words, IIA string theory should be dual to M-theory

E-mail addresses: [email protected] (J. Dai), [email protected] (Y.S. Wu).

0550-3213/$ – see front matter  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2004.02.013 76 J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99 compactified on a circle, and the hidden dimension opens up as a new flat dimension in the strong-coupling limit. For IIB/M duality, the story is obviously more complicated [2–4]. IIB string theory is conjectured to be dual to M-theory compactified on a 2-torus. The IIB string coupling depends on the area of the 2-torus, in such a way that the weak coupled IIB string theory should be recovered when the area of the hidden 2-torus shrinks to zero. A great advantage of this scenario is that the mysterious SL(2, Z) duality in IIB string theory can be interpreted as the geometric, modular transformation of the hidden 2-torus. But wait, there is apparently a mismatch of dimensionality here: with the compactified two dimensions diminished in M-theory, it seems that only eight spatial dimensions remain, but IIB string theory lives in nine, not eight, dimensional space. The clever solution [2] to this puzzle is that in M-theory there are topological soliton-like states, corresponding to wrapping a membrane around the compactified 2-torus. Because of the conjectured SL(2, Z) invariance, the energy of the wrapping membrane should only depend on its wrapping number w. With the torus shrinking to zero, a tower of the wrapping membrane states labeled by w gives rise to the momentum states in a new, flat, emergent dimension. The picture looks perfect. But can we really formulate it in a mathematical formalism with above-mentioned ideas demonstrated explicitly? Certainly this needs a non- perturbative formulation of M-theory. Though a fully Lorentz invariant formulation of M-theory is not available yet, fortunately we have had a promising candidate (or conjecture), namely the BFSS M(atrix) theory [5], in a particular kinematical limit, the infinite momentum frame (IMF), or in the discrete light-cone quantization (DLCQ) [6]. It was shown that M(atrix) theory compactified on an n-torus (for n 3) is just an (n + 1)- dimensional supersymmetric Yang–Mills (SYM) theory [7]. Facilitated by this fact, the IIA/M duality was verified soon after BFSS conjecture. Namely M(atrix) theory compactified on a circle was explicitly shown, via the so-called “9–11 flip”, to give rise to a description of (second quantized) IIA strings [8]. As to IIB/M duality, again the story was not so simple as IIA/M duality. The statement that M(atrix) theory compactified on a 2-torus (with one side in the longitudinal direction) is dual to IIB string theory compactified on a circle was checked [9] for the action of D-string [10] and the energy of (p, q)-strings, etc. An emergent dimension with full O(8) invariance together with seven ordinary dimensions were argued to come about in the (2 + 1)-dimensional SYM that results from compactifying M(atrix) theory on a transverse torus [11–13]. These arguments invoked the conjectured SL(2, Z) duality of (3 + 1)-dimensional SYM. However, the explicit construction of wrapping membrane states and a proof of the SL(2, Z) invariance of these states, which plays a pivotal role in the IIB/M duality, are still lacking. Here let us try to scrutinize the origin of the difficulty for constructing a wrapping membrane state in M(atrix) theory. First recall that in the usual membrane theory, a nonlinear sigma model with continuous world-volume, a membrane state wrapping on a compactified 2-torus is formulated by 1 x (p, q) = R1(mp + nq) + (oscillator modes), 2 x (p, q) = R2(sp + tq)+ (oscillator modes), (1) where p and q are two affine parameters of the membrane, two independent cycles on the compactified 2-torus have R1, R2 as their sizes and x1, x2 their affine parameters, m, n, s, t J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99 77 four integers characterizing linearly how the membrane wraps on the torus. However, in M(atrix) theory the target space coordinates are lifted to matrices, and the basis of the membrane world-volume functions, eip and eiq, are transcribed to noncommutative, (so- called) clock and shift matrices U and V of finite rank. Moreover, when the theory is compactified on a torus, the quotient conditions for compactification require the matrix coordinates of the target space to become covariant derivatives on the dual torus. Thus, the difficulty in constructing the matrix analog of the wrapping membrane states in Eq. (1) lies in how to extract the affine parameters p,q from their finite matrix counterparts U and V , as well as to keep the linear relation containing the information of wrapping in Eq. (1). In a word, the problem is a mismatch between the linear coordinates of the target torus and the nonlinear coordinates of the membrane degrees of freedom. To our knowledge, no explicit construction to overcome this mismatch is known in the literature. The main goal of this paper is to fill this gap, namely we want to develop a M(atrix) theory description of IIB string theory, which makes IIB/M duality and SL(2, Z)-duality more accessible to analytic treatments. Our strategy to change the mismatch to be a match is to simply let both sides to be nonlinear; technically, we deconstruct the (2 + 1)-dimensional SYM that results from compactifying M(atrix) theory on a torus and then, within the resulting framework of quiver matrix mechanics, to construct the matrix membrane wrapping states. In our previous paper [14], we have been able to deconstruct the matrix string theory for type IIA strings, which is a (1 + 1)-dimensional SYM. Here we generalize this deconstruction to type IIB strings. The basic idea is to achieve compactification by orbifolding the M(atrix) theory, like in Ref. [14]. There, to compactify the theory on a circle, we took the orbifolding (or quotient) group to be ZN ; but here to compactify the theory on a 2- Z ⊗ Z Z2 torus we need to take a quotient with the group N N (or shortly N ). This leads to a supersymmetric quiver matrix mechanics with a product gauge group and bi-fundamental matter. By assigning nonzero vacuum expectation values (VEV) to bi-fundamental scalars, the quiver theory looks like a theory on a planar triangular lattice, whose continuum limit gives rise to the desired (2 + 1)-dimensional SYM. From the point of view of D-particles in target space, this is nothing but “compactifying via orbifolding”, or “deconstruction of compactification”. (The use of the literary term and analytic technique of “deconstruction” in high-energy physics was advocated in [15]. Similar applications of the deconstruction method can also be found in [16–21].) It is in this approach we have been able to accomplish a down-to-earth construction of wrapping membrane states in M(atrix) theory, a modest step to better understand IIB/M duality. In this paper we will carry out our deconstruction for the simplest case with a right triangular lattice, and concentrate on the construction of the membrane states. The general case with a slanted triangular lattice and a detailed study of SL(2, Z) duality are left to the subsequent paper(s). We organize this paper as follows. In Section 2, we will give a brief review of deconstruction of the matrix string theory, to stipulate our notations and demonstrate the idea to approximate a compactification by an orbifolding sequence (in the so-called theory space). Then in Section 3, extending this approach, we achieve the deconstruction of the (2 + 1)-dimensional SYM that describes IIB strings by means of orbifolding M(atrix) theory by ZN ⊗ ZN , resulting in a supersymmetric quiver matrix mechanics. Metric aspects of this (de)construction after assigning VEVs is explored in 78 J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99

Section 4; then the desired SYM is shown to emerge in the continuum limit with N →∞ upon choosing the simplest (right) triangular lattice. Section 5 is devoted to the central issue of this paper, namely explicit construction of topologically nontrivial states in quiver mechanics that in the continuum limit give rise to the membranes wrapping on the compactified torus. The energy spectrum of these states is shown to depend only on the SL(2, Z)-invariant wrapping number w. This provides an explicit verification of SL(2, Z) invariance, as well as the correct membrane tension in our discrete approach. Moreover, the spectrum of these states is such that it can be identified as the spectrum of momentum states in an emergent flat transverse dimension when the size of the compactified torus shrinks to zero. Finally in Section 6 discussions and perspectives are presented.

2. A brief review: matrix string (de)construction

This preparative section serves two purposes: to establish our notations and to review brieﬂy the physical aspects of our previous (de)construction of the matrix string theory.

2.1. Preliminaries and notations

By now it is well known that at strong couplings of IIA string theory, a new spatial dimension emerges so that the 10-dimensional IIA theory is dual to an 11-dimensional theory, dubbed the name M-theory [1]. In accordance with the M(atrix) theory conjecture [5] and as shown by Seiberg [22] and Sen [23], the same dynamics that governs N low- energy D-particles in a ten-dimensional Minkowski spacetime actually captures all the information of M-theory in the discrete light-cone quantization. If we label the spatial coordinates by y1,y2,...,y9, then the basic idea of M(atrix) theory is that, to incorporate open strings stretched between N D-particles, one has to lift the D-particle coordinates to N-by-N matrices, Y I (I = 1,...,9), as the basic dynamical variables. Formally the matrix theory a la BFSS can be formulated as a dimensional reduction of d = 9 + 1, = 1 supersymmetric Yang–Mills theory: 10 1 MN i T 0 M S = d x − Tr FMNF − Tr λ Γ Γ DM λ , 4g2 2g2

s s s s where the Majorana–Weyl spinor λ, whose components will be labeled as λ 0 1 2 3 (sa = ∗ 1, 2, a = 0, 1, 2, 3), satisfy the spinor constraint equations λ =− Γ 11λ, λ = λ.The 10 2 2 dimensional reduction follows the procedure below: d x → dt V9; g˜ := g /V9; I I 2 2 1/2 2 2 2 I 1/3 1/2 I then A =: −X /α , g˜ α = gs α , λ /g˜ →−λ and X =: gs α Y where I = 1,...,9. Note that in accordance with the IIA/M duality, the M-theory parameters 1/3 are related to those of IIA string theory by R = gs ls , lp = gs ls ,inwhichlp is the eleven- dimensional Planck length and R the radius of the hidden M-circle; in our convention, = 2 α ls . Time variable and M-circle radius can be rescaled in units of Planck length to become dimensionless: t = lpτ , R = R11lp, respectively. J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99 79

After all these efforts, the BFSS action reads 1 I 2 R11 I J 2 i T S = dτ Tr Dτ ,Y + Y ,Y − λ [Dτ ,λ] 2R11 4 2 R11 T I + λ γ [YI ,λ] , (2) 2 0 in which eleven-dimensional Planck length is taken to be unity, Dτ = d/dτ − i[Y , ·] is the covariant time derivative, and Y 0 is the gauge connection in the temporal direction. A representation of the gamma matrices γ I in Eq. (2) is listed below for later uses: γ 0 := 1 ⊗ 1 ⊗ 1 ⊗ 1,γ1 = ⊗ ⊗ ⊗ , 2 3 γ = τ1 ⊗ 1 ⊗ ⊗ , γ = τ3 ⊗ 1 ⊗ ⊗ , 4 5 γ = ⊗ τ1 ⊗ 1 ⊗ , γ = ⊗ τ3 ⊗ 1 ⊗ , 6 7 γ = 1 ⊗ ⊗ τ1 ⊗ , γ = 1 ⊗ ⊗ τ3 ⊗ , 8 9 γ = 1 ⊗ 1 ⊗ 1 ⊗ τ1,γ= 1 ⊗ 1 ⊗ 1 ⊗ τ3, (3) where = iτ2 and τa (a = 1, 2, 3) are Pauli matrices. In the following, the N-by-N clock and shift matrices will be used frequently. We = 2 N = i2π/N denote them by UN diag(ωN ,ωN ,...,ωN ),whereωN e ,and   00··· 01  ···   10 00  ···  VN :=  01 00. (4)  .  .. 00··· 10 Quite often we will just write U, V when the rank N is understood.

2.2. Compactiﬁcation via orbifolding

In this subsection we give a brief review of our previous work [14], to show how the matrix theory compactiﬁcation on a circle was achieved via orbifolding and how the worldsheet/target-space duality emerges in a natural manner. 2 Consider a group of K D-particles moving in a C /ZN orbifold background; the − orbifolding is performed√ via equivalence relations√ z1 ∼ ei2π/Nz1, z2 ∼ e i2π/Nz2,where z1 = (y6 + iy7)/ 2, z2 = (y8 + iy9)/ 2. The complex√ coordinates z1, z2 √can be 1 i(ϑ+ϕ) 2 i(ϑ−ϕ) parameterized in a polar coordinate form, z = ρ1e / 2, z = ρ2e / 2; the quotient conditions, in this parametrization, are simply ϕ ∼ ϕ + 2π/N. To consider K D-particles on the orbifold, we must lift the D-particle coordinate to KN-by-KN matrices, for which we will use upper-cased letters Z1, Z2. To satisfy the orbifolding conditions a a these matrices are of the special form z VN ,wherez (a = 1, 2) are in the form of N × N block-diagonal matrix with each block being a K-by-K matrix and the unities in VN (see Eq. (4)) understood as 1K . Besides orbifolding, we push the D-particles away from the singularity by assigning nonzero√ VEV to√z1 and z2. Note that the moduli of√ the 1 = 2 = 1 = D-particle coordinates, z c1/ 2, z c2/ 2, can be rotated to z c1/ 2, 80 J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99

2 Fig. 1. Worldsheet/target-space duality in C /ZN orbifold. = | |2 +| |2 2 = C2 c1 c1 c2 with z 0, by an element of SU(2) actingonthe .(The orbifolding results in a circular quiver diagram with N sites. The Z-fields live on the links connecting neighboring sites, indicated by the presence of the block-matrix VN .) So essentially only one modulus is needed to characterize the VEV. This modulus can be interpreted in two different pictures: = I: 2πc1 NL, (5) = II: N/c1 2πΣ. (6) On the one hand, L is interpreted as the size of the circle on the orbifold in the target space, which is orthogonal to the radial direction and at the radius c1. Viewed from the target space, D-particles live around this circle, so it can be viewed as the compactified circle, or the M-circle for IIA theory. On the other hand, Σ in Eq. (5) can be viewed as the size of the circle that is dual to the target circle L. It is both amazing and amusing to see that with the fluctuations in Z1 and Z2 included, the orbifolded BFSS action can be equivalently written as an action on a one-dimensional lattice with lattice constant a = 2πΣ/N.Furthermore, in the limit N →∞, this quiver mechanics action approaches to the d = 1 + 1SYM(on the dual circle Σ) with 16 supercharges, known to describe the matrix string theory, which was previously obtained by compactifying M(atrix) theory on the circle L directly. (Note that the so-called “9–11 flip”, that was necessary to give an appropriate interpretation for the compactified matrix theory on L, is not necessary at all in the orbifolding context.) Thus, what we have done is actually to (de)construct the (type IIA) string worldsheet via a circular quiver diagram, and the relations (5) and (6) between the geometric parameters is a clear demonstration of the worldsheet/target-space duality: namely Eq. (5) is over target space orbifold, in which c prescribes the location of the M-circle; Eq. (6) is over 1 worldsheet cycle, in which c1 is the inverse of the lattice constant. The dual relation between the two interpretations and the compactification scale is shown in Fig. 1. The methodology underlying the above deconstruction of IIA/M duality and the relationship between the matrix theory, type II string theory and our quiver quantum J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99 81

Fig. 2. Relation between quiver mechanics and matrix theory, type II string theory. mechanics are demonstrated in a commuting diagram (Fig. 2). In subsequent sections we will show that this diagram applies to the deconstruction of IIB strings as well. Additional complications from higher dimensions will emerge; and we will show how to deal with them in this and sequential paper.

3. ZN ⊗ ZN orbifolded matrix theory

Now we proceed to make and justify our proposal that the ZN ⊗ZN orbifolded M(atrix) theory, being a supersymmetric quiver matrix mechanics, provides a promising candidate for the nonperturbative formulation of IIB string theory. 3 3 a Consider the orbifold√ C /ZN ⊗ ZN , where the complex coordinates of C are z = (y2a+2 + iy2a+3)/ 2fora = 1, 2, 3 and the quotient conditions for orbifolding are

− I: z1 ∼ e i2π/Nz1,z2 ∼ ei2π/Nz2,z3 ∼ z3; (7) − II: z1 ∼ e i2π/Nz1,z2 ∼ z2,z3 ∼ ei2π/Nz3. (8)

As did in the last section, we lift√ the D-particle coordinates to KN2-by-KN2 matrix variables Za = (Y 2a+2 + iY2a+3)/ 2 (a = 1, 2, 3), and subject them to the following orbifolding conditions:

− − ˆ † a ˆ = M45 M67 a ˆ † a ˆ = M45 M89 a U1 Z U1 ωN Z , U2 Z U2 ωN Z , (9) in which MIJ are rotational generators on (IJ )-plane for vectors in nine-dimensional ˆ ˆ transverse space, and U1 := 1K ⊗ UN ⊗ 1N , U2 := 1K ⊗ 1N ⊗ UN embed the action of rotations into the gauge group, U(KN2), of M(atrix) theory. Similarly introduce complexiﬁed fermionic coordinates for the Majorana–Weyl spinor λ in M(atrix) theory. For each real spinor index s = 1, 2forλ, introduce complex spinor index t =±through 1 + λ = λ = √1 11 2 T − ,T . λ λ 2 −ii Therefore, (λt )† = λ−t . In addition,

† † † T τ1T =−τ2,Tτ2T =−τ3,Tτ3T = τ1. 82 J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99

Table 1 Quantum numbers of variables in the BFSS matrix theory upon orbifolding

J45 J67 J89 J45 − J67 J45 − J89 Y 0,1,2,3 0000 0 Z1 −100−1 −1 Z2 0 −10 1 0 Z3 00−10 1 +++ λs0 1/21/21/20 0 ++− λs0 1/21/2 −1/20 1 +−+ λs0 1/2 −1/21/21 0 −++ λs0 −1/21/21/2 −1 −1

After using this T to change the basis for the last three spinor indices for fermionic coordinates, the gamma matrices become

0 1 γ = 1 ⊗ 1 ⊗ 1 ⊗ 1,γ=−τ2 ⊗ τ3 ⊗ τ3 ⊗ τ3, 2 3 γ =−τ1 ⊗ 1 ⊗ τ3 ⊗ τ3,γ=−τ3 ⊗ 1 ⊗ τ3 ⊗ τ3, 4 5 γ =−τ2 ⊗ τ2 ⊗ 1 ⊗ τ3,γ= τ2 ⊗ τ1 ⊗ 1 ⊗ τ3, 6 7 γ = 1 ⊗ τ3 ⊗ τ2 ⊗ τ3,γ=−1 ⊗ τ3 ⊗ τ1 ⊗ τ3, 8 9 γ =−1 ⊗ 1 ⊗ 1 ⊗ τ2,γ= 1 ⊗ 1 ⊗ 1 ⊗ τ1. (10) The sixteen-component fermionic coordinate λ is thus complexified, as an eight- component complex spinor. The orbifolding conditions for fermionic variables read − − ˆ † ˆ = σ45 σ67 ˆ † ˆ = σ45 σ89 U1 λU1 ω λ, U2 λU2 ω λ, (11) J I where σIJ = i[γ ,γ ]/4 are the rotational generators for spinors: 1 σ 45 = 1 ⊗ τ ⊗ 1 ⊗ 1, 2 3 1 σ 67 = 1 ⊗ 1 ⊗ τ ⊗ 1, 2 3 1 σ 89 = 1 ⊗ 1 ⊗ 1 ⊗ τ . (12) 2 3 Quantum numbers of each variable are summarized in Table 1. In this table, JIJ = MIJ s ,t ,t ,t † s ,−t ,−t ,−t for vectors, JIJ = σIJ for spinors, and remember that (λ 0 1 2 3 ) = λ 0 1 2 3 . Schematically, the field contents of our orbifolded M(atrix) theory is encoded by a quiver diagram in the form of a planar triangular lattice. Fig. 3 illustrates six cells in the quiver diagram. Each variable is a KN2-by-KN2 matrix, which can also be viewed as a K-by-K matrix field living on the sites or links in the quiver diagram with N2 sites. The matrix block form for the N2-by-N2 indices of these variables is dictated by the quiver +++ diagram. From either Table 1 or Fig. 3, we can read off that Y 0,1,2,3 and λs0 , hence −−− λs0 , are defined on the sites, while Z1,2,3 and other fermionic coordinates, as well as their hermitian conjugate, on the links [19]. In the following, we will often suppress the K- by-K indices, while using a pair of integers (m, n), m, n = 1, 2,...,N, to label the lattice J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99 83

C3 Z2 Fig. 3. Quiver diagram for the matrix theory on / N . sites in the quiver diagram. Note that for convenience the “background” in the diagram is drawn like a regular lattice, but remember that at this moment no notion of metric has been deﬁned. The orbifolded action, as a quiver mechanical model, reads 1 i 2 1 m 2 S = dτ Tr Dτ ,Y + Dτ ,Y 2R11 2R11 R R R + 11 Y i,Yj 2 + 11 Y i,Ym 2 + 11 Y m,Yn 2 4 2 4 i † R11 † i R11 † m − λ [Dτ ,λ]+ λ γ [Yi,λ]+ λ γ [Ym,λ] , (13) 2 2 2 where i, j run from 1 to 3, m, n from 4 to 9. Or equivalently in terms of the complex coordinates Za , 1 i 2 R11 i j 2 1 a a† S = dτ Tr Dτ ,Y + Y ,Y + Dτ ,Z Dτ ,Z 2R11 4 R11 R + 11 Za,Za † Za†,Za + Za,Za Za†,Za † 2 i a i a† i † R11 † i + R11 Y ,Z Y ,Z − λ [Dτ ,λ]+ λ γ [Yi,λ] 2 2 + R√11 † ˜ a +˜† a† λ γa Z ,λ γa Z ,λ . (14) 2 84 J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99

2 2 2a+2 In this action, the trace Tr is taken on a KN -by-KN matrix, and γ˜a = (γ − iγ2a+3)/2, with

1 γ˜ =− ⊗ τ− ⊗ 1 ⊗ τ3, 2 γ˜ = i1 ⊗ τ3 ⊗ τ− ⊗ τ3, 3 γ˜ =−i1 ⊗ 1 ⊗ 1 ⊗ τ−, (15) and τ− = (τ1 − iτ2)/2. Recall that the number of surviving supersymmetries corresponds to the number of +++ fermionic variables on each site; so we have four supercharges corresponding to λs0 s −−− and λ 0 with (s0 = 1, 2). Relabel fermionic variables according their statuses in the s = s+++ s = s−++ s = s+−+ s = s++− quiver diagram, Λ0 λ , Λ1 λ , Λ2 λ , Λ3 λ . Hence, the Yukawa interactions in the action (14) can be recast into a trilinear form: √ = − † 1 + † 2 − † 3 SBF dτ 2R11 Tr Λ1 Z ,Λ0 iΛ2 Z ,Λ0 iΛ3 Z ,Λ0 1 2 3 − Z ,Λ2 Λ3 − iΛ1 Z ,Λ3 + iΛ1 Λ2,Z + h.c. . (16) As well known in the literature on deconstruction, the allowed Yukawa interactions are such that a closed triangle (which may a degenerate one) is always formed by the two fermionic legs and one bosonic leg or site, due to the orbifolded gauge invariance.

4. Constructing d = 2 + 1 world-volume theory

Previously in M(atrix) theory on a 2-torus, the compactiﬁed coordinates that solve the quotient conditions were expressed as covariant derivatives, resulting in a (2 + 1)- dimensional SYM on the dual torus with 16 supersymmetries. In this section, to provide a (de)construction of this d = 2 + 1 SYM, in the same spirit as advocated in Ref. [14] and in Section 2, we will assign nonvanishing VEV to the bosonic link variables in our quiver mechanical model with the action (14), and show that it will approach in the continuum limit (i.e., a large N limit) to the desired SYM. Though the outcome may be expected in advance (see for example [19]), to see how the limit is achieved with a generic triangular lattice and how it could be reduced to a rectangular lattice should be helpful for our later discussion for wrapping matrix membranes.

4.1. Moduli and parametrization of ﬂuctuations

To proceed, we ﬁrst note that the orbifolding conditions (9) require the bi-fundamental a bosonic matrix variables be of a particular block form in the site indices: (Z )mn,mn = a ˆ z (m, n)(Va)mn,mn , with ˆ (V2)mn,mn = (VN )m,m (1N )n,n , ˆ (V3)mn,mn = (1N )m,m (VN )n,n , (17) J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99 85

ˆ := ˆ † ˆ † a = and V1 V2 V3 . Each z (m, n) (a 1, 2, 3) (for ﬁxed (m, n))isaK-by-K matrix. (Since ˆ ˆ V2, V3 can be viewed as the generators of the discrete group ZN ⊗ ZN , the orbifolded Z2 matrices are of the form of the so-called crossed product of Mat(K) with N [24,25], expressing the properly projected form of the matrices upon orbifolding.) In our approach, the key step after orbifolding is to assign nonzero VEV to each element za(m, n): za = za +˜za, √ a with z =ca/ 2, ca complex numbers to be speciﬁed later. Later in this paper we will take the following moduli:

c1 = 0,c2 = NR2/2π, c3 = NR3/2π. (18) Accordingly, the fluctuations can be parameterized as √ z˜1 = (φ + iφ )/ 2, 1 1 √ ˜2 = − z (φ2 iR2A2)/√2, 3 z˜ = (φ3 − iR3A3)/ 2. (19) We remark that all the new variables appearing in the parametrization are K-by-K matrices and that the parametrization (19) is intimately related with our choice (18) for the moduli. The above decomposition has an obvious physical interpretation: za are the modulus part and zã the fluctuations. For the modulus part, as in Eq. (6) for our previous deconstruction of IIA string, we expect that the lattice constants are simply inversely proportional to the VEV of the bi-fundamental boson variable. Schematically a ∝ 1/c2,3 such that aN ∼ 1. Indeed we will see that the spatial geometry, after taking the continuum limit, emerges from three modulus parameters za (a = 1, 2, 3); in other words, a world- volume metric will be constructed out of them. Our choice (18) of moduli will greatly simplify the complexity that arises from our quiver diagram being a two-dimensional triangular lattice. For the fluctuations, the parametrization in Eq. (19) corresponds to the infinitesimal form of the polar-coordinate decomposition of za. Note that the Yang–Mills fields come from the imaginary (or angular) part of the bi-fundamental fluctuations. Before proceeding to consider the continuum limit, we make the following remarks on ˆ algebraic properties of the matrices Va.Letf be a diagonal matrix in the site indices: fmn,mn = f(m,n)δmm δnn ,then ˆ † ˆ = Va f Va Saf, (20) where Sa is the shift operator by a unit along a-direction, i.e.,

S1f(m,n)= f(m− 1,n− 1), (21)

S2f(m,n)= f(m+ 1,n), (22)

S3f(m,n)= f(m,n+ 1). (23) Subsequently, one can easily derive the following relations for manipulations on a site function f : ˆ ˆ † = −1 ˆ − =[ ˆ ] ˆ † =−[ ˆ ]† Vaf Va Sa f, Va(Saf f) f,Va , f,Va f,Va . (24) 86 J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99

4.2. World-volume geometry from deconstruction

In the following two subsections, we will examine the bosonic part of the action, 1 R S = dτ Tr D ,Za 2 − 11 Za,Za † 2 + Za,Za 2 B R τ 11 2 1 i 2 i a 2 R11 i j 2 + Dτ ,Y − R11 Y ,Z + Y ,Y . (25) 2R11 4 With the parametrization proposed above, the quadratic part for Y i becomes 1 i 2 i a 2 SY = dτ Tr Dτ ,Y − R Y ,Z 2R 11 11 1 = dτ Tr Y˙ i + i Y ,Yi 2 2R 0 11 a 2 i 2πSa(z ) i a − R11∂aY + Y ,Sa z˜ , (26) N where we have introduced the lattice partial derivative,

∂af := N(Saf − f)/2π. (27) First we see the information of the metric on the would-be world-volume is contained in the term 2| a |2 = 1 ˙ i 2 − (2π) Sa(z ) i 2 SYK dτ R11 Tr 2 Y 2 ∂aY . (28) 2R11 N Recall that the target space index i runs from 1 to 3 and that the index a also from 1 to 3. Note that ∂1 is not independent of ∂2,3, because of the relation =− −1 −1 − −1 ∂1 S2 S3 ∂2 S3 ∂3. Changing the unit for time τ := R11τ and then suppressing the prime for τ , Eq. (28) changes into 1 00 ˙ i 2 22 i 2 33 i 2 SYK = dτ Tr g Y − g ∂2Y − g S2∂3Y 2 23 i i − g ∂2Y ,S2∂3Y , (29) from which we can read off the contravariant metric on world-volume 8π2(|z1|2 +|S (z2)|2) g00 = 1,g22 = 2 , N2 − 8π2(|z1|2 +|S 1(z3)|2) 8π2|z1|2 g33 = 1 ,g23 = (30) N2 N2 and g0k = 0 for the spatial index k = 2, 3. In this way, we see how the world-volume naturally metric arises from our (de)construction. J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99 87

Now we are ready to take the continuum limit. First regularize the trace Tr to be 2 2 (2π) tr /N κ; with our choice (18) of the moduli, we take κ = R2R3. To see how world- volume geometry comes out of this construction, we consider the continuum limit with 2 N →∞,thenTr→ d σ tr /R2R3, where tr is the trace over K-by-K matrices. Here we have taken the spatial world-volume coordinates σ 2,3 to run from 0 to 2π. Subsequently, the action in the continuum limit reads 2 1 1 αβ i i SYK =− dτ d σ tr g ∂αY ∂β Y , (31) R2R3 2 in which the contravariant metric on (2 + 1)-dimensional world-volume is αβ = − 2 2 g diag 1,R2,R3 . (32) In fact, we have specified the behavior of fluctuations zã, in the continuum limit N →∞, as O(1); namely the contribution from the fluctuations to the world-volume metric in Eq. (30) smears or smooths in the large-N limit. As another fact, due to our choice c1 = 0, world-volume metric (30) becomes diagonal, as shown in Eq. (32). Accordingly, 2 the factor κ, taken to be R2R3, multiplied by the coordinate measure d σ , is nothing but the invariant world volume measure, namely κ = −det(gαβ ),where = − 2 2 (gαβ ) diag 1, 1/R2, 1/R3 . (33) A further check shows that the area of the world-volume torus is (2π)2 Area of world-volume torus = −det(gαβ ) × coordinate area = . (34) R2R3

Now for the continuum limit of the full SY , it is easy to show that = 2 1 − αβ i i + i 2 + i 2 SY dτ d σ tr g DαY Dβ Y φ1,Y φa,Y , (35) 2R2R3 0 where Dα = ∂α + i[Aα, ·]and A0 = Y . In this subsection we have constructed, from our quiver mechanics, the toroidal geometry on the spatial world-volume, as well as the standard gauge interactions of world- volume scalar fields. (Incidentally let us observe that besides the local world-volume geometry, the modular parameter that describes the shape of the compactified 2-torus in target space is also dictated by the moduli za that also fix the shape of the triangular unit cell in the quiver diagram. However, in this paper we will concentrate on our special choice (18) for the moduli, and leave the analysis of more general moduli to the sequential paper.)

4.3. Yang–Mills ﬁeld construction

Now we consider the action of bi-fundamental ﬁelds; our goal here is to construct the desired SYM. Collecting the relevant terms in the action, we have 1 a 2 R11 a a† 2 a a 2 SZ = dτ Tr Dτ ,Z − Z ,Z + Z ,Z . (36) R11 2 88 J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99

− − [ a † a]= ˆ † ˆ = a † a − 1 a 1 a † Write Z ,Z Va Pa aVa, with Pa a z z Sa z Sa z . Separating the VEV and fluctuations, − − 1/2 ¯ + 1 ã † 1/2 +ã −1 a 2 ca Sa z 2 ca z −1 a† a† a P = S ∂ z˜ + S ∂az˜ + z˜ , z˜ . a a a a (2π)−1N (2π)−1N a − † = [ a a ]= ˆ ˆ = a 1 a − Therefore, Paa Paa . Moreover, Z ,Z Qa aVaVa , with Qa a z Sa z a −1 a z Sa z ; thus − − − −1 1/2 + 1 ã 1/2 + ã −1 a 2 ca Sa z 2 ca Sa z −1 a Q = S ∂ z˜ − S ∂az˜ a a a a (2π)−1N (2π)−1N a + −1 ã ã Sa z , z .

Hence Qaa =−Qaa . The action (36) is recast into 1 2 R11 = a − 2 + | |2 +| |2 SZ dτ Tr Dτ ,Z Paa 2 Pa a Qa a R11 2 a a

Here the gauge field strength Fαβ is defined in the usual way. Subsequently, we have, in the continuum limit, 2 1 1 αβ 1 αβ M M 1 M N 2 SZ = dτ d σ tr − F Fαβ − g Dαφ Dβ φ + φ ,φ , (52) R2R3 4 2 4 M in which φ include φ1,2,3 and φ1. In summary, Eqs. (35) and (52) leads to the complete continuum bosonic action: 2 1 1 αβ 1 αβ I I 1 I I 2 SB = dτ d σ tr − F Fαβ − g DαΦ Dβ Φ + Φ ,Φ , (53) R2R3 4 2 4 where ΦI include both φM and Y i . Eq. (53) is the dimensional reduction from the bosonic part of either four-dimensional = 4 SYM or ten-dimensional = 1 SYM. Therefore, with appropriate rescaling of world- i volume fields, the action (53) possesses an O(7) symmetry for fields Y , φa and φ1. We remark that the first two terms in the continuum action (53) contain one and the same world-volume metric. This cannot be a mere coincidence; it is dictated by the stringent internal consistency of our deconstruction procedure.

4.4. Fermion construction

The deduction of the continuum action for fermionic variables can be carried out in a similar way. We only emphasize that the phase of D-particle moduli can be absorbed int redeﬁnition of fermionic variables in Eq. (16). Indeed after assigning nonzero VEV (or moving the D-particles collectively away from the orbifold singularity), the free fermionic action becomes 2 (2π) 1† 0 0 S = dτ tr λ (−)c λ + + − λ FF N2κ m,n 1 m 1,n 1 m,n m,n 2† 0 0 3† 0 0 + λ ic2 λ − − λ + λ (−i)c3 λ − − λ m,n m 1,n m,n m,n m,n 1 m,n 2 2 3 1 3 3 − λ + − λ − c1λ + iλ c2 λ + + − λ + m 1,n m,n 1 m,n m,n m 1,n 1 m,n 1 + 1 3 − 3 + iλm,nc3 λm+1,n+1 λm+1,n h.c. , (54) ˆ where we have introduced λa by writing (Λa)mn,mn = (λa)mn(Va)mn,mn for a = ˆ 0, 1, 2, 3, with V0 the unit matrix. With the choice (18) and κ = R2R3, the continuum limit reads = 2 1 † − + † SFF dτ d σ tr λ2( i)R2∂2λ0 λ3iR3∂3λ0 R2R3 + λ1iR2∂2λ3 + λ1iR3∂3λ2 + h.c. . (55) Comparing with our deﬁnition of gamma matrices, (10) and (15), we get 2 1 † 7 9 SFF = dτ d σ tr λ i γ R2∂2 + γ R3∂3 λ /2. (56) R2R3 90 J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99

The appearance of the seventh and ninth gamma matrices can be easily understood from the parametrization of the fluctuations in Eq. (19), since the gauge fields arise from the fluctuations in the imaginary part of complex coordinates zã. The continuum limit of the complete action for fermionic variables is thus 2 1 i † i † 7 SF = dτ d σ tr − λ [Dτ ,λ]+ λ γ R2[D2,λ] R2R3 2 2 i 1 + λ†γ 9R [D ,λ]+ λ†γ I ΦI ,λ . (57) 2 3 3 2 Summarily, Eq. (53) plus Eq. (57) constitute precisely the desired d = 2 + 1 SYM with sixteen supercharges. This outcome in the continuum limit verifies our idea that M(atrix) theory compactification can be deconstructed in terms of orbifolding and quiver mechanics, so that we are on the right track to IIB/M duality. Moreover, the constructed world-volume geometry, in view of world-volume/target-space duality, provides the platform for the discussion in next section on the matrix membranes wrapping on the compactified 2-torus in target space.

5. Wrapping membranes and their spectrum

In the above we have “(de)constructed” the M(atrix) theory compactified on a 2-torus, by starting from orbifolding and then assigning nonzero VEVs to bi-fundamental bosons. According to the IIB/M duality conjecture, M(atrix) theory compactified on a 2-torus is dual to IIB string theory on a circle. In this section, we will show that the quiver mechanics resulting from our deconstruction procedure indeed possesses features that are characteristic to the IIB/M duality. More concretely, we will explicitly construct matrix states in our quiver mechanics, which in the continuum limit correspond to membranes wrapping over compactified a 2-torus in the target space. And we will show a SL(2, Z) symmetry for these states and the emergence from these states of a new flat dimension when the compactified 2-torus shrinks to zero. As mentioned in Section 1, these features are central for verifying the IIB/M duality conjecture. The existence of these states and their properties were discussed in the literature of M(atrix) theory in the context of d = 2 + 1 SYM, but to our knowledge the explicit matrix construction of these states is still absent in the literature.

5.1. Classical wrapping membrane

To motivate our construction, we recall that a closed string winding around a circle is described by a continuous map from the worldsheet circle to the target space circle, satisfying the periodic condition: x(σ + 2π) = x(σ) + 2πwR,wherew is the winding 2 number. For a C /ZN orbifold, a winding string is a state in the twisted sector satisfying z(σ + 2π)= ei2πw/Nz(σ). J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99 91

In the same way, in the nonlinear sigma model for a toroidal membrane wrapping on a compactiﬁed 2-torus, we have the boundary conditions

x1(p + 2π,q) = x1(p, q) + 2πmR1,

x1(p, q + 2π)= x1(p, q) + 2πnR1,

x2(p + 2π,q) = x2(p, q) + 2πsR2,

x2(p, q + 2π)= x2(p, q) + 2πtR2 (58) where (p, q) (0 p,q 2π) parameterize the membrane. The solutions to these conditions are given in Eq. (1). They contain four integers (m,n,s,t). One combination of them is SL(2, Z) invariant, i.e., the winding/wrapping number: w = mt − ns. (59) (This is a slight modiﬁcation of Schwarz’s original construction [2,26]; note that this combination appeared also in the context of noncommutative torus, see for example Eq. (4.2) in [27].) The geometric signiﬁcance of w is simply the ratio of the pullback area of the membrane to the area of the target torus. In IIB/M duality, w is related to the Kaluza–Klein momentum in the newly emergent dimension. C3 Z2 For a membrane wrapping on the orbifold / N , the twisted boundary conditions become z2(p + 2π,q) = ei2πm/Nz2(p, q), z2(p, q + 2π)= ei2πn/Nz2(p, q), z3(p + 2π,q) = ei2πs/Nz3(p, q), z3(p, q + 2π)= ei2πt/Nz3(p, q), (60) as well as − + z1(p + 2π,q) = e i2π(m s)/Nz1(p, q), + z1(p, q + 2π)= ei2π(n t)/Nz2(p, q). (61) Their solutions are of the form 1 −1/2 −i[(m+s)p+(n+t)q]/N z (p, q; τ)= 2 c1(τ)e × (oscillator modes), 2 −1/2 i(mp+nq)/N z (p, q; τ)= 2 c2(τ)e × (oscillator modes), 3 −1/2 i(sp+tq)/N z (p, q; τ)= 2 c3(τ)e × (oscillator modes). (62)

Here the coefﬁcients ca(τ) describe the center-of-mass degrees of freedom. Again, the solutions contain a quadruple, (m,n,s,t), of integers, giving rise to an SL(2, Z) invariant wrapping number w = mt − ns. Now let us consider a stretched membrane (62), with the oscillator modes suppressed (by setting the corresponding factor to unity). The membrane stretching energy density is known to be proportional to {Y I ,YJ }{Y I ,YJ }, in terms of Poisson bracket with respect to the canonical symplectic structure on the membrane spatial world-volume. (See, e.g., Ref. [28] for a latest review and also for a comprehensive list of references.) It is easy to 2 3 show that with choosing c1 = 0, c2,3 ∝ NR2,3, the Poisson bracket of z (p, q) and z (p, q) is 2 3 z (p, q), z (p, q) ∝ wR2R3. (63) 92 J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99

2 So the energy of this wrapping conﬁguration is proportional to (wR2R3) ,havingthe correct dependence on w and on the area of the 2-torus in target space, because the choice of ca corresponds to a regular target torus. We note that to show IIB/M duality in the context of M(atrix) theory, we need a construction of wrapping membranes either in the setting of d = 2 + 1 SYM or in that of our quiver mechanics. But the above construction (62) is in neither, so it is not what we are looking for. However, we may view it as a sort of the classical and continuum limit of the wrapping matrix membrane we are looking for, pointing to the direction we should proceed.

5.2. Wrapping matrix membranes

Now we try to transplant the above continuum scenario into the notion of matrix membrane. It is well known that for the finite matrix regularization of a membrane, whose topology is a torus, one needs to prompt the membrane coordinates into the clock and shift ip iq matrices, namely to substitute e → UK and e → VK [28], where K is the number of D-particles that constitute the membrane. The key property of UK and VK for this substitution to work is m n = mn n m UK VK (ωK ) VK UK , (64) i2π/K with arbitrary integers m, n,andωK = e . In the classical or continuum limit, it gives to the correct Poisson bracket between p and q. Eq. (62) motivates us to deal with the fractional powers of the clock and shift matrices. The guide line to formulate the arithmetics of the fractional powers is to generalize Eq. (64) to be valid for fractional powers. Below we model a simplest substitution scheme ip/N → =: 1/N iq/N → =: 1/N e UKN2 UK ,e VKN2 VK , (65) which is enough at the current stage to enable us to explore the physics of IIB/M duality. In accordance with these substitutions, we introduce the following ansatz for the matrix configuration of a wrapping membrane: 1 = −1/2 −(m+s) −(n+t) ⊗ ˆ Z(mem) 2 c1U V V1, 2 = −1/2 m n ⊗ ˆ Z(mem) 2 c2U V V2, 3 = −1/2 s t ⊗ ˆ Z(mem) 2 c3U V V3, (66) := := with omitted subscripts U UKN2 ,V VKN2 . These matrices are KN4-by-KN4, of the form of a direct product of three factors, with the rank KN2, N and N respectively. In our quiver mechanics, the first factor describes Z2 K D-particles on the N orbifold while the last two factors describe the deconstructed ˆ 2 torus coordinates. The form of the last two factors in ansatz (66), say V2 in Z(mem),is m n 2 dictated by the orbifolding conditions. The form of the first factor, say√ U V in Z(mem) m n is motivated by Eqs. (62). In other words, the expression c2U V / 2 can be viewed as 2 polar coordinate decomposition of Z(mem),inwhichc2 is interpreted as the distance of the membrane on the orbifold to the singularity, while the unitary matrix U mV n shows how J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99 93 the membrane constituents wrap in the angular direction of Z2. A similar interpretation 3 1 holds for Z(mem) and Z(mem). Though the above intuitive picture looks satisfactory, the real justification for the ansatz (66) to describe wrapping membranes on a compactified 2-torus should come the detailed study of the physical properties of these states in accordance with M(atrix) theory, which will be carried out in next a few subsections.

5.3. Spectrum and SL(2, Z) invariance

a From now on, we will suppress the subscript in Z(mem). First let us verify that the equations of motion, that follow from the action (14), R2 Z¨ a + 11 Zb, Zb†,Za + Zb†, Zb,Za = 0 (67) 2 can be satisfied by the ansatz (66). Indeed, from the experience of the equation of motion for the spherical matrix membrane, to solve the time-dependence of the scalar coefficients (ca here) is by far kind of involved [29,30]; for the toroidal case, however, this situation is simplified because the coefficient functions are complex. It is a straightforward calculation −iω τ to show that the equations of motion (67) are satisfied if we take ca(τ) = ca0e a , with 2 2 2πw 2 ω = R 1 − cos |cb0| . (68) a 11 KN2 b =a

Consistent with our previous choice (18) for the moduli, hereafter we will just take c10 = 0, ck0 = NRk/2π for (k = 2, 3). Accordingly, we have − 2πw ω2 = (2π) 1NR R 2 1 − cos , (69) 2 11 3 KN2 − 2πw ω2 = (2π) 1NR R 2 1 − cos . (70) 3 11 2 KN2 Moreover concerning the membrane dynamics, one can easily check an additional constraint equation Z˙ a,Za† + Z˙ a†,Za = 0, (71) which plays the role of the gauge ﬁxing of the membrane world-volume diffeomorphism symmetry; see, e.g., Ref. [28]. Next let us examine the energy of the wrapping membrane states (66). By a direct calculation, we ﬁnd that the potential (or stretching) energy density (before taking the trace Tr) is proportional to the unit matrix, 1N4 : a b† 2 a b 2 V = R11 Z ,Z + Z ,Z a

Including the kinetic energy due to the τ -dependence of ca, the total energy of the membrane is given by 3 3 = 1 ˙ a2 + = 1 |˙ |2 + = Hmem Tr Z V ca Tr 1N4 Ew 2Ew. (74) R11 2R11 a=1 a=2 A characteristic feature of Eqs. (73) and (74) is that though the states given by ansatz (66) contain four integers (m, n; s,t), their energy only depends on the wrapping number w = mt − ns, which is known invariant under the SL(2, Z) transformations m ab m s ab s = , = , (75) n cd n t cd t where a,b,c,d are integers, satisfying ad − bc = 1. This result is one chief evidence to justify our matrix construction (66), showing the SL(2, Z) symmetry of the spectrum of the wrapping membranes directly and explicitly in a constructive manner; this SL(2, Z) in Eqs. (75) is right the modular symmetry of the target torus, as well being interpreted as the S-duality in IIB string theory.

5.4. Emergent ﬂat dimension

In last subsection, we have seen that the stretched energy Ew in Eq. (73) contains a factor (1 − cos(2πw/KN2)). Such factor is a common praxis in the lattice (gauge) ﬁeld theory for the spectrum of the lattice Laplacian. In the large-N limit, when w is small 2 2 2 2 2 4 compared to N , one has (1 − cos(2πw/KN )) → 2π w /K N . Hence Ew recovers 2 2 the w (R2R3) behavior of the Poisson bracket calculation for continuous membranes in previous Subsection 5.1. (See Eq. (63) and discussion below it.) More precisely, we have

2 2 Ew = w R11(R2R3) /2K. (76) Consequently, it is amusing to note that this stretched energy can be rewritten as the light-cone energy due to a transverse momentum pw: 2 pw Ew = ,pw = w · R R , (77) 2P + 2 3 + in which the light-cone mass P = K/R11! This energy is of the form of the energy for an excitation with Kaluza–Klein momentum w on a transverse circle with the radius 1/R2R3. 3 (After recovering the Planck length, this radius is lp/R2R3.) We assert that the spectrum of wrapping membranes is identical to that of excitations on a transverse circle. In fact, though there exists the apparent degeneration due to the states with the same wrapping number w for different combinations of (m,n,s,t),however,as noted by Schwarz [26], this degeneration can be easily eliminated by considering F–D J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99 95 string spectrum and SL(2, Z) equivalence. Therefore, with wrapping membrane states viewed as momentum states, a new ﬂat dimension opens up when the compactiﬁed 2- torus, on which membranes wrap, shrinks to zero (i.e., R2R3 → 0). As mentioned in the introduction, this is exactly what the IIB/M duality requires for a M-theory formulation of IIB string theory.

5.5. Membrane tension

To verify that the membrane has the correct tension, we note that the light cone energy in M-theory for a stretched transverse membrane is given by

2 Mw Em = , (78) 2P + = = 2 in which Mw wTmemAT 2 . Tmem 1/(2π) is the membrane tension, with Planck length taken to be unity. AT 2 is the area of the compactiﬁed 2-torus in target space, which in our = 2 case is regular with sides 2πR2 and 2πR3;soAT 2 R2R3(2π) . Therefore we have 2 2 Em = w R11(R2R3) /2K. (79) So in the large-N limit, we have the equality:

Ew = Em. (80) This is a perfect and marvelous match! An equivalent statement is that our wrapping matrix membrane indeed has the correct tension as required by M-theory. An astute reader may have noticed that the energy (74) of our membrane conﬁguration contains a kinetic part, which also equals Ew. We are going to clarify its origin in the next section.

5.6. Adding center-of-mass momentum

To better understand the origin of the kinetic part in the energy (74), which is equal to the stretching energy Ew, we try to add a generic center-of-mass momentum to the membrane states (66). Let us first examine a closed string. A winding state on a compactified circle satisfies xw(σ + 2π,τ) = xw(σ, τ) + 2πwR, with R the radius of the circle. The periodic boundary condition is a linear homogeneous relation, therefore the momentum quantum number can be added as a zero-mode part x0, which is linear in τ , such that xw + x0 still satisfies the same equation. Now we turn to the orbifold description at large N of the same fact, in which x/R becomes the angular part of a complex coordinate z and the radius of this circle is NR.At large N we can make the following substitution:

ixw/NR z = NR + i(xw + x0) ↔ NRe + ix0. (81)

The equation of motion for the string on the C/ZN orbifold gives the solution xw/R = w(σ − τ),forw>0, xw/R = w(σ + τ),forw<0andx0 = pτ. Suppose w>0for 96 J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99 definiteness. One finds the energy density is given by w h =|˙z|2 +|z |2 = p2 + (Rw)2 − 2pRw cos (σ − τ) + (Rw)2. (82) N It is amusing to observe that in the large-N limit, h approaches to (p − Rw)2 + (Rw)2. With p = 0, the kinetic energy is equal to the stretching energy! This feature is the same as we have encountered in Eq. (74), indicating that this is a common phenomenon in approaching compactification via orbifolding. Giving a bit more details, we note that the orbifolding solution, in the large-N limit, becomes z → NR + i(p − Rw)τ + iRwσ. Accordingly, we should interpret the linear combination p − Rw as the momentum in the continuum limit, though naively one might have expected that p would be identified as the center-of-mass momentum. The same analysis can be applied to the continuous membrane wrapping on an orbifold: 2 −1/2 i(N−1(mp+nq)−ω τ) z (p, q; τ)= 2 ik2τ + NR2e 2 , −1 3 −1/2 i(N (sp+tq)−ω3τ) z (p, q; τ)= 2 ik3τ + NR3e . (83) A similar analysis shows again that the momenta in the continuum limit are shifted by the wrapping number: pa = ka − NRaωa ,fora = 2, 3. Thus with ka = 0, a membrane wrapping on a torus is not at rest! The states in Eqs. (81) and (83) can be generalized to matrix membranes, with center- of-mass momentum added: − Z2 = 2 1/2 iR k τ/2πK + c (τ)U mV n ⊗ Vˆ , 11 2 2 2 3 −1/2 s t ˆ Z = 2 iR11k3τ/2πK + c3(τ)U V ⊗ V3, (84) where, for a regular torus in target space, we have set Z1 = 0. As in the previous section, we −iω τ take ca = ca0e a with ωa (a = 2, 3) in Eq. (69). In large-N limit, the matrix membrane configuration in Eq. (84) is expected to be equivalent to the continuous configuration in Eq. (83) up to a normalization. := −1| ˙ |2 The kinetic energy density in the discrete setting, T R11 Za , can be easily computed: 1 R k 2 R k = 11 a + 2| |2 − 11 a ma na + ⊗ T ωa ca ωacaU V h.c. 1N2 , 2R11 2πK 2πK (85) in which the summation is over a = 2, 3, m2 = m, n2 = n, m3 = s, n3 = t. Adding the T in (85) to the V in Eq. (72), we get the total energy density, which results in the total energy of a wrapping membrane configuration with momenta and wrapping: 2 = R11 − 2πK | | + Ek2,k3;w ka ωa ca Ew. (86) 2K R11 We are glad, just for convenience, to rewrite √ −1 2 ω2 = ( 2π) NR11R3 sin πw/KN , (87) J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99 97 √ −1 2 ω3 = ( 2π) NR11R2 sin πw/KN , (88)

|ca|=NRa /2π. (89) Accordingly, the genuine momentum is identiﬁed to be √ √ −1 2 2 pa = ka − ( 2π) KN R2R3 sin πw/KN → ka − R2R3w/ 2. In closed string theory, the quantization of momentum is a consequence of the single- ˆ valuedness of the translation operator ei2πRapa (nosummationona). In our present case, we can also impose the quantization condition at large N;asaresult√ pa = la/Ra where la is an integer (or equivalently, one has ka = la/Ra + R2R3w/ 2). Note that the domain of (p2,p3) is just the lattice dual to the target torus; therefore, the above equation is once Z more a manifestation of the covariance for Ek2,k3;w under the SL(2, ) transformations over the target torus. In summary, our explicit construction of wrapping membrane states, the appearance of SL(2, Z) symmetry in their spectrum, and an emergent ﬂat dimension as well, all these combined together, constitute strong evidence for our quiver matrix mechanical model to be a nonperturbative formulation of IIB string theory, which naturally exhibits IIB/M duality.

6. Discussions and perspectives

Problems that remain for future study and some perspectives are collected in this section. Some of them are just slightly touched in this work. (As mentioned before, we will leave to a sequential paper the discussions on generic moduli for the VEV of Za (a = 1, 2, 3), which would allow both the SYM world-volume torus and the compactiﬁed target space torus nonregular, and would explicitly demonstrate the SL(2, Z) duality of our approach to IIB string theory.)

(1) There have been three different ideas to deal with IIB/M duality, i.e., M-theory on a 2-torus should be dual to IIB string theory on a circle: namely the wrapping membrane states, the vector–scalar duality in three dimensions and the magnetic charges of membranes, respectively (see for example [12]). We did not explore the latter two in the present work. In fact, a naive definition of magnetic charge such as Tr[Za,Za ] for finite matrix configurations vanishes identically. (2) For IIA/M duality, there is a complete dictionary of the correspondence between the spectrum, as well as operators, between IIA D0 brane and M-theory objects, in the matrix string theory a la DVV [8]. As for IIB/M duality a comprehensive dictionary for spectrum and operators between the two sides remains to be worked out. Moreover, in this work we recovered only part of the SL(2, Z)-invariant spectrum. A more detailed study to demonstrate the full SL(2, Z) symmetry is, in principle, possible in the present framework for IIB strings, and we leave it for future research. How to relate this approach to other nonperturbative IIB string theory, such as IIB matrix model [31] or that based on the D-string action [9], is another interesting issue to address. 98 J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99

(3) A discrete approach, the string bit model, has been proposed to IIB string theory before [32]. The difference between our quiver mechanics approach and the string bit model is the latter discretizes the nonlinear sigma model for string theory on sites of a lattice, while we deconstruct SYM with part of matter fields living on links. Recently the string bit model gained revived interests [33] in the context of the BMN correspondence [34]. On the other hand, BMN have devised a (massive) matrix model in PP-wave background. How to do orbifolding with this M(atrix) model remains a challenge. (4) Probing spacetime with strings has revealed T-duality of spacetime; i.e., stringy dynamics gives rise to new features of spacetime geometry. In the same spirit, one expects that probing spacetime with membranes would expose new, subtle properties or structures in spacetime geometry too. Actually it has been suggested [13] that M(atrix) theory compactified on circle, on two- and three-torus are tightly related to each other. We leave the exploration in the framework of quiver mechanical deconstruction to the future. (5) The appearance of nonunitary fields living on links is a generic phenomenon in discrete models. In addition to our previous work [35], a few other authors also paid attention to the effects of nonunitary link fields [36–38]. Here we would like to emphasize the significant role of nonunitary link fields in dynamics of geometry, which is worthwhile to explore in depth.

Acknowledgements

Y.S.W. thanks the Interdisciplinary Center for Theoretical Sciences, Chinese Academy of Sciences (Beijing, China) for warm hospitality during his visit, when the work was at the beginning stage. J.D. thanks the High Energy Astrophysics Institute and Department of Physics, the University of Utah, and Profs. K. Becker, C. DeTar and D. Kieda for warm hospitality and ﬁnancial support; he also appreciates the helpful discussion with B. Feng and K. Becker.

References

[1] E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B 443 (1995) 85, hep-th/9503124. [2] J.H. Schwarz, An SL(2, Z) multiplet of type IIB superstrings, Phys. Lett. B 360 (1995) 13–18, hep- th/9508143; J.H. Schwarz, Phys. Lett. B 364 (1995) 252, Erratum. [3] J.H. Schwarz, The power of M theory, Phys. Lett. B 367 (1996) 97–103, hep-th/9510086. [4] P.S. Aspinwall, some relationships between dualities in string theory, Nucl. Phys. (Proc. Suppl.) 46 (1996) 30–38, hep-th/9508154. [5] T. Banks, W. Fischler, S.H. Shenker, L. Susskind, M-theory as a matrix model: a conjecture, Phys. Rev. D 55 (1997) 5112, hep-th/9610043. [6] L. Susskind, Another conjecture about M(atrix) theory, hep-th/9704080. [7] W. Taylor, D-brane ﬁeld theory on compact spaces, Phys. Lett. B 394 (1997) 283, hep-th/9611042. [8] R. Dijkgraaf, E. Verlinde, H. Verlinde, Matrix string theory, Nucl. Phys. B 500 (1997) 43, hep-th/9703030. [9] P.M. Ho, Y.S. Wu, IIB/M duality and longitudinal membranes in M(atrix) theory, Phys. Rev. D 57 (1998) 2571, hep-th/9703016. J. Dai, Y.S. Wu / Nuclear Physics B 684 (2004) 75–99 99

[10] M. Agianagic, J. Park, C. Popescu, J. Schwarz, Dual D-brane actions, Nucl. Phys. B 496 (1997) 215, hep- th/9702133. [11] L. Susskind, T-duality in M(atrix) theory and S-duality in field theory, hep-th/9611164. [12] S. Sethi, L. Susskind, Rotational invariance in the M(atrix) formulation of type IIB theory, Phys. Lett. B 400 (1997) 265, hep-th/9702101. [13] W. Fischler, E. Halyo, A. Rajaraman, L. Susskind, The incredible shrinking torus, Nucl. Phys. B 501 (1997) 409–426, hep-th/9703102. [14] J. Dai, Y.-S. Wu, Quiver mechanics for deconstructed matrix string, Phys. Lett. B 576 (2003) 209–218, hep-th/0306216. [15] N. Arkani-Hamed, A.G. Cohen, H. Georgi, (De)Constructing dimensions, Phys. Rev. Lett. 86 (2001) 4757, hep-th/0104005. [16] I. Rothstein, W. Skiba, Mother moose: generating extra dimensions from simple groups at large N, Phys. Rev. D 65 (2002) 065002, hep-th/0109175. [17] N. Arkani-Hamed, A.G. Cohen, D.B. Kaplan, A. Karch, L. Motl, Deconstructing (2, 0) and little string theories, JHEP 0301 (2003) 083, hep-th/0110146. [18] S. Mukhi, M. Rangamani, E. Verlinde, Strings from quivers, membranes from moose, JHEP 0205 (2002) 023, hep-th/0204147. [19] D.B. Kaplan, E. Katz, M. Ünsal, Supersymmetry on a spatial lattice, JHEP 0305 (2003) 037, hep- lat/0206019; A. Cohen, D. Kaplan, E. Katz, M. Ünsal, Supersymmetry on a euclidean spacetime lattice I: a target theory with four supercharges, JHEP 0308 (2003) 024, hep-lat/0302017. [20] N.R. Constable, J. Erdmenger, Z. Guralnik, I. Kirsch, (De)Constructing intersecting M5-branes, Phys. Rev. D 67 (2003) 106005, hep-th/0212136. [21] I. Kirsch, D. Oprisa, Towards the deconstruction of M-theory, hep-th/0307180. [22] N. Seiberg, Why is the matrix model correct?, Phys. Rev. Lett. 79 (1997) 3577, hep-th/9710009. [23] A. Sen, D0 branes on T n and matrix theory, Adv. Theor. Math. Phys. 2 (1998) 51, hep-th/9709220. [24] A. Connes, Noncommutative Geometry, Academic Press, 1994, Appendix C. [25] M.R. Douglas, Two lectures on D-geometry and noncommutative geometry, hep-th/9901146. [26] J.H. Schwarz, Lectures on superstring and M-theory dualities, Nucl. Phys. B (Proc. Suppl.) 55 (1997) 1–32, hep-th/9607201. [27] J.C. Varilly, An introduction to noncommutative geometry, physics/9709045. [28] W. Taylor, M(atrix) theory: matrix quantum mechanics as a fundamental theory, Rev. Mod. Phys. 73 (2001) 419–462, hep-th/0101126. [29] D. Kabat, W. Taylor, Spherical membranes in matrix theory, Adv. Theor. Math. Phys. 2 (1998) 181–206, hep-th/9711078. [30] S.-J. Rey, Gravitating M(atrix) Q-balls, hep-th/9711081. [31] N. Ishibashi, H. Kawai, Y. Kitazawa, A. Tsuchiya, A large-N reduced model as superstring, Nucl. Phys. B 498 (1997) 467, hep-th/9612115. [32] O. Bergmann, C.B. Thorn, String bit models for superstring, Phys. Rev. D 52 (1995) 5997, hep-th/9506125. [33] H. Verlinde, Bits, matrices and 1/N, hep-th/0206059; D. Vaman, H. Verlinde, Bit strings from N = 4 gauge theory, hep-th/0209215. [34] D. Berenstein, J.M. Maldacena, H. Nastase, Strings in flat space and pp waves from N = 4 super-Yang– Mills, JHEP 0204 (2002) 013, hep-th/0202021. [35] J. Dai, X.-C. Song, Wilson action of lattice gauge fields with an additional term from noncommutative geometry, hep-th/0101184. [36] C. Lehmann, G. Mack, Gauge theories with nonunitary parallel transporters: a soluble Higgs model, hep- lat/0305026. [37] C. Lehmann, G. Mack, T. Prüstel, Origin of quark masses and Cabbibo–Kobayashi–Maskawa matrix in a gauge theory with nonunitary parallel transporters, hep-ph/0305331. [38] V. Jejjala, R.G. Leigh, D. Minic, The cosmological constant and the deconstruction of gravity, Phys. Lett. B 556 (2003) 71–79, hep-th/0212057; V. Jejjala, R.G. Leigh, D. Minic, Deconstructing the cosmological constant, gr-qc/0305072. Nuclear Physics B 684 (2004) 100–124 www.elsevier.com/locate/npe

D-branes of covariant AdS superstrings

Makoto Sakaguchi, Kentaroh Yoshida

Theory Division, High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan Received 6 November 2003; received in revised form 5 February 2004; accepted 13 February 2004

Abstract 5 We consider D-branes of open superstrings in the AdS5 × S background. The possible configurations of D-branes preserving half of supersymmetries are classified by analyzing the κ- invariance of an open superstring in a covariant manner. We also revisit the classification of D-branes 5 in the pp-wave background. It is shown that Penrose limits of the possible D-branes in the AdS5 × S give all of the D-branes in the pp-wave. In addition, a 1/4 supersymmetric D-string, which is related to the D-string preserving 8 dynamical supersymmetries in the pp-wave, is presented. We also discuss the relation between our result and the AdS branes in a brane probe analysis.  2004 Elsevier B.V. All rights reserved.

PACS: 11.25.-w; 11.30.Pb

Keywords: D-branes; AdS string; AdS branes; Wess–Zumino term; κ-symmetry; Penrose limit; pp-wave

1. Introduction

The discovery of D-branes [1] has played an important role for revealing non- perturbative aspects of superstring theories and M-theory [2]. Recent interest in studies of D-branes is the classification of possible configurations in non-trivial backgrounds. Among other non-trivial backgrounds, the maximally supersymmetric type IIB pp-wave background [3] has attracted great interests because the Green–Schwarz superstring on this background was shown to be exactly solvable [4,5]. The configurations of D-branes on pp-wave backgrounds were examined in many works [6–22]. In particular, covariant analyses of D-branes of an open string in pp-waves were given in [14,19] by applying a method developed by Lambert and West [23].

E-mail addresses: [email protected] (M. Sakaguchi), [email protected] (K. Yoshida).

It is also possible to apply this method for analyzing Dirichlet branes of an open supermembrane. In the eleven-dimensional flat spacetime, Dirichlet p-branes were shown to be allowed for the values p = 1, 5, 9 [24,25]. The p = 5 case corresponds to M5-brane, while the p = 9 case describes the end-of-world 9-brane in the Horava–Witten theory [26]. Studies of open supermembrane theories are expected to promote our understanding of features of M-branes. Dirichlet branes of an open supermembrane on the maximally supersymmetric pp-wave background (Kowalski–Glikman solution)1 were classified in [30,32]. In the work [32], we provided a covariant classification of the possible D-brane configurations by combining methods in Refs. [24,25] and Ref. [23]. We found 1/2 supersymmetric M5-brane and 9-brane configurations sitting at the origin of the pp- wave. These configurations are not 1/2 supersymmetric outside the origin while 1-brane configurations are 1/2 supersymmetric at and outside the origin. Also, 1/2 supersymmetric 9-brane configurations obtained in Ref. [32] are expected to be available for a study of a heterotic matrix model [33]. If we consider a double dimensional reduction of open supermembrane on the pp-wave background to a type IIA string theory [13] through a compactification of one of transverse dimensions, then M5- and 9-branes in eleven dimensions wrapping an S1-circle become possible configurations of D4-branes and D8- branes in the resulting type IIA pp-wave background [13,16,19,20]. This correspondence strongly supports the weak-strong duality between M-theory and type IIA string theory even in pp-wave backgrounds, and thus we can say that a consistency of superstring theories on a pp-wave has been checked. In our previous work [34], Dirichlet branes of an open supermembrane in the AdS4/7 × S7/4 backgrounds were examined, and possible configurations were classified. It was shown that under the Penrose limit [35] possible Dirichlet branes are mapped one-to- one to those on the pp-wave background, as expected from the fact that the Kowalski– 7/4 Glikman solution [36] can be obtained from the AdS4/7 × S backgrounds [37]. The 7/4 allowed Dirichlet branes are also related to AdS branes in the AdS4/7 × S backgrounds. Motivated by our previous works [32,34], we will consider D-branes of an AdS string by analyzing the covariant Wess–Zumino term. First, we classify D-brane configurations of 5 an open superstring in the AdS5 × S background. The possible D-brane configurations can be compared to the AdS brane configurations. Secondly, we extend the covariant classification of D-branes in the pp-wave [14] by including the cases that the null directions (+, −) are Dirichlet ones. Thirdly, we examine Penrose limits of the possible D-branes 5 in the AdS5 × S background, and directly show that all of D-branes in the pp-wave are realized in these limits. In addition, we present an example of 1/4 supersymmetric D-string 5 configurations in the AdS5 × S background, which preserves 1/4 supersymmetries even outside the origin. This configuration reduces to the 1/4 supersymmetric D-string in the pp-wave background found in the work [14] through the Penrose limit. From the viewpoint 5 of embedded branes in the AdS5 × S background, this D-string may possibly correspond to R × S1.

1 Supermembrane on the pp-wave background is related to the pp-wave matrix model [27] through the matrix regularization [28]. This relation in the pp-wave case is discussed in [29–31]. 102 M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124

The organization of our paper is as follows: in Section 2, we introduce the action of an 5 open superstring on the AdS5 × S background, and the covariant Wess–Zumino term is presented. In Section 3, we classify possible 1/2 supersymmetric D-brane configurations of the AdS string by investigating the vanishing conditions of the κ-variation surface terms of the covariant Wess–Zumino term. In addition, we find a 1/4 supersymmetric D-string configuration. This D-string preserves a quarter of supersymmetries both at and outside the origin. In Section 4, we classify D-brane configurations preserving half of supersymmetries in the pp-wave background which include the cases that the null directions (+, −) satisfy the Dirichlet conditions. In Section 5 we consider the Penrose limit of the result obtained in Section 3. All of the resulting configurations after the Penrose limit are shown to be included in the list presented in Section 4. We also discuss the relation between our result 5 and AdS branes in the AdS5 × S background. Section 5 is devoted to a conclusion and discussions. In Appendix A, our notation and convention are summarized.

2. Covariant Wess–Zumino term for the AdS string

In this section, we will introduce the Green–Schwarz action of a superstring on the 5 AdS5 ×S background (called AdS string below), and explain some characteristics relevant to our later considerations. In this paper, we do not utilize any speciﬁc gauges but discuss in a covariant fashion, because the covariant expression of the Wess–Zumino term is available for the covariant classiﬁcation of D-branes of the AdS string. First of all, the action of AdS string we consider is written as 2 S = d σ [LNG + LWZ], LNG =− −g(X,θ). (2.1)

The Nambu–Goto part of this Lagrangian is represented in terms of the induced metric gij , which is given by (for notation and convention, see Appendix A) ˆ g = EM EN G = EAEB η ,g= detg ,EA = ∂ ZM EA , (2.2) ij i j MN i j AB ij i i Mˆ Mˆ M α¯ A 5 where Z = (X ,θ ) and E are supervielbeins of the AdS5 × S background. For D- Mˆ strings, g is replaced with det(gij + Fij ) where F is deﬁned by F = dA − B with the Born–Infeld U(1) gauge ﬁeld A and the pull-back of the NS–NS two-form B. The Wess–Zumino term is described as2 1 ˆ A ¯ ˆ LWZ =−2i dt E θΓAσE, (2.3) 0 where Eˆ A ≡ EA(tθ) and Eˆ α ≡ Eα(tθ). When we consider a fundamental string (F-string), the matrix σ is given by σ3. If we consider a D-string, then σ is represented by σ1.Since we would like to discuss boundary surfaces for both of fundamental string and D-string,

2 Alternative superstring actions have been proposed in [38] for superstrings in the AdS background and in [39,40] for those in the pp-wave background. M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124 103 we do not explicitly fix σ in our consideration. It is known that κ-invariance of the action is equivalent to supergravity equations of motion [41,42]. In the case of an open string, the κ-variation leads to surface terms. One can show that the surface terms originating Mˆ A A from SNG vanish, because they include δκ Z E = δκ E = 0, by using the definition of Mˆ A the κ-variation δκ E = 0. In addition, the κ-variation of F does not lead to surface terms because δκ F does not include a derivative of the κ-variation of a field. Thus non-vanishing surface terms originate from SWZ only. 5 In this section, we consider an open string in the AdS5 × S background. The concrete expressions of the supervielbein and the spin connection are given in Appendix B where the coset construction is briefly reviewed for this background. We examine the κ-variation surface terms up to and including the fourth order of θ in this paper.3 After short calculation, the Wess–Zumino term can be rewritten as

= 0 + spin + M SWZ SWZ SWZ SWZ, where each part in the above decomposition is defined as, respectively, 0 2 ij 1 ¯ λ ˆ M B N A S ≡−2i d σ − θΓAσ ∂i θ + ΓBiσ θ∂iX e ∂j X e WZ 2 2 2 M N i ¯ A λ ˆ M B ¯ λ ˆ N C + θΓ ∂i θ + ΓBiσ θ∂i X e · θΓAσ ∂j θ + ΓC iσ θ∂j X e , 4 2 2 M 2 2 N (2.4) spin 2 ij 1 ¯ BC M N A S ≡−2i d σ − θΓAσΓBCθω ∂iX ∂j X e WZ 8 M N i ¯ A BC M ¯ λ ˆ N D + θΓ ΓBCθω ∂iX · θΓAσ ∂j θ + ΓDiσ θ∂j X e 16 M 2 2 N i ¯ A λ ˆ M B ¯ CD N + θΓ ∂iθ + ΓBiσ θ∂iX e · θΓAσΓCDθω ∂j X 16 2 2 M N i ¯ A BC M ¯ DE N + θΓ ΓBCθω ∂iX · θΓAσΓDEθω ∂j X , (2.5) 64 M N M 2 ij 1 ¯ 2 M A S ≡−2i d σ − θΓAσM Di θ∂j X e . (2.6) WZ 24 M This is the covariant Wess–Zumino term, which will be used for a covariant classification 5 of D-branes in the AdS5 × S background in the next section. The sketch of our strategy is as follows: We will consider surface terms of κ-variation of the above Wess–Zumino term, and investigate the boundary conditions under which all of the surface terms vanish. Then these boundary conditions lead to a classification of possible D-branes. It is worth mentioning that the surface terms do not cancel out each other. Hence, without loss of generality, we can examine the κ-variation of surface terms separately in each part.

3 It is expected that the fourth order analysis is sufﬁcient as we will brieﬂy comment later. 104 M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124

3. Classiﬁcation of supersymmetric D-branes of AdS string

Here let us investigate the boundary conditions under which the κ-invariance holds. Then we will classify D-brane conﬁgurations preserving half of supersymmetries.

3.1. Boundary conditions of covariant string

Before going to the concrete analysis, we need to introduce the boundary conditions. The open-string world-sheet Σ has one-dimensional boundary ∂Σ. We can impose the Neumann condition and Dirichlet condition on this boundary ∂Σ. These boundary conditions are represented by A¯ ≡ M A¯ = ∂nX ∂nX eM 0 (Neumann condition), (3.1) A ≡ M A = ∂tX ∂tX eM 0 (Dirichlet condition), (3.2) ¯ wherewehaveusedtheoverlineasAi (i = 0,...,p) for the indices of Neumann = + coordinates and the underline as Aj (j p 1,...,9) for the indices of Dirichlet coordinates. The operator ∂t is a tangential derivative ∂/∂τ and ∂n is a normal derivative ∂/∂σ on the boundary ∂Σ, respectively. We impose the boundary condition on the fermionic variable θ:

± ± 1 P θ = θ, P = (I + M). (3.3) 2 The gluing matrix M is a product of the SO(1, 9) gamma matrices tensored with Pauli matrices, and its concrete form will be ﬁxed so that the κ-variation surface terms vanish.

3.2. D-branes of AdS strings at the origin

0 In this subsection, we examine the κ-variation of SWZ.Theκ-variation is deﬁned by A δκ E = 0, i.e., M ¯ M 4 δκ X =−iθΓ δκ θ + O θ . (3.4) 0 One ﬁnds that the κ-variation of SWZ is 0 = − ¯ · M A¯ δκ SWZ dξ iθΓA¯ σδκ θ ∂tX eM ∂Σ 1 ¯ A ¯ ¯ ¯ A + θΓ δκ θ · θΓAσ + θΓAσδκ θ · θΓ ∂ θ 2 t − λ ¯ ˆ · ¯ B¯ − ¯ ˆ · ¯ B¯ 2 θΓA¯ ΓB¯ σiσ2θ θΓ δκ θ θΓB¯ ΓA¯ σiσ2θ θΓ δκ θ 4 + ¯ B¯ · ¯ ˆ − ¯ B ˆ · ¯ M A¯ θΓ δκ θ θΓB¯ ΓA¯ σiσ2θ θΓ ΓA¯ iσ2θ θΓBσδκ θ ∂tX eM + O θ 6 . (3.5) M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124 105

First, let us consider the ﬁrst line. In order for this line to vanish, the following two conditions have to be satisﬁed:

¯ = ¯ A = θΓA¯ σδκ θ 0andθΓ δκ θ 0. (3.6) We find that the condition (3.6) is satisfied if we define the projection operator (3.3) with gluing matrices

m ⊗ iσ ,d= 2 (mod 4), p =−1, 3, 7, M = 2 m ⊗ ρ, d = 4 (mod 4), p = 1, 5, 9,

1, for X0:Neumann, m = sΓ A1 ···Γ A d ,s= i, for X0: Dirichlet,

σ , when σ = σ , ρ = 1 3 (3.7) σ3, when σ = σ1, where the inclusion of the factor i for the case that the time direction X0 is a Dirichlet one is equivalent to performing the Wick rotation and considering in the Euclidean formulation. This procedure does not affect the analysis below. These conditions imply that, for the σ = σ3 case, Dp-branes exist for the values p =−1, 1, 3, 5, 7, 9, while if we consider the σ = σ1 case, then Dp-branes with p = 1 and 5 are replaced with F1- and NS5-branes, respectively.4 Hence, we obtain the well-known conditions for branes in type IIB string theory in ﬂat spacetime. It is not clear whether the S-duality holds for superstrings in the 5 AdS5 × S and pp-wave or not, but it should exist for branes even in these backgrounds. Therefore, we will discuss the D-string case in parallel as well as the fundamental string in the following consideration. Next, we consider the second and third lines. Because these lines are proportional to the parameter λ characterizing the AdS geometry, these give conditions intrinsic to the branes 5 in the AdS5 × S background. In order to make these lines vanish, we have to impose two additional conditions:

¯ ˆ = ¯ B ˆ = θΓA¯ ΓB¯ σiσ2θ θΓ ΓA¯ iσ2θ 0. (3.8) Let us classify the conﬁgurations satisfying the boundary conditions (3.8). For the d = 2 (mod 4) case, the conditions (3.8) are satisﬁed for the following case:

0 4 • The number of Dirichlet directions in the AdS5 coordinates (X ,...,X ) is even, and the same condition is also satisﬁed for the S5 coordinates (X5,...,X9).

On the other hand, for the d = 4 (mod 4) case, the following conditions should be imposed:

0 4 • The number of Dirichlet directions in the AdS5 coordinates (X ,...,X ) is odd, and the same condition is also satisﬁed for the S5 coordinates (X5,...,X9).

4 For p = 9, the boundary condition is θ1 = θ2 when σ = σ3, while θ2 = 0whenσ = σ1. 106 M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124

Table 1 5 The possible 1/2 supersymmetric D-branes in AdS5 × S sitting at the origin D-instanton D-string D3-brane D5-brane D7-brane D9-brane (0, 0)(0, 2), (2, 0)(1, 3), (3, 1)(2, 4), (4, 2)(3, 5), (5, 3) absent

These conditions restrict the directions to which a brane world-volume can extend, while (3.6) restricts the dimension of the world-volume. The D-brane configurations satisfying the conditions (3.6) and (3.8) are summarized in Table 1. We found that D-strings are allowed to exist as 1/2 supersymmetric objects, which correspond to 1/2 supersymmetric D-strings in the pp-wave background as will be seen in Section 5. On the other hand, it is known that there exist 1/4 supersymmetric D-strings preserving 8 dynamical supersymmetries in the pp-wave [14]. In Section 3.4, we will present a 1/4 supersymmetric D-string which is the AdS origin of the 1/4 supersymmetric D-string found there. We found that D9-branes are not allowed to exist. As will be seen in Section 4, D9-branes are not also allowed in the pp-wave [6,7]. Since our discussion contains the cases that the time direction satisfies the Dirichlet condition, we were able to find the D-instanton configuration. 5 Notably, our result agrees with the classification of branes embedded in the AdS5 × S background (called AdS branes) [8], and thus all of D-branes obtained here possibly correspond to the AdS branes. Such a correspondence emerges in the case of Dirichlet 7/4 branes of an open supermembrane in the AdS4/7 × S backgrounds, where the classification of Dirichlet branes [34] agrees with that of AdS branes obtained in [43]. 0 We have discussed the κ-variation of SWZ until now. We have to take account of the spin M κ-variations of SWZ and SWZ. As we will see later, these parts have no effect on the above classification at the origin. Therefore we can say that the classification of the possible D-brane configurations at the origin has been completely accomplished. For D-branes spin sitting outside the origin, the κ-variation of SWZ leads to additional conditions. On the M other hand, SWZ does not affect the results in both at and outside the origin. The discussion concerning these points will be given in the following subsection.

3.3. D-branes of AdS strings outside the origin

spin spin Here we will consider the contribution of SWZ part. The κ-variation of SWZ is written as

spin i i ¯ A¯ ¯ ¯ A ¯ BC D¯ δκ S = − θΓ δκ θ · θΓ ¯ σΓBCθ − θΓ ΓBCθ · θΓAσδκ θ ω ¯ dX WZ 2 4 A D ∂Σ i ¯ ¯ D¯ BC A¯ − θΓ ¯ σΓBCθ · θΓ δκ θ · ω ¯ dX . (3.9) 2 A D In order to make the above surface terms vanish, we need the following conditions:

BC A BC θΓ¯ ¯ Γ σθ · ω = θΓ¯ Γ θ · ω = 0. (3.10) A BC D¯ BC D¯ M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124 107

These conditions are trivially satisfied at the origin (i.e., XA = 0), while no configurations, except for D-instantons, satisfy these conditions outside the origin. We found that 1/2 supersymmetric Dp-branes (p>0) are not allowed except for branes sitting at the origin. This is because the homogeneity is not manifest in the coordinate system we took here. The same situation emerges in the case of branes in the pp-wave. There, a flat brane sitting at (outside) the origin of the Brinkmann coordinate system is mapped to a planer (less supersymmetric curved) brane in the Rosen coordinate system [9]. So, it is expected that a 1/2 supersymmetric brane sitting outside the origin has a non-trivial shape in general. On the other hand, we found that D-instantons are allowed to sit even outside the origin as in flat spacetime. This may be related to the fact that a D-instanton world-volume is a point in any coordinate system. The same situation arises in the pp-wave case as we will see later. We can expect less supersymmetric D-branes such as 1/4 supersymmetric D-branes to exist even outside the origin as discussed in Refs. [8,9]. We will present this type of D-string later. M The remaining task is to examine whether the SWZ part affects the classification at and M outside the origin or not. The κ-variation of SWZ is given by M λ ¯ ˆ ¯ B¯ δκ S = θΓ ¯ σΓ ¯ iσ θ · θΓ δκ θ WZ 12 A B 2 ∂Σ 1 ¯ ¯ ˆ BC A¯ − θΓ ¯ σΓBCθ · θΓ iσ δκ θ dX . (3.11) 2 A 2 The above surface terms vanish under the following conditions:

¯ ˆ = ¯ ˆ BC¯ = θΓA¯ ΓB¯ σiσ2θ θΓ iσ2δκ θ 0, (3.12) 0 which are nothing but those obtained in the analysis of the κ-variation of the SWZ part. 0 Thus no new conditions arise, and hence the SWZ part has no effect on our classification. As a final remark in this section, we comment on higher order terms. Until now we have studied the D-brane configurations up to and including fourth order in θ. We expect that the higher order terms do not lead to any new additional conditions. In other words, the conditions we have obtained in this paper are expected to be sufficient and all surface terms will vanish under these conditions even in the full theory including all orders of θ. In fact, some arguments for the higher order terms were given in Ref. [14].

3.4. 1/4 supersymmetric D-string

We can obtain 1/4 supersymmetric D-branes in addition to those preserving half of supersymmetries. Here we present a conﬁguration of 1/4 supersymmetric D-string as an example. Since we would like to study a D1-brane conﬁguration, let us consider the following gluing matrix:

1...8 M = m ⊗ σ1,m≡ Γ (3.13) 108 M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124 in order to pass the conditions in ﬂat space (3.6). This gluing matrix does not automatically satisfy the conditions (3.8) speciﬁc to the AdS geometry, and so we need to impose additional conditions: ˆ ˆ Γ0σ2θ = θ, Γ9σ2θ = θ. (3.14) Here it should be noted that the above two conditions are not independent because of the condition θ = Mθ. We can easily rewrite these conditions as

1...8 1...8 Γ θ1 = θ1,Γθ2 = θ2. (3.15) +−1...8 11 11 By the use of the relations Γ = Γ and Γ θ1,2 = θ1,2, we obtain − + − + Γ Γ θ1 = 0,ΓΓ θ2 = 0. (3.16) = 1 − + + 1 + − ≡ When the spinors θ1 and θ2 are decomposed as θ1,2 2 Γ Γ θ1,2 2 Γ Γ θ1,2 (+) + (−) (+) = = (−) θ1,2 θ1,2 , the conditions (3.16) implies θ1,2 0andθ1,2 θ1,2 . Finally, we obtain the expressions of additional conditions: + + Γ θ1 = Γ θ2 = 0. (3.17) This condition reduces the number of the remaining supersymmetries, and the D-string configuration we considered here preserves quarter of supersymmetries. M Moreover, we can see that the SWZ part has no effect on this D-string configuration since the conditions derived from this part is identical with (3.8). We can also check spin that the κ-variation of SWZ vanishes. The two conditions (3.10) ensure that the surface spin 1···8 ⊗ terms originating from the κ-variation of SWZ vanish. By the use of the relation (Γ I2)θ = θ, we can see that the first one in (3.10) is satisfied, while the second one is not ¯ satisfied. However, the term θΓAσδκ θ that couples to the second condition vanishes under 1...8 ⊗ I = spin the condition (Γ 2)θ θ, and hence the surface terms from SWZ vanish. Thus, we 5 have found a 1/4 supersymmetric D-string configuration in the AdS5 × S background. In particular, the supersymmetries preserved by this configuration are not changed even if it is slid from the origin. If we consider the Penrose limit of this D-string, then we can recover the D-string configuration originally found in the work [14], which is 1/4 5 supersymmetric even outside the origin. The D-string in the AdS5 ×S might correspond to R × S1 mentioned in the works on the brane probe analysis given in [8]. It is an interesting future work to delve into other branes preserving 1/4 or less supersymmetries.

4. D-branes of pp-wave strings revisited

In this section we will discuss the classification of D-branes in the pp-wave background. In the work [14], the allowed 1/2 supersymmetric D-brane configurations have been already classified for the case that the light-cone directions (+, −) satisfy the Neumann conditions, by using the same method in this paper. Here we extend the discussion given in [14] by including the case that (+, −)-directions are the Dirichlet directions. In particular, we find the 1/2 supersymmetric D-instanton configurations at and outside the origin. M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124 109

5 By taking the Penrose limit [35] around a certain null geodesic in the AdS5 × S background, we obtain the maximally supersymmetric IIB pp-wave background:

8 2 = + − − 2 2 +···+ 2 − 2 + m 2 ds 2 dX dX µ X1 X8 dX dX , (4.1) m=1 F+1234 = F+5678 = 4µ, (4.2) where FA1...A5 is a constant Ramond–Ramond self-dual five-form flux. Our strategy is almost the same as in the AdS case. We define the projection operator (3.3) with the gluing matrix

m ⊗ iσ ,d= 2 (mod 4), p =−1, 3, 7, M = 2 m ⊗ ρ, d = 4 (mod 4), p = 1, 5, 9,

1, for X+,X−:Neumann, m = sΓ A1 ···Γ A d ,s= i, for X+,X−: Dirichlet,

σ , when σ = σ , ρ = 1 3 (4.3) σ3, when σ = σ1. We examine the κ-variation surface terms of the action (2.1) in the pp-wave background. As before, it turns out to be enough to consider the Wess–Zumino term (2.3). The supervielbein and the spin connection are given in Appendix C. We again divide the Wess– Zumino action into the three individual parts as follows: M S = Sµ + S + Sspin, (4.4) WZ WZ WZ WZ µ 1 A ¯ − µ m µ ˆ S =−2i e θΓAσ dθ + e (f + g)iσ θ + e Γmiσ θ WZ 2 2 2 2 2 Σ i ¯ A − µ m µ ˆ + θΓ dθ + e (f + g)iσ θ + e Γmiσ θ 8 2 2 2 2 ¯ − µ m µ ˆ × θΓAσ dθ + e (f + g)iσ θ + e Γmiσ θ , (4.5) 2 2 2 2 M 1 A ¯ 2 S =−2i e θΓAσM Dθ , (4.6) WZ 24 Σ spin 1 A ¯ m µ S =−2i e θΓAσ e∗ ΓmΓ+θ WZ 2 2 Σ i ¯ A − µ m µ ˆ ¯ m µ + θΓ dθ + e (f + g)iσ θ + e Γ iσ θ · θΓ σ e∗ Γ Γ+θ 8 2 2 2 m 2 A 2 m i ¯ A m µ ¯ − µ m µ ˆ + θΓ e∗ ΓmΓ+θ · θΓAσ dθ + e (f + g)iσ θ + e Γmiσ θ 8 2 2 2 2 2 i ¯ A m µ ¯ m µ + θΓ e∗ ΓmΓ+θ · θΓAσ e∗ ΓmΓ+θ . (4.7) 8 2 2 110 M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124

We examine the surface terms under the κ-variation of these three parts in turn. µ The vanishing conditions of the κ-variation surface terms coming from SWZ lead to the classification of possible branes sitting at the origin. The vanishing conditions of the κ- spin variation surface terms of SWZ do not affect the classification at the origin, but lead to the additional conditions for the brane configurations outside the origin. The surface terms M which come from SWZ vanish for the obtained configurations at and outside the origin. A The κ-variation is defined by δκ E = 0, which means in this background M i ¯ M 4 δκ X =− θΓ δκ θ + O θ . (4.8) 2 µ When we perform the κ-variation on SWZ, the surface terms are obtained as µ 1 ¯ M A δκ S =−2i θΓAσδκ θdX e WZ 2 M ∂Σ i ¯ A ¯ ¯ ¯ A + θΓ δκ θ · θΓAσ + θΓAσδκ θ · θΓ dθ 8 µi ¯ − ¯ M A − θΓ δκ θ · θΓAσ(f + g)iσ θdX e 8 2 M µi ¯ m ¯ ˆ M A − θΓ δκ θ · θσΓmiσ θdX e 8 2 M − µi ¯ A · ¯ + M − + ˆ M m θΓ δκ θ θΓAσ (f g)iσ2θdX eM Γmiσ2θdX eM 16 µi ¯ A ¯ M − ˆ M m + θΓ σδκ θ · θΓA (f + g)iσ θdX e + Γmiσ θdX e . (4.9) 16 2 M 2 M The vanishing conditions for the first line read

¯ A¯ ¯ A θΓ σδκ θ = 0andθΓ δκ θ = 0, (4.10) which are satisfied by the projector given above. We thus have rederived the well-known condition, p = odd, for IIB Dp-branes in flat background. The additional conditions which come from the second, third and fourth lines are ¯ + = ¯ + = −∈ θΓA¯ (f g)σiσ2θ θΓA(f g)iσ2θ 0or Dirichlet, (4.11) and ¯ ˆ = ¯ ˆ = θΓA¯ Γm¯ σiσ2θ θΓAΓm¯ iσ2θ 0. (4.12) The condition (4.11) is satisfied for p = 3 (1) mod 4 cases by one of the followings:

• The number of Dirichlet directions in (X1,X2,X3,X4) is even (odd), and the same condition is also satisﬁed for (X5,X6,X7,X8), •−∈Dirichlet, while the second condition (4.12) is satisﬁed if M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124 111

Table 2 The possible 1/2 supersymmetric conﬁgurations of D-branes sitting at the origin in the pp-wave background D-instanton D-string D3-brane D5-brane D7-brane D9-brane (+, −; 0, 2), (+, −; 1, 3), (+, −; 2, 4), (+, −; 2, 0), (+, −; 3, 1), (+, −; 4, 2),absent (0, 0)(0, 2), (2, 0)(1, 3), (3, 1)(2, 4), (4, 2)

Table 3 The possible 1/2 supersymmetric conﬁgurations of D-branes sitting outside the origin in the pp-wave background D-instanton D-string D3-brane D5-brane D7-brane D9-brane (0, 0)(0, 2), (2, 0)(1, 3), (3, 1)(2, 4), (4, 2) absent absent

• The number of Dirichlet directions in (X+,X1,X2,X3,X4) is even (odd), and the same condition is also satisﬁed for (X+,X5,X6,X7,X8).

From these conditions, one can classify the possible branes in the pp-wave background. We denote branes with the world-volume extending along m directions in (1, 2, 3, 4) and n directions in (5, 6, 7, 8) as (m, n)-branes. In addition, when the directions (+, −) are also Neumann directions, we denote as (+, −; m, n)-branes. We summarize in Table 2 the classification of 1/2 supersymmetric D-branes sitting at the origin. spin Next we examine the surface terms of the κ-variation of SWZ . We show that the surface terms vanish for branes sitting at the origin, but lead to additional conditions for branes sitting outside the origin. The κ-variation surface terms are 2 spin µ ¯ A¯ ¯ − δκ S =− θΓ δκ θ · θΓ ¯ σΓmΓ+θdX WZ 8 A ∂Σ − M A¯ + 2θΓ¯ δ θ · θΓ¯ ¯ σΓ Γ+θdX e κ A m M ¯ ¯ A − m − θΓAσδκ θ · θΓ ΓmΓ+θdX X , (4.13) which vanish when −∈Dirichlet or Xm = 0. Thus we found that for branes sitting at the origin the surface terms vanish, while for branes sitting outside the origin (−)-direction has to be a Dirichlet direction. We summarize in Table 3 the possible branes sitting outside the origin. M Finally we examine the κ-variation surface terms of SWZ and show that the surface terms vanish for the brane configurations classified above. The surface terms become

M µ ¯ ¯ − ¯ ˆ ¯ m¯ δκ S =− θΓ ¯ σ(f + g)iσ θ · θΓ δκ θ + θΓ ¯ σΓm¯ iσ θ · θΓ δκ θ WZ 24 A 2 A 2 ∂Σ ¯ ¯ i ¯ ¯ i + θΓ ¯ σΓiΓ+θ · θΓ fiσ2δκ θ − θΓ ¯ σΓi Γ+θ · θΓ giσ2δκθ A A − ¯ · ¯ ˆ mn M A¯ θΓA¯ σΓmnθ θΓ iσ2δκ θ dX eM . (4.14) 112 M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124

One can show that these surface terms vanish when one of the followings is satisﬁed for p = 3 (1) mod 4 cases:

•−∈Dirichlet, and both (X1,X2,X3,X4) and (X5,X6,X7,X8) contain odd (even) number of Dirichlet directions, •−∈Neumann, and both (X1,X2,X3,X4) and (X5,X6,X7,X8) contain even (odd) number of Dirichlet directions.

These conditions are satisfied by the brane configurations classified above. The longitudinal D-brane configurations agree with those found in [6,14]. We found that there exist transverse D-branes sitting at and outside the origin. D-instantons preserving half of supersymmetries are allowed to freely sit in the pp-wave as in flat space and 5 AdS5 × S . D-instanton correspond to a point in the pp-wave spacetime while the existence of 1/2 supersymmetric D-instantons sitting outside the origin seems to be consistent to the homogeneity of the pp-wave geometry.

5. Penrose limits of D-branes of AdS string

In this section we will consider Penrose limits [35] of the possible D-brane conﬁgu- rations in the AdS background obtained in Section 3, and show that all of the D-branes in the pp-wave obtained in Section 4 is recovered as Penrose limits of those in the AdS background. The Penrose limit is taken as follows. One makes a set of light-cone coordinate (X+,X−) from (X0,X9),5 scales X+ as X+ → Ω2X+ and then take the limit Ω → 0. We distinguish the cases depending on the boundary conditions of the light-cone coordinate (X+,X−) as (N, N) for X± ∈ Neumann directions, and so on. For the fermionic coordinates, we scale θ+ → Ωθ+ and take the limit Ω → 0. Some relevant aspects of the Penrose limit are explained in Appendix D. We examine Penrose limits of (N, N)-and(D, D)-cases. 5.1. Penrose limit of D9-brane

There are no allowed 1/2 supersymmetric D9-brane conﬁgurations both in the AdS5 × S5 and in the pp-wave. Hence, if we want to consider the correspondence of D9-branes in the AdS and those in the pp-wave, then we need to consider the less supersymmetric conﬁgurations of D9-branes. 5.2. Penrose limit of D7-brane

The possible D7-brane conﬁgurations are (3, 5)-and(5, 3)-branes. Now we cannot 5 consider (D, D)-case since there is no Dirichlet direction in the S and AdS5, respectively.

5 If we make a pair of light-cone coordinates only in the AdS space or sphere, then ﬂat spacetime is obtained as discussed in [37]. That is, in order to obtain the pp-wave background, we need to construct the light- cone coordinates by choosing one direction from the AdS coordinates and the other direction from the sphere coordinates. M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124 113

First, we consider the Penrose limit of the (3, 5)-branes, whose boundary condition is ab sΓ ⊗ iσ2θ = θ.Inthe(N, N)-case the resulting boundary condition after the Penrose limit is given by

ij sΓ ⊗ iσ2θ = θ. (5.1) That is, this conﬁguration describes the (+, −; 2, 4)-type D7-branes, which is included in the list of the allowed D-branes in the pp-wave case. In the same way as in the (3, 5)-brane case, we can investigate the Penrose limit of (5, 3)-branes. The boundary condition after the Penrose limit is

ij sΓ ⊗ iσ2θ = θ, (5.2) which is the (+, −; 4, 2)-brane boundary condition in the pp-wave. Thus, two types of D7- brane conﬁgurations in the pp-wave are rederived as the Penrose limit of D7-branes in the AdS background.

5.3. Penrose limit of D5-brane

The allowed D5-brane conﬁgurations are (2, 4)-and(4, 2)-branes. First, we consider (2, 4)-type D5-branes whose boundary condition is sΓ a1a2a3a ⊗ ρθ = θ with ρ = σ1. When we choose the (N, N)-boundary condition, the above condition becomes

 Γ i1i2i3i ⊗ ρθ = θ, (5.3) after the Penrose limit. That is, we obtain (+, −; 1, 3)-type D5-brane in the pp-wave. If we take the (D, D)-condition, then we get the boundary condition:

+− iΓ ij ⊗ ρθ = θ, (5.4) and hence (2, 4)-type D5-branes are obtained. The resulting (+, −; 1, 3)-and(2, 4)-type D5-branes are possible in the pp-wave case. Secondly, we study (4, 2)-type D5-branes. In this case the boundary condition is given aa a a by sΓ 1 2 3 ⊗ ρθ = θ. After taking the Penrose limit, we obtain, according to the choice of light-cone coordinates, the conditions:

ii i i Γ 1 2 3 ⊗ ρθ = θ for (N, N)-case, +− iΓ i j ⊗ ρθ = θ for (D, D)-case. For (N, N)-and(D, D)-cases, we obtain (+, −; 3, 1)-and(4, 2)-type branes, respectively. These conﬁgurations are also allowed in the pp-wave. In the above discussion we have assumed the fundamental string case. It is also possible to consider the D-string by taking ρ = σ3, and simultaneously D5-branes are replaced with NS5-branes. The only difference in our consideration of the Penrose limit is only a choice of σ and the discussion in the D-string case is exactly the same as the above one. 114 M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124

5.4. Penrose limit of D3-brane

The possible D3-brane conﬁgurations are (1, 3)-and(3, 1)-branes. a ...a ab The boundary condition of (1, 3)-type D3-brane is given by sΓ 1 4 ⊗ iσ2θ = θ, which becomes, according to the choice of the light-cone coordinates,

i1...i4i j Γ ⊗ iσ2θ = θ for (N, N)-case, +−i i i i iΓ 1 2 3 ⊗ iσ2θ = θ for (D, D)-case. aba ···a When we consider the (3, 1)-type D3-brane whose boundary condition is sΓ 1 4 ⊗ iσ2θ = θ, the resulting conditions after taking the Penrose limit are given by

ij i ...i Γ 1 4 ⊗ iσ2θ = θ for (N, N)-case, +−ii i i iΓ 1 2 3 ⊗ iσ2θ = θ for (D, D)-case. Thus, we have obtained the D3-branes in the pp-wave as Penrose limits of D3-branes in 5 the AdS5 × S background.

5.5. Penrose limit of D1-brane

We can take (0, 2)-and(2, 0)-branes as possible D1-brane conﬁgurations. In this case we cannot take (N, N)-type boundary condition. The boundary conditions for the (0, 2)-and(2, 0)-type D-strings are

a ...a abc a a a a ...a sΓ 1 5 ⊗ ρθ = θ and sΓ 1 2 3 1 5 ⊗ ρθ = θ, (5.5) with ρ = σ1, respectively. When we consider the (D, D)-type boundary condition, these conditions become, after taking the Penrose limit,

+−i ...i ij +−ij i ...i iΓ 1 4 ⊗ ρθ = θ and iΓ 1 4 ⊗ ρθ = θ, (5.6) and these conditions imply (0, 2)-and(2, 0)-type D-string conﬁgurations in the pp-wave, respectively. If we take ρ = σ3,then(0, 2)-and(2, 0)-type F-string conﬁgurations in the pp-wave are obtained.

5.6. Penrose limit of D-instanton

In this case we can consider the (D, D)-type boundary condition only. The condition for 5 01...9 D-instanton in the AdS5 × S background is iΓ ⊗ iσ2θ = θ. After taking the Penrose limit, we obtain the following condition:

+−1...8 iΓ ⊗ iσ2θ = θ. (5.7) This condition is nothing but that of D-instanton in the pp-wave. Finally, we summarize the results obtained above in Table 4. We also brieﬂy comment on (N, D)-or(D, N)-cases. If we consider the Penrose limit in these cases, the boundary condition becomes θ− = 0 for every D-brane. This condition M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124 115

Table 4 Penrose limit of D-branes of AdS string D7-branes AdS (3, 5) (5, 3) 2 N2 2 N2 Penrose D D

pp-wave − (+, −; 2, 4) − (+, −; 4, 2) D5-branes AdS (2, 4) (4, 2) D2 N2 D2 N2 Penrose

pp-wave (2, 4)(+, −; 1, 3) (4, 2)(+, −; 3, 1) D3-branes AdS (1, 3) (3, 1) 2 N2 2 N2 Penrose D D

pp-wave (1, 3)(+, −; 0, 2) (3, 1)(+, −; 2, 0) D1-branes AdS (0, 2) (2, 0) D2 N2 D2 N2 Penrose

pp-wave (0, 2) − (2, 0) − −: we cannot take this boundary condition. implies the two types of conditions as follows:

+− 1···8 M = Γ ⊗ I2 or M = Γ ⊗ I2. However, the boundary conditions associated with these gluing matrices do not eliminate the κ-variation surface terms. In fact, these matrices are not included in (4.3). It may be the case that we may not take (N, D) and (D, N)-boundary conditions.

5.7. Penrose limit of 1/4 supersymmetric D-string

Here we will consider the Penrose limit of D-string conﬁguration preserving quarter of supersymmetries. This D-string is a (1, 1)-type D-string. When we consider the (N, N)- case, we obtain the (+, −; 0, 0)-type D-string in the pp-wave. This D-string conﬁguration is nothing but the D-string found in the work [14]. If we take the (D, D)-boundary condition, we obtain the (1, 1)-type D-string which is expected to be a 1/4 supersymmetric D-string in the pp-wave.

6. Conclusion and discussion

5 We have classified possible configurations of D-branes of a superstring in the AdS5 × S background. In contrast to the pp-wave case, the D-string configuration is realized as ausual1/2 supersymmetric configuration. Notably, our classification result agrees with that of AdS brane configurations of the brane probe analysis. We have also extended the classification of allowed 1/2 supersymmetric D-branes in the pp-wave [14] by including the case that both of light-cone directions satisfy the Dirichlet conditions. In particular, we 116 M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124 have found that 1/2 supersymmetric D-instantons can exist at and outside the origin in both AdS and pp-wave backgrounds. The fact that D-instanton can survive even outside the origin seems to reflect the homogeneity of these backgrounds. In addition, we have 5 investigated Penrose limits of the allowed D-branes in the AdS5 × S background, and it has been shown that possible 1/2 supersymmetric configurations in the pp-wave case can 5 be recovered as Penrose limits of those in the AdS5 × S . We have mainly examined the standard configurations preserving half of supersymmetries, but it is interesting to approach 1/4 or less supersymmetric D-branes of AdS string in our covariant formulation. In fact, as an example, we have presented a 1/4 supersymmetric 5 D-string configuration in the AdS5 × S background. This configuration has been shown to reduce to the 1/4 supersymmetric D-string in the pp-wave preserving 8 dynamical supersymmetries. D-branes of this type are called D+-branes in the works [8]. On the other hand, the standard D-branes are called D−-branes. In this paper we have clarified the AdS origin of D−-branes. It may be possible to find the AdS origin of D+-branes. Of course, we can also seek for the AdS origin of oblique D-branes, curved D-branes and intersections of D-branes in the pp-wave case (for a work in this direction, [44]). In addition, giant gravitons6 or baryon vertex operators may be investigated in our covariant formulation if 5 we treat these objects in the AdS5 × S by the use of our formulation. It is also interesting to examine Dirichlet boundaries of an open D3-brane constructed in [46] and compare the result with that obtained here. Another direction is to study D-branes on other AdS back- 5 1,1 grounds such as AdS5 × S /ZN and AdS5 × T [47]. We will report all of these problems in another place in the near future [48]. It is also an interesting problem to apply our results to defect conformal field theories via the AdS/CFT correspondence [49]. Recall that a stringy nature in pp-wave string theories played an important role in studies of the AdS/CFT correspondence at the stringy level beyond supergravity approximation [27]. Our classification results and our methods are expected to promote the understanding of AdS/CFT correspondence.

Acknowledgements

We would like to thank Machiko Hatsuda and Katsuyuki Sugiyama for useful discussion.

Appendix A. Notation and convention

In this place we will summarize miscellaneous notation and convention used in this paper.

A.1. Notation of coordinates

5 For the superstring in the ten-dimensional curved spacetime: AdS5 × S and pp- wave backgrounds, we use the following notation of supercoordinates for its superspace

6 The matrix model on the pp-wave also has supersymmetric fuzzy sphere solutions (giant graviton) [27]. Giant gravitons in the pp-wave matrix model are discussed in [29,45]. M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124 117

(XM ,θα¯ ): 5 M = (µ, µ ), µ = (0, 1, 2, 3, 4) ∈ AdS5,µ= (5, 6, 7, 8, 9) ∈ S , (A.1) and the background metric is expressed by GMN. The coordinates in the Lorentz frame are denoted by (XA,θα): A = (a, a ), a = 0, 1, 2, 3, 4,a = 5, 6, 7, 8, 9 for AdS case, A = (+, −,i,i ), i =, 1, 2, 3, 4,i = 5, 6, 7, 8,m= (i, i ) for pp-wave case, and its metric is described by ηAB = diag(−1, +1,...,+1) with η00 =−1. The light- cone coordinates are deﬁned by X± ≡ √1 (X9 ± X0). The coordinates of world-sheet are 2 parameterized by (σ 1,σ2) = (τ, σ). The induced metric on the world-sheet is represented by gij (i, j = 1, 2).

A.2. SO(1, 9) gamma matrices

We denote two 16-component Majorana–Weyl spinors as θ1 and θ2,andtheSO(1, 9) Clifford algebra is written as

Γ A,ΓB = 2ηAB, Γ M ,ΓN = 2GMN, A ≡ A M M ≡ M A Γ eM Γ ,ΓeA Γ , where the light-cone components of the SO(1, 9) gamma matrices are

1 + − Γ± = √ (Γ9 ± Γ0), Γ ,Γ = 2I16. 2 The chirality operator and the Dirac conjugate are deﬁned as 11 ¯ T Γ ≡ Γ0...9, θ ≡ θ C, where the charge conjugation matrix C satisﬁes the relations: CT =−C, C−1 Γ M T C =−Γ M . T In this paper we mainly use the 32-component representation θ = (θ1,θ2).

5 Appendix B. Coset construction of supervielbein in the AdS5 × S

5 Here we will brieﬂy review the coset construction of supervielbein in the AdS5 × S 5 5 background. The super-AdS5 × S -algebra is regarded as su(2, 2|4),andtheAdS5 × S manifold can be constructed as a coset SU(2, 2|4)/(SO(1, 4) × SO(5)). 5 The bosonic part of the super AdS5 × S -algebra is so(2, 4) × so(6) given by 2 2 [Pa,Pb]=λ Mab, [Pa ,Pb ]=−λ Mab ,

[Pa,Mbc]=ηabPc − ηacPb, [Pa ,Mbc ]=ηab Pc − ηac Pb ,

[Mab,Mcd ]=ηbcMad + 3terms, [Mab ,Mcd ]=ηbc Mad + 3terms. 118 M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124

5 The parameter λ characterizes the AdS5 × S geometry, and if we take the limit λ → 0then the geometry reduces to the ten-dimensional Minkowski spacetime. Now we decompose the SO(1, 9) gamma matrices Γ A’s as follows:

a a Γ = γ ⊗ I ⊗ τ1 (a = 0, 1, 2, 3, 4), a a Γ = I ⊗ γ ⊗ τ2 (a = 5, 6, 7, 8, 9), 11 Γ = Γ0 ···Γ9 = I ⊗ I ⊗ τ3, where the 2 × 2 matrices τi ’s are the standard Pauli matrices. The gamma matrices of the a AdS5 part γ ’s are i 4 γ (i = 0, 1, 2, 3) and γ ≡ iγ0123 4 2 5 a and hence (γ ) = 1andiγ01234 =+1. For the gamma matrices of the S part γ ’s, we use

i 9 γ (i = 5, 6, 7, 8) and γ ≡ γ5678, and then γ 56789 =+1. The charge conjugation matrix C is defined as C ≡ C ⊗ C ⊗ iτ2, 5 where C and C are the charge conjugation matrices in the AdS5 and S , respectively. = 1 ± 11 We introduce chirality projection operators h± 2 (1 Γ ) and Majorana–Weyl supercharges QI (I = 1, 2) by QI h+ = QI . By the use of these quantities, the fermionic 5 part of the super-AdS5 × S algebra given in [50] is rewritten as A ab {QI ,QJ }=2iCΓ (I)IJh+PA + iλCΓ I(iσ2)IJh+Mab ab − iλCΓ J (iσ2)IJh+Mab , λ λ [QI ,Pa]=+ QJ (iσ )JIΓaI, [QI ,P ]=− QJ (iσ )JIΓ J , 2 2 a 2 2 a 1 01234 56789 [QI ,MAB ]=− QI ΓAB, I ≡ Γ , J ≡ Γ , 2 where the Pauli matrices (σi)IJ act on the two-component supercharges QI . Now we consider G = gx gθ defined as a a θ X Pa+X P Qθ 1 gx = e a ,gθ = e ,Q= (Q1,Q2), θ = . θ2 The supervielbeins EA and Eα, and super spin connection EAB are defined by the relations

−1 A 1 AB α G dG= E PA + E MAB + QαE , 2 −1 A 1 AB g dgx = e PA + ω MAB, x 2 A AB 5 where e and ω are the vielbein and the spin connection of the AdS5 × S . We can derive the expressions for the (super)vielbein and the (super)spin connection by expanding M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124 119 the l.h.s. of the above relations. As a result, we obtain the following expressions sinh(M/2) 2 EA = eA + iθΓ¯ A Dθ, M/2 sinh M α Eα = Dθ , M sinh(M/2) 2 EAB = ωAB − iλθ¯Γˆ ABiσ Dθ, 2 M/2 where we have introduced the quantities: 2 ˆ ¯ A 1 ¯ ˆ AB M = iλ ΓAiσ θ · θΓ − ΓABθ · θΓ iσ , 2 2 2

λ A ˆ 1 AB Dθ = dθ + e ΓAiσ θ + ω ΓABθ, 2 2 4 ˆ ˆ ΓA ≡ (−ΓaI,Γa J ), ΓAB ≡ (−ΓabI,Γab J ). 5 The vielbein and the spin connection of the AdS5 × S background are given by a a sinh Y sinh Y ea = dX ,ea = dX , Y Y ] sinh(Y/2) 2 b ωab =−λ2X[a dX , Y/2 2 b ] sinh(Y /2) ωa b =+λ2X[a dX , Y /2 2a = 2 2 a − a 2a =− 2 2 a − a Y b λ X δb X Xb ,Yb λ X δb X Xb . The above expressions of the spin connection are used in considering the D-branes sitting outside the origin.

Appendix C. Coset construction of supervielbein in the pp-wave

∗ The super-pp-wave algebra is generated by momenta, P− and Pm, boost, Pm,Lorentz generators of SO(4) × SO(4), Mrs and Mrs , and supercharge, QI ,as ∗ ∗ 2 ∗ 2 [P−,P ]=P , P−,P =−µ P , P ,P =−µ δ P+, m m m m m n mn [ ]= − ∗ = ∗ − ∗ Mmn,Pp δnp Pm δmp Pn, Mmn,Pp δnp Pm δmp Pn , [Mmn,Mpq]=δnp Mmq + 3terms, µ µ [Pr ,QI ]=− QJ Γr Γ+f(iσ )JI, [P ,QI ]= QJ Γ Γ+g(iσ )JI, 2 2 r 2 r 2 2 µ ∗ µ [P−,QI ]= QJ (f + g)(iσ )JI, P ,QI = QI ΓmΓ+, 2 2 m 2 120 M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124

1 [Mmn,QI ]= QI Γmn, 2 + − m {QI ,QJ }=iCΓ IIJh+P+ + iCΓ IIJh+P− + iCΓ IIJh+Pm

i r ∗ i r ∗ + CΓ f(iσ ) h+P , − CΓ g(iσ ) h+P µ 2 IJ r µ 2 IJ r rs rs + iµCΓ Γ+f(iσ2)IJh+ Mrs − iµCΓ Γ+g(iσ2)IJh+Mrs , where f ≡ Γ 1234 and g ≡ Γ 5678. It is known that this superalgebra is an Inonu–Wigner 5 7/4 contraction of the super-AdS5 × S algebra [51] (for AdS4/7 × S cases, see [52]). We parameterize the group manifold by

+ − m X P+ X P− X Pm QI θI G = gx gθ ,gx = e e e ,gθ = e , and deﬁne Maurer–Cartan one-forms by

−1 + − m 1 mn m ∗ α G dG= E P+ + E P− + E Pm + E Mmn + E∗ P + QαE . 2 m After simple algebraic calculation, one obtains the supervielbein described by i sinh(M/2) 2 sinh M α EA = eA + θΓ¯ A Dθ, Eα = Dθ , 2 M/2 M where we have introduced several quantities:

2 µ ¯ − µ ˆ ¯ m µ ¯ r M = i (f + g)iσ θ · θΓ + i Γmiσ θ · θΓ + i Γr Γ+θ · θΓ fiσ 2 2 2 2 2 2 µ ¯ r µ ¯ ˆ mn − i Γ Γ+θ · θΓ giσ − i Γmnθ · θΓ iσ , 2 r 2 2 2 2 − µ m µ ˆ m µ Dθ = dθ + e (f + g)iσ θ + e Γmiσ θ + e∗ ΓmΓ+θ, 2 2 2 2 2 ˆ ˆ Γm = (−Γr Γ+f, Γr Γ+g), Γmn = (−ΓrsΓ+f, Γrs Γ+g), 2 + + µ − − − e = dX − Xm 2 dX ,e= dX , 2 m m m m − e = dX ,e∗ = X dX . In this parameterization, the pp-wave metric becomes the standard form + − + − − ds2 = 2e e + em 2 = 2 dX dX − µ2 Xm 2 dX 2 + dXm 2. The constructed supervielbein is used in Section 4 for the classiﬁcation of D-branes in the pp-wave background.

Appendix D. Penrose limit

It is known that the pp-wave background is obtained via the Penrose limit from the 5 AdS5 × S background. The Penrose limit is taken by rescaling coordinates as + 2 + m m X → Ω X ,X→ ΩX ,θ+ → Ωθ+, M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124 121

= P P = 1 where θ+ is deﬁned by θ+ +θ with + 2 Γ+Γ−, and then by bringing Ω to zero. 5 After the Penrose limit, the scale λ of AdS5 or S can be regarded as the scale µ of the pp-wave.7 Noting that the super-pp-wave algebra is an Inonu–Wigner contraction of 5 5 the super-AdS5 × S algebra [51], the Penrose limit of the AdS5 × S group manifold XaP +Xa P parameterized by gx = e a a turns to be the pp-wave group manifold parameterized + − X P++X P−+XmP by gx = e m . In this parameterization, the metric is calculated to be + − ds2 = 2e e + em 2, − + + sin µX X − − e = dX + 1 − m X dXm − Xm dX , µX− (X−)2 − = − e dX , − − m sin µX sin µX X − em = dXm + 1 − dX . µX− µX− X− In order to make the metric to be of the standard form (4.1), we perform the coordinate transformation µ µX− − sin µX− cosµX− Xˆ + = X+ + Xˆ m 2, 2 sin2 µX− − − − sin µX Xˆ = X , Xˆ m = Xm, µX− under which the metric becomes the standard form + − − ds2 = 2 dXˆ dXˆ − µ2 Xˆ m 2 dXˆ 2 + dXˆ m 2.

This coordinate transformation reveals the fact that Xm = 0 is mapped to Xˆ m = 0and X+ = X− = 0 corresponds to Xˆ + = Xˆ − = 0.

References

[1] J. Polchinski, Dirichlet-branes and Ramond–Ramond charges, Phys. Rev. Lett. 75 (1995) 4724, hep- th/9510017. [2] T. Banks, W. Fischler, S.H. Shenker, L. Susskind, M-theory as a matrix model: a conjecture, Phys. Rev. D 55 (1997) 5112, hep-th/9610043. [3] M. Blau, J. Figueroa-O’Farrill, C. Hull, G. Papadopoulos, A new maximally supersymmetric background of IIB superstring theory, JHEP 0201 (2002) 047, hep-th/0110242. [4] R.R. Metsaev, Type IIB Green–Schwarz superstring in plane wave Ramond–Ramond background, Nucl. Phys. B 625 (2002) 70, hep-th/0112044.

7 The scale λ ∼ 1/R can be absorbed and the scale µ can be introduced by a ﬁeld redeﬁnition

2 2 + λ Ω + − − X → X ,X→ µX ,Xm → λΩXm. µ This reveals the fact that the Penrose limit Ω → 0 is equivalent to the limit R →∞which is considered in the literature. 122 M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124

[5] R.R. Metsaev, A.A. Tseytlin, Exactly solvable model of superstring in plane wave Ramond–Ramond background, Phys. Rev. D 65 (2002) 126004, hep-th/0202109. [6] A. Dabholkar, S. Parvizi, Dp branes in pp-wave background, Nucl. Phys. B 641 (2002) 223, hep-th/0203231. [7] M. Billo, I. Pesando, Boundary states for GS superstrings in an Hpp wave background, Phys. Lett. B 536 (2002) 121, hep-th/0203028. [8] K. Skenderis, M. Taylor, Branes in AdS and pp-wave spacetimes, JHEP 0206 (2002) 025, hep-th/0204054; K. Skenderis, M. Taylor, Open strings in the plane wave background, I: quantization and symmetries, Nucl. Phys. B 665 (2003) 3, hep-th/0211011; K. Skenderis, M. Taylor, Open strings in the plane wave background, II: superalgebras and spectra, JHEP 0307 (2003) 006, hep-th/0212184; K. Skenderis, M. Taylor, An overview of branes in the plane wave background, Class. Quantum Grav. 20 (2003) S567, hep-th/0301221; D.Z. Freedman, K. Skenderis, M. Taylor, Worldvolume supersymmetries for branes in plane waves, hep- th/0306046. [9] P. Bain, P. Meessen, M. Zamaklar, Supergravity solutions for D-branes in Hpp-wave backgrounds, Class. Quantum Grav. 20 (2003) 913, hep-th/0205106. [10] O. Bergman, M.R. Gaberdiel, M.B. Green, D-brane interactions in type IIB plane-wave background, JHEP 0303 (2003) 002, hep-th/0205183; M.R. Gaberdiel, M.B. Green, The D-instanton and other supersymmetric D-branes in IIB plane-wave string theory, Ann. Phys. 307 (2003) 147, hep-th/0211122; M.R. Gaberdiel, M.B. Green, S. Schafer-Nameki, A. Sinha, Oblique and curved D-branes in IIB plane-wave string theory, JHEP 0310 (2003) 052, hep-th/0306056. [11] S. Seki, D5-brane in anti-de Sitter space and Penrose limit, Phys. Lett. B 542 (2002) 162; S. Seki, Phys. Lett. B 542 (2002) 165, hep-th/0205266; D. Mateos, S. Ng, Penrose limits of the baryonic D5-brane, JHEP 0208 (2002) 005, hep-th/0205291. [12] Y. Michishita, D-branes in NSNS and RR pp-wave backgrounds and S-duality, JHEP 0210 (2002) 048, hep-th/0206131. [13] K. Sugiyama, K. Yoshida, Type IIA string and matrix string on pp-wave, Nucl. Phys. B 644 (2002) 128, hep-th/0208029; S. Hyun, H. Shin, N = (4, 4) type IIA string theory on pp-wave background, JHEP 0210 (2002) 070, hep- th/0208074. [14] P. Bain, K. Peeters, M. Zamaklar, D-branes in a plane wave from covariant open strings, Phys. Rev. D 67 (2003) 066001, hep-th/0208038. [15] H. Takayanagi, T. Takayanagi, Notes on giant gravitons on pp-waves, JHEP 0212 (2002) 018, hep- th/0209160. [16] S. Hyun, H. Shin, Solvable N = (4, 4) type IIA string theory in plane-wave background and D-branes, Nucl. Phys. B 654 (2003) 114, hep-th/0210158. [17] Y. Hikida, S. Yamaguchi, D-branes in pp-waves and massive theories on worldsheet with boundary, JHEP 0301 (2003) 072, hep-th/0210262. [18] R.R. Metsaev, Supersymmetric D3 brane and N = 4 SYM actions in plane wave backgrounds, Nucl. Phys. B 655 (2003) 3, hep-th/0211178. [19] S. Hyun, J. Park, H. Shin, Covariant description of D-branes in IIA plane-wave background, Phys. Lett. B 559 (2003) 80, hep-th/0212343. [20] H. Shin, K. Sugiyama, K. Yoshida, Partition function and open/closed string duality in type IIA string theory on a pp-wave, Nucl. Phys. B 669 (2003) 78, hep-th/0306087; Y. Kim, J. Park, Boundary states in IIA plane-wave background, Phys. Lett. B 572 (2003) 81, hep- th/0306282. [21] J. Kim, B.H. Lee, H.S. Yang, Superstrings and D-branes in plane wave, Phys. Rev. D 68 (2003) 026004, hep-th/0302060; K.S. Cha, B.H. Lee, H.S. Yang, Intersecting D-branes in type IIB plane wave background, hep-th/0307146; K.S. Cha, B.H. Lee, H.S. Yang, A complete classiﬁcation of D-branes in type IIB plane wave background, hep-th/0310177. [22] C.S. Chu, P.M. Ho, Noncommutative D-brane and open string in pp-wave background with B-ﬁeld, Nucl. Phys. B 636 (2002) 141, hep-th/0203186; M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124 123

A. Kumar, R.R. Nayak, Sanjay, D-brane solutions in pp-wave background, Phys. Lett. B 541 (2002) 183, hep-th/0204025; M. Alishahiha, A. Kumar, D-brane solutions from new isometries of pp-waves, Phys. Lett. B 542 (2002) 130, hep-th/0205134; M. Alishahiha, M.A. Ganjali, A. Ghodsi, S. Parvizi, On type IIA string theory on the PP-wave background, Nucl. Phys. B 661 (2003) 174, hep-th/0207037; A.Biswas,A.Kumar,K.L.Panigrahi,p–p branes in pp-wave background, Phys. Rev. D 66 (2002) 126002, hep-th/0208042; O.J. Ganor, U. Varadarajan, Nonlocal effects on D-branes in plane-wave backgrounds, JHEP 0211 (2002) 051, hep-th/0210035; M. Alishahiha, O.J. Ganor, Twisted backgrounds, pp-waves and nonlocal field theories, JHEP 0303 (2003) 006, hep-th/0301080; R.R. Nayak, D-branes at angle in pp-wave background, Phys. Rev. D 67 (2003) 086006, hep-th/0210230; L.F. Alday, M. Cirafici, An example of localized D-branes solution on pp-wave backgrounds, JHEP 0305 (2003) 006, hep-th/0301253; K. Ito, Y. Sekino, Penrose limit and enhancon geometry, Phys. Rev. D 67 (2003) 126005, hep-th/0302127; K.L. Panigrahi, Sanjay, D-branes in pp-wave spacetime with nonconstant NS–NS flux, Phys. Lett. B 561 (2003) 284, hep-th/0303182; N. Ohta, K.L. Panigrahi, Sanjay, Intersecting branes in pp-wave spacetime, hep-th/0306186; R.R. Nayak, K.L. Panigrahi, More D-branes in plane wave spacetime, hep-th/0310219. [23] N.D. Lambert, P.C. West, D-branes in the Green–Schwarz formalism, Phys. Lett. B 459 (1999) 515, hep- th/9905031. [24] K. Ezawa, Y. Matsuo, K. Murakami, Matrix regularization of open supermembrane: towards M-theory five- brane via open supermembrane, Phys. Rev. D 57 (1998) 5118, hep-th/9707200. [25] B. de Wit, K. Peeters, J.C. Plefka, Open and closed supermembranes with winding, Nucl. Phys. (Proc. Suppl.) 68 (1998) 206, hep-th/9710215. [26] P. Horava, E. Witten, Heterotic and type I string dynamics from eleven dimensions, Nucl. Phys. B 460 (1996) 506, hep-th/9510209; P. Horava, E. Witten, Eleven-dimensional supergravity on a manifold with boundary, Nucl. Phys. B 475 (1996) 94, hep-th/9603142. [27] D. Berenstein, J.M. Maldacena, H. Nastase, Strings in flat space and pp waves from N = 4 super-Yang– Mills, JHEP 0204 (2002) 013, hep-th/0202021. [28] B. de Wit, J. Hoppe, H. Nicolai, On the quantum mechanics of supermembranes, Nucl. Phys. B 305 (1988) 545. [29] K. Dasgupta, M.M. Sheikh-Jabbari, M. Van Raamsdonk, Matrix perturbation theory for M-theory on a PP- wave, JHEP 0205 (2002) 056, hep-th/0205185. [30] K. Sugiyama, K. Yoshida, Supermembrane on the pp-wave background, Nucl. Phys. B 644 (2002) 113, hep-th/0206070. [31] K. Sugiyama, K. Yoshida, BPS conditions of supermembrane on the pp-wave, Phys. Lett. B 546 (2002) 143, hep-th/0206132; N. Nakayama, K. Sugiyama, K. Yoshida, Ground state of the supermembrane on a pp-wave, Phys. Rev. D 68 (2003) 026001, hep-th/0209081. [32] M. Sakaguchi, K. Yoshida, Dirichlet branes of the covariant open supermembrane on a pp-wave background, Nucl. Phys. B 676 (2004) 311, hep-th/0306213. [33] L. Motl, A. Neitzke, M.M. Sheikh-Jabbari, Heterotic plane wave matrix models and giant gluons, hep- th/0306051. 7 [34] M. Sakaguchi, K. Yoshida, Dirichlet branes of the covariant open supermembrane in AdS4 × S and 4 AdS7 × S , Nucl. Phys. B 681 (2004) 137, hep-th/0310035. [35] R. Penrose, Any spacetime has a plane wave as a limit, in: Differential Geometry and Relativity, Reidel, Dordrecht, 1976, pp. 271–275; R. Gueven, Plane wave limits and T-duality, Phys. Lett. B 482 (2000) 255, hep-th/0005061. [36] J. Kowalski-Glikman, Vacuum states in supersymmetric Kaluza–Klein theory, Phys. Lett. B 134 (1984) 194. [37] M. Blau, J. Figueroa-O’Farrill, C. Hull, G. Papadopoulos, Penrose limits and maximal supersymmetry, Class. Quantum Grav. 19 (2002) L87, hep-th/0201081. 124 M. Sakaguchi, K. Yoshida / Nuclear Physics B 684 (2004) 100–124

[38] M. Hatsuda, K. Kamimura, M. Sakaguchi, Nondegenerate super-anti-de Sitter algebra and a superstring action, Phys. Rev. D 62 (2000) 105024, hep-th/0007009; M. Hatsuda, M. Sakaguchi, Wess–Zumino term for the AdS superstring and generalized Inoenue–Wigner contraction, Prog. Theor. Phys. 109 (2003) 853, hep-th/0106114; M. Hatsuda, M. Sakaguchi, Wess–Zumino term for AdS superstring, Phys. Rev. D 66 (2002) 045020, hep- th/0205092. [39] M. Hatsuda, K. Kamimura, M. Sakaguchi, Gauge invariant action for superstring in Ramond–Ramond plane- wave background, Nucl. Phys. B 644 (2002) 40, hep-th/0207157. [40] N. Berkovits, J. Maldacena, N = 2 superconformal description of superstring in Ramond–Ramond plane wave backgrounds, JHEP 0210 (2002) 059, hep-th/0208092; N. Berkovits, N = 2 sigma models for Ramond–Ramond backgrounds, JHEP 0210 (2002) 071, hep- th/0210078. [41] M. Cederwall, A. von Gussich, B.E. Nilsson, A. Westerberg, The Dirichlet super-three-brane in ten- dimensional type IIB supergravity, Nucl. Phys. B 490 (1997) 163, hep-th/9610148; M. Cederwall, A. von Gussich, B.E. Nilsson, P. Sundell, A. Westerberg, The Dirichlet super-p-branes in ten-dimensional type IIA and IIB supergravity, Nucl. Phys. B 490 (1997) 179, hep-th/9611159. [42] E. Bergshoeff, P.K. Townsend, Super-D-branes, Nucl. Phys. B 490 (1997) 145, hep-th/9611173. [43] N. Kim, J.T. Yee, Supersymmetry and branes in M-theory plane-waves, Phys. Rev. D 67 (2003) 046004, hep-th/0211029. [44] G. Sarkissian, M. Zamaklar, Diagonal D-branes in product spaces and their Penrose limits, hep-th/0308174. [45] D. Bak, Supersymmetric branes in PP wave background, Phys. Rev. D 67 (2003) 045017, hep-th/0204033; K. Sugiyama, K. Yoshida, Giant graviton and quantum stability in matrix model on PP-wave background, Phys. Rev. D 66 (2002) 085022, hep-th/0207190; H. Shin, K. Yoshida, One-loop ﬂatness of membrane fuzzy sphere interaction in plane-wave matrix model, Nucl. Phys. B 679 (2004) 99, hep-th/0309258; Y.X. Chen, J. Shao, Giant graviton in type IIA pp-wave background, hep-th/0310062; W.H. Huang, Thermal instability of giant graviton in matrix model on pp-wave background, hep-th/0310212; H. Shin, K. Yoshida, Thermodynamics of fuzzy spheres in PP-wave matrix model, hep-th/0401014. 5 [46] R.R. Metsaev, A.A. Tseytlin, Supersymmetric D3 brane action in AdS5 × S , Phys. Lett. B 436 (1998) 281, hep-th/9806095. [47] N. Itzhaki, I.R. Klebanov, S. Mukhi, PP wave limit and enhanced supersymmetry in gauge theories, JHEP 0203 (2002) 048, hep-th/0202153; J. Gomis, H. Ooguri, Penrose limit of N = 1 gauge theories, Nucl. Phys. B 635 (2002) 106, hep-th/0202157; L.A. Pando Zayas, J. Sonnenschein, On Penrose limits and gauge theories, JHEP 0205 (2002) 010, hep- th/0202186. [48] M. Sakaguchi, K. Yoshida, in preparation. [49] D. Berenstein, E. Gava, J.M. Maldacena, K.S. Narain, H. Nastase, Open strings on plane waves and their Yang–Mills duals, hep-th/0203249; V. Balasubramanian, M.X. Huang, T.S. Levi, A. Naqvi, Open strings from N = 4 super-Yang–Mills, JHEP 0208 (2002) 037, hep-th/0204196; P. Lee, J. Park, Open strings in PP-wave background from defect conformal ﬁeld theory, Phys. Rev. D 67 (2003) 026002, hep-th/0203257. [50] R.R. Metsaev, A.A. Tseytlin, Type IIB superstring action in AdS(5) × S(5) background, Nucl. Phys. B 533 (1998) 109, hep-th/9805028. 5 [51] M. Hatsuda, K. Kamimura, M. Sakaguchi, From super-(AdS5 × S ) algebra to super-pp-wave algebra, Nucl. Phys. B 632 (2002) 114, hep-th/0202190. [52] M. Hatsuda, K. Kamimura, M. Sakaguchi, Super-PP-wave algebra from super-(AdS × S) algebras in eleven- dimensions, Nucl. Phys. B 637 (2002) 168, hep-th/0204002. Nuclear Physics B 684 (2004) 125–161 www.elsevier.com/locate/npe

Infrared-ﬁnite amplitudes for massless gauge theories

Darren A. Forde, Adrian Signer

Institute for Particle Physics Phenomenology, University of Durham, Durham, DH1 3LE, England, UK Received 6 November 2003; received in revised form 8 January 2004; accepted 20 February 2004

Abstract We present a method to construct infrared-ﬁnite amplitudes for gauge theories with massless fermions. Rather than computing S-matrix elements between usual states of the Fock space we construct order-by-order in perturbation theory dressed states that incorporate all long-range interactions. The S-matrix elements between these states are shown to be free from soft and collinear + − singularities. As an explicit example we consider the process e e → 2 jets at next-to-leading order in the strong coupling. We verify by explicit calculation that the amplitudes are infrared-ﬁnite and + − recover the well-known result for the total cross section e e → hadrons.  2004 Elsevier B.V. All rights reserved.

PACS: 11.10.Jj; 11.15.Bt

1. Introduction

Soft and collinear singularities which arise in gauge theories such as QED and QCD are usually dealt with by summing up physically indistinguishable cross sections [1,2]. This results in a cancellation of these singularities for so-called infrared-safe observables and therefore reliable theoretical predictions can be made for such observables. Even though this is a perfectly valid approach to the problem it is useful to investigate the origin of these singularities and to explore the possibility of avoiding them altogether. The origin of the problem lies in the long-range nature of the interactions. As a consequence the usual in and out states do not evolve in time asymptotically according to the free Hamiltonian. It is this breakdown of the standard assumption that results in the non- existence of the scattering operator. Thus, if we want to avoid infrared singularities from

E-mail address: [email protected] (A. Signer).

0550-3213/$ – see front matter  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2004.02.024 126 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 the outset we have to construct true asymptotic states and compute transition amplitudes between them [3,4]. This program has been carried out initially for QED with massive fermions [5] and then many steps have been made to extend it to soft singularities in non-abelian theories [6–8]. It is possible to construct asymptotic states (generalized coherent states) which include multiple soft gluon emission to all orders in the coupling [9]. It can be shown that the S-matrix between such states is free of soft singularities [10,11]. Apart from the more complicated structure of the soft singularities due to the self-interaction of the gauge bosons there is the additional complication of collinear singularities in a non-abelian gauge theory. The inclusion of collinear singularities makes the asymptotic Hamiltonian more complicated [12–15] and the prospect of being able to include these effects to all orders in perturbation theory are not very promising. But the idea of constructing an asymptotic Hamiltonian that takes into account the asymptotic dynamics and using the corresponding evolution operator to dress the usual states [16,17] can still be applied and is not tied to any particular theory. In particular, four-point interactions that are present in non-abelian gauge theories can be incorporated [18]. In the present work we investigate the practical feasibility of constructing asymptotic states whose S-matrix elements are free from soft and collinear singularities. We are not so much interested in general considerations but rather try to establish a method to define and explicitly compute infrared-finite amplitudes order-by-order in perturbation theory. Apart from the conceptional advantage of avoiding divergent amplitudes such a method would have a variety of practical advantages. Obviously, the finiteness of the amplitudes would facilitate a completely numerical approach to the calculation of amplitudes. This also applies to the combination of fixed-order results with parton shower Monte Carlo programs. After some general remarks about infrared singularities in Section 2 we discuss how to construct the dressed states in Section 3. This discussion will be completely general. In order to test the practicability of our method we consider a simple process in Section 4. We compute the infrared-finite amplitudes for e+e− → 2 jets at next-to-leading order in the strong coupling and verify that upon integration over the phase space they reproduce the well-known total cross section for e+e− → hadrons. Technical details of this section are relegated to Appendix A. Finally, in Section 5, we present a summary and suggestions on how to improve the method to make applications to more complicated processes more feasible.

2. Infrared singularities

We are concerned with gauge theories with massless fermions. Such theories are plagued by infrared singularities. Infrared singularities are related to either arbitrarily soft gauge bosons or arbitrarily collinear gauge bosons and/or fermions.1 More precisely, if we deﬁne our external states in the usual way by acting with creation operators on the

1 We use the term “infrared singularities” for both, soft and collinear singularities. D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 127 vacuum, then higher-order S-matrix elements between such external states contain infrared divergences. Many attempts have been made to define amplitudes that are well-defined, i.e., do not contain such singularities. However, most attempts were restricted to soft singularities. In particular, it has been shown that in an abelian gauge theory with massive fermions it is possible to define external states whose S-matrix elements are free from infrared singularities to all orders in perturbation theory [5]. An abelian gauge theory with massive fermions does not contain collinear singularities. This simplifies the situation considerably. As soon as we consider the non-abelian case, however, we cannot avoid the appearance of collinear divergences. The reason is that in the non-abelian case a massless gauge boson can split into two arbitrarily collinear gauge bosons. Such a splitting results in a collinear singularity. Thus, giving the fermions a mass does not protect us from collinear singularities. The aim of this work is to investigate the possibility of defining infrared-finite amplitudes for a massless, non-abelian gauge theory. The application we have in mind is, of course, QCD. In most applications of perturbative QCD the quarks (at least the light flavors) are treated as massless. Since we will have to deal with collinear singularities anyway in a non-abelian theory, we might as well take the common approach and treat the quarks as massless.

2.1. The conventional approach

Before we tackle the problem of defining infrared-finite amplitudes in a situation where collinear singularities have to be taken into account as well, it might be useful to remind ourselves how infrared singularities are dealt with in the conventional approach. In the conventional approach, which in the following we shall call the cross- section approach, we evaluate amplitudes between conventional external states that are obtained simply by acting with creation operators on the vacuum. Since such amplitudes contain infrared singularities we regularize them. In virtually all applications dimensional regularization is used, whereby the amplitudes are evaluated in D ≡ 4 − 2 dimensions. The infrared singularities then reveal themselves as poles in 1/. In order to relate these amplitudes to measurable quantities, we will have to integrate the (squared) amplitudes over the phase space, weighted by a measurement function that defines the quantity we are interested in. The key point is that we can only obtain theoretical predictions for quantities that are infrared-safe. An infrared-safe quantity is one that does not depend on whether or not a parton emits an arbitrarily soft gluon. Also it must not depend on whether or not a parton splits into two collinear partons. For such quantities the infrared singularities present in the amplitudes are cancelled by infrared singularities that appear due to the phase space integration and after the cancellation the regulator can be removed. Even though there are various general methods available to perform such calculations at next-to-leading order [19] the appearance of infrared singularities make the phase-space integration a non-trivial problem. 128 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161

2.2. Asymptotic interactions

One of the basic assumptions in the derivation of Greens functions or transition amplitudes is that in the limit t →±∞the external states are free. Thus, it is assumed that all interactions vanish rapidly enough so that they can be neglected in the distant past and future. More precisely, we assume that for any exact state vector |Ψ(t) (in the Schrödinger picture) of the full Hamiltonian H we can ﬁnd a corresponding state vector |Φ(t) of the free Hamiltonian H0 such that − − |Ψ(t)=e itH |Ψ(0)→|Φ(t)=e itH0 |Φ(0), (1) for t →±∞. We then specify the in- and out-states |Φi and Φf | by a complete set of quantum numbers of H0 and compute the transition amplitudes † Φf |Ω−Ω+|Φi≡Φf |S|Φi, (2) where we expressed the scattering operator S in terms of the usual Møller operators

iτH −iτH Ω∓ ≡ lim e e 0 . (3) τ→±∞

Since S commutes with H0 energy is conserved and the amplitude, Eq. (2), is proportional to δ(Ei − Ef ). This is a point we will come back to below. The important point is that in a gauge theory with conventional external states the above mentioned assumption is simply incorrect. The interactions due to massless gauge bosons are long-range interactions and do not vanish rapidly enough for t →±∞.Itisthis violation of our basic assumption that results in ill-defined, infrared-singular amplitudes. In other words, given an exact state |Ψ(t), in general there does not exist a free state |Φ(t) such that Eq. (1) is satisfied. If we want to tackle the problem at its root, we will have to compute amplitudes between modified external states. The long-range interactions that cause the problem will have to be included in the external states themselves. This amounts to replacing our basic assumption with the following: we can find an asymptotic Hamiltonian HA such that for any exact state vector |Ψ(t) we can find an asymptotic state |Ξ(t) such that − − |Ψ(t)=e itH |Ψ(0)→|Ξ(t)=e itHA |Ξ(0), (4) for t →±∞. We then define modified Møller operators

iτH −iτHA ΩA∓ ≡ lim e e , (5) τ→±∞ and compute modiﬁed S-matrix elements | † | ≡ | | Ξf ΩA−ΩA+ Ξi Ξf SA Ξi . (6) Once all true long-range interactions are included in the deﬁnition of the asymptotic Hamiltonian we are guaranteed that S-matrix elements between such asymptotic states are free from infrared singularities [10,11]. These S-matrix elements in turn are related to measurable quantities. In the same way as several (squared) amplitudes contribute to a physical cross section in the conventional approach, there will be several (squared) D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 129

S-matrix elements between asymptotic states contributing to an observable. The crucial difference is that contrary to the cross-section method all these contributions are separately ﬁnite. To ﬁnd the asymptotic Hamiltonian we have to split the interaction Hamiltonian into a “hard” and a “soft” piece

HI = HH (∆) + HS(∆), (7) and define HA(∆) ≡ H0 + HS(∆). The separation of the Hamiltonian into two pieces is by no means unique. The only requirement is that HS(∆) includes all true long-range interactions. Thus, the emission of a soft gauge boson and the splitting of a parton into two collinear partons has to be included. But there is a lot of freedom on how precisely we make the split between soft and hard interactions. To indicate this arbitrariness we include the parameter ∆ in the notation. Later, we will often use the abbreviated notation H∆ ≡ HS(∆). Let us stress that even though we will call H∆ ≡ HS(∆) the soft Hamiltonian, it includes all long-range interactions that potentially give rise to infrared singularities. In particular, the soft Hamiltonian also includes the splitting of a parton into two collinear partons. In what follows we will define HA and the asymptotic states more carefully. We then show how infrared-finite S-matrix elements are related to conventional amplitudes. Finally, we consider a simple explicit example, e+e− → 2 jets at next-to-leading order.

3. Infrared-ﬁnite amplitudes

In order to obtain infrared-finite amplitudes we have to find true asymptotic states and evaluate the (modified) S-matrix elements between such states, as given in Eq. (6). Measurable cross sections then are constructed out of these infrared-finite matrix elements. Since SA commutes with HA we conclude from Eq. (6) that energy is also conserved for the modified transition amplitudes. However, given an asymptotic Hamiltonian it is generally not possible to find the corresponding eigenstates |Ξi. First of all, being an eigenstate of the asymptotic Hamiltonian which includes all soft interactions, the true states |Ξi would correspond to quasi hadrons. In our strict perturbative approach we will never be able to describe bound states and, at each order in perturbation theory, the states |Ξi will correspond to some sort of “jet-like” states. In particular, these states are colored and will have to be related to hadronic states using a hadronization model. This is as in the cross-section approach and is an issue that we do not address in this paper. Still, we have to relate the asymptotic states |Ξi to conventional free states |Φi order-by-order in perturbation theory. We are then led to compute matrix elements

Mfi ≡Φf |SA|Φi (8) order-by-order in perturbation theory and relate them to physical (infrared-safe) cross sections. It has been argued previously that matrix elements as given in Eq. (8) are also free of soft singularities [10,11]. In the present article we extend this to include collinear singularities (see also [14]). For more details on this issue we refer to Section 3.5. 130 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161

3.1. Notation and conventions

Before we proceed let us fix our notation and conventions. Whereas part of the discussion so far was done in the Schrödinger picture we will now turn to the interaction picture. Thus all operators and states are now to be understood to be given in the interaction picture. To start with we construct the usual states of the Fock space † † † |qi(pi) ···q ¯j (pj ) ···gk(pk) ···≡ b (pi) d (pj ) a (pk)|0, (9) i j k where b†,d† and a† denote the creation operators for fermions, antifermions and gauge bosons respectively, and we suppressed the helicity labels. We will generically denote such states by |i and f |. Of course, we have to keep in mind that the states as given in Eq. (9) are not normalizable and we tacitly assume they have been smeared with test functions. Thus, we are really concerned with wave packets. However, we assume they are sharply peaked around a certain value of the momentum such as to represent a particle beam with (nearly) uniform, sharp momentum. The creation and annihilation operators satisfy the usual (anti)commutation relations a(λ , k ), a†(λ , k ) =−(2π)3 2ω(k )g δ(k − k ), (10) 1 1 2 2 1 λ1λ2 1 2 b(r , k ), b†(r , k ) = (2π)3 2ω(k )δ δ(k − k ), (11) 1 1 2 2 1 r1r2 1 2 † = 3 − d(r1, k1), d (r2, k2) (2π) 2ω(k1)δr1r2 δ(k1 k2), (12) with ω(ki) ≡|ki|. Note that the ordering used in Eq. (9) implies a certain phase convention. Of course, all amplitudes are only defined up to such a convention. The field operators are given by ˜ −ikx † +ikx Ψα = dk uα(r, k)b(r,k)e + vα(r, k)d (r, k)e , (13) ¯ ˜ † +ikx −ikx Ψα = dk u¯α(r, k)b (r, k)e +¯vα(r, k)d(r,k)e , (14) = ˜ −ikx + ∗ † +ikx Aµ dk εµ(λ, k)a(λ,k)e εµ(λ, k)a (λ, k)e , (15) where we defined dD−1k dk˜ ≡ (16) D−1 (2π) 2ω(k) 1,2 and the sum is over the two helicities of the fermions or gauge bosons, respectively. Once the interaction Hamiltonian is given we can compute the evolution operator and obtain in the interaction picture t

U(t,t0) ≡ T exp −i dt HI (t) . (17)

t0 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 131

The Møller operators are given by Ω± = U(0, ∓∞) and, thus, the scattering operator S is related to the evolution operator † S = Ω−Ω+ = U(+∞, 0)U(0, −∞). (18) This allows us to ﬁnd the S-matrix elements between some initial and ﬁnal state +∞

f |S|i=f |T exp −i dt HI (t) |i, (19) −∞ where |i and f | are states as deﬁned in Eq. (9). Inserting the explicit form of HI into Eq. (19) allows us to compute S-matrix elements. Of course, in practice such a calculation is nothing but the computation of the corresponding Feynman diagrams.

3.2. Deﬁnition of infrared-ﬁnite amplitudes

In analogy to Eq. (17) we deﬁne a soft evolution operator t

U∆(t, t0) ≡ T exp −i dt H∆(t) , (20)

t0 where we only include the soft Hamiltonian H∆(t) ≡ HS(∆, t). Acting on a certain state, the soft evolution operator modiﬁes this state by allowing for soft and collinear emissions. Then, the usual Feynman–Dyson scattering matrix S can be decomposed as = +∞ −∞ ≡ † S U( , 0)U(0, ) Ω∆−SA(∆)Ω∆+, (21) where we have introduced the soft Møller operators Ω∆± ≡ U∆(0, ∓∞). More explicitly, we have ∞ † ≡ − Ω∆− T exp i dt H∆(t) 0 ∞ ∞ t 2 = 1 − i dt H∆(t) + (−i) dt dt H∆(t)H∆(t ) +··· 0 0 0 ∞ ∞ ∞ 2 (−i) = 1 − i dt H∆(t) + dt dt T {H∆(t)H∆(t )}+···. (22) 2! 0 0 0

Eq. (21) defines a modified scattering operator SA(∆). This operator has the crucial property that it includes at least one hard interaction and, therefore, matrix elements f |SA(∆)|i of this operator with ordinary external initial and final states as defined in Eq. (9) have no infrared singularities. If we define dressed initial and final states, |{i} and {f }| according to |{ } ≡ † | i Ω∆+ i , (23) {f }| ≡ f |Ω∆−, (24) 132 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 then

{f }|S|{i} = f |SA(∆)|i. (25) Thus, the S-matrix elements of dressed states are free of infrared singularities. We should stress that dressed states are not asymptotic states, i.e., they are not eigenstates of the asymptotic Hamiltonian. Let us look at a dressed final state somewhat more carefully. We obtain a dressed final state by acting with Ω∆− on a final state as defined in Eq. (9). We denote this dressed state by adding curly brackets.

f {q(pi) ···q(p ¯ j ) ···g(pk) ···}|=q(pi ) ···¯q(pj ) ···g(pk) ···|Ω∆−. (26) Once the asymptotic Hamiltonian is fixed Eq. (26) is a unique relation, order-by-order in perturbation theory, between an ordinary final state f | and the corresponding dressed final state {f }|. A similar relation holds for dressed initial states: |{ ··· ¯ ··· ···} = † | ···¯ ··· ··· q(pi) q(pj ) g(pk) i Ω∆+ q(pi) q(pj ) g(pk) . (27) In what follows we will suppress the labels f and i but keep in mind that the states |{q(pi) ···q(p ¯ j ) ···g(pk) ···} and {q(pi) ···¯q(pj ) ···g(pk) ···}| are not conjugates of each other. Also, we would like to stress that all these states are states in the usual Fock space. Of course, this implicitly assumes that we use some kind of regularization for the infrared singularities in intermediate steps. The soft Møller operators dress the usual non-interacting external states with a cloud of soft and collinear partons. Since the infrared behavior of H∆ and the full interaction Hamiltonian are the same by construction, this dressing generates infrared singularities that cancel those generated by the full scattering operator. There are two main differences between the soft(/collinear) Møller operator, Eq. (22), and the usual scattering operator, Eq. (19). Firstly, Ω∆± involve only the soft part H∆ of the interaction Hamiltonian. Secondly, the time integration in the soft Møller operator runs only from 0 to ∞ rather than from −∞ to ∞. The fact that the time integration is restricted to t>0 is related to the loss of Lorentz invariance in the amplitudes Mfi, Eq. (8). This is to be expected since SA does not commute with H0 and, therefore, Mfi is generally not proportional to an energy- conserving δ(Ei − Ef ). Instead, individual parts of the amplitude will have δ-functions with different energy arguments (see Eq. (36)). The difference between these arguments determines the amount by which energy conservation can be violated in Mfi and is related to the parameter ∆. In the limit ∆ → 0 the amount by which energy can be violated tends to 0. Thus, the parameter ∆ determines how much the initial wave packets are distorted through the evolution with the soft Møller operators. We will come back to these issues in Section 3.7.

3.3. Factorization of modiﬁed S-matrix elements

We now turn to the question on how to compute the infrared-ﬁnite amplitudes deﬁned in Eq. (25) and how they are related to ordinary amplitudes. D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 133

A possible approach is to start from the right-hand side of Eq. (25). This would involve using the explicit form of SA, given below in Eq. (36) to compute the amplitudes. As we argue in Section 3.5 the structure of SA is such that no infrared singularities occur. This opens up the possibility of evaluating the amplitudes numerically. We have to keep in mind, however, that there are still ultraviolet singularities which will have to be removed by renormalization. In order to take an entirely numerical approach the renormalization procedure would have to be done at the integrand level [20]. We will take a somewhat different approach in that we start from the left-hand side of Eq. (25). We relate the infrared-finite amplitude to ordinary amplitudes by inserting a complete set of states twice { }| |{ } = | † | = | | ⊗ | | ⊗ | † | f S i f Ω∆−SΩ∆+ i f Ω∆− f f S i i Ω∆+ i . (28) Note that in the final expression all states are ordinary Fock space states as defined in Eq. (9). In this way, the infrared-finite amplitude is split into three pieces. First, there is an ordinary S-matrix element, f |S|i . The other two factors are dressing factors for the initial and final state. All these pieces are infrared-divergent and only the complete amplitude is infrared-finite, order-by-order in perturbation theory. The ultraviolet singularities appear only in f |S|i and are dealt with as usual by renormalization. The symbol ⊗ denotes integration over all momenta and summation over all helicities of the state under consideration. Thus, for say

f |=q(p1,r1)q(p¯ 2,r2)g(p3,r3)|≡qp1q¯p2gp3| we have 

|f ⊗f |= dp˜1 dp˜2 dp˜3 |qp1q¯p2gp3qp1q¯p2gp3|. (29) r1r2r3 We should stress that Eq. (28) implies that the dressing is not done for each external parton separately. The dressing factors f |Ω∆−|f do contain terms that factorize into separate contributions for each parton, but they also contain color correlated contributions.

3.4. Dressing factors

As we have seen in Eq. (28) infrared-finite amplitudes are composed of three factors. First, there is an ordinary amplitude, f |S|i , computed in the usual way using ordinary Feynman rules. Then there are the two dressing factors, one for the initial and one for the final state. The calculation of these dressing factors is somewhat different from the calculation of ordinary amplitudes and it is useful to look at this in some more detail. For concreteness we consider the calculation of a final state dressing factor. The starting point is Eq. (22). Let us stress again that since the time integration in Eq. (28) is from 0 to ∞ we break Lorentz invariance right from the beginning. Of course, in the final result for a physical quantity Lorentz invariance will be restored. In fact, the calculation has many features of (old-fashioned) time-ordered perturbation theory. Most notably, all particles will be on-shell. Three-momentum will be conserved in all vertices, but energy will not be conserved. 134 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161

A typical term of the (asymptotic) Hamiltonian that gives rise to an n-point interaction has the form n ˜ dx dki V(ki)Θ(∆) exp ix · σi ki exp −it σi ω(ki) , (30) i=1 where ω(kj ) ≡|kj | denotes the energy of the particles and the sign σi is positive (negative) for incoming (outgoing) particles. V(ki) is made up of creation and annihilation operators, eventually accompanied by spinors and/or polarization vectors and a certain power of the coupling constant. The range of integration over the momenta is restricted to the singular regions. This is indicated in the notation by Θ(∆). The precise form of this function is not important at the moment. After performing the dx integration we obtain the momentum- D−1 (D−1) conserving delta function (2π) δ ( σiki). However, since the t integration is restricted to t 0 we do not obtain an energy-conserving δ function. Rather we have to introduce the usual adiabatic factor 0+ > 0 and use ∞ ∞ − − − + −i dte iωt → dte iωte t0 = . (31) ω − i0+ 0 0 Of course, if the t integration was restricted to t 0 we would have

0 0 − − + + i dte iωt → dte iωte t0 = , (32) ω + i0+ −∞ −∞ and the sum of Eq. (31) and Eq. (32) indeed results in 2πδ(ω). To summarize, for an n-point vertex in the calculation of a dressing factor for a ﬁnal statewehavetouse n − − Θ(∆) dk˜ (2π)D 1δ(D 1) σ k V(k ). (33) i i i − + i i=1 σi ω(ki) i0 Were it not for the Θ(∆)function and the restriction of the t-integration to t 0 this would lead to the standard Feynman rule.

3.5. Finiteness of modiﬁed S-matrix elements

In this subsection we substantiate our claim that matrix elements as deﬁned in Eq. (8) or Eq. (25) are free from collinear and soft singularities. We start from the deﬁnition = † SA(∆) Ω∆−SΩ∆+, (34) and use the explicit form of the soft Møller operator and S to express SA(∆) in terms of H∆ and HH . Furthermore, we observe that according to Eq. (30) the time dependence of the Hamiltonian H(tj ) is given by

−itj j −itj j HH (tj ) = hj e ; H∆(tj ) = sj e , (35) D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 135 where j ≡ σi ω(ki ) is the sum of the energies of the particles associated with the corresponding n-point vertex and hj and sj are time-independent. Performing the algebra and the t-integrations we obtain up to third order

SA = 1 − 2iπ h1 δ(1)

δ(1 + 2) δ(1) − δ(1 + 2) + 2iπh1h2 + 2iπ[h1,s2] 1 2

δ(1 + 2 + 3) + 2iπh1(h2 + s2)h3 13 δ(1 + 2) − δ(1 + 2 + 3) δ(1) − 2iπ [h1,s2],s3 + 13 2(2 + 3)

δ(2 + 3) − δ(1 + 2 + 3) − 2iπs1h2h3 13

δ(1 + 2) − δ(1 + 2 + 3) − 2iπh1h2s3 . (36) 13

First of all we notice that all the purely soft terms s1s2 ··· vanish. This holds to all orders and is crucial to ensure that SA is free from infrared singularities. Infrared singularities potentially arise if i → 0. This corresponds to either a soft or collinear emission at the corresponding vertex. Let us now go through the terms in Eq. (36) and check that for none of them such a singularity can occur. For this to be true we have to define hi such that it vanishes for i → 0. This can be achieved by choosing the Θ(∆) in Eq. (30) accordingly. We start by looking at the second-order terms, given in the second line of Eq. (36). The only potential singularity in the first term is 1 → 0. This is harmless since h1 = 0inthis limit. In the second term we have the potential singularity 2 → 0 which is not prevented by s2. However, in this limit the term is proportional to δ(1) and the same argument as for the first-order term applies. The arguments for the third-order terms, given in the third to sixth line of Eq. (36) are similar. The only dangerous limits in the third line term for example are 1 → 0and 3 → 0. Both of these are prevented by the presence of h1 and h3. Considering the term in the fourth line, we first note that δ(1)h1 = 0. As a result there is no problem with the limit 2 → 0and2 →−3. Furthermore, the singularity in the limit 1 → 0isprevented by the presence of h1 and the limit 3 → 0 is made harmless by the combination of δ functions. Similarly, the terms in the fifth and sixth line are finite in the limit 1 → 0and 3 → 0. Thus we see that (up to this order) there are no singularities in SA as long as hi is chosen to vanish for → 0. We mention again that SA does not only contain terms proportional to δ(1 + 2 + 3) but also terms with “incomplete” δ-functions. These are the energy-violating terms mentioned above. We also remark that the absence of terms containing 1/(i + j ) in SA (the corresponding term in the fourth line of Eq. (36) vanishes) justifies our initial claim that all infrared singularities are related to limits i → 0. 136 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161

3.6. Construction of infrared-ﬁnite amplitudes

The expression given in Eq. (28) is a (double) sum over all possible intermediate states |f f | and |i i |. However, if we compute the amplitude to a certain order in the coupling constant, only a very limited number of intermediate states contribute. It is, for 0 example, clear that at order O(g ) the dressing factor f |Ω∆−|f is zero, unless f = f . From this we see that at leading order in perturbation theory the amplitude {f }|S|{i} is thesameasf |S|i. Including higher-order corrections this identity will, of course, not hold any longer. At order O(g1) the states f and f can be different. To get a non-vanishing contribution they must be related either by adding a (soft or collinear) gluon or by exchanging a quark– antiquark pair by a gluon. In order to illustrate this in more detail, let us consider a concrete process. To simplify matters we consider a case with no partons in the initial state. What we have in mind is, for example, the process e+e− → γ → jets. As long as we treat this process at leading order in the electromagnetic coupling but at higher order in the strong coupling, g, we encounter only final state singularities. Thus, for the purpose of understanding how the dressing removes the infrared singularities we can restrict ourselves to the final state partons and treat the initial state simply as |0. Before writing down Eq. (28) more explicitly for the process under consideration, let us introduce a somewhat more compact notation. We will denote the momenta and helicities of the partons in the intermediate state f by qi and si respectively, and use the notation qqi ≡ q(qi,si ), etc. The momenta and helicities of the partons in the final state f ,onthe n other hand, will be denoted by pi and ri and we use qpi ≡ q(pi,ri ). The order O(g ) terms of the dressing factors are then denoted by gnW(n)(q , q¯ ,g ,...; q , q¯ ,...) p1 p2 p3 q1 q2 ≡ ¯ ···| −| ¯ ··· q(p1,r1)q(p2,r2)g(p3,r3) Ω∆ q(q1,s1)q(q2,s2) gn . (37) Similarly, we denote the order O(gn) terms of the amplitude by gnA(n) q(q ,s ), q(¯ q ,s ), g(q ,s ),...; γ 1 1 2 2 3 3 = ¯ ···| | ≡ n A(n) ¯ ; q(q1,s1)q(q2,s2)g(q3,s3) S 0 gn g (qq1, qq2,gq3,... γ), (38) and we introduce a notation for the infrared-finite amplitudes gnA(n) {q (p ,r ), q¯ (p ,r ), g (p ,r ),...}; γ 1 1 1 2 2 2 3 3 3 ≡{ ¯ ···}| | q(p1,r1)q(p2,r2)g(p3,r3) S 0 gn n (n) ≡ g A ({qp1, q¯p2,gp3,...}; γ). (39)

We always make use of the convention that the helicity associated with momentum pi is ri whereas the helicity associated with momentum qi is si . Let us now use Eq. (28) to write down the infrared-ﬁnite amplitude

{q(p1,r1)q(p¯ 2,r2)}|S|0 order-by-order in perturbation theory. At leading order we have D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 137 A(0) { ¯ }; ≡{ ¯ }| | ( qp1, qp2 γ) q(p1,r1)q(p2,r2) S 0 g0 (0) (0) = W (qp1, q¯p2; qq1, q¯q2) ⊗ A (qq1, q¯q2; γ) (0) = A (qp1, q¯p2; γ), (40) where in the last step we used

(0) W (qp1, q¯p2; qq1, q¯q2) = 3 − 3 − (2π) 2ω(p1)δr1s1 δ(p1 q1)(2π) 2ω(p2)δr2s2 δ(p2 q2). (41) 0 Eq. (41) is simply obtained by noting that Ω∆− = 1atO(g ), Eq. (22), and using the (anti)commutation relations Eqs. (10), (11) and (12). At O(g) the amplitude is zero because for every intermediate state f either the dressing factor f |Ω∆−|f or the amplitude f |S|0 vanishes. At O(g2) the situation is more interesting. We have A(2) { ¯ }; ≡{ ¯ }| | ( qp1, qp2 γ) q(p1,r1)q(p2,r2) S 0 g2 (0) (2) = W (qp1, q¯p2; qq1, q¯q2) ⊗ A (qq1, q¯q2; γ) (2) (0) + W (qp1, q¯p2; qq1, q¯q2) ⊗ A (qq1, q¯q2; γ) (1) (1) + W (qp1, q¯p2; qq1, q¯q2,gq3) ⊗ A (qq1, q¯q2,gq3; γ). (42) The ﬁrst term on the right-hand side of Eq. (42) is nothing but the usual one-loop amplitude multiplied by the O(g0) dressing factor and, using Eq. (41), can be written as

(0) (2) (2) W (qp1, q¯p2; qq1, q¯q2) ⊗ A (qq1, q¯q2; γ)= A (qp1, q¯p2; γ). (43) The second term is also a two-particle cut term, but this time it is the usual tree-level amplitude multiplied by the next-to-leading order dressing factor. These two terms are shown in Fig. 1. The third term in Eq. (42) is of a somewhat different nature as it is a three-particle cut (1) diagram, as illustrated in Fig. 2. The dressing factor W (qp1, q¯p2; qq1, q¯q2,gq3) is zero unless the gluon gq3 is either soft or collinear to the quark or antiquark. Thus, the dressing factor projects out the infrared-singular piece of the bremsstrahlung amplitude. This is (2) exactly the piece that is needed to render the full amplitude A ({qp1, q¯p2}; γ)ﬁnite. In the next section we will calculate this amplitude explicitly and check that the infrared singularities present in the three terms of Eq. (42) cancel in the sum.

Fig. 1. Cut diagrams for 2-particle intermediate state. The term W(0) ⊗ A(2) of Eq. (43) corresponds to j = 0, i = 2andW(2) ⊗ A(0) corresponds to j = 2, i = 0. 138 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161

The construction of the amplitude at higher orders in g follows the same pattern. For any odd power of g the amplitude vanishes for the same reason as it vanishes at O(g).At O(g4) it is given by

(4) A ({qp1, q¯p2}; γ) (0) (4) = W (qp1, q¯p2; qq1, q¯q2) ⊗ A (qq1, q¯q2; γ) (2) (2) + W (qp1, q¯p2; qq1, q¯q2) ⊗ A (qq1, q¯q2; γ) (4) (0) + W (qp1, q¯p2; qq1, q¯q2) ⊗ A (qq1, q¯q2; γ) (1) (3) + W (qp1, q¯p2; qq1, q¯q2,gq3) ⊗ A (qq1, q¯q2,gq3; γ) (3) (1) + W (qp1, q¯p2; qq1, q¯q2,gq3) ⊗ A (qq1, q¯q2,gq3; γ) (2) (2) + W (qp1, q¯p2; qq1, q¯q2,gq3,gq4) ⊗ A (qq1, q¯q2,gq3,gq4; γ) (2) (2) + W (qp1, q¯p2; qq1, q¯q2,qq3, q¯q4) ⊗ A (qq1, q¯q2,qq3, q¯q4; γ). (44) The separate terms in Eq. (44) are infrared-divergent but in the sum all these divergences cancel. This can be seen by looking at a particular Feynman diagram, for example, the one shown in Fig. 3, and realizing that Eq. (44) is nothing but the sum over all possible cuts. Since the dressing factors are constructed such that in the infrared limit they correspond to (4) the usual amplitudes it is clear that the infrared singularities in A ({qp1, q¯p2}; γ)have to cancel in the same way as they cancel in ordinary cut diagrams. The ﬁrst term of Eq. (44) corresponds to the ordinary two-loop amplitude and is represented by cut 1. The other two-particle cuts, the second and third term, are represented by cut 2 and 3. There are two three-particle cut terms, term 4 and 5. Finally, for the diagram under consideration, there is one four-particle cut contribution, namely, term 6. For a certain Feynman diagram not

Fig. 2. Cut diagram for 3-particle intermediate state.

Fig. 3. All possible cuts of a Feynman diagram representing the various terms of Eq. (44). D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 139 all terms of Eq. (44) are present. In our case, the last term of Eq. (44) which is another four-particle cut contribution is missing. We should stress that our approach to construct infrared-finite amplitudes is by no means restricted to amplitudes with final state singularities only. Initial state singularities are dealt with by dressing the initial state, as can be seen in Eq. (28). In fact, the dressing of the initial state would even be needed for processes as discussed above, i.e., with say only a γ in the initial state. Above and in the rest of this paper we have excluded any QED vertices from the soft Hamiltonian even though there is a potential collinear singularity at this vertex. We do this because we treat the incoming photon as off-shell and so it will generate no infrared singularities. If we were to include such vertices in the soft Hamiltonian then we would generate many more diagrams with non-vanishing initial-state dressing factors such as W(q(q1,s1), q(¯ q2,s2), g(q3,s3); γ). We would find though, that all such extra contributes would cancel as all diagrams with purely soft vertices cancel as described in Section 3.5.

3.7. From amplitudes to cross sections

Once the infrared-finite amplitudes have been computed, they can be used to compute cross sections for observables related to these amplitudes. The procedure to obtain cross sections from amplitudes depends to some extent on the external states we use and deserves some further considerations. In the cross-section approach we usually deal with amplitudes that are proportional to a four-dimensional delta function. Upon taking the absolute value squared, this leaves us with the problem of interpreting the square of a delta function. Usually this is dealt with in a rather non-rigorous manner by putting the system in a four-dimensional box of size V ·T . The square of the delta function is then interpreted as V · T times a single delta function. The factor V · T is cancelled by taking into account the normalization of the states and the flux factor, which leaves us with a cross section proportional to a single four-dimensional delta function, expressing conservation of four-momentum. The appearance of the square of the delta function is of course related to the fact that we usually work with non-normalizable states with a sharp value of momentum and energy. In a more rigorous treatment within the cross-section approach the in and out states would have to be written as wave packets, sharply peaked around a certain value of momentum and energy. It can then be shown that the spreading of the wave packet during the scattering process can be safely neglected [21]. As mentioned in Section 2.1 the precise definition of the measurable quantity is given in terms of a measurement function. This is a function of the partonic momenta. If we are dealing with wave packets rather than sharp-momentum states, the measurement function has to be defined in terms of these wave packets. However, as long as we deal with wave packets whose spread is well below any experimental resolution, we can simply use the normal measurement function with the partonic momentum replaced by the central value of the wave packet and we get the same result as in the above mentioned, less rigorous approach [21]. Let us now turn to the situation we encounter if we work with infrared-finite amplitudes, defined in Eq. (8). As mentioned before, the amplitude is then not proportional to an energy-conserving delta function, even if we were to start with the usual non-normalizable 140 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 states. Following the proper treatment with wave packets, we think of the states |i and f | (or |Φi and Φf |) as sharply peaked wave packets. The states |{i} and {f }| as defined in Eqs. (23) and (24) are also wave packets. Through the action of the Møller operators, their spread is larger than the spread of |i and f | and depends crucially on the parameter ∆. If we choose ∆ small enough such that the spread of the wave packets related to the states |{i} and {f }| is still smaller than any experimental resolution, we can still compute any measurable cross section by using the standard measurement function with the partonic momenta replaced by the central value of the wave packet. The important point is that we must be able to express any measurable quantity in terms of the states |{i} and {f }|. However, since the states |{i} and {f }| differ from |i and f | only by soft and collinear interactions, this is nothing but the requirement that the quantity we are dealing with is infrared-safe. Indeed, the requirement of infrared-safety states that the quantity must not depend on whether or not a parton emits another arbitrarily soft or collinear parton. But in the limit ∆ → 0 the soft Møller operators do precisely this. Thus, choosing ∆ small enough ensures that the construction of the measurable quantity in terms of the partonic momenta is not affected by the soft Møller operators. This solves the problem on how to obtain differential cross sections, once the infrared- finite amplitudes are known, in principle. In practice, the explicit implementation of this programme is far from trivial and requires further investigations. We mention for example that choosing ∆ very small might result in numerical problems, similar to the so-called binning problem in the standard approach. If, on the other hand, we choose ∆ too large (relative to the experimental resolution) the infrared-finite amplitudes are too inclusive to allow the computation of any possible physical quantity. It has been advocated before that the most convenient choice of HA is the one that precisely corresponds to the experimental resolution [14]. While this might be true in principle, we think that this is not a practicable way to proceed, since then the asymptotic Hamiltonian would depend on the details of the experiment.

4. An example e+e− → 2jetsatNLO

We consider the process e+e− → γ(P)→ 2 jets. At leading order there is only one partonic process that contributes, e+e− → qq¯. However, at next-to-leading order there is also the process e+e− → qqg¯ . Since the initial state does not interact strongly we can restrict our considerations to the process γ ∗(P ) → 2jets.

4.1. The conventional approach

In the conventional cross-section approach we compute the amplitudes for the two partonic processes (0) 2 (2) A qp1, q¯p2; γ(P) = A qp1, q¯p2; γ(P) + g A qp1, q¯p2; γ(P) + O(g4) (45) D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 141 and (1) 3 A qp1, q¯p2,gp3; γ(P) = gA qp1, q¯p2,gp3; γ(P) + O(g ). (46) Upon squaring the amplitude and integration over the phase space we obtain

2 2 4 dσ = dσ0 + g dσqq¯ + g dσqqg¯ + O(g ), (47) where dσ ∼ A(0) q , q¯ ; γ(P) 2, (48) 0 p1 p2 (0) (1)∗ dσqq¯ ∼ 2Re A qp1, q¯p2; γ(P) A qp1, q¯p2; γ(P) , (49) (1) 2 dσqqg¯ ∼ A qp1, q¯p2,gp3; γ(P) . (50)

The virtual cross section, dσqq¯ and the real cross section, dσqqg¯ both contain infrared singularities and only when combined to form an infrared-safe observable do these divergences cancel. For the total cross section, for example, we obtain 2 2 4παemeq Nc σ0 = , (51) 3s 2 αs 2 3 19 π σ ¯ = σ C c + + − , (52) qq 0 F π Γ 2 2 4 2 2 αs 2 3 π σqqg¯ = σ CF cΓ − − − 4 + , (53) 0 π 2 2 2 = + O = 2 − = where cΓ 1 () and CF (Nc 1)/(2Nc) 4/3. Thus, at next to leading order the total cross section is given by αs σ = σqq¯ + σqqg¯ = σ 1 + 3CF . (54) 1 0 4π

4.2. The infrared-ﬁnite amplitudes

In terms of infrared-finite amplitudes, at next-to-leading order a cross section is also made up of two contributions. The two amplitudes that contribute are those with the final states {qp1q¯p2}| and {qp1q¯p2gp3}|. However, the crucial point is that both these amplitudes are infrared-finite. Up to the order in g required they are given by

A({qp1, q¯p2}; γ) ≡{q q¯ }|S| p1 p2 0 (0) (0) = dq˜1 dq˜2 W (qp1, q¯p2; qq1, q¯q2) × A qq1, q¯q2; γ(P) 2 (0) (2) + dq˜1 dq˜2 g W (qp1, q¯p2; qq1, q¯q2) × A qq1, q¯q2; γ(P) 2 (2) (0) + dq˜1 dq˜2 g W (qp1, q¯p2; qq1, q¯q2) × A qq1, q¯q2; γ(P) 142 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 2 (1) (1) + dq˜1 dq˜2 dq˜3 g W (qp1, q¯p2; qq1, q¯q2,gq3) × A qq1, q¯q2,gq3; γ(P)

+ O(g4) (55) and

A({qp1, q¯p2,gp3}; γ) ≡{q q¯ g }|S|0 p1 p2 p3 (1) (0) = dq˜1 dq˜2 gW (qp1, q¯p2,gp3; qq1, q¯q2) × A qq1, q¯q2; γ(P) (0) + dq˜1 dq˜2 dq˜3 gW (qp1, q¯p2,gp3; qq1, q¯q2,gq3) (1) × A qq1, q¯q2,gq3; γ(P) + O(g3), (56) where a sum over the spin/helicities of the intermediate particles is understood to be included in dq˜i , Eq. (16).

4.3. The asymptotic Hamiltonian

Before we can proceed with the calculation of the infrared-finite amplitudes we have to define the asymptotic Hamiltonian H∆.OncewehaveH∆ we can obtain the Møller operator, Eq. (22), and use it to construct the dressed states, Eq. (26), order-by-order in perturbation theory. The only condition on H∆ is that it includes all long-range interactions from the original Hamiltonian. In order to separate these long range soft and collinear emission terms from the hard emission terms we need to introduce (at least) one parameter which we denote by ∆. The dependence of the asymptotic Hamiltonian on this parameter is indicated in the notation H∆. Once these terms are incorporated into the asymptotic Hamiltonian we are free to include any other terms from the original Hamiltonian that we wish, as these will only produce finite ∆-dependent contributions to the two final amplitudes. It is clear from Eq. (28) that for the final result this ∆ dependence has to cancel. In our case the only term of the interaction Hamiltonian we wish to include in H∆ is the quark–gluon interaction vertex, ≡ : ¯ a µ : a HI g dx ΨT γ Ψ Aµ. (57)

Using Eqs. (13), (14) and (15) we see that HI consists of eight terms 8 a ˜ ˜ ˜ HI = gT dk1 dk2 dk3 Vi(k1, k2, k3) = i 1 3 3 (D−1) × exp −it σij ω(kj ) δ σij kj , (58) j=1 j=1 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 143 where (suppressing the helicity and color labels)

† V1 = b (k1)b(k2)a(k3) ·¯u(k1)/ε(k3)u(k2), † † V2 = b (k1)d (k2)a(k3) ·¯u(k1)/ε(k3)v(k2), V3 = d(k1)b(k2)a(k3) ·¯v(k1)/ε(k3)u(k2), † V4 =−d (k1)d(k2)a(k3) ·¯v(k2)/ε(k3)v(k1), † † ∗ V5 = b (k1)b(k2)a (k3) ·¯u(k1)/ε (k3)u(k2), † ∗ V6 = d(k1)b(k2)a (k3) ·¯v(k1)/ε (k3)u(k2), † † † ∗ V7 = b (k1)d (k2)a (k3) ·¯u(k1)/ε (k3)v(k2), † † ∗ V8 =−d (k1)d(k2)a (k3) ·¯v(k2)/ε (k3)v(k1). (59)

The sign factors σij are +1 (−1) for incoming (outgoing) particles. As we only have to include terms in H∆ that contribute in the singular regions we are free to exclude the Vi in Eq. (59) for which σiω(ki ) can never equal zero with all ω(ki) 0 such that not all of the ω(ki) = 0. We can see from Eq. (33) that such terms will always be finite. From the remaining terms we choose only those that give a singularity in the physically relevant soft or collinear regions. This means that for our example we can exclude V2 and V6, as these only go singular when the two incoming or outgoing quarks from the vertex are collinear. We emphasize that for more general processes these terms have to be included in the asymptotic Hamiltonian. We can confine the remaining terms even further as we are free to choose the form of the finite part of H∆. We restrict the integration of the momenta k1, k2 and k3 to just the potentially singular regions. This restriction is achieved here by including a theta function, Θ(∆i(k1, k2, k3)) in each Vi from Eq. (59) which will appear in H∆. The form of ∆i(k1, k2, k3) is completely arbitrary as long as Θ → 1 in the soft and collinear limits. The form of the Θ function that we will take for this example is, Θ ∆i(k1, k2, k3) ≡ Θ ∆ − σij ω(kj ) . (60) j This choice of ∆(k1, k2, k3) is particularly appropriate because as we see in Eq. (33), j σij ω(kj ) is the exact form that the singular terms take. This theta function therefore restricts the integral to just the regions close to these singular limits. By splitting up the covariant vertex into pieces and restricting the integration to just the singular regions we are removing the manifest Lorentz and gauge invariance from the amplitudes. Physical observables will though be Lorentz- and gauge-invariant as we are effectively just performing a unitary transformation (as we have regulated the Ω± operators) on a known Lorentz- and gauge-invariant result. To summarize, for our asymptotic Hamiltonian we take just the vertices V1, V4, V5 and V8,giving 144 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 3 ˜ ˜ ˜ H∆ = g dk1 dk2 dk3 Vi(k1, k2, k3) exp −it σij ω(kj ) i=1,4,5,8 j=1 3 (D−1) × δ σij kj Θ ∆ − σij ω(kj ) . j=1 j (61) 4.4. Diagrammatic rules for the asymptotic regions

We can now take Eq. (61) and use it in Eq. (22) to form the asymptotic operator. We could then go on to calculate the dressed states of Eq. (37) with this operator defined between suitable in and out states by using the commutation relations between a, b, d, a†, b†, d† and time-ordered perturbation theory. However, it can be shown that there are a set of diagrammatic rules for the asymptotic region which behave in a similar way to Feynman diagrams in normal perturbative field theory. Using these we rules we can simplify the calculation. These diagrammatic rules consist of vertex and propagator-‘like’ objects, but unlike normal Feynman diagrams we must take all time-orderings of the vertices into account. This is because we base the evaluation of the amplitude in the asymptotic region on time- ordered perturbation theory. As mentioned before, energy is not conserved at each vertex and since the range of the time integration in the Møller operators is from 0 to ∞ there is no overall energy conservation. As there is a time-ordering to the vertices we have both absorption and emission rules. These are defined in Fig. 4 with time flowing from right to left. We form propagator-‘like’ objects from the spin sums of fermion spinors and an associated energy denominator. Although they are not real propagators in the normal field theory sense of inverted off-shell two-point Green functions, they do represent the transition from one vertex to another. The rules for these are shown in Fig. 5, where p¯µ ≡ (p0, −p).

(3) + − a µ δ (p3 p2 p1) ≡ (ig)T γ + Θ ∆ − ω(p3) + ω(p2) − ω(p1) ω(p3) + ω(p2) − ω(p1) − i0

(3) + − a µ δ (p1 p2 p3) ≡ (−ig)T γ + Θ ∆ − ω(p1) + ω(p2) − ω(p3) ω(p1) + ω(p2) − ω(p3) − i0

Fig. 4. The diagrammatic rules for vertices.

ip/ ≡ . 2ω(p) ab δ pµp¯ν + pν p¯µ ≡ −gµν + . 2ω(p) (pp)¯

Fig. 5. The diagrammatic rules for propagators. D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 145

As with ordinary ﬁeld theory we must integrate over all internal momenta and so for each propagator in the asymptotic region we must integrate over its momentum dD−1p/(2π)D−1 in D − 1 = 3 − 2 dimensions. The rules for external particles are exactly the same as for QED or QCD and so do not need to be reproduced here. Finally, we must include a factor of 1/n! with each diagram, where n represents the order in the coupling in the asymptotic region. As stated before the soft Møller operators are not necessarily gauge-invariant and neither Lorentz-invariant. Infrared singularities though will only occur in the region where = σiω(ki ) = 0. In this limit Lorentz invariance is restored and so the structure of the singularities will also be Lorentz-invariant. Given that our amplitudes will not be gauge-invariant, we will perform all calculations including the second term of the gluon propagator which ensures that we sum over physical polarizations only.

4.5. The amplitude A({q(p1), q(p¯ 2)}; γ)

Let us start with the amplitude A({qp1, q¯p2}; γ) given in Eq. (55). This amplitude consists of four terms and we will look at each of them in turn. The ﬁrst and second term will be dealt with in Section 4.5.1 and 4.5.2, respectively. For the third term (2) of Eq. (55) we need the dressing factor W (qp1, q¯p2; qq1, q¯q2). There are various combinations of interaction terms of the asymptotic Hamiltonian that give rise to non- vanishing contributions to W(2). These can be found using the diagrammatic rules of Section 4.4. We ﬁnd that there are four contributing diagrams. These four diagrams can be split into two classes, two self-interaction terms and two one-gluon exchange terms. We consider the former in Section 4.5.3 and the latter in Section 4.5.4. Finally, the last term of Eq. (55) will be computed in Section 4.5.5.

4.5.1. The Born term The ﬁrst term is (0) (0) dq˜1 dq˜2 W (qp1, q¯p2; qq1, q¯q2) × A qq1, q¯q2; γ(P) , (62) and is of order g0. As discussed previously, Eq. (40), this term corresponds precisely to the tree-level amplitude. (0) α D (D) A {qp1, q¯p2}; γ(P) = (−ie)δij p1|γ |p2(2π) δ (P − p1 − p2) (0) = A qp1, q¯p2; γ(P) .

We use a notation where pi | represents the spinor of a massless outgoing fermion with momentum pi and similarly |pj represents the spinor of a massless incoming fermion of momentum pj . Of course, these spinors depend on the helicity of the fermion, but we suppress this dependence in the notation. The delta function as usual ensures energy– momentum conservation for the process and the δij represents the color ﬂow through the diagram. 146 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161

Fig. 6. Cut diagram for self interaction with 2-particle intermediate state.

4.5.2. The virtual term In the same way we see that the second term of Eq. (55) corresponds to the one-loop amplitude (2) A qp1, q¯p2; γ(P) 2 2 αs µ 1 3 cR π (0) = CF − − − 4 + + A qp , q¯p ; γ(P) . (63) 2π s 2 2 2 12 1 2 The infrared singularities appearing in Eq. (63) will be cancelled by infrared singularities of the third and fourth term of Eq. (55). We should mention that the ﬁnite term in Eq. (63) depends on the regularization scheme used. The result in conventional dimensional regularization is obtained by setting cR = 0 whereas in dimensional reduction we set cR = 1.

4.5.3. The self-interaction terms The two self-interaction terms are obtained by taking the interacting terms V1,V5 and V4,V8 of the asymptotic Hamiltonian as given in Eq. (59). Since there is a symmetry between these two contributions we only need to calculate one of the pair of diagrams. The self interacting term resulting from the vertices V1, V5 isshowninFig.6andisgivenby {2,0} ≡ ˜ ˜ 2W(2) ¯ ; ¯ × A(0) ¯ ; a15 dq1 dq2 g 15 (qp1, qp2 qq1, qq2) qq1, qq2 γ(P) − 2 D−1 ( ie)g D−1 D−1 d q3 a b =− d q1 d q2 T T 2 (2π)D−1 ik kj ab µ ¯ν + ν ¯µ δ µν q3 q3 q3 q3 α × −g + p1|γµq/2γνq/1γ |p2 2ω(q3) (q3q¯3) −| + − | −| + − | × Θ(∆ ω(q2) ω(q3) ω(p1) )Θ(∆ ω(q2) ω(q3) ω(q1) ) 2ω(q1)(ω(q2) + ω(q3) − ω(p1)) 2ω(q2)(ω(q2) + ω(q3) − ω(q1)) (D−1) (D−1) × δ (q2 +q3 −p1)δ (q2 +q3 −q1) D (D) × (2π) δ (P − q1 − p2). (64) D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 147

Note that this expression contains a D-dimensional delta function coming from A(0) and two (D − 1)-dimensional delta functions coming from 3-momentum conservation of the vertices in the dressing factor. We now proceed to perform the integrals over q1 and q2, removing the two (D − 1)- dimensional delta functions. There is an important subtlety here. Since the delta functions are (D − 1)-dimensional, only the spatial part of the 4-vectors is altered. All 4-vectors in the asymptotic region though must be on-shell and so we are forced to modify the energy component of these 4-vectors to preserve this property. Although these modified 4-vectors are 4 component objects they no longer transform as tensors. This is simply a manifestation of the breaking of Lorentz invariance that occurs in time-ordered perturbation theory. To denote such objects we place curly brackets of the type {}around them, i.e., we define {p1 − q3}≡ ω(p1 −q3), p1 −q3 . (65) We then have 2 { } (−ie)g 2,0 =− a a D (D) − − a15 TikTkj (2π) δ (P p1 p2) 2 µ ¯ ν + ν ¯µ µν q3 q3 q3 q3 × dq˜3 −g + Θ ∆ −|ρ(q3, p1 −q3)| (q3q¯3) p |γ {p/ − q/ }γ p/ γ α|p × 1 µ 1 3 ν 1 2 2 , (66) 2ω(p1)2ω(p1 −q3)ρ(q3, p1 −q3) where we defined ρ(k1, k2) ≡ ω(k1) + ω(k2) − ω(k1 + k2). (67)

This diagram contains infrared singularities coming from the region where q3 is soft and/or collinear to p1. We discuss its evaluation in Appendix A. Multiplying by two to take into account both of the self-interaction diagrams we get the ﬁnal result {2,0} = ˜ ˜ 2W(2) ¯ ; ¯ × A(0) ¯ ; 2a15 2 dq1 dq2 g 15 (qp1, qp2 qq1, qq2) qq1, qq2 γ(P) 2 αs µ 1 5 cR = CF − − + g1(∆) + 2π s 2 2 2 (0) + dq˜3 Θ ∆ −|ρ(q3, p1 −q3)| f1(p1,p2,q3) A qp1, q¯p2; γ(P) (68) with 7 5 ∆ 2 7π2 7 ∆ 1 ∆ 2 ∆ g1(∆) =− − − + + + log 2 2 2 12 2 2 2 2 2 ∆ ∆ 2 + log2 + 2log2 1 + + 4Li , (69) 2 2 2 2 + ∆ and where f1(p1,p2,q3) is a function that is free from singularities when integrated over dq˜3. The explicit form is given in Eq. (A.7). Note that we took care to sum over the physical polarizations of the gluon only and evaluated the diagram in the center-of-mass frame p1 =−p2. 148 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161

Fig. 7. Cut diagram for the 2-particle cut diagram with one-gluon exchange in the asymptotic region.

4.5.4. The one-gluon exchange terms We now look at the one-gluon exchange diagrams. There are two such diagrams, one for each time-ordering of the two vertices. One diagram is obtained from taking the vertices V1, V8 of Eq. (59), the other from taking the vertices V4, V5. These diagrams are symmetric under exchange of all momenta and so we need only calculate one of them. The diagram shown in Fig. 7 gives us {2,0} = ˜ ˜ 2W(2) ¯ ; ¯ × A(0) ¯ ; a18 dq1 dq2 g 18 (qp1, qp2 qq1, qq2) qq1, qq2 γ(P) 2 D−1 (−ie)g − − d q = dD 1q dD 1q 3 T a T b 2 1 2 (2π)D−1 ik kj ab µ ¯ν + ν ¯µ δ µν q3 q3 q3 q3 × −g + p1|γµq/1γαq/2γν |p2 2ω(q3) (q3q¯3) −| + − | −| + − | × Θ(∆ ω(q1) ω(q3) ω(p1) )Θ(∆ ω(p2) ω(q3) ω(q2) ) 2ω(q1)(ω(q1) + ω(q3) − ω(p1)) 2ω(q2)(ω(p2) + ω(q3) − ω(q2)) (D−1) (D−1) D (D) × δ (q1 +q3 −p1)δ (p2 +q3 −q2)(2π) δ (P − q1 − q2). (70)

We again integrate over q1 and q3 with the delta functions and introduce the on-shell momenta {p/1 − q/3} and {p/2 + q/3} to obtain 2 D−1 { } (−ie)g d q a 2,0 = T a T a 3 (2π)Dδ(D) P −{p − q }−{p + q } 18 2 ik kj (2π)D−1 1 3 2 3 µ ¯ν + ν ¯µ 1 µν q3 q3 q3 q3 × −g + p1|γµ{p/1 − q/3}γα{p/2 + q/3}γν|p2 2ω(q3) (q3q¯3) Θ(∆−|ρ(q , p −q )|)Θ(∆ −|ρ(q , p )|) × 3 1 3 3 2 . (71) 2ω(p1 −q3)ρ(q3, p1 −q3)2ω(p2 +q3)ρ(q3, p2)

Looking at the denominator ρ(q3, p1 −q3)ρ(q3, p2) it appears that there are collinear singularities for q3 p2,q3 p1 and a soft singularity q3 → 0. However, the denominator µ ¯ν + ν ¯µ µν q3 q3 q3 q3 −g + p1|γµ{p/1 − q/3}γα{p/2 + q/3}γν|p2 (72) (q3q¯3) D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 149

Fig. 8. Cut diagram for 3-particle intermediate state. vanishes in the collinear regions q3 p2 and q3 p1. Thus, this diagram has only a soft singularity. {2,0} We delegate the explicit evaluation of a18 to Appendix A. Multiplying by two to account for both one-gluon exchange diagrams we have the ﬁnal result {2,0} = ˜ ˜ 2W(2) ¯ ; ¯ × A(0) ¯ ; 2a18 2 dq1 dq2 g 18 (qp1, qp2 qq1, qq2) qq1, qq2 γ(P) 2 αs µ 1 (0) = CF + g (∆) A qp , q¯p ; γ(P) 2π s 2 1 2 − − − (−ie)δ p |γ α|p (2π)(D 1)δ(D 1)(P −p −p ) ij 1 2 1 2 × dq˜3 Θ ∆ −|ρ(q3, p1 −q3)| Θ ∆ −|ρ(q3, p2)| f2(p1,p2,q3) , (73) where again we have not performed the ﬁnite f2 integral analytically and ∆ g (∆) = 2log2− 2log . (74) 2 2

The explicit form of f2 is given in Eq. (A.9).

4.5.5. 3 particle cut diagram Let us now turn to the forth part of Eq. (55). For this term we need the dressing factor (1) W (qp1, q¯p2; qq1, q¯q2,gq3). Again we use the diagrammatic rules of Section 4.4. There are two possible diagrams as the gluon can be absorbed either by the quark or antiquark line. The two diagrams are obtained by taking either the vertex V1 or V4 and they are symmetric under exchange of momenta. So we need only calculate one of them. For the diagram shown in Fig. 8 we get {1,1} = ˜ ˜ ˜ 2W(1) ¯ ; ¯ × A(1) ¯ ; a1 dq1dq2 dq3 g 1 (qp1, qp2 qq1, qq2,gq3) qq1, qq2,gq3 γ(P) dD−1q δab = (−ie)g2 dD−1q dD−1q 3 T a T b 1 2 D−1 ik kj (2π) 2ω(q3) 150 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 qµq¯ ν + qνq¯ µ Θ(∆−|ω(q ) + ω(q ) − ω(p )|) × −gµν + 3 3 3 3 1 3 1 ¯ + − (q3q3) 2ω(q1)(ω(q1) ω(q3) ω(p1)) ν α α ν γ (/q1 + q/3)γ γ (/q2 + q/3)γ ×p1|γµq/1 − |p2 2(q1q3) 2(q2q3) (D−1) (D−1) D (D) × δ (q1 +q3 −p1)δ (p2 −q2)(2π) δ (P − q1 − q2). (75)

After integration over q1 and q2 using the delta functions we observe that there are collinear singularities q3 p1 and soft singularities q3 → 0. There are, however, no collinear (1) singularities q3 p2. This is expected since the amplitude A (qq1, q¯q2,gq3; γ(P)) has W(1) ¯ ; only an integrable square-root singularity for q3 q2 and the dressing factor 1 (qp1, qp2 qq1, q¯q2,gq3) is regular for q3 q2. As for the other diagrams we have to multiply by two to take into account both pairs of diagrams and we get the ﬁnal result {1,1} = ˜ ˜ ˜ 2W(1) ¯ ; ¯ 2a1 2 dq1 dq2 dq3 g 1 (qp1, qp2 qq1, qq2,gq3) × A(1) q , q¯ ,g ; γ(P) q1 q2 q3 2 αs µ 2 3 (0) = CF + + g (∆) − cR A qp , q¯p ; γ(P) 2π s 2 3 1 2 − − + (−ie)δ p |γ α|p (2π)(D 1)δ(D 1)(P −p −p ) ij 1 2 1 2 × dq˜3 Θ ∆ −|ρ(q3, p1 −q3)| f3(p1,p2,q3) , (76) where ∆ 2 7π2 ∆ ∆ 2 ∆ g3(∆) = 7 + + + −3 + 2 − log 2 6 2 2 2 ∆ ∆ 2 − 2log2 − 4log2 1 + − 8Li . (77) 2 2 2 2 + ∆

The function f3 is given in Eq. (A.11) and does not produce any infrared singularity upon integration over dq˜3.

4.5.6. An infrared-finite amplitude We have now calculated all terms contributing to the amplitude A({q,q¯}; γ), Eq. (55), at next-to-leading order. Using Eqs. (63), (68), (73) and (76) to assemble the amplitude we get A({q , q¯ }; γ) p1 p2 2 αs π (0) = 1 + CF g (∆) + g (∆) + g (∆) − 4 + A qp , q¯p ; γ(P) 2π 1 2 3 12 1 2 + − | α| (D−1) (D−1) − − ( ie)δij p1 γ p2 (2π) δ (P p1 p2) √ × dq˜3 f1(p1,p2,q3)Θ ∆ −|ρ(q3, p1 −q3)| δ s − ω(p1) − ω(p2) D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 151 + f2(p1,p2,q3)Θ ∆ −|ρ(q3, p1 −q3)| Θ ∆ −|ρ(q3, p2)| + f3(p1,p2,q3)Θ ∆ −|ρ(q3, p1 −q3)| (78) up to order αs in the coupling. The functions g1, g2, g3 are given in Eqs. (69), (74) and (77) and the functions f1, f2 and f3 are given in Eqs. (A.7), (A.9) and (A.11), respectively. We see that this result is completely free of infrared singularities. We are only left with some finite ∆-dependent terms, gi and some finite terms, fi which will in general need to be numerically integrated. Even though the amplitude A({qp1, q¯p2}; γ)depends on ∆ this dependence will disappear when we combine the various amplitudes to calculate physical observables.

4.6. The amplitude A({q(p1), q(p¯ 2), g(p3)}; γ)

We are now going to calculate the amplitude A({qp1, q¯p2,gp3}; γ) given in Eq. (56). There are only two terms to calculate for this amplitude and there is no integration over the ﬁnal state gluon as it is now a real ﬁnal state particle. Again we calculate these terms using the diagrammatic rules from Section 4.4. Let us start with the diagrams where the gluon is emitted in the dressing factor. Fig. 9 shows one of the two possible diagrams, the other is exactly the same but with all momenta interchanged. So for both diagrams we have ˜ ˜ W(1) ¯ ; ¯ × A(0) ¯ ; dq1 dq2 g 1 (qp1, qp2,gp3 qq1, qq2) qq1, qq2 γ(P) − = (−ie)gT a(2π)Dδ(D 1)(P −p −p −p )p | ij 1 2 3 1 ε/ {p/ +p/ }γ α √ × − p3 1 3 Θ ∆ −|r | δ s − ω(p ) − ω(p ) − ω(p ) + r + 1 1 2 3 1 2ω(p1 p3)r1 α{ + } √ γ p/2 p/3 /εp3 + Θ ∆ −|r2| δ s − ω(p1) − ω(p2) − ω(p3) + r2 |p2, 2ω(p2 +p3)r2 (79) where we used the notation

r1 ≡ ρ(p1, p3) = ω(p1) + ω(p3) − ω(p1 +p3),

r2 ≡ ρ(p2, p3) = ω(p2) + ω(p3) − ω(p2 +p3), (80) with ρ deﬁned in Eq. (67).

Fig. 9. Cut diagram for the 3-particle asymptotic region with a 2-particle intermediate state. 152 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161

Fig. 10. Cut diagram for 3-particle asymptotic region with a 3-particle intermediate state.

(1) The second contribution is just the usual A (qp1, q¯p2,gp3; γ(P)) amplitude. The three external particles of this amplitude do not interact in the asymptotic region and so we simply have the diagram as shown in Fig. 10, this gives (0) (1) dq˜1 dq˜2 gW (qp1, q¯p2,gp3; qq1, q¯q2,gq3) × A qq1, q¯q2,gq3; γ(P) ε/ (/p +p/ )γ α γ α(/p +p/ )/ε = − a | p3 1 3 − 2 3 p3 | ( ie)gTij p1 p2 2(p1p3) 2(p2p3) D (D) × (2π) δ (P − p1 − p2 − p3). (81) We now assemble Eq. (56) to find A {qp1, q¯p2,gp3}; γ a = (−ie)gT p1| ij { + } α √ ε/p3 p/1 p/3 γ × − Θ ∆ −|r1| δ s − ω(p1) − ω(p2) − ω(p3) + r1 2ω(p1 +p3)r1 + α √ ε/p3 (/p1 p/3)γ + δ s − ω(p1) − ω(p2) − ω(p3) 2(p1p3) γ α{p/ +p/ }/ε √ + 2 3 p3 Θ ∆ −|r | δ s − ω(p ) − ω(p ) − ω(p ) + r 2ω(p +p )r 2 1 2 3 2 2 3 2 α + √ γ (/p2 p/3)/εp3 − δ s − ω(p1) − ω(p2) − ω(p3) |p2 2(p2p3) D (D−1) × (2π) δ (P −p1 −p2 −p3). (82) This amplitude splits up into two pairs. The first (last) two terms are due to the gluon being emitted from the leg p1 (p2). Looking at the first two terms shows that for r1 >∆ the contribution from the asymptotic region disappears. We are then left with the normal amplitude, A(qp1, q¯p2,gp3; γ).Forr1 <∆the term from the asymptotic region does contribute and will cancel any potential infrared singularities. We can see this by taking the limit ∆ → 0, we have D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 153

ω(p +p )r → (p p ), {p/ +p/ }→(/p +p/ ), √1 3 1 1 3 1 3 √1 3 δ s − ω(p1) − ω(p2) − ω(p3) + r1 → δ s − ω(p1) − ω(p2) − ω(p3) . With these we can see that the terms from the asymptotic region approach those of the normal amplitude in the soft and collinear limits, but with the opposite sign. So the two terms will cancel in the ∆ → 0 limit, leaving us with an amplitude that is infrared-ﬁnite when integrated over the phase space.

4.7. Calculation of the total cross section

In the previous sections we computed the two infrared-finite amplitudes that contribute to the process γ ∗(P ) → 2 jets at next-to-leading order. In this section we would like to check our results by computing the total cross section, starting from the infrared-finite amplitudes, Eqs. (78) and (82). Of course, we have to recover the well-known result, Eq. (54). Let us stress that the idea of our approach is to compute the amplitudes numerically and perform the phase-space integration also numerically. It is for the sole purpose of checking our results and facilitating the comparison with Eq. (54) that in this section we compute the total cross section analytically. Usually a non-zero value of ∆ would be chosen for a numerical calculation and we would expect all ∆ dependence to cancel between the contributions of the two amplitudes (squared) to the cross section. Here though to simplify the analytical calculation we will take the limit ∆ → 0. In this limit even the infrared-finite amplitudes are proportional to a four-dimensional delta function and we can use the standard procedure to obtain the total cross section from the amplitudes. However, since the amplitudes are singular for ∆ → 0 we must be careful in taking this limit and leave it until the end of the calculation. The fn finite terms of Eq. (78) which we would usually have to calculate numerically will all go to zero in this limit. This simplification occurs because the region of integration shrinks to zero as ∆ → 0 and as these terms are finite they can no longer give a contribution. We now use our infrared-finite amplitudes Eq. (55) and Eq. (56) instead of Eq. (45) and Eq. (46) and square them in the usual way to obtain

σ = σ{qq¯} + σ{qqg¯ }, (83) where 2 σ{qq¯} = dΦ2 A({qp1, q¯p2}; γ) , (84) 2 σ{qqg¯ } = dΦ3 A({qp1, q¯p2,gp3}; γ) . (85)

Here we integrate Eq. (84) over the two particle phase space and Eq. (85) over the three particle phase space. First we rewrite the three-particle ﬁnal state amplitude, Eq. (82), in a more convenient form 154 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161

A({qp1, q¯p2,gp3}; γ) = (−ie)gT a p | ij 1 { + } α + α ε/p3 p/1 p/3 γ /εp3 (/p1 p/3)γ × − δ(E + r1) + δ(E) Θ ∆ −|r1| 2ω(p1 +p3)r1 2(p1p3) + α /εp3 (/p1 p/3)γ + δ(E)Θ |r1|−∆ 2(p1p3) γ α{p/ +p/ }/ε γ α(/p +p/ )/ε + 2 3 p3 δ(E + r ) − 2 3 p3 δ(E) Θ ∆ −|r | ω(p +p )r 2 (p p ) 2 2 2 3 2 2 2 3 α + γ (/p2 p/3)/εp3 − δ(E)Θ |r2|−∆ |p2 2(p2p3) D (D−1) × (2π) δ (P −p1 −p2 −p3), (86) √ where E = s − ω(p1) − ω(p2) − ω(p3). Taking Eq. (86) we then square it in the usual way and sum over the gluon polarizations using pµp¯ν + pν p¯µ µ ν =−gµν + 3 3 3 3 , p3 p3 (p3p¯3) where p¯3 = (ω(p3), −p3). This is because the amplitude is no longer gauge-invariant as we are using dressed states. At this point we drop any terms multiplied by Θ(∆−|r1|) or Θ(∆−|r2|). These terms are ﬁnite and therefore can be shown to go to zero in the ∆ → 0 limit after we have performed the three particle phase space integral in a similar way to the fn terms. After integrating one of the phase space integrals using the delta function we are left with A({q , q¯ ,g }; γ)2 p1 p2 p3 3−2D dΩD−1 = 4CF (2π) dΩD−2 2D−1 ∆/2 0 2y − y (y + y )2 × dy dy 23 13 13 23 13 23 2 y13(y13 + y23) 0 1−y13 ∆ 1−∆/2 2 3 2 2 y + y y23 + 2y13(y23 − 1) − dy dy 23 13 13 23 2 y23(y13 + y23) 0 1−y13 1−∆/2 ∆/2 2 − y 2 − y 4 + dy dy 2 − 23 − 13 + , (87) 13 23 2 y13 y23 (y13 + y23) ∆/2 1−y13 where we deﬁned

2(pipj ) y ≡ . (88) ij ξ 2 p1 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 155

We perform the final two integrals and then prematurely take the ∆ → 0 limit everywhere except in the log(∆) terms, as these diverge in this limit. The log(∆) terms will cancel later in the final result. This then leaves 1 αs 5 3 ∆ 2 ∆ σ{qqg¯ } = CF − log4 + log + log , (89) σ0 π 4 2 2 2 where σ0 is the total Born cross section as given in Eq. (48). 2 Now we calculate |A({qp1, q¯p2}; γ)| . Again, we take the ∆ → 0 limit early except for the log(∆) pieces of the gn terms in the finite part of Eq. (78). As stated before the fn(p1,p2,q3) terms go to zero and so we have 2 A({qp1, q¯p2}; γ) (0) 2 = A qp1, q¯p2; γ(P) 2 αs 1 3 ∆ 2 ∆ × 1 + CF − + log 4 − log − log . (90) 2π 2 2 2 2 After integrating over the two particle phase space we get σ{ ¯} qq = + αs −1 + − 3 ∆ − 2 ∆ + O 2 1 CF log4 log log αs . (91) σ0 π 2 2 2 2 Putting Eq. (89) and Eq. (91) together gives finally σ = + αs 3 + O 2 1 CF αs . (92) σ0 π 4 We have recovered the well-known result for the total γ → qq¯ cross section and all the ∆ dependence of the amplitudes has disappeared including the log(∆) terms, justifying our taking of the ∆ → 0 limit early.

5. Summary and outlook

We have presented a method on how to construct infrared-finite amplitudes and applied it to the case of e+e− → 2 jets at next-to-leading order in the strong coupling. The idea is to separate from the Hamiltonian a part that describes the asymptotic dynamics. This asymptotic Hamiltonian is then used to asymptotically evolve the usual states of the Fock space. In this way we construct dressed states, Eqs. (23) and (24), such that the transition amplitudes between these states are free from infrared singularities. Contrary to most of the previous work done in this field we are not so much interested in obtaining all-order resumed results taking into account soft emission of an arbitrary number of gauge bosons from external partons. Our aim is to construct dressed states explicitly order-by-order in perturbation theory and use them to do explicit calculations. In this paper we have done this for a particularly simple final state up to next-to-leading order. In the future we would like to expand this to more complicated external states and higher orders. 156 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161

The reason that we cannot obtain all-order results is that we include the collinear singularities as well. In non-abelian theories these singularities cannot be avoided. The additional complications due to the collinear singularities make it impossible to obtain exact solutions to the asymptotic dynamics. Collinear singularities have been considered previously [12–14] but to the best of our knowledge the amplitudes presented in this work are the first infrared-finite amplitudes for a realistic scattering process in QCD. As for the standard approach, physical cross sections obtain in general contributions from more than one partonic process. However, in our case all these contributions are separately finite. They depend on a parameter, ∆, that determines the precise split of the Hamiltonian into an asymptotic Hamiltonian and the remainder. The result for any physical quantity is independent of this parameter as long as it is smaller than any experimental resolution. For any finite value of ∆ the amplitude contains a part that is not proportional to an energy-conserving delta function which represents the spread of the initial wave packet due to the asymptotic evolution. For any physical cross section at any order in perturbation theory we will get the same answer using the standard cross-section method or infrared-finite amplitudes. Thus, one might wonder what has been gained using this approach. Apart from the conceptional benefit that the S-matrix between dressed states is well-defined there are also practical advantages. First of all, the avoidance of infrared singularities facilitates the use of numerical methods. This might not be apparent in the approach we have taken. In fact, using Eq. (28) to split the infrared-finite amplitudes into separately divergent factors still requires us to use an infrared regulator (dimensional regularization in our case) and revert to analytical calculations. However, since the final amplitude is infrared-finite it is feasible to compute it directly in a numerical way, avoiding the split into separately divergent pieces. Once the amplitudes have been obtained, the integration over the phase space is trivial and no sophisticated method is needed. This also opens up the possibility of combining fixed-order calculations directly with a parton shower approach. Needless to say that the explicit example we considered, e+e− → 2 jets has many simplifying features. To start with, the non-abelian nature of QCD does not really enter. Secondly, we only considered the amplitudes at next-to-leading order. Furthermore, the initial state does not interact strongly. The last point simply results in the fact that there is no need to dress the initial state. While this is a simplification concerning the amount of computations to be performed, there is no conceptual problem associated with more complicated initial states. If the initial state contains hadrons a physical cross section is obtained by folding the partonic cross section with parton densities. In the conventional approach these parton densities are associated with the probability of finding a certain partonic state within a hadron. In our case, we would have to use modified parton densities that are related to the probability of finding a certain dressed state within a hadron. Thus the global analyzes of extracting the parton densities would have to be modified and repeated. The fact that the non-abelian nature of QCD does not really show up in the explicit example we considered results in a particularly simple asymptotic Hamiltonian. In fact, the asymptotic Hamiltonian we use involves only quark–gluon interactions and is basically the same that was used many times previously [14]. Again, this results in a technical simplification of the computation and facilitates the explicit construction of the D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 157 asymptotic Hamiltonian. In more complicated examples the full non-abelian structure of the asymptotic Hamiltonian will enter the problem and its construction will be much more involved. However, the only crucial feature is that the asymptotic Hamiltonian reproduces the full asymptotic dynamics, i.e., it has to reproduce the soft and collinear behavior of the full theory. There are no further requirements and the construction of dressed states presented in this paper can be taken over directly. However, it is clear that the construction used so far is rather cumbersome. In order to exploit the advantage of the infrared-finiteness a systematic numerical approach should be developed. This will become particularly important if the method is to be extended beyond next-to-leading order.

Acknowledgements

D.A.F. acknowledges support from a PPARC studentship.

Appendix A

In this appendix we give some details concerning the evaluation of the diagrams mentioned in Section 4.5. {2,0} We consider ﬁrst with the self-interaction term a15 of Section 4.5.3. We start from Eq. (66), substitute p/1 − q/3 + /r for {p/1 − q/3},wherer = (r0, 0) with r0 = ρ(q3, p1 −q3) and then expand the numerator to obtain − 2 {2,0} = ( ie)g a a D (D) − − ˜ −| | a15 TikTkj (2π) δ (P p1 p2) dq3 Θ ∆ r0 4 4(p1q3)(p1q¯3) 2r0(p1q3)(p2q3) × (D − 2) (p1q3) − (p1r) − − (q3q¯3) ω(p1)(q3q¯3) p |γ α|p × 1 2 . (A.1) − 2 ω(p1)ω(p1 q3)r0

This expression contains infrared singularities coming from the region where q3 is soft and/or collinear to p1. In order to evaluate the expression, Eq. (A.1) we choose to parameterize the momenta in the center-of-mass frame. The momenta are all on-shell and are deﬁned as √ P = s(1, 0, 0),

ξp1 ξp1 p1 = (1, 0, 1), p2 = (1, 0, −1), 2 2 ξ p1 2 q3 = z 1, 1 − y eT ,y , 2 ξp1 2 2 {p − q }= 1 − 2zy + z , −z 1 − y eT , 1 − zy , (A.2) 1 3 2 where 0 is the null vector in a (2 − 2)-dimensional space, eT is a unit vector in the (2 − 2)-dimensional transverse-momentum space and we have 0 z ∞, −1 y 1. 158 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161

The singular limits are then given by the limits z → 0 for soft singularities, y → 1for q3 p1 singularities and y →−1forq3 p2 singularities. As the asymptotic region does not conserve energy we ﬁnd that the upper limit of z goes to ∞. This would suggest the possibility of UV singularities in the asymptotic regions. However, we will see that the Θ function will restrict this upper limit to a ﬁnite value, removing the need to renormalize these regions. The integral measure is given by

dq˜3 Θ(∆− r0) 2 ξ 2 µ 1 p1 1−2 − − → z (1 − y) (1 + y) dydzdΩ( − ), (A.3) s 2(2π)3−2 4 2 2 where we have three separate integration regions for the z and y integrals,

2 (2 + ∆) 2 − ∆2 0 z with − 1 y , ∆ (1 − y + ∆) 2 2 − ∆2 0 z 1 with y 1, 2 2 (2 + ∆) 2 − ∆2 1 z with y 1. ∆ (1 − y + ∆) 2 The infrared singularities are in the ﬁrst two regions whereas the last region will give a ﬁnite contribution. The remaining angular integral is given by 2π1− dΩ − = . (A.4) (2 2) (1 − )

We now turn back to Eq. (A.1) and notice that the infrared singularities q3 soft and/or collinear to p1 come from the region z = 0andy = 1 but not y =−1. We use the subtraction method to isolate these singularities and evaluate them analytically. Writing the integrand schematically as a function F(z,y)we write F(z,y)= F(0,y)+ F(z,1) − F(0, 1) + F(z,y)− F(0,y)− F(z,1) + F(0, 1) . (A.5)

The ﬁrst term contains all the divergent pieces whereas the second term will give a ﬁnite contribution upon integration over dq˜3. Applying this method to Eq. (A.1) we obtain 2 {2,0} 2 a a µ 1 D (D) a = (−ie)g T T (2π) δ (P −p1 −p2) 15 ik kj s 2(2π)3−2 α 1−2 − − ×p1|γ |p2 dΩ(2−2) dy dzz (1 − y) (1 + y) (2 − D) (2z − 1 − y) × + + f (p ,p ,q ) (A.6) 4(1 − y) 2(1 − y)z2 1 1 2 3 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 159 where

f (p ,p ,q ) 1 1 2 3 ω(p )2 (p q ) (p q )(p q ) r = 1 1 − 1 3 − 1 3 2 3 2 − 0 2ω(p −q )r ω(p )r ω(p )r (q q¯ ) ω(p ) 1 3 0 1 0 1 0 3 3 1 2 − ω(p1) − ω(q3) − (p2q3) + 2 2 . (A.7) 2(p1q3) ω(p1) ω(q3) Integrating the singular terms and expanding around = 0 we obtain Eq. (68). Note that the D in the ﬁrst term of Eq. (A.6) arises from the γ -matrix algebra. Thus we write it as D = 4 − 2 + cR2 to obtain the expressions in conventional dimensional regularization (cR = 0) and in dimensional reduction (cR = 1). {2,0} Let us now turn to the evaluation of a18 needed in Section 4.5.4. We start with the {2,0} expression Eq. (71) and proceed in the same way as for a15 . We introduce the on-shell { − }= − + { + }= + − = momenta p/1 q/3 p/ 1 q/3 /r and p/2 q/3 p/2 q/3 /r ,wherer (r0, 0) = = + − + with r0 ρ(q3, p2) ω(q3) ω(p2) ω(p2 q3). In order to proceed we subtract the soft singularity in Eq. (71) and add it back to produce an integrand that results in a non-singular term. In the soft limit the D-dimensional delta function becomes the usual (D) δ (P − p1 − p2) which can be pulled out from the integral. We use the same momentum parametrization as for the self-interacting case, but because of the extra Θ function the integration ranges change to 2 (2 + ∆) ∆ 0 z with − 1 y , ∆ (1 − y + ∆) 2 2 (2 − ∆) ∆ 0 z − with y 1. ∆ (1 + y − ∆) 2 The remaining angular integral is as given in Eq. (A.4). Using this momentum parametrization and expanding around the soft region gives 2 { } µ 1 − a 2,0 =−T a T a (−ie)g2 (2π)Dδ(D 1)(P −p −p ) 18 ik kj 3−2 1 2 s 2(2π) α 1−2 − − ×p1|γ |p2 dΩ(2−2) dy dzz (1 − y) (1 + y) 1 √ × δ s − ω(p ) − ω(p ) + f (p ,p ,q ) (A.8) 2z2 1 2 2 1 2 3 with

f2(p1,p2,q3) ω(p )2 = 1 (p p ) + (p q ) − (p q ) − + 1 2 1 3 2 3 2ω(p1 q3)r0ω(p2 q3)r0 r r (p q )r + − − 0 0 + (p1q3)r0 + 2 3 0 − (p1q3)(p2q3) ω(p1)r0 ω(p1)r0 2 2 2ω(p1) 2ω(p1) 2ω(p1) 160 D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161

1 (p q )3 r (p q )3 r − 1 3 1 + 0 − 2 3 1 − 0 2 2 2(q3q¯3) ω(p1) 2ω(p1) ω(p1) 2ω(p1) r r r + + r0 − 0 − 0 0 2 + 2 2 2 (p1q3) (p2q3) ω(p1) ω(p1) ω(p1) √ × − − − + δ s ω(p1) ω(p2) r0 r0 2 √ − ω(p1) − − 2 δ s ω(p1) ω(p2) . (A.9) 2ω(q3) Upon performing the integration of the singular terms explicitly and expanding in we get Eq. (73). In this case the expression is the same in conventional dimensional regularization and dimensional reduction. {1,1} Finally we turn to the evaluation of a1 needed in Section 4.5.5, proceeding as in the previous cases. We subtract the soft and collinear singular parts and integrate them analytically. In both limits the D-dimensional delta function takes its (D) usual form δ (P − p1 − p2). Thus, the delta function is independent of the integration variables and can be taken outside the integral, as in the one-gluon exchange terms. We can use the same momentum parametrization and integration regions as the self- interacting case as we have the same Θ function in both cases. Taking the z → 0and y → 1 limits of the above terms we obtain 2 { } µ 1 − a 1,1 = (−ie)g2T a T a (2π)Dδ(D 1)(P −p −p ) 1 ik kj s 3−2 1 2 2(2π) α 1−2 − − ×p1|γ |p2 dΩ(2−2) dy dzz (1 − y) (1 + y) 4 − 4z + (D − 2)z2 √ × δ s − ω(p ) − ω(p ) + f (p ,p ,q ) , 2(1 − y)z2 1 2 3 1 2 3 (A.10) where

f (p ,p ,q ) 3 1 2 3 ω(p )2 = 1 r2 + 2r ω(p ) 2ω(p −q )r ({p − q }q ) 0 0 1 1 3 0 1 3 3 r2 − + r0 + (p1q3)(p2q3) + 2r0 + 0 (p1q3) 2 2 2 ω(p1) (q3q¯3) ω(p1) 2ω(p1) ω(p )2 r (p q ) + 1 2(p p ) + (p q ) 2 + 0 − 2 3 1 2 1 3 2 2ω(p1 −q3)r0(p2q3) ω(p1) ω(p1)

− 2(p2q3) + 2r0ω(p1) 2 √ − (p1q3) + r0 (p1q3) + r0 − − − 2 2 1 δ s ω(p1) ω(p2) r0 (q3q¯3) ω(p1) ω(p1) 2ω(p1) D.A. Forde, A. Signer / Nuclear Physics B 684 (2004) 125–161 161 (p p ) ω(q ) (p q ) − ω(p )2 1 2 + 3 + 2 3 1 2 (p1q3)(p2q3) ω(p1)(p1q3) 2(p1q3)ω(q3) (p q ) 2 √ − 1 3 − δ s − ω(p ) − ω(p ) . (A.11) 2 1 2 2(p2q3)ω(q3) (p1q3)

Integrating the singular terms with D = 4 − 2 + cR2 and expanding around = 0we obtain Eq. (76).

References

[1] F. Bloch, A. Nordsieck, Phys. Rev. 52 (1937) 54. [2] T. Kinoshita, J. Math. Phys. 3 (1962) 650; T.D. Lee, M. Nauenberg, Phys. Rev. 133 (1964) B1549. [3] V. Chung, Phys. Rev. 140 (1965) B1110. [4] D. Zwanziger, Phys. Rev. D 7 (1973) 1082; D. Zwanziger, Phys. Rev. D 11 (1975) 3481. [5] P.P. Kulish, L.D. Faddeev, Theor. Math. Phys. 4 (1970) 745. [6] M. Greco, F. Palumbo, G. Pancheri-Srivastava, Y. Srivastava, Phys. Lett. B 77 (1978) 282. [7] D.R. Butler, C.A. Nelson, Phys. Rev. D 18 (1978) 1196; C.A. Nelson, Nucl. Phys. B 181 (1981) 141; C.A. Nelson, Nucl. Phys. B 186 (1981) 187. [8] M. Ciafaloni, Phys. Lett. B 150 (1985) 379; S. Catani, M. Ciafaloni, G. Marchesini, Phys. Lett. B 168 (1986) 284; S. Catani, M. Ciafaloni, G. Marchesini, Nucl. Phys. B 264 (1986) 588. [9] S. Catani, M. Ciafaloni, Nucl. Phys. B 249 (1985) 301. [10] J. Frenkel, J.G. Gatheral, J.C. Taylor, Nucl. Phys. B 194 (1982) 172. [11] G. Giavarini, G. Marchesini, Nucl. Phys. B 296 (1988) 546. [12] F.N. Havemann, PHE-85-14. [13] V. Del Duca, L. Magnea, G. Sterman, Nucl. Phys. B 324 (1989) 391. [14] H.F. Contopanagos, M.B. Einhorn, Phys. Rev. D 45 (1992) 1291. [15] L.V. Prokhorov, Phys. Usp. 42 (1999) 1099, Usp. Fiz. Nauk 169 (1999) 1199 (in Russian). [16] M. Lavelle, D. McMullan, Phys. Rep. 279 (1997) 1, hep-ph/9509344. [17] E. Bagan, M. Lavelle, D. McMullan, Ann. Phys. 282 (2000) 471, hep-ph/9909257. [18] R. Horan, M. Lavelle, D. McMullan, J. Math. Phys. 41 (2000) 4437, hep-th/9909044. [19] W.T. Giele, E.W.N. Glover, Phys. Rev. D 46 (1992) 1980; S. Frixione, Z. Kunszt, A. Signer, Nucl. Phys. B 467 (1996) 399, hep-ph/9512328; S. Catani, M.H. Seymour, Nucl. Phys. B 485 (1997) 291, hep-ph/9605323; S. Catani, M.H. Seymour, Nucl. Phys. B 510 (1997) 503, Erratum. [20] Z. Nagy, D.E. Soper, JHEP 0309 (2003) 055, hep-ph/0308127. [21] M.L. Goldberger, K.M. Watson, Collision Theory, Wiley, New York, 1964. Nuclear Physics B 684 (2004) 162–182 www.elsevier.com/locate/npe

Ginsparg–Wilson relation and ’t Hooft–Polyakov monopole on fuzzy 2-sphere

Hajime Aoki a,b, Satoshi Iso c, Keiichi Nagao c

a Department of Physics, Stanford University, Stanford, CA 94305-4060, USA b Department of Physics, Saga University, Saga 840-8502, Japan c Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan Received 29 December 2003; accepted 6 February 2004

Abstract We investigate several properties of Ginsparg–Wilson fermion on fuzzy 2-sphere. We first examine chiral anomaly up to the second order of the gauge field and show that it is indeed reduced to the correct form of the Chern character in the commutative limit. Next we study topologically nontrivial gauge configurations and their topological charges. We investigate ’t Hooft–Polyakov monopole type configuration on fuzzy 2-sphere and show that it has the correct commutative limit. We also consider more general configurations in our formulation.  2004 Elsevier B.V. All rights reserved.

1. Introduction

Various matrix models have been proposed toward nonperturbative formulations of the superstring theory. In matrix models like the type IIB model [1], space–time itself is described by matrices and thus noncommutative (NC) geometries [2] naturally appear [3,4]. Small fluctuations around the classical background give matter degrees of freedom and hence space–time and matter are unified in the same matrices. However it is not clear how space–time and matter are embedded in matrices, for example, how metric, topology, etc. are described in matrices. A construction of configurations with nontrivial indices in finite NC geometries has been an important subject not only from the mathematical interest

E-mail addresses: [email protected], [email protected] (H. Aoki), [email protected] (S. Iso), [email protected] (K. Nagao).

Table 1 The properties of three types of Dirac operators on fuzzy 2-sphere are summarized. Each Dirac operator represents Watamuras’ operator DWW [9], Grosse et al.’s operator DGKP [10] and Ginsparg–Wilson Dirac operator DGW Dirac op. Chiral symmetry No doublers Counterpart in LGT

DWW DWWΓ + ΓDWW = 0 ×naive fermion + = O 1 × DGKP DGKPΓ ΓDGKP ( L ) Wilson fermion ˆ DGW DGWΓ + ΓDGW = 0 GW fermion

but also from the physical point of view such as the Kaluza–Klein compactification of extra dimensions with nontrivial indices to realize four-dimensional chiral gauge theories. Topologically nontrivial configurations in finite NC geometries have been constructed based on algebraic K-theory and projective modules in Refs. [5–8] but the relations to local forms of chiral anomaly or indices of Dirac operators are not very clear because Dirac operators on the fuzzy sphere considered so far [9–11] are not suitable to discuss these problems in this kind of system with finite degrees of freedom. This is summarized later in this section. The most suitable framework will be to utilize the Ginsparg–Wilson (GW) relation [12] developed in lattice gauge theory (LGT), because the GW relation enables us to have the exact index theorem [13,14] at a finite lattice spacing using the GW Dirac operator [15] and the modified chiral symmetry [14,16]. The formulation of NC geometries in Connes’ prescription is based on the spectral triple (A, H, D), where a chirality operator and a Dirac operator which anticommute are introduced [2]. In Ref. [17], we generalized the algebraic relation to the GW relation in general gauge field backgrounds and provided prescriptions to construct chirality operators and Dirac operators which satisfy the GW relation, so that we can define chiral structures in finite NC geometries. As a concrete example we considered fuzzy 2-sphere [18] and constructed a set of chirality and Dirac operators. We showed that the Chern character is obtained by evaluating chiral anomaly up to the first order of the gauge field [17]. The evaluation of the chiral anomaly is also considered in Refs. [19,20]. Fuzzy 2-sphere is one of the simplest compact NC geometry and it can be regarded as the classical solution of the matrix model with a Chern–Simons term [11]. Since it has finite degrees of freedom and UV/IR cutoffs, its stability can be studied analytically [21] or numerically [22]. We here summarize various Dirac operators on fuzzy 2-sphere. There had been known two types of Dirac operators, DWW [9] and DGKP [10,11]. Doubling problems of these operators are studied [23,24] and based on this, these authors introduced two sets of chirality operators and a free Dirac operator satisfying GW relation. In [17], we generalized these free operators to an interacting case in general gauge field background configurations. Incorporation of gauge fields is essential in the GW formalism. Denoting this third Dirac operator as DGW, various properties of these three types of the Dirac operators are summarized in Table 1. DWW has no chiral anomaly because of the doublers. Both of DGKP and DGW have no doublers and the chiral currents satisfy Ward identities with chiral anomaly, whose local forms were evaluated explicitly in Refs. [17,25], respectively. The global form of the chiral anomaly of DGKP was studied in Refs. [6,26,27]. While in lattice gauge theories, the chiral symmetry and the no-doubler condition have been shown to be 164 H. Aoki et al. / Nuclear Physics B 684 (2004) 162–182 incompatible with some reasonable assumptions [28], Table 1 implies the existence of analogous no-go theorem in finite NC geometries or matrix models. In the present paper we proceed to study the chiral and topological properties of the GW fermion on fuzzy 2-sphere. After an introduction of fuzzy 2-sphere in Subsection 2.1, we provide a set of GW Dirac operator and chirality operators on the fuzzy 2-sphere in Subsection 2.2. We also define a topological charge and provide an index theorem on fuzzy 2-sphere. We prove the index theorem in a more general formulation in Appendix A which can be applied to any NC geometries. The topological charge we defined takes only integer values by definition, and we also show in Appendix B that its value does not change under any small fluctuation of any parameters or any fields in the theory. We further need to show that it has the correct commutative limit, and also that it takes nonzero integer values for topologically-nontrivial configurations. The purpose of this paper is to study these two points. In Subsection 2.3, we evaluate chiral anomaly by calculating the nontrivial Jacobian under the local chiral transformation. In NC geometries, there is an ambiguity to define local transformations since the transformation parameter do not commute with the fields and the operators in the theory. We here consider two chiral transformations and corresponding chiral currents, which transform covariantly and invariantly under the gauge transformation, respectively. We show that the correct form of the Chern character is obtained for the covariant current case by calculating the Jacobian up to the second order in gauge fields, and taking commutative limit. This confirms the previous result up to first order in the gauge field in [17]. Since the topological charge we defined on the fuzzy 2- sphere has the same form as the anomaly term, it has the correct commutative limit. In Section 3 we study topologically nontrivial gauge configurations and their topological charges. Even in the theories on the commutative sphere, after integrating the topological charge density over the sphere, the topological charge vanishes identically for any configurations. For the Dirac monopole, we need to introduce the notion of patches in the sphere and obtain nonzero values for the topological charge. For the ’t Hooft–Polyakov (TP) monopole, we need to introduce the idea of spontaneous symmetry breaking, insert projector to pick up unbroken U(1) component into the topological charge, and then obtain nonzero topological charge. These ideas correspond to introducing some projections into matrices in the NC theories since both space–time and gauge group space are embedded in matrices. In Subsection 3.1 we construct a TP monopole configuration on the fuzzy sphere and show that it becomes the correct form in the commutative limit. The normal component of the gauge field plays the role of the Higgs field. We also review the topological charge for the TP monopole in the commutative theory. In Subsection 3.2 we define a topological charge for the TP monopole on the fuzzy 2-sphere by introducing a projector. In the commutative limit this projector becomes the one to pick up the unbroken U(1) component, and thus this topological charge has the correct commutative limit. In Subsection 3.3 we investigate TP monopole configurations with higher isospin. Since it is important that these configurations satisfy SU(2) algebra, we study its meaning. We also define another topological charge with the Higgs field inserted, which has the correct commutative limit with the projector to pick up unbroken U(1) component. We H. Aoki et al. / Nuclear Physics B 684 (2004) 162–182 165 then see that the winding number of π2(SU(2)/U(1)) for these configurations is 1. In Subsection 3.4 we consider more general configurations. Similar study was done from mathematical points of view in [5–8,20]. In this paper we give physical interpretation by studying its commutative limit. Also, we believe that our derivation is simpler and more transparent. Section 4 is devoted to conclusions and discussions.

2. Ginsparg–Wilson fermion on fuzzy 2-sphere

In the paper [17] we proposed a general prescription to construct chirality operators and Dirac operators satisfying the GW relation on general finite NC geometries and gave a simple example on the fuzzy 2-sphere. In this section we first review this construction and calculate the local form of anomaly up to the second order of the gauge field.

2.1. Brief review of fuzzy 2-sphere and DGKP

We ﬁrst brieﬂy explain fuzzy 2-sphere and the Dirac operator DGKP, which will be used when we construct the GW Dirac operator DGW in the next subsection. NC coordinates of the fuzzy 2-sphere are given by

xi = αLi , (2.1) where α is a NC parameter, and Li ’s are (2L + 1)-dimensional irreducible representation matrices of SU(2) algebra. Thus

[xi,xj ]=iαij k xk, (2.2) 2 2 (xi) = α L(L + 1), (2.3) and ρ = α L(L + 1) (2.4) gives the radius of the fuzzy sphere. Wave functions on fuzzy 2-sphere are composed of (2L + 1) × (2L + 1) matrices, ˆ and can be expanded in terms of NC analogues of the spherical harmonics Ylm.They are traceless symmetric products of the NC coordinates. There is an upper bound for the ˆ angular momentum l in Yl,m: l 2L. Derivatives along the Killing vectors on the sphere are given by the adjoint action of Li as L =[ ]= L − R i M Li,M Li Li M, (2.5) where M is any wavefunction and the superscript L (R)inLi means that this operator acts from the left (right) on matrices. An integral over the 2-sphere is expressed by a trace over matrices: 1 dΩ tr ↔ . (2.6) 2L + 1 4π 166 H. Aoki et al. / Nuclear Physics B 684 (2004) 162–182

The fermionic action with the Dirac operator DGKP is given by [10,11] ¯ SGKP = tr(ψDGKPψ), (2.7)

DGKP = σi (Li + ρai) + 1. (2.8)

Here i runs from 1 to 3, σi ’s are Pauli matrices, and the fermion ﬁeld ψ and the gauge ﬁeld ai are expressed as (2L + 1) × (2L + 1) Hermitian matrices. The free part of this theory does not have doubler modes [23,25]. SGKP is invariant under the following gauge transformation: ψ → Uψ, ψ¯ → ψU¯ †, (2.9) 1 a → Ua U † + UL U † − L , (2.10) i i ρ i i since a combination

Ai ≡ Li + ρai (2.11) transforms covariantly under the gauge transformation as † Ai → UAiU , (2.12) and the fermion ψ transforms as the fundamental representation. The normal component of ai to the sphere can be interpreted as the scalar field on the sphere. We define it covariantly as A2 − L(L + 1) 1 φ φ = i = . (2.13) 2L + 1 ρ ρ Here we also defined a normalized scalar field φ for later convenience. The commutative limit can be taken by α → 0,L →∞with ρ fixed. Then, DGKP becomes a Dirac operator on the commutative 2-sphere, ˜ Dcom = σi(Li + ρai) + 1, (2.14) where ˜ Li =−iij k xj ∂k. (2.15)

The scalar ﬁeld (2.13) becomes the one on the commutative 2-sphere, aini ,whereni = xi/ρ. More detailed explanations are given in Refs. [11,17,25].

2.2. Ginsparg–Wilson fermion on fuzzy 2-sphere

In this subsection we review the construction of the GW Dirac operator [17]. We ﬁrst introduce two Hermitian chirality operators Γ R and Γˆ satisfying Γˆ 2 = (Γ R)2 = 1 as follows: σ LR − 1 R = R − 1 = i i 2 Γ a σi Li , (2.16) 2 R − 1 2 (σiLi 2 ) H. Aoki et al. / Nuclear Physics B 684 (2004) 162–182 167

H Γˆ = √ , (2.17) H 2 where 1 a = (2.18) + 1 L 2 R is introduced as a NC analogue of a lattice-spacing. The superscript R of Li means that this operator acts from the right on matrices. We define the Hermitian operator H as 1 H = a σ A + (2.19) i i 2 R = Γ + aDGKP (2.20) = L + L Γ aρσiai , (2.21) where σ LL + 1 L = L + 1 = i i 2 Γ a σiLi , (2.22) 2 L + 1 2 (σi Li 2 ) is the Hermitian chirality operator introduced in [24] and satisfies (Γ L)2 = 1. In L L Eqs. (2.21), (2.22) the superscript L in ai and Li means that this operator acts from the left on matrices. We thus see that Γˆ in Eq. (2.17) becomes Γ L for vanishing gauge fields. We also note R L that both of the two chirality operators Γ and Γ become the chirality operator γ = σi ni in the commutative limit. This was discussed in [24] for the free case without gauge field backgrounds. We next define the GW Dirac operator as −1 R R ˆ DGW =−a Γ 1 − Γ Γ . (2.23) The fermionic action ¯ SGW = tr(ΨDGWΨ) (2.24) R is invariant under the gauge transformation (2.9), (2.12) because DGW is composed of Γ and Ai . The free part of the theory has no doublers. In the commutative limit DGW becomes the commutative Dirac operator without coupling to the scalar field φ, R R DGW DGKP − Γ ,DGKP Γ /2 + O(1/L) ˜ → σi(Li + ρPij aj ) + 1, (2.25) where Pij is a projection operator on the sphere,

Pij = δij − ni nj . (2.26) 2 This satisfies (P )ij = Pij and ni Pij = 0. We can see from the definition (2.23) that DGW satisfies the GW relation: R ˆ Γ DGW + DGWΓ = 0. (2.27) 168 H. Aoki et al. / Nuclear Physics B 684 (2004) 162–182

Then, as we show in Appendix A, we can prove the following index theorem: 1 R indexD ≡ (n+ − n−) = T r Γ + Γˆ , (2.28) GW 2 where n± are the numbers of zero eigenstates of DGW with a positive (or negative) chirality (for either Γ R or Γˆ )andT r is a trace of operators acting on matrices. We also prove in Appendix B that T r(Γ)ˆ is invariant under a small deformation of any parameter or any 1 T R + ˆ configuration such as gauge field in the operator H .Furthermore, 2 r(Γ Γ) takes only integer values since both Γ R and Γˆ have a form of sign operator by the definitions (2.16), (2.17). Therefore, we may call this a topological charge. In the next subsection we will investigate the commutative limit of this topological charge, and show that it becomes the Chern character in the commutative limit. In Section 3 we will investigate topologically nontrivial configurations and their topological charges.

2.3. Chiral anomaly

In this subsection we calculate the local form of the chiral anomaly. The fermionic action (2.24) is invariant under the global chiral transformation,

δΨ = iΓΨ,ˆ δΨ¯ = iΨΓ¯ R, (2.29) due to the GW relation (2.27). For a local transformation, however, we need to specify the ordering of the chiral transformation parameter λ, the fermion field, and the chirality operator, since they are not commutable. This ambiguity is specific to the NC field theories and makes the analysis of the Ward–Takahashi (WT) identity complicated [17,25,29]. Here we consider two types of local chiral transformations. The first type of chiral transformation is defined as

δΨ = iλΓΨ,ˆ δΨ¯ = iΨλΓ¯ R, (2.30) where the chiral transformation parameter λ should transform covariantly as λ → UλU† under the gauge transformation (2.9), (2.10). The associated chiral current transforms covariantly. Another chiral transformation is deﬁned as

δΨ = iΓΨλ,ˆ δΨ¯ = iλΨΓ¯ R, (2.31) where the chiral transformation parameter λ is assumed to be invariant under gauge transformations, so is the associated chiral current. In the WT identity for the gauge-covariant current, the variation of the action (2.24) under (2.30) gives the current-divergence term, and the variation of the integration measure gives the anomaly term, L ˆ L R 2qcov(λ) ≡ T r λ Γ + λ Γ = tr(1) Tr(λΓ)ˆ + tr(λ) Tr Γ R 2 = Tr(λΓ)ˆ − 2tr(λ), (2.32) a H. Aoki et al. / Nuclear Physics B 684 (2004) 162–182 169 where in the ﬁrst line, T r is a trace of operators acting on matrices, and the superscript L (R) means that this operator acts from the left (right) on matrices. In the second and third lines, Tr is a trace over matrices and spinors, tr is a trace over matrices, and Γ R and Γˆ are considered as mere matrices instead of operators acting on matrices. Similarly, in the WT identity for the gauge-invariant current, the variation of the measure under (2.31) gives the anomaly term, 2q (λ) ≡ T r λRΓˆ + λRΓ R inv = tr(λ) Tr(Γ)ˆ + tr(1) Tr λΓ R = tr(λ) Tr(Γ)ˆ − 2tr(λ). (2.33)

For a global chiral transformation, that is, λ = 1 case, both qcov(λ) and qinv(λ) become the topological charge defined in Eq. (2.28). When background gauge fields vanish, Γˆ = Γ L, and qcov(λ) and qinv(λ) vanish. For the covariant case (2.32), the chiral transformation parameter λ and the gauge field ˆ ai in Γ are inserted in the same trace, while for the invariant case (2.33), λ and ai are inserted in different traces. Since traces are replaced by the integrations in the commutative limit, this indicates that local WT identity can be written down for the covariant current, while some nonlocality must be introduced in the WT identity for the invariant current. This is consistent with the previous results [25,29]. Gauge invariant operators can be defined only by taking traces over matrices, and thus this introduces nonlocality in NC spaces. We now consider weak gauge field configurations, and see if the topological charge density qcov(λ) reduces to the Chern character in the commutative limit. Expanding the topological charge density (2.32) with respect to the gauge field ai up to the second order, and taking a trace over σ matrices, we obtain a2ρ2 = [ ] qcov(λ) i tr λ Li,ai α 2 3 4 2 2 ρ 2 ρ + tr λ a ρ [Li,ai] − 4 (a ) + 4i Li [Lj ,aj ],a 8 α i α i 2 ρ ρ − 8i L a a − a2ρ2i [a ,a ] + O a 3 , (2.34) α ij k i j k α i i i where α a = ij k (Lj ak + akLj ) (2.35) i 2ρ is the tangential component of the gauge field ai . The gauge field ai are decomposed into the tangential component ai and the normal component, that is, the scalar field φ. In the commutative limit, the second line of (2.34) vanishes, and we obtain 2 dΩ xi q (λ) → ρ tr λij k Fjk , (2.36) cov 4π ρ where the trace is taken over the non-Abelian gauge group, Fjk is the field strength defined = − − [ ] = as Fjk ∂j ak ∂kaj i aj ,ak ,andai ij k xj ak/ρ.FortheU(1) gauge theory, the trace 170 H. Aoki et al. / Nuclear Physics B 684 (2004) 162–182 in (2.36) is not necessary and the field strength is defined as such. All contributions from higher order terms in the gauge field ai vanish in the commutative limit. It is now confirmed that the Chern character is reproduced in all orders in ai , which was previously checked up to the first order of ai in [17]. This topological charge density is nothing but the magnetic flux density penetrating the 2-sphere, and we obtain the correct form of anomalous WT identity in the commutative limit. Since the topological charge we defined in (2.28) has the form of qcov(λ) with λ = 1 in (2.32), our topological charge (2.28) becomes the Chern character in the commutative limit, at least locally. However, for λ = 1, even in the commutative theories, (2.36) vanishes identically for any configurations after the integration over the sphere, if the gauge field ai is a single- valued function on the sphere for Abelian gauge theory, or if we naively take the trace for non-Abelian gauge theory. In order to describe topologically-nontrivial configurations and classify them by some topological charge, we need to introduce such ideas as patches in the Dirac monopole, or spontaneous symmetry breakings in the TP monopole. In NC theories, these ideas correspond to introducing some kind of projections into matrices, since both space–time and gauge group space are embedded in matrices in NC theory. We will study this subject in the next section.

3. ’t Hooft–Polyakov monopole on fuzzy 2-sphere

In this section we construct topologically nontrivial configurations which correspond to the ’t Hooft–Polyakov (TP) monopole in the commutative theory. We then define a topological charge for nontrivial configurations by introducing projection operators. We show that the topological charge has the correct commutative limit. Similar study was done by using projective modules [5–8,20]. In the following we will give physical interpretation of the previous mathematical settings by studying its commutative limit.

3.1. TP monopole conﬁguration on fuzzy 2-sphere

Let us consider the following conﬁguration in the SU(2) gauge theory on fuzzy 2- sphere:

1 τi ai = 1 L+ ⊗ , (3.1) ρ 2 1 2 where the ﬁrst and second factors represent the NC space and the gauge group space, respectively. Then, the combination (2.11) becomes

L τi Ai = L ⊗ 1 + 1 L+ ⊗ . (3.2) i 2 2 1 2 We note here that Ai ’s satisfy the SU(2) algebra:

[Ai,Aj ]=iij k Ak. (3.3) We will use this property for constructing a nontrivial topological charge on the fuzzy sphere in the next subsection. Here we will first show that the configuration (3.1) H. Aoki et al. / Nuclear Physics B 684 (2004) 162–182 171 corresponds to the section of the TP monopole configuration on S2 in the commutative theory. We will also see that the scalar field φ, the normal component of the gauge field ai , has a role of the Higgs field in the SU(2) adjoint representation. In the commutative limit, (3.1) becomes

1 τi ai(x) = , (3.4) ρ 2 or 1 aa(x) = δ , (3.5) i ρ ia a a if decomposed by ai = a τ /2. We further decompose it into the tangential component on i 2-sphere, ai and normal component φ by a = n a , i ij k j k (3.6) φ = ni ai, =− + ai ij k nj ak ni φ, (3.7) where ni = xi/ρ. Then we obtain

a 1 a = ij a xj , (3.8) i ρ2 1 φa = n , (3.9) ρ a for the configuration (3.5). These are nothing but the TP monopole configurations. This configuration satisfies a =− abc b c ai φ ∂iφ , (3.10) where a a φ φ = = na (3.11) (φa)2 a 2 = is the normalized scalar field which satisfy a(φ ) 1. For (3.10), the covariant derivative of φ and the field strength become a ≡ a + abc b c (Di φ ) ∂iφ ai φ (3.12) = 0, (3.13)

a a a abc b c F ≡ ∂i a − ∂j a + a a (3.14) ij j i i j abc b c bcd a b c d =−2 ∂i φ ∂j φ + φ φ ∂i φ ∂j φ . (3.15) If we extend this configuration to 3-dimensional space, by regarding ρ as the radial coordinate, we obtain the following asymptotic behavior:  2 →  (φa) 1, → →∞  Di φ 0, for ρ , (3.16) 2 Fij → O(1/ρ ), 172 H. Aoki et al. / Nuclear Physics B 684 (2004) 162–182 which assures the finiteness of the energy defined in 3-dimensions. We next consider the winding number of π2(SU(2)/U(1)). The magnetic flux of unbroken U(1) component penetrating the 2-sphere is written as ρ2 Q = dΩ tr(Pτ ij k ni Fjk) (3.17) 4π 2 S 2 ρ a a = dΩ ij k ni φ F , (3.18) 8π jk S2

1+τi ni where Pτ = is the projector to pick up the unbroken U(1) component. We note here 2 that this is the Chern character (2.36) with the projection operator Pτ or φ inserted. In the configuration (3.10), using (3.15), we obtain 2 ρ abc a b c Q =− dΩ ij k ni φ ∂j φ ∂kφ . (3.19) 8π S2 − a 2 → 2 2 = Thus Q is the degree of mapping of φ (xi) : Sx Sφ , π2(S ) Z. Inserting (3.11), this gives 1 Q =− dΩ =−1. (3.20) 4π S2 Thus the configuration (3.1) corresponds to the TP monopole configuration in the commutative theory.

3.2. Topological charge of TP monopole on fuzzy 2-sphere

In this subsection we study the topological charge for the conﬁguration (3.1) by introducing a projection operator into the topological charge in (2.28). We see that this topological charge corresponds to that in commutative theory which was studied at the end of the previous subsection. Since Ai ’s in (3.2) satisfy SU(2) algebra, we can decompose Ai into irreducible representations using some unitary matrix U as L(1) = i † Ai U (2) U , (3.21) Li (1) (2) (1) = + 1 (2) = − 1 where Li and Li are L L 2 and L L 2 representations, respectively. (1) (2) H(1) H(2) We denote the Hilbert spaces on which the operator Li and Li act as and , respectively. Each Hilbert space can be picked up by the following projection operators: (A )2 − L(2)(L(2) + 1) P (1) = i (3.22) L(1)(L(1) + 1) − L(2)(L(2) + 1) 1 + Γ = τ (3.23) 2 H. Aoki et al. / Nuclear Physics B 684 (2004) 162–182 173

L + 1 = ρφ + 4 , (3.24) 2L + 1 L(1)(L(1) + 1) − (A )2 P (2) = i (3.25) L(1)(L(1) + 1) − L(2)(L(2) + 1) 1 − Γ = τ (3.26) 2 L + 3 =−ρφ + 4 , (3.27) 2L + 1 where the Hermitian operator 1 Li τi + Γ = 2 (3.28) τ + 1 L 2 2 satisfies (Γτ ) = 1 and becomes τini in the commutative limit. The scalar field φ in (3.24), (3.27) is defined in (2.13). We define the chirality operator Γˆ as in (2.17). This has the following block diagonal form when Ai are decomposed into irreducible representations: Γˆ (1) Γˆ = U U †, (3.29) Γˆ (2) where (a) 1 σi L + Γˆ (a) = i 2 , (3.30) (a) + 1 L 2 for a = 1, 2. We now consider the topological charge (2.28) with the projector (3.22)–(3.27) 1 1 T r P (a) Γ R + Γˆ = T r Γ R + Γˆ (a) , (3.31) 2 2 (a) T − R where the trace r is taken over the whole Hilbert space on which Ai , Li , σi /2 operate. (a) The trace T r(a) is taken over the restricted Hilbert space H . We note that, since Ai’s satisfy the SU(2) algebra, the projector P (a) commutes with Γ R and Γˆ . We can see that the topological charge (3.31) is invariant under any small perturbations and take only integer values since this has the form of a sign operator as we mentioned below (2.28). 1 If we take the weak gauge limit and the commutative limit of (3.31) as we did in (2.36), we obtain 1 (a) R ˆ 2 dΩ (a) T r P Γ + Γ → ρ tr P ij k niFjk . (3.32) 2 4π By using (3.23), (3.24) for P (1), this becomes (3.17), (3.18), while by using (3.27) for P (2), this becomes a product of −1 and (3.18). We thus see that the topological charge (3.31) becomes the desired form in the commutative limit.

1 We can also deﬁne a GW Dirac operator as (2.23) multiplied by the projection operator P (a),andshow that the index theorem is satisﬁed between the index for this Dirac operator and the topological charge (3.31). However, the meaning of this Dirac operator is not clear. 174 H. Aoki et al. / Nuclear Physics B 684 (2004) 162–182

We will now evaluate the toplogical charge (3.31). It was done in Ref. [20], but we will 1 T R give simpler evaluation below. For evaluating the ﬁrst term, 2 r(a) Γ , we introduce an = σi − R operator Ji 2 Li . The Casimir operator of Ji is given by

2 3 R (Ji ) = J(J + 1) = L(L + 1) + − σi L . (3.33) 4 i Denoting the degeneracy as m, we obtain

• = + 1 R =− = + (a) + for J L 2 , Γ 1, m (2L 2)(2L 1); • = − 1 R = = (a) + for J L 2 , Γ 1, m 2L(2L 1), and thus 1 T r Γ R =− L(a) + 1 . (3.34) 2 (a) 1 T ˆ (a) Similarly, for evaluating the second term, 2 r(a) Γ , we introduce another operator = σi + (a) Ji 2 Li . The Casimir operator of Ji is calculated as 2 (a) (a) 3 (a) (J ) = J (J + 1) = L L + 1 + + σi L . (3.35) i 4 i Then we obtain

• = (a) + 1 ˆ (a) = = (a) + + for J L 2 , Γ 1, m (2L 2)(2L 1); • = (a) − 1 ˆ (a) =− = (a) + for J L 2 , Γ 1, m 2L (2L 1), and 1 T r Γˆ (a) = 2L + 1. (3.36) 2 (a) From Eqs. (3.34), (3.36), we obtain 1 1 T r P (a) Γ R + Γˆ = T r Γ R + Γˆ (a) = 2 L − L(a) . (3.37) 2 2 (a) For a = 1, 2, L(1) = L + 1/2,L(2) = L − 1/2, and 1 1 T r P (1) Γ R + Γˆ = T r ρφ Γ R + Γˆ =−1, (3.38) 2 2 1 1 T r P (2) Γ R + Γˆ =− T r ρφ Γ R + Γˆ = 1. (3.39) 2 2 This result agrees with the monopole charge Q in the commutative theory which was calculated in (3.20) in the previous subsection. We have thus shown that the non- trivial configuration (3.1) can be interpreted as the TP monopole configuration, and the topological charge (3.31) for this configuration gives the correct value. H. Aoki et al. / Nuclear Physics B 684 (2004) 162–182 175

3.3. Higher isospin

In this subsection we consider a configuration coupled to a fermion with higher isospin T [5,6,20]: 1 a = 1 + ⊗ T , (3.40) i ρ 2L 1 i where Ti ’s are the (2T + 1)-dimensional representation of SU(2) algebra. Since the combination = L ⊗ + ⊗ Ai Li 12T +1 12L+1 Ti (3.41) satisfies SU(2) algebra, it can be decomposed into the irreducible representations of SU(2) as  (1)  Li  (2)   L   i  Ai  .  , (3.42) .. (2T +1) Li (a) = + (a) = + + − where Li ’s (a 1,...,2T 1) denote the L L T 1 a representations, respectively. (a) Projectors to pick up each Hilbert space on which Li acts can be defined as in (3.22), (3.25), (A )2 − L(b)(L(b) + 1) P (a) = i . (3.43) L(a)(L(a) + 1) − L(b)(L(b) + 1) b =a Then we define the same topological charge as (3.31), and obtain the same result as (3.37). Both of the configurations (3.2), (3.41) satisfy the SU(2) algebra (3.3). Since it is essential in the calculations of the topological charge (3.31), we here study its meaning. By inserting (2.11) into (3.3), taking the commutative limit, using (2.15), and decomposing ai into ai and φ by (3.7), we obtain 1 n n F + (n − n )n (D φ) + n φ = 0. (3.44) ilm jpq l p mq i jlm j ilm l m ρ ij k k Since (3.44) is a second rank antisymmetric tensor with indices i, j, we can decompose it into normal–tangential and tangential–tangential components on the sphere, and obtain

ij k nj (Dkφ) = 0, (3.45) 2 n F + φ = 0. (3.46) ij k i jk ρ

Eq. (3.45) means that the tangential component of Di φ vanishes, while (3.46) means that the normal component of the flux is equal to the scalar field. We see that the finite energy = 2 = condition (3.16) is satisfied by (3.45), (3.46) since φ φ /ρ and (φa) 1. Also, by (3.46) 176 H. Aoki et al. / Nuclear Physics B 684 (2004) 162–182 the topological charge (3.32) becomes dΩ −2 tr P (a)φ . (3.47) 4π For T = 1/2 case, by using (3.24), (3.27) for P (a), we obtain the same result as (3.38), (3.39). For T 1, P (a) is given by (3.43). By using (2.13), taking commutative limit of (3.43), and inserting it into (3.47), we can obtain the same result as (3.37). We now consider the commutative limit of the configuration (3.40). We regard it as a configuration in the SU(2) gauge theory coupled with the fermion in (2T +1)-dimensional representation, although we could also regard it as a configuration in the SU(2T + 1) gauge theory coupled with the fermion in the fundamental representation. Then we obtain the same commutative limit as we did in Subsection 3.1. Eqs. (3.8), (3.9) imply that the configuration (3.40) has the winding number of 1. We now define another topological charge: 1 T r φ Γ R + Γˆ , (3.48) 2 where we insert φ of (2.13) instead of inserting the projection operator as in (3.31). On the fuzzy 2-sphere it is evaluated as

T + 21 L(a)(L(a) + 1) − L(L + 1) 2 2 L − L(a) =− T(T + 1)(2T + 1) (3.49) 2L + 1 3 a=1 for the configuration (3.40). On the other hand, by taking the weak gauge limit and the commutative limit, this becomes 1 R ˆ 2 dΩ T r φ Γ + Γ → ρ tr[φ ij k ni Fjk], (3.50) 2 4π and then, by using SU(2) condition (3.46), it is calculated as dΩ 2 −2 tr φ 2 =− T(T + 1)(2T + 1), (3.51) 4π 3 which agrees with (3.49). Also, for the configurations (3.10), (3.50) becomes 2 2 ρ abc a b c − T(T + 1)(2T + 1) dΩ ij k ni φ ∂j φ ∂kφ , (3.52) 3 8π S2 which is the winding number of π2(SU(2)/U(1)). Comparing with the above values, we see that the winding number is 1 for the configuration (3.40), and for configurations which satisfy the SU(2) algebra (3.3) in general. We thus obtain a configuration with winding number 1, by suitably combining the coordinates of fuzzy 2-sphere and the gauge configuration, both of which satisfy the SU(2) algebra. It will be an interesting attempt to construct configurations with general winding numbers by generalizing the procedure. H. Aoki et al. / Nuclear Physics B 684 (2004) 162–182 177

3.4. Other conﬁgurations

In this subsection we consider other configurations. First we consider the case where the fermion is in m-dimensional representation of the gauge group, for example, U(m) gauge theories coupled with the fermion in the fundamental representation. Then the Hilbert space on which Ai acts is m(2L + 1)- dimensional. We here assume that Ai satisfy the SU(2) algebra. Then Ai is decomposed into irreducible representations as  (1)  Li  (2)   L   i  Ai  .  , (3.53) .. (r) Li (a) (a) = where Li ’s are L representations of SU(2) for a 1,...,r. The projection operator (a) to pick up the Hilbert space on which Li acts is defined to be (3.43). Then we have the r (a) + = same topological charge as (3.31) and the same result as (3.37). Since a=1(2L 1) m(2L + 1),wehave r 1 1 T r(Γ + Γ)ˆ = T r P (a)(Γ + Γ)ˆ 2 2 a=1 r = 2 L − L(a) a=1 = (r − m)(2L + 1), (3.54) which means that the topological charge without the projection operator gives a multiple number of 2L + 1. Next we consider a NC analogue of U(1) Dirac monopoles. Let us consider the configuration  (1)  Li  (1)   c   i  Ai  . , (3.55) .. (s) ci (1) (1) (a) = where Li is L representation of SU(2) algebra, and ci ’s (a 1,...,s) are some numbers (1 × 1 matrices). We note here that the Ai does not satisfy the SU(2) algebra, but does satisfy the equation of motion in the reduced model of 3-dimensional Yang–Mills theory with the Chern–Simons term [11]: [Ai, [Ai,Aj ]] = −ijkl[Ak,Al]. The topological charge is calculated as 1 T r (Γ + Γ)ˆ = 2 L − L(1) = s. (3.56) 2 (1) We may be able to regard this solution as “D2 with s D0’s” [30], or “Dirac monopole with s Dirac strings” [31]. But in the latter interpretation there are several problems which should be resolved. 178 H. Aoki et al. / Nuclear Physics B 684 (2004) 162–182

4. Discussion

In this paper, after we briefly review the construction of GW Dirac operator and chirality operators on fuzzy 2-sphere [17], we further investigate several topological properties. First we calculated the chiral anomaly from the nontrivial Jacobian up to the second order in gauge fields and showed that the correct form of the Chern character is reproduced in the commutative limit. This result completed the first order calculation of the chiral anomaly in [17]. Thus we saw that the topological charge we defined on the fuzzy sphere has the correct commutative limit. We then studied topologically nontrivial configurations. Even in the theories on the commutative sphere, if we naively integrate the topological charge density over the sphere, the topological charge vanishes identically for any configurations. In the ’t Hooft–Polyakov (TP) monopole, we introduce the idea of spontaneous symmetry breaking, insert the projector to pick up the unbroken U(1) component into the topological charge, and obtain nonzero values for it. These idea and procedure correspond to introducing some projections into matrices in the noncommutative (NC) theories. We considered the TP monopole configurations and their topological charges on the fuzzy sphere, and showed that they have the correct commutative limit. We further studied other nontrivial configurations and their topological charges. Projections into matrices may be necessary not only for describing the topologically nontrivial configurations, but for describing the smooth gauge configurations√ themselves, and the smooth space–time itself. Since the denominator of Γˆ is given by H 2, we need to avoid zero eigenvalues of H , which corresponds to the admissibility conditions in the lattice gauge theories. Also, since space–time and field-theoretical degrees of freedom are considered to be embedded in the near-diagonal elements of the matrices, we need to project these elements from the full matrices to describe smooth field-configurations and space–time. Studies of embeddings of classical configurations in matrices [32,33] may be useful for this study. It is also interesting to construct GW fermions on various NC geometries, such as NC torus [34,35] and NC CPn, and consider the above mentioned problems in these models. Also, it may be interesting to study the Seiberg–Witten map [36] on fuzzy 2-sphere [37] to investigate nontrivial configurations and topological charges in commutative and NC theories. We expect that our formalism can provide a clue for classifying the space–time topology as well as the topology of gauge field space, since space–time and matter are indivisible in matrix models or NC field theories. We hope that our formalism based on the GW relation will have important roles for considering topological structures of space–time and matter in these theories.

Acknowledgements

We would like to thank U. Carow-Watamura and S. Watamura for useful discussions. The content of the paper was presented at the string and ﬁeld theory workshop at KEK H. Aoki et al. / Nuclear Physics B 684 (2004) 162–182 179 on March 18–20, 2003 by S. Iso, and at the conference Lattice 2003 at Tsukuba on July 15–19, 2003 by K. Nagao [38].

Appendix A. Proof of index theorem

In this appendix we prove the index theorem (2.28). We deﬁned the chirality operator and the GW Dirac operator more generally in [17], so here we prove the index theorem in this general formulation. It is much easier to prove it in the formulation of this paper. We deﬁned in [17] the GW Dirac operator DGW as ˆ f(a,Γ)DGW = 1 − Γ Γ, (A.1) where Γ and Γˆ are the generalization of Γ R and Γˆ of this paper, which are Hermite, satisfy Γ 2 = Γˆ 2 = 1, and become the commutative chirality operator in the commutative limit. The prefactor f(a,Γ) can be any function of a and Γ , but must be of the order of a, and must be invertible. Hermite conjugate of (A.1) gives † † = − ˆ DGWf(a,Γ) 1 ΓΓ. (A.2) By multiplying (A.1) by Γ from the left, multiplying another (A.1) by Γˆ from the right, and summing them, we obtain the following GW relation: ˆ ΓDGW + DGWΓ = 0, (A.3) † + ˆ † = DGWΓ ΓDGW 0. (A.4) Also, since (A.1) multiplied by Γ both from the left and the right becomes (A.2), † = †−1 DGW Γf (a, Γ )DGWΓf(a,Γ) , (A.5) = −1 † † DGW f(a,Γ) ΓDGWf(a,Γ) Γ. (A.6) Similarly, by multiplying (A.1) by Γˆ , we obtain † = ˆ ˆ †−1 DGW Γ f (a, Γ )DGWΓf(a,Γ) , (A.7) = −1 ˆ † † ˆ DGW f(a,Γ) ΓDGWf(a,Γ) Γ. (A.8) Now we introduce Fock space H for the spinor matrices ψ. In the case of 2-sphere, H is an ensemble of (2L + 1) × (2L + 1) Hermitian matrices. The above GW Dirac operator ˆ DGW and the chirality operators Γ , Γ act on this space. We then decompose H into the † spaces of zero and nonzero eigenmodes for DGW,andalsoforDGW: ¯ H = H0 ⊕ H0 (A.9) = H ⊕ H¯ 0 0, (A.10) where H ={ψ ∈ H | D ψ = 0}, (A.11) 0 GW H = ∈ H | † = 0 ψ DGWψ 0 , (A.12) 180 H. Aoki et al. / Nuclear Physics B 684 (2004) 162–182

H¯ H¯ and 0, 0 are their complementary space. First we show the following statement:

H = H H¯ = H¯ 0 0, 0 0, ∈ H = H ⇒ = ˆ ∈ H = H ψ 0 0 Γψ Γψ 0 0. (A.13) ˆ ˆ Its proof is as follows: if ψ ∈ H0, then due to (A.3), Γψ∈ H0. (A.1) leads to Γψ= Γψ. ∈ H † = ∈ H H ⊂ H Then Γψ 0. Thus, due to (A.5), DGWψ 0, and then ψ 0. Hence, 0 0. ∈ H ∈ H = Similarly, if ψ 0, then due to (A.4), Γψ 0. Thus, due to (A.6), DGWψ 0, and ∈ H H ⊂ H H = H H¯ = H¯ then ψ 0. Hence, 0 0. Therefore 0 0. 0 0 is its contraposition. Next we show

∈ H¯ = H¯ ⇒ ˆ ∈ H¯ = H¯ ψ 0 0 Γψ, Γψ 0 0. (A.14) ∈ H † = We prove its contraposition as follows: if Γψ 0, DGWΓψ 0. Then, due to (A.4), ˆ † = ˆ † = ∈ H ΓDGWψ 0. Multiplying it by Γ from the left we have DGWψ 0. Thus ψ 0. ˆ ˆ Similarly, if Γψ ∈ H0, DGWΓψ = 0. Then, due to (A.3), ΓDGWψ = 0. Multiplying it by Γ from the left we have DGWψ = 0. Thus ψ ∈ H0. Finally,

∈ H¯ = H¯ ⇒ ˆ ψ 0 0 Γψ, Γψ anti-pairing (A.15) =± ˆ † =− † =∓ † since if Γψ ψ, then due to (A.4), Γ(DGWψ) DGWΓψ (DGWψ). Similarly, ˆ ˆ if Γψ=±ψ, then due to (A.3), Γ(DGWψ) =−DGWΓψ=∓(DGWψ). From (A.13), (A.14), (A.15), we can prove the index theorem:

T r(Γ + Γ)ˆ = T rH (Γ + Γ)ˆ + T r ¯ (Γ + Γ)ˆ 0 H0 = T + ˆ rH0 (Γ Γ) = 2(n+ − n−)

= 2 index(DGW). (A.16)

Appendix B. Proof of δ(T r Γ)ˆ = 0

In this appendix we show that if we define Γˆ as in (2.17), T r(Γ)ˆ is invariant under any small deformation of any parameter or any configuration such as gauge field in the operator H . Under the infinitesimal deformation of H , H → H + δH, T r(Γ)ˆ varies as 1 δ(T r Γ)ˆ = δ T r H √ 2 H 1 1 1 = T r δH √ + T r H − T r H √ . H 2 (H + δH)2 H 2 H. Aoki et al. / Nuclear Physics B 684 (2004) 162–182 181

The second term can be evaluated as 1 T r H (H + δH)2 ∞ dt 1 = T r H π t2 + H 2 + HδH + (δH )H + (δH )2 −∞ ∞ dt 1 = T r H π t2 + H 2 −∞ ∞ dt 1 1 − T r H (H δH + δHH) + O (δH )2 π t2 + H 2 t2 + H 2 −∞ 1 1 = T r H √ − T r √ δH + O (δH )2 , (B.1) H 2 H 2 where we have utilized the cyclic property in T r and the following identities, ∞ 1 dt 1 √ = , (B.2) 2 π t2 + X2 X −∞ ∞ 1 dt 1 √ = . (B.3) 2 2 π (t2 + X2)2 2X X −∞ Therefore we obtain

δ(T r Γ)ˆ = 0. (B.4)

References

[1] N. Ishibashi, H. Kawai, Y. Kitazawa, A. Tsuchiya, Nucl. Phys. B 498 (1997) 467, hep-th/9612115; For a review, see H. Aoki, S. Iso, H. Kawai, Y. Kitazawa, A. Tsuchiya, T. Tada, Prog. Theor. Phys. Suppl. 134 (1999) 47, hep-th/9908038. [2] A. Connes, Noncommutative Geometry, Academic Press, 1990. [3] A. Connes, M.R. Douglas, A. Schwarz, JHEP 9802 (1998) 003, hep-th/9711162. [4] H. Aoki, N. Ishibashi, S. Iso, H. Kawai, Y. Kitazawa, T. Tada, Nucl. Phys. B 565 (2000) 176, hep-th/9908141. [5] H. Grosse, C. Klimcik, P. Presnajder, Commun. Math. Phys. 178 (1996) 507, hep-th/9510083; S. Baez, A.P. Balachandran, B. Ydri, S. Vaidya, Commun. Math. Phys. 208 (2000) 787, hep-th/9811169; G. Landi, J. Geom. Phys. 37 (2001) 47, math-ph/9905014. [6] A.P. Balachandran, S. Vaidya, Int. J. Mod. Phys. A 16 (2001) 17, hep-th/9910129. [7] P. Valtancoli, Mod. Phys. Lett. A 16 (2001) 639, hep-th/0101189. [8] H. Steinacker, hep-th/0307075. [9] U. Carow-Watamura, S. Watamura, Commun. Math. Phys. 183 (1997) 365, hep-th/9605003; U. Carow-Watamura, S. Watamura, Commun. Math. Phys. 212 (2000) 395, hep-th/9801195. 182 H. Aoki et al. / Nuclear Physics B 684 (2004) 162–182

[10] H. Grosse, J. Madore, Phys. Lett. B 283 (1992) 218; H. Grosse, P. Presnajder, Lett. Math. Phys. 33 (1995) 171; H. Grosse, C. Klimcik, P. Presnajder, Commun. Math. Phys. 185 (1997) 155, hep-th/9507074; H. Grosse, C. Klimcik, P. Presnajder, hep-th/9603071. [11] S. Iso, Y. Kimura, K. Tanaka, K. Wakatsuki, Nucl. Phys. B 604 (2001) 121, hep-th/0101102. [12] P.H. Ginsparg, K.G. Wilson, Phys. Rev. D 25 (1982) 2649. [13] P. Hasenfratz, Nucl. Phys. (Proc. Suppl.) 63 (1998) 53, hep-lat/9709110; P. Hasenfratz, V. Laliena, F. Niedermayer, Phys. Lett. B 427 (1998) 125, hep-lat/9801021. [14] M. Lüscher, Phys. Lett. B 428 (1998) 342, hep-lat/9802011. [15] H. Neuberger, Phys. Lett. B 417 (1998) 141, hep-lat/9707022; H. Neuberger, Phys. Rev. D 57 (1998) 5417, hep-lat/9710089; H. Neuberger, Phys. Lett. B 427 (1998) 353, hep-lat/9801031. [16] F. Niedermayer, Nucl. Phys. (Proc. Suppl.) 73 (1999) 105, hep-lat/9810026. [17] H. Aoki, S. Iso, K. Nagao, Phys. Rev. D 67 (2003) 085005, hep-th/0209223. [18] J. Madore, Class. Quantum Grav. 9 (1992) 69. [19] B. Ydri, JHEP 0308 (2003) 046, hep-th/0211209. [20] A.P. Balachandran, G. Immirzi, Phys. Rev. D 68 (2003) 065023, hep-th/0301242. [21] T.A. Imai, Y. Kitazawa, Y. Takayama, D. Tomino, Nucl. Phys. B 665 (2003) 520, hep-th/0303120; T. Imai, Y. Kitazawa, Y. Takayama, D. Tomino, hep-th/0307007. [22] T. Azuma, S. Bal, K. Nagao, J. Nishimura, in preparation. [23] A.P. Balachandran, T.R. Govindarajan, B. Ydri, Mod. Phys. Lett. A 15 (2000) 1279, hep-th/9911087. [24] A.P. Balachandran, T.R. Govindarajan, B. Ydri, hep-th/0006216. [25] H. Aoki, S. Iso, K. Nagao, Phys. Rev. D 67 (2003) 065018, hep-th/0209137. [26] H. Grosse, P. Presnajder, hep-th/9805085; H. Grosse, P. Presnajder, Lett. Math. Phys. 46 (1998) 61. [27] P. Presnajder, J. Math. Phys. 41 (2000) 2789, hep-th/9912050. [28] H.B. Nielsen, M. Ninomiya, Nucl. Phys. B 185 (1981) 20; H.B. Nielsen, M. Ninomiya, Nucl. Phys. B 195 (1982) 541, Erratum; H.B. Nielsen, M. Ninomiya, Nucl. Phys. B 193 (1981) 173; H.B. Nielsen, M. Ninomiya, Phys. Lett. B 105 (1981) 219. [29] F. Ardalan, N. Sadooghi, Int. J. Mod. Phys. A 16 (2001) 3151, hep-th/0002143; J.M. Gracia-Bondia, C.P. Martin, Phys. Lett. B 479 (2000) 321, hep-th/0002171; F. Ardalan, N. Sadooghi, Int. J. Mod. Phys. A 17 (2002) 123, hep-th/0009233; R. Banerjee, S. Ghosh, Phys. Lett. B 533 (2002) 162, hep-th/0110177; A. Armoni, E. Lopez, S. Theisen, JHEP 0206 (2002) 050, hep-th/0203165. [30] K. Hashimoto, K. Krasnov, Phys. Rev. D 64 (2001) 046007, hep-th/0101145. [31] D. Karabali, V.P. Nair, A.P. Polychronakos, Nucl. Phys. B 627 (2002) 565, hep-th/0111249. [32] J. Kiskis, R. Narayanan, H. Neuberger, Phys. Rev. D 66 (2002) 025019, hep-lat/0203005. [33] Y. Kikukawa, H. Suzuki, JHEP 0209 (2002) 032, hep-lat/0207009; T. Inagaki, Y. Kikukawa, H. Suzuki, JHEP 0305 (2003) 042, hep-lat/0305011; T. Inagaki, Y. Kikukawa, H. Suzuki, hep-lat/0309067. [34] J. Nishimura, M.A. Vazquez-Mozo, JHEP 0108 (2001) 033, hep-th/0107110; J. Nishimura, M.A. Vazquez-Mozo, JHEP 0301 (2003) 075, hep-lat/0210017. [35] S. Iso, K. Nagao, Prog. Theor. Phys. 109 (2003) 1017, hep-th/0212284. [36] N. Seiberg, E. Witten, JHEP 9909 (1999) 032, hep-th/9908142. [37] K. Hayasaka, R. Nakayama, Y. Takaya, Phys. Lett. B 553 (2003) 109, hep-th/0209240; J.M. Grimstrup, T. Jonsson, L. Thorlacius, hep-th/0310179. [38] K. Nagao, hep-th/0309153, Nucl. Phys. (Proc. Suppl.), in press. Nuclear Physics B 684 (2004) 183–208 www.elsevier.com/locate/npe

Phase structure of black holes and strings on cylinders

Troels Harmark, Niels A. Obers

The Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark Received 30 September 2003; received in revised form 28 January 2004; accepted 19 February 2004

Abstract We use the (M, n) phase diagram recently introduced in hep-th/0309116 to investigate the phase structure of black holes and strings on cylinders. We first prove that any static neutral black object on a cylinder can be put into an ansatz for the metric originally proposed in hep-th/0204047, generalizing a result of Wiseman. Using the ansatz, we then show that all branches of solutions obey the first law of thermodynamics and that any solution has an infinite number of copies. The consequences of these two results are analyzed. Based on the new insights and the known branches of solutions, we finally present an extensive discussion of the possible scenarios for the Gregory–Laflamme instability and the black hole/string transition.  2004 Elsevier B.V. All rights reserved.

1. Introduction

In [1] a new type of phase diagram for neutral and static black holes and black strings on cylinders Rd−1 ×S1 was introduced.1 In this paper we continue to examine various aspects of the phase structure of black holes and strings on cylinders using this phase diagram. We furthermore establish new results that deepen our understanding of the phase structure. The phase structure of neutral and static black holes and black strings on cylinders Rd−1 × S1 has proven to be very rich, but is also largely unknown. We can categorize the possible phase transitions into four types:

• Black string → black hole;

E-mail addresses: [email protected] (T. Harmark), [email protected] (N.A. Obers). 1 Recently, the paper [2] appeared which contains similar considerations.

• Black hole → black string; • Black string → black string; • Black hole → black hole.

Since the event horizons of the black string and black hole have topologies Sd−2 × S1 and Sd−1, respectively, we see that the first two types of phase transitions involve changing the topology of the horizon, while the last two do not. Gregory and Laflamme [3,4] discovered that light uniform black strings are classically unstable. In other words strings that are symmetric around the cylinder, are unstable to linear perturbations when the mass of the string is below a certain critical mass. This was interpreted to mean that a light uniform string decays to a black hole on a cylinder since that has higher entropy, thus being a transition of the first type above. However, Horowitz and Maeda argued in [5] that the topology of the event horizon cannot change through a classical evolution. This means that the first two types of transitions above cannot happen through classical evolution. Therefore, Horowitz and Maeda conjectured that the decay of a light uniform string has an intermediate step: the light uniform string decays to a non-uniform string, which then eventually decays to a black hole.2 Thus, we first have a transition of the third type, then afterwards a transition from non-uniform black string to a black hole which is of the first type. The conjecture of Horowitz and Maeda in [5] prompted a search for non-uniform black strings, and a branch of non-uniform string solutions was found in [7,8].3 However, this new branch of non-uniform string solutions was discovered to have lower entropy than uniform strings of the same mass, thus a given non-uniform string will decay to a uniform string (i.e., a process of the third type above). If we consider black holes on a cylinder, it seems likely that they cannot exist beyond a certain critical mass, since at some point the event horizon might become too large to fit on the cylinder. Therefore, there should be a transition from large black holes to black strings, i.e., a transition of the second type above. However, it is not known how this transition precisely works. Finally, as an example of a transition of the fourth type above, one can consider two black holes on a cylinder which are put opposite to each other. This configuration exists as a classical solution but it is unstable. Thus, the two black holes will decay to one black hole. Several suggestions for the phase structure of the Gregory–Laflamme instability, the black hole/string transitions and the uniform/non-uniform string transitions have been put forward [5,7,10–15] but it is unclear which of these, if any, are correct.4 The outline of this paper is as follows. In Section 2 we collect the main results of [1]. We start by reviewing the two parameters of the new phase diagram, the mass M and a new asymptotic quantity called the relative binding energy n. The plot of the known branches in the (M, n) phase diagram is then given. We also recall the new Smarr formula of [1] and some of its consequences for the thermodynamics, as well as the Uniqueness Hypothesis.

2 Note that the classical decay of the Gregory–Laﬂamme instability was studied numerically in [6]. 3 This branch of non-uniform string solutions was in fact already discovered by Gregory and Laﬂamme in [9]. 4 Other recent and related work includes [16–23]. T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208 185

In Section 3 we show that any static and neutral black hole or string on a cylinder can be put into the ansatz originally proposed in [10], generalizing the result of Wiseman [13] for d = 5. This is done in two steps by ﬁrst deriving the three-function conformal ansatz, and subsequently showing that one can transform to the two-function ansatz of [10]. The latter ansatz was shown to be completely determined by one function only, after explicitly solving the equations of motion. We then comment on the boundary conditions for this one function and their relation to the asymptotic quantities for black holes/strings on cylinders introduced in [1]. In Section 4 we use the ansatz of the previous section to prove that any curve of solutions always obeys the ﬁrst law of thermodynamics

δR δM = TδS+ nM T . (1.1) RT

Here the last term is due to the pressure in the periodic direction, and confirms the fact that n is a meaningful physical quantity. Moreover, for fixed value of the circle radius RT , this means that δM = TδS for all solutions in the (M, n) phase diagram. We analyze the consequences of this for the black hole/string phase transitions. In Section 5 we further develop an argument of Horowitz [11] that a given non-uniform black string solution can be “unwrapped” to give a new non-uniform black string solution. We write down explicit transformations for our ansatz that generate new solutions and compute the resulting effect on the thermodynamics. We then explore the consequences of this solution generator for both the non-uniform black strings as well as the black hole branch. In particular, we prove the so-called invertibility rule. Under certain assumptions, this rule puts restrictions on the continuation of Wiseman’s branch. Furthermore, in Section 6 we discuss the consequences of the new phase diagram for the proposals on the phase structure of black holes and strings on cylinders. We first go through the three existing proposal for the phase structure, and subsequently introduce three new proposals for how the phase structure could be. Finally, in Section 7 we make the phenomenological observation that the non-uniform string branch of Wiseman [8] has a remarkable linear behavior when plotting the quantity TSversus the mass M. We have the discussion and conclusions in Section 8. Two appendices are included providing some more details and derivations. Appendix A contains a refinement of the original proof [13] that for d = 5 the conformal ansatz can be mapped onto the ansatz proposed in [10]. A useful identity for static perturbations in the ansatz of [10] is derived in Appendix B.

2. New type of phase diagram

In this section we review the paper [1] and summarize the formulas and results that are important for this paper. 186 T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208

We begin by reviewing the new phase diagram of [1]. We first define the parameters of the phase diagram.5 Consider some localized, static and neutral Newtonian matter on a cylinder Rd−1 × S1. Let the cylinder space–time have the flat metric −dt2 + dz2 + dr2 + 2 2 r dΩd−2.Herez is the period coordinate with period 2πRT . We then consider Newtonian matter with the non-zero components of the energy momentum tensor being

T00 = , Tzz =−b. (2.1) We deﬁne then the mass M and the relative binding energy n as 1 M = dd x ρ(x), n = dd x b(x). (2.2) M n is called the relative binding energy since it is the binding energy per unit mass. We can measure M and n by considering the leading part of the metric for c c g =−1 + t ,g= 1 + z , (2.3) 00 rd−3 zz rd−3 for r →∞.Intermsofct and cz we have Ωd−22πRT ct − (d − 2)cz M = (d − 2)ct − cz ,n= . (2.4) 16πGN (d − 2)ct − cz Eqs. (2.3), (2.4) can then be used to ﬁnd the mass M and the relative binding energy n for any black hole or string solution on a cylinder. Three known phases of black holes and strings on cylinders are depicted in Fig. 1 for the case d = 5. The uniform black string phase has n = 1/(d − 2) and exists for all

Fig. 1. (M, n) phase diagram for d = 5 containing the uniform string branch and the non-uniform string branch of Wiseman. The black hole branch is sketched.

5 Similar asymptotic quantities as the ones given here have been previously considered in [24,25]. T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208 187 masses M. For the non-uniform black string phase that starts at the Gregory–Laflamme point (M, n) = (MGL, 1/(d − 2)) we have depicted the solutions found by Wiseman in [8] for d = 5 (see [1] for details on how the data of [8] are translated into the (M, n) phase diagram). The black hole branch is sketched in Fig. 1. The branch starts at (M, n) = (0, 0) but apart from that the precise curve is not known. A useful result for studying the thermodynamics of solutions in the (M, n) phase diagram is the Smarr formula [1]6 d − 2 − n TS= M. (2.5) d − 1 A curve of solutions obeying the first law δM = TδS is called a branch. For branches of solutions, it then follows from (2.5) that δ logS 1 d − 1 = , (2.6) δM M d − 2 − n(M) so that given the curve n = n(M), one can integrate to obtain the entire thermodynamics. Another important consequence of the Smarr formula is the intersection rule. This states that for two intersecting branches we have the property that for masses below an intersection point the branch with the lower relative binding energy has the highest entropy, whereas for masses above an intersection point it is the branch with the higher relative binding energy that has the highest entropy. In [1] we also commented on the uniqueness of black holes/strings on cylinders. It seems conceivable that the higher Fourier modes (as defined in [1]) are determined by M and n for black holes/strings. Therefore we proposed in [1] the following:

Uniqueness Hypothesis. Consider solutions of the Einstein equations on a cylinder Rd−1 × S1. For a given mass M and relative binding energy n there exists at most one neutral and static solution of the Einstein equations with an event horizon.

It would be very interesting to examine the validity of the above Uniqueness Hypothesis. If the higher Fourier modes of the black hole/string instead are independent of M and n,it would suggest that one should have many more dimensions in a phase diagram to capture all of the different black hole/string solutions.

3. Derivation of ansatz for solutions

3.1. Derivation of ansatz

In this section we derive that any neutral and static black hole/string on a cylinder d−1 1 R × S (of radius RT ) can be written in a very restrictive ansatz, originally proposed in [10]. This generalizes an argument of Wiseman in [8].

6 In Ref. [1] the numerical data of Wiseman was used to explicitly check that the Smarr formula holds to very high precision for the non-uniform black string branch. 188 T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208

Note that it is assumed in the following that any static and neutral black string or hole on a cylinder Rd−1 × S1 is spherically symmetric in Rd−1.

The black string case We begin by considering the black string case, i.e., where the topology of the horizon is Sd−2 × S1. The black hole case is treated subsequently. Using staticity and spherical symmetry in Rd−1, the most general form of the metric for a black string is 2 =− 2B 2 + a b + 2D 2 ds e dt gab dx dx e dΩd−2, (3.1) a,b=1,2 1 2 where B, D, gab are functions of x and x only. This ansatz involves 5 unknown functions, and the location of the event horizon is given by the relation eB(x1,x2) = 0. Since we are considering black strings wrapped on the cylinder, i.e., with an event horizon of topology Sd−2 × S1, we can think of the two coordinates x1 and x2 as coordinates on a two-dimensional semi-infinite cylinder of topology R+ × S1 and with metric gab, where the boundary of the cylinder is the location of the event horizon of the black string.7 Now, it is a mathematical fact that any two-dimensional Riemannian manifold is conformally flat. We can thus find coordinates r, z so that a b 2C 2 2 gab dx dx = e dr + dz . (3.2) a,b=1,2

Since dr2 + dz2 now describes a Ricci-flat semi-infinite cylinder of topology R+ × S1 we can choose z as the periodic coordinate for the cylinder and r to be the coordinate for R+, i.e., r = 0 is the boundary of the cylinder and r 0. With this, we have shown that the metric (3.1) can be put in the conformal form 2 =− 2B 2 + 2C 2 + 2 + 2D 2 ds e dt e dr dz e dΩd−2, (3.3) involving three functions only. Moreover, z is a periodic coordinate of period 2πRT and r 0, with r = 0 being the location of the horizon. The form (3.3) of the black string metric is called the conformal ansatz, which was also derived and used in Refs. [7,8] for d = 4, 5. We now wish to further reduce the three-function conformal ansatz to the two-function ansatz of Ref. [10]. For d = 5 this was done in [13] as reviewed in Appendix A, where we also add a small refinement. Here we generalize the argument to arbitrary d. The two- function ansatz reads 2 2 2 −1 2 A 2 2 2 ds =−fdt + R f AdR + dv + KR dΩ − , T Kd−2 d 2 − Rd 3 f = 1 − 0 , (3.4) Rd−3

7 We use here and in the following the notation that R+ =[0, ∞[. T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208 189 with A and K functions of R and v. We ﬁrst perform one coordinate transformation and a redeﬁnition of A and K

2(d−4) − − − − r¯ Rd 3 = Rd 3 +¯rd 3, Aˆ = f 1AR2 , 0 T R

d−2 2(d−4) − K r¯ Kˆ d 2 = , (3.5) f R after which we ﬁnd r¯d−3 1 ds2 =− dt2 + Aˆ dr¯2 + dv2 d−3 +¯d−3 ˆ d−2 R0 r K 5−d − − 3 − + 2 ˆ ˆ d−2 d 3 +ˆd 3 d−2 d 2 RT Kr R0 r dΩ2 . (3.6) ˆ = 2a 2 ˆ = 2k We then write A e and RT K e and change to the conformal form (3.3) by the coordinate transformation r¯ = g(r,z), v = h(r, z), (3.7) −(d−2)k −(d−2)k ∂r g = e ∂zh, ∂zg =−e ∂r h. (3.8) Matching the lapse function of the resulting metric with that in (3.3) gives the condition

d−3 2B − R e gd 3 = 0 , (3.9) 1 − e2B while identifying the rest of the metric yields

2C 2 − e R 2(d 5) B 2 e2a = ,e2k = T e2De (d−2)(d−3) 1 − e2B d−3 . (3.10) 2 + 2 2 (∂r g) (∂zg) R0 However, the system (3.8) implies an integrability condition on g(r,z) which reads 2 + 2 + − + = ∂r ∂z g (d 2)(∂r g∂r k ∂zg∂zk) 0. (3.11) Inserting in this equation the form of g in (3.9) and k from (3.10) results in the second order differential equation 2 + 2 + 2 + 2 + − + = ∂r ∂z B (∂r B) (∂zB) (d 2)(∂r B∂r D ∂zB∂zD) 0. (3.12)

This turns out to be precisely the equation of motion Rtt = 0 in the ansatz (3.3). As a consequence we have shown that the ansatz (3.4) is a consistent ansatz, so that in particular, the four differential equations for the two functions A(R, v) and K(R,v) (see Eqs. (6.1)–(6.4) of [10]) are a consistent set of differential equations, as was conjectured in [10].8 To complete the argument, we also need to consider the location of the horizon and the periodicity in the compact direction. In the conformal ansatz these are given by r = 0, and

8 In [10] it was shown that the four equations (6.1)–(6.4) of [10] are consistent to second order in a perturbative expansion of these equations for large R. 190 T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208 z z + 2πRT , respectively. As far as the horizon is concerned, it follows immediately from (3.9) that in the two-function ansatz the horizon will be at r¯ = 0, as desired (since this means R = R0 by (3.5)). We moreover want that A(R, v) and K(R,v) are periodic in v with period 2π.9 A necessary requirement is that v → v + 2π when z → z + 2πRT . From (3.7) we see this is true provided

h(r, z + 2πRT ) − h(r, z) = 2π. (3.13) We ﬁrst prove that the left-hand side of (3.13) is independent of r. This is proven by noting that since g(r,z) is periodic in z by (3.9) then ∂zg is periodic in z and from (3.8) it follows that then ∂r h is also periodic in z, which proves the statement. We then write the left-hand side of (3.13) as

2πRT 2πRT (d−2)k dz∂zh = dze ∂r g, (3.14) 0 0 where we used (3.8) in the last step. Since we know that (3.14) is independent of r we may evaluate the integrand for r →∞. We know that k → 0 in this limit, so this leaves ∂r g. 2B −(d−3) However, from (2.3) we see that 1 − e = ct r to leading order so using (3.9) we d−3 ﬁnally get from (3.14) that (3.13) is true provided that ct = (R0RT ) , which also follows from (3.4) since RRT /r → 1 asymptotically. We can think of the coordinate transformation from (r, z) to (R, v) as a map from R+ × S1 to R+ × S1. This map is non-singular and invertible since we are dealing with the black string case where the curves {R = const} have S1 topology in the R+ × S1 cylinder given by the (r, z) coordinates. Thus, that {R = const} has S1 topology means that it has a periodicity in v, and in order for the total coordinate map to be continuous, the periodicity must be the same everywhere. Hence, since the periodicity is 2π asymptotically, it is 2π everywhere. We therefore conclude that A(R, v) and K(R,v) are periodic functions in v with period 2π.

The black hole case The proof that any static and neutral black hole on a cylinder Rd−1 × S1 can be written in the ansatz (3.4) follows basically along the same lines as the black string argument above, though with some notable differences. Any static and neutral black hole on a cylinder Rd−1 × S1 can be written in the ansatz (3.1). The location of the event horizon is still specified by eB = 0. Topologically we can again think of (x1,x2) as being coordinates on a semi-infinite cylinder R+ ×S1 with metric gab. The metric gab, a,b = 1, 2, for the two-dimensional cylinder is non-singular away from the boundary eB = 0 and we can get (3.3) again by using that any two-dimensional Riemannian manifold is conformally flat. We furthermore require again that z is periodic with period 2πRT . Using the same coordinate transformations (3.5)–(3.8) as for the string

9 The reason for requiring 2π as period is that A,K → 1 asymptotically and then v has circumference 2πRT from (3.4). T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208 191 case we find that (3.3) can be transformed into (3.4) since the equation of motion Rtt = 0 again proves the consistency of the transformation. From (3.9) it is furthermore evident that the horizon for the black hole is at R = R0, as it should. In conclusion, we have proven that any static and neutral black hole solution on a cylinder Rd−1 × S1 can be written in the ansatz (3.4). The only remaining issue is the periodicity in v. Clearly, we still have that v → v + 2π when z → z + 2πRT . But, the coordinate map from (r, z) to (R, v) is not without singularities. The qualitative understanding of this is basically that the curves {R = const} still have topology S1 in the (r, z) coordinates when R is sufficiently large. But for small R the curve {R = const} ends on the boundary of R+ × S1. However, as explained in [10] for a specific choice of R and v coordinates, the functions A(R, v) and K(R,v) are still periodic in v with period 2π for small R.

3.2. Ansatz for black holes and strings on cylinders

Assuming spherical symmetry on the Rd−1 part of the cylinder, we have derived in Section 3.1 that the metric of any static and neutral black hole/string on a cylinder can be put in the form 2 2 2 −1 2 A 2 2 2 ds =−fdt + R f AdR + dv + KR dΩ − , T Kd−2 d 2 − Rd 3 f = 1 − 0 , (3.15) Rd−3 with A = A(R, v) and K = K(R,v). This ansatz was proposed and studied in [10] as an ansatz for black holes on cylinders. The ansatz (3.15) has two functions A(R, v) and K(R,v) to determine. However, in [10] it was shown that A(R, v) can be written explicitly in terms of K(R,v) and its partial derivatives (see Eq. (6.6) of [10]). Thus, the ansatz (3.15) is completely determined by only one function. We impose the following boundary condition on K(R,v), − 1 − (d − 2)n Rd 3 K(R,v) = 1 − 0 +···. (3.16) (d − 3)(d − 2 − n) Rd−3 Which implies, using the equations of motion [10], the same boundary condition on A(R, v). Note that the fact that K → 1forR0/R → 0 means that R = r/RT and v = z/RT when R →∞. Moreover, the condition (3.16) ensures that we have the right asymptotics at r →∞, according to Section 2. We can then use (3.16) in (2.4), to compute the mass in terms of R0 and n − − Ωd−22πRT d−3 (d 1)(d 3) M = (R0RT ) , (3.17) 16πGN d − 2 − n which means we can find R0 as function of M and n. Define now the flat-space limit of K(R,v) ≡ | K0(R, v) K(R,v) R0=0. (3.18) 192 T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208

2 = 2 =− + d−3 d−3 Since r KR and g00 1 R0 /R , we then notice that we can write d−3 R d−3 2 8πGN nM 0 2 =− − d−3 K0(r, z) 2Φ(r,z) d−3 , (3.19) r (d − 1)(d − 3) Ωd−22πRT r with Φ(r,z) given by

− ∞ =− d 2 8πGN 1 kr kz Φ(r,z) d−3 h cos k. (3.20) (d − 1)(d − 3) 2πRT r RT RT k=0

Here k are the Fourier modes defined in [1], which characterize the z dependence of the black hole/string. Therefore, we see that given M, n and k, k 1, we obtain K0(r, z) and thereby K0(R, v). Thus, imposing the two boundary conditions (3.16) and (3.19) corresponds to imposing specific values for M, n and k. As shown in [10], one of the nice features of the ansatz (3.15) is that it has a Killing horizon at R = R0 with a constant temperature, provided K(R,v) is finite for R = R0. Thus, unless K(R,v) diverges for R → R0, the ansatz corresponds to a black hole or string. For use below, we recall here the corresponding expressions for temperature and entropy [10] √ − d 3 Ωd−22πRT Ah d−2 T = √ ,S= (R0RT ) , (3.21) 4π Ah R0RT 4GN where Ah ≡ A(R0,v).

4. First law of thermodynamics

In this section we derive the ﬁrst law of thermodynamics for static and neutral black holes/strings on a cylinder, and discuss its consequences for the (M, n) phase diagram.

4.1. Proof of the ﬁrst law

The three central ingredients of our derivation of the first law of thermodynamics are (i) the Smarr law (2.5); (ii) the identity derived in Appendix B; and (iii) the ansatz (3.15) describing black holes/strings on the cylinder. We first note that by variation of the Smarr law (2.5) we find the relation d − 2 − n 1 SδT + TδS= δM − Mδn. (4.1) d − 1 d − 1 Our aim is now to use the identity (B.3), (B.4) to obtain another differential relation involving δM. To this end we first rescale the ansatz (3.15) so that it becomes 2 2 2 2 −1 2 A 2 2 2 2 ds =−fdt + R R f AdR + dv + R KR dΩ − , T 0 Kd−2 0 d 2 1 f = 1 − . (4.2) Rd−3 T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208 193

In this way of writing the ansatz the horizon is always located at R = 1. Consider now a given time t = t and take S1 to be the (d − 1)-dimensional null surface Sh deﬁned t = t by R = 1andS2 as the (d − 1)-dimensional space-like surface S∞ speciﬁed by t = t and R = Rm with Rm →∞. Putting this into the identity (B.3) along with the metric (4.2) we get

2π 2π = = = d−2 = = dvA(R 1, v)Γ (R 1,v) Rm dvA(R Rm, v)Γ (R Rm,v), (4.3) v=0 v=0 RR 2 RR µν α Γ(R,v)= g ∂RδgRR − g g Γ δgαν µR − RR µν R − RR µν g g ΓµνδgRR g ∂R g δgµν . (4.4) We now use the rescaled ansatz (4.2) to compute Γ(R,v) and substitute this into (4.3). After some algebra we obtain the relation δR 1 δA (d − 3)(1 + n) δR d − 1 0 + h = 0 + δn, (4.5) 2 R0 2 Ah d − 2 − n R0 (d − 2 − n) where Ah = A(R = 1,v). We recall here that it follows from the equations of motion that this quantity is constant on the horizon [10]. To obtain the left-hand side we have expressed the variation of the metric on the horizon in terms of δR0 and δAh (the variation δK does not appear, nor does the quantity K(R = 1,v) which is generally not constant on the horizon). The right-hand side is obtained by expressing the variation of the metric at inﬁnity in terms of δR0 and δn, where we have used the asymptotic form (3.16) for K(R,v) and the same form of A(R, v) as is required by the equations of motion. To reexpress the identity (4.5) in terms of thermodynamic quantities we use the expressions (3.21) for the temperature T and entropy S of the ansatz (3.15) along with the mass in (3.17). The identity (4.5) can then be put in the form 1 + n 1 −SδT = δM + Mδn. (4.6) d − 1 d − 1 Adding this to the differential relation (4.1) that followed from Smarr, we immediately obtain

δM = TδS, (4.7) which is the first law of thermodynamics.10 The first law indeed holds for all known branches in the phase diagram. For the uniform black string branch and small black holes on the cylinder this is of course obvious. For the non-uniform black string branch, one can also verify the validity to high precision (see Appendix C of [1]). So far we have kept the radius RT of the periodic direction fixed. It is a simple matter to repeat the analysis above by also allowing RT to vary. The result is that the first law

10 Note that indeed the relation in Eq. (7.13) of [10],√ which was shown to hold iff the ﬁrst law holds, coincides = = 1−(d−2)n precisely with the relation (4.5) when using γ 1/ Ah and χ (d−3)(d−2−n) . 194 T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208 acquires an extra work term and becomes

δR δM = TδS+ nM T . (4.8) RT The extra term could in fact be expected on general physical grounds. This is because from (2.2) we see that the average pressure is p =−nM/V and since δRT /RT = δV/V we have nMδRT /RT =−pδV ,givingδM = TδS− pδV . This observation also conﬁrms the fact that n really is a meaningful physical quantity and it provides a nice thermodynamic consistency check on the connection between n and the negative pressure p in the periodic direction.

4.2. Consequences for space of solutions

We have proven above that any curve of solutions in the (M, n) diagram has to obey the first law δM = TδS. The most important consequence of this is that there cannot exist solutions for all masses M 0 and relative binding energies 0 n 1/(d − 2). This is perhaps surprising, since using the ansatz (3.15) one can easily come up with a solvable system of equations and asymptotic boundary conditions11 (at R →∞)that would correspond to a given M and n. The resolution of this is of course then that the horizon is only regular for special choices of M and n. In terms of the ansatz (3.15) this means that only for special choices of M and n we have that A(R, v) and K(R,v) are 12 finite on the horizon R = R0. The above proof that any curve of solutions obeys the first law moreover makes it highly doubtful that there is any set of solutions with a two-dimensional measure in the (M, n) phase diagram. Thus, for any two given branches of solutions, we do not expect there to be a continuous set of branches in between them. The main reason for this is that if one had two different points in the (M, n) phase diagram with same mass, the first law δM = TδS would imply they had the same entropy since it would be possible to go from one point to the other varying only n. In fact, it is clearly true that the uniform branch, the non-uniform branch that is connected to the Gregory–Laflamme point (M, n) = (MGL, 1/(d − 2)) and the black hole branch, cannot be continuously connected in this way.13 For the Horowitz–Maeda conjecture [5] that there exist light non-uniform strings with entropy larger than that of a uniform string of equal mass, this means that those solutions should be a one-dimensional set of solutions which describe the end point of the decay of a light uniform string. The intermediate classical solutions that the classical decay of the uniform string evolves through (see [6] for numerical studies of the decay) should therefore be non-static.

11 One can, for example, use the asymptotic boundary conditions described in [10]. 12 Note that this shows that the mechanism proposed in [10] to choose which curve in the (M, n) diagram corresponds to the black hole branch is then proven wrong, since in that paper it was proposed that one could use the first law δM = TδSto find the black hole branch. 13 The same is true for any two copies (as defined in Section 5) of the non-uniform branch connected to the Gregory–Laflamme point (M, n) = (MGL, 1/(d − 2)). T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208 195

5. Copies of solutions

5.1. Solution generator

In [11] Horowitz argued that for any non-uniform string solution one can “unwrap” this solution to get a new solution which is physically different. The argument is essentially that if one takes a solution with mass M and circumference L of a cylinder Rd−1 × S1, then one can change the periodicity of the cylinder coordinate z from L to kL, with k a positive integer. This will produce a new solution with mass M˜ = kM and cylinder circumference L˜ = kL.SinceM/Ld−2 is dimensionless, we see that the new solution has M/˜ L˜ d−2 = k−(d−3)M/Ld−2. In the following we ﬁrst write down an explicit “solution generator” for our ansatz (3.15) that directly implements Horowitz’s “unwrapping” and we then explore the consequences for black strings and black holes in Sections 5.2 and 5.3, respectively. The speciﬁc transformation that takes any black hole/string solution on the cylinder and transforms it to a new black hole/string solution on the cylinder with the same radius, is as follows:

Consider a black hole/string solution on a cylinder with radius RT and horizon radius R0, and with given functions A(R, v) and K(R,v) in (3.15). Then for any k ∈{2, 3,...} we obtain a new solution as R A(R,˜ v) = A(kR, kv), K(R,v)˜ = K(kR,kv), R˜ = 0 . (5.1) 0 k This gives a new black hole/string solution on a cylinder of the same radius. The statement above may be veriﬁed explicitly from the equations of motion (see Eqs. (6.1)–(6.4) of [10]). One can also easily implement this transformation in the ansätze (3.1) and (3.3) instead, but we choose here for simplicity to write the transformation in the ansatz (3.15).14 We can now read off from (2.3), (2.4), (3.17) and (3.21) that the new solution (5.1) has mass, relative binding energy, temperature and entropy given by M S M˜ = , n˜ = n, T˜ = kT, S˜ = . (5.2) kd−3 kd−2 We see that the transformation rule for the mass is precisely that of Horowitz in [11]. Note also that the rule is in agreement with the Smarr formula (2.5).

5.2. Consequences for non-uniform black strings

We can now use the transformation (5.2) on the known branches of solutions. For the uniform string branch we merely go from one point in the branch to another, i.e., the transformation does not create any new branch of solutions. In fact, using the entropy

14 In the ansatz (3.3) the transformation is instead B(r,z)˜ = B(kr,kz), C(r,z)˜ = C(kr,kz) and D(r,z)˜ = D(kr,kz). 196 T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208

Fig. 2. Part of the (M, n) phase diagram for d = 5 with copies of the non-uniform string branch of Wiseman. expression for the uniform black string

1 d−2 d−2 16πG d−3 − − − N SBS(M) = d1M d 3 ,d1 = 4π(d − 3) d 3 , (5.3) Ωd−22πRT it is easy to verify from (5.2) that the entropy of the kth copy is the same as that of the original branch. Considering instead the non-uniform string branch that is connected to the Gregory– Laﬂamme point (M, n) = (MGL, 1/(d − 2)) we see that we do create new branches, in fact inﬁnitely many new branches. We have depicted this in Fig. 2 using Wiseman’s data for d = 5. Note that by the Intersection Rule [1] reviewed in Section 2 all the copies in Fig. 2 also have the property that the entropy for a solution is less than that of a uniform string of same mass. If we take two copies which both exist for the same mass, we furthermore see from the Intersection Rule that the branch which lies to the left in the (M, n) diagram will have less entropy for a given mass. We are now capable of addressing an important question regarding Wiseman’s branch (see Fig. 5 of [1] for a more detailed plot): is Wiseman’s branch really starting to have increasing n at a certain point? This question can be addressed using the following rule that we prove below.

Invertibility Rule. Consider a curve of solutions in the (M, n) phase diagram. There are no intersections between the curve and its copies, or between any of the copies, if and only if the curve is described by a well-deﬁned function M = M(n) (i.e., if a given n only corresponds to one M).

We call this the Invertibility Rule since we normally would describe a branch by a function n = n(M), and then the Invertibility Rule implies such a function is invertible. For the non-uniform string branch in Fig. 1 the Invertibility Rule implies that the small increase in n in the end of the branch can be ruled out if we also assume the Uniqueness Hypothesis [1] (recalled in Section 2) to be valid. To see this note that the latter assumption implies that an intersection of two branches means in particular that the two branches intersect in the same solution and not just in the same point in the (M, n) diagram. Thus, using the Intersection Rule (see Section 2) we cannot have two branches intersecting twice, T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208 197 and if the non-uniform branch was not invertible, then they clearly would intersect twice since they already both intersect the uniform string branch. It then follows that the non- uniform branch must continue to decrease in n, and the small increase that one can see must be from inaccuracies in the numerically computed data. Although it seems reasonable to assume that the above stated conditions hold, it would be interesting to further examine the behavior of the Wiseman branch at the end. We conclude by proving the Invertibility Rule. On the one hand, it is clearly true that a curve described by a well-deﬁned function M = M(n) will not have intersections between the curve and its copies, or between any of the copies. On the other hand, consider a continuous curve in the (M, n) phase diagram that is not invertible, so that there are two points (M1,n) and (M2,n), M1 M1/q . Thus, we have intersections if we can ﬁnd integers p d−3 d−3 and q so that M2/M1 >p /q > 1. But that is always possible. This completes the proof.

5.3. Consequences for black hole branch

We ﬁnally consider the consequence of the transformation (5.2) for black holes on the cylinder, i.e., the branch that starts at (M, n) = (0, 0). We ﬁrst comment on what, according to (5.1) and (5.2), the physical meaning is of the kth copy for the black hole branch. From (5.1) it is easy to see that the kth copy of a black hole is k black holes on a cylinder put at equal distance from each other. While these solutions clearly exist they are not classically stable. If we take k = 2, for instance, it is clear that this solution consisting of two black holes is unstable against perturbation of the relative distance between the black holes. The solution will then obviously decay to a single black hole on a cylinder. More generally, k black holes on a cylinder will decay to a fewer number of black holes when perturbed. Thus, all these solutions with k>1 are in an unstable equilibrium. At the level of the entropy this implies that we expect that SBH,k(M) < SBH,k (M) for k>k. To see this explicitly, note that the thermodynamics for the small black hole branch is given by

1 d−1 d−1 16πG d−2 − − − N SBH(M) = c1M d 2 ,c1 = 4π(d − 1) d 2 , (5.4) Ωd−1 since it is equal to that of a black hole in (d + 1)-dimensional Minkowski space. We can then use (5.2) and (5.4) to compute that the entropy function of the kth copy of the small black hole branch is

d−1 1 − − − SBH,k(M) = ckM d 2 ,ck = c1k d 2 . (5.5) Since then ck >ck for k Sk (M) for k0 amounts to determining which intuition is right: that the behavior of black holes as objects are given by the shape of the singularity or the shape of the event horizon. Note also that if n = 0 for the entire black hole branch the Uniqueness Hypothesis [1] which is recalled in Section 2 must be wrong since then we have an inﬁnite number of different solutions with the same M and n.

6. Possible scenarios and phase diagrams

6.1. Review of three proposed scenarios

As mentioned in the introduction, Horowitz and Maeda argued in [5] that a classically unstable uniform black string cannot become a black hole in a classical evolution. More precisely they showed that an event horizon cannot have a circle that shrinks to zero size in a finite affine parameter. This led them to conjecture that there exists a new branch of solutions which are non-uniform neutral black strings with entropy greater than that of the uniform black string with equal mass. The conjecture of Horowitz and Maeda motivated Gubser in [7] to look for a branch of non-uniform black string solutions, connected to the uniform black string solution in the Gregory–Laflamme critical point where M = MGL, by using numerical techniques. The numerical data of Gubser were later improved by Wiseman in [8].15 As mentioned above, we depicted the non-uniform branch in the (M, n) phase diagram of Fig. 1, in the case d = 5. However, as noted by [7], the piece of the non-uniform branch found by [7,8] cannot be the conjectured Horowitz–Maeda non-uniform solutions. Firstly, it was shown in both [7] and [8] that the entropy of a non-uniform black string on the branch is smaller than that of a uniform black string of the same mass. Secondly, all the new solutions in the branch have masses bigger than the Gregory–Laflamme mass MGL. Clearly we need non- uniform solutions with masses M

15 Note that in [7] the case d = 4 was studied while in [8] it was the case d = 5 instead. T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208 199 branch simply terminates when the mass reaches the critical value when the black horizon reaches all around the cylinder. Clearly then it should be so that the entropy of a black hole just below the critical mass should be less than that of the uniform black string, so that one can have a first order transition from the black hole to a uniform string of the same mass. In the following we refer to this combined scenario as “scenario I”. In [10] it was instead suggested that the black hole phase should go directly into a non-uniform string phase which connects to the uniform string phase at M =∞, i.e., n → 1/(d − 2) for M →∞. Alternatively the branch could instead approach some other limiting value, i.e., n → n for M →∞. In this type of combined scenario we thus have two non-uniform black string branches, one that connects to the black hole branch and another that connects to the Gregory–Laflamme point (M, n) = (MGL, 1/(d − 2)).We refer to this type of scenario as “scenario II”. An entirely different scenario was proposed by Kol in [12], where it was conjectured that the non-uniform string branch found in [7,8] should be continued all the way to the black hole branch. Since this scenario requires for small masses that the non-uniform black string branch changes topology and becomes the black hole branch, it would in any case not be able to account for the conjectured Horowitz–Maeda non-uniform black strings for small masses. Therefore, the unstable uniform string should instead decay classically in infinite “time”, i.e., affine parameter on the horizon, and become arbitrarily near the black hole branch but never reach it classically. We call Kol’s scenario “scenario III” below. A final relevant issue is whether the non-uniform branch is classically stable or unstable, which is presently not known. However, since these solutions were found by applying numerical methods that work by perturbing a solution which then relaxes to a new fixed point, it is likely that the Wiseman branch is classically stable.16 In this connection it is also worth considering the copies of the Wiseman branch discussed in Section 5.2. On the one hand, the fact that for increasing number k the entropy at a given mass decreases, could suggest that the copies are classically unstable. On the other hand, any pair of copied branches has a different mass range, as can be seen in Fig. 2. Moreover, contrary to the black hole copies which constitute separate objects, in this case all copies of non-uniform strings have the same topology. As a consequence we view the classical stability of the Wiseman copies as another open problem.

6.2. Phase diagrams for the three scenarios

We can now try to draw how scenarios I–III should look in the (M, n) phase diagram for d = 5. When drawing the scenarios in the following we use the (M, n) phase diagram in Fig. 1 as the starting point. However, in that diagram (see also Appendix C of [1]) we see that the non-uniform branch starts having increasing n, just before it ends. We argued in Section 5 that, under reasonable assumptions, the non-uniform branch in fact has to continue decreasing in n. The small increase one can see in Fig. 1 could thus possibly be

16 We thank Roberto Emparan for this observation. 200 T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208

Fig. 3. Scenario I. Gubser’s proposal for the non-uniform string branch plus a black hole branch that terminates at the critical point. due to inaccuracies of the numerics. In the following we have therefore ignored these last few data points and assumed that n keeps decreasing. It is also important to note that when we construct the (M, n) phase diagrams below we are following the Intersection Rule [1] reviewed in Section 2 which, among other things, has the consequence that one cannot have two branches intersecting twice, at least not when the branches are described by well-deﬁned functions n = n(M). In scenario I that we described above we clearly need to let the non-uniform string branch turn around and reach M = 0 with n>0 since we do not want to cross the black hole branch, and we need n 0. We have sketched this scenario in Fig. 3. It looks though unlikely that the curve should make such a complicated turn, perhaps one can imagine that the curve has a cusp. This would, however, imply at least a second order phase transition in the point of the cusp which would seem peculiar since the solution does not change phase. In Fig. 3 we have also depicted the black hole phase which in scenario I terminates at a critical mass when the black hole horizon reaches around the horizon. Note that since we require the entropy at the critical point to be higher than that of a uniform string of the same mass, and since one can integrate up the entropy [1] given a curve in the (M, n) diagram (see Eq. (2.6)), we can in fact put bounds on the curve.17 In Fig. 4 we have depicted scenario II. As described above this scenario again realizes Gubser’s proposal for the non-uniform black string branch that starts at the Gregory– Laﬂamme point (M, n) = (MGL, 1/(d − 2)). But in this scenario the black hole branch continues into another non-uniform string branch that continues for arbitrarily high masses (following the proposal of [10]). In Fig. 5 we have depicted scenario III. This is Kol’s scenario where the non-uniform string branch turns around and meets the black hole branch. A smooth turn again seems problematic, but in this case it seems natural to suppose that there is a cusp precisely at the point where the phase transition between the non-uniform string phase and the black hole phase occurs.18

17 In particular, this enables us to obtain a lower bound on the critical mass by equating the entropy of a black hole in ﬂat Minkowski space to that of a black string. Using the GL masses listed in Table 1 of [1] this yields for 4 d 9thevaluesMc/MGL = 4.23, 4.53, 4.09, 3.98, 3.93, 3.66. 18 We thank Toby Wiseman for suggesting this to us. T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208 201

Fig. 4. Scenario II. Gubser’s proposal for the non-uniform string branch plus a black hole branch that continues into another non-uniform string branch which continues to arbitrarily large masses.

Fig. 5. Scenario III. The scenario of Kol where the black hole branch is connected to the non-uniform branch.

Evidence in support of this scenario was given in Ref. [15], by numerically showing that for sufﬁcient non-uniformity the non-uniform string branch has the property that the local geometry about the minimal horizon sphere approaches the cone metric. They argue furthermore that this implies that the non-uniform branch will not continue beyond the endpoint of Wiseman’s data at M 2.3MGL. If true, this would imply that the other scenarios presented in this section are not viable.

6.3. Other possible scenarios

Each of the three scenarios described above suffers from possible shortcomings. Kol’s scenario in Fig. 5 does not have the needed Horowitz–Maeda non-uniform string phase that Horowitz and Maeda argued for in [5]. That can of course possibly turn into a virtue in that the recent paper [6] failed to find an endpoint of the classical decay, suggesting that a stable endpoint in fact does not exists. On the other hand, the scenarios I and II in Figs. 3 and 4 suffer from having a seemingly unnatural U-turn on the curve. One can instead imagine other scenarios which fit with the data in Fig. 1. For example, one can consider the new scenario depicted in Fig. 6. Note that this scenario is constructed so that it has the Horowitz–Maeda non-uniform string branch and so that the above- mentioned U-turn is absent. Note also that the black hole branch is disconnected from the non-uniform string branch originating from the Gregory–Laflamme point (M, n) = (MGL, 1/(d − 2)). 202 T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208

Fig. 6. Scenario IV.

Fig. 7. Scenario V.

Fig. 8. Scenario VI.

Finally, we present the scenarios V and VI in Figs. 7 and 8. These two scenarios both have two non-uniform branches, each of which continues for M →∞. Scenario V does not have the conjectured Horowitz–Maeda strings while scenario VI does.

7. A remarkable near-linear behavior

In this section we report on a remarkable nearly linear behavior of the non-uniform string branch, if one plots TSversus M. In Fig. 9 we have plotted Wiseman’s data in a TS versus M diagram, and we have ﬁtted a line to the data points in order to demonstrate how near the points falls to a line. T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208 203

Fig. 9. TS versus M plot for the non-uniform string branch. The crosses are the data for Wiseman’s solutions. The line is a linear ﬁt to the data.

Fig. 9 shows that the TS versus M data falls very near the line

TS M − 1 = x − 1 ,x= 1.12. (7.1) TGLSGL MGL

Here TGL and SGL are the temperature and entropy at the Gregory–Laflamme point (M, n) = (MGL, 1/(d − 2)). Note that we have chosen the line so that it fits with the first half of the set of data points. Using now Smarr’s formula (2.5) with d = 5 we see that (7.1) corresponds to the curve

8 M 9 − 8x n = (x − 1) GL + , (7.2) 3 M 3 in the (M, n) diagram. The explicit form of n(M) above, enables us to integrate (2.6) to ﬁnd the entropy19

3 M 2x Sw = SBS(MGL) x − 1 + 1 , (7.3) MGL where the constant of integration is ﬁxed by the intersection point with the uniform string branch. The temperature then also follows easily from Smarr’s formula (2.5). We have plotted the curve (7.2) along with Wiseman’s data in Fig. 10. As one can see, Wiseman’s data deviate more and more from (7.2) as the mass increases. However, it is not clear whether Wiseman’s data is accurate enough to rule out that the exact solutions can fall on a curve of the form (7.2). Of course the above observations are of purely phenomenological nature. However, it seems reasonable to expect that there should be some theoretical reasoning leading to the fact that the non-uniform string branch have the linear behavior in Fig. 9, at least to a good approximation. This would be interesting to investigate further.

19 More generally, we note that any branch represented by a line in the (M, T S) plot implies that n(M) = c α + β/M, which in turn gives an entropy of the form S(M) = S0(aM + b) . 204 T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208

Fig. 10. The non-uniform branch of Wiseman [8] (dotted curve) compared to the curve (7.2) (solid curve).

8. Discussion and conclusions

In this paper, we have reported on several new results that enhance the understanding of black holes and strings on cylinders. This development builds partly on the new phase diagram and other tools introduced in Ref. [1]. We presented three main results. Firstly, we proved that the metric of any neutral and static black hole/string solution on a cylinder can be written in the ansatz (3.15), an ansatz that was proposed in [10]. This proof is a generalization of the original proof of Wiseman for d = 5 in Ref. [13]. The ansatz (3.15) is very simple since it only depends on one function of two variables, which means that most of the gauge freedom of the metric is fixed. Secondly, we showed that the first law of thermodynamics δM = TδS holds for any neutral black hole/string on a cylinder. This was proven using the ansatz (3.15) and also the Smarr formula (2.5) of [1]. That any curve of solutions obeys δM = TδS implies that we do not expect a continuous set of solutions between two given branches in the phase diagram. The third result is the explicit construction of a solution generator that takes any black/hole string solution and transforms it to a new solution, thus giving a far richer phase diagram. This is a further development of an idea of Horowitz [11] that a black hole/string on a cylinder can be “unwrapped” to give new solutions. We constructed the solution generator using the ansatz (3.15) which furthermore made it possible to write down explicit transformation laws for all the thermodynamic quantities. Beyond these new results, we have presented a comprehensive analysis of the proposals for the phase structure in light of the new results presented here and in Ref. [1]. In addition to the three previously proposed scenarios, we have listed three new scenarios for the phase structure, all of which could conceivably be true. We have also commented on the viability of the various scenarios. Finally, we presented a phenomenological observation: Wiseman’s numerical data [8] for the non-uniform string branch lie very near a straight line in a TS versus M diagram. If the true non-uniform branch really lies on a line, that would clearly suggest that only scenarios IV–VI would have a chance to be true (in principle also scenario III if the line terminates). Moreover, it would present the tantalizing possibility that perhaps all branches of black holes/strings on cylinders are lines in the TS versus M diagram. That would T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208 205 suggest that the black hole branch lies on the curve given by n = 0inthe(M, n) phase diagram. It will be highly interesting to uncover the final and correct (M, n) phase diagram for neutral black holes and strings on cylinders. In the path of finding that, we are bound to learn more about several important questions regarding black holes in General Relativity. We can get a better understanding of the uniqueness properties of black holes, i.e., how many physically different solutions are there for a given M and n. We can learn about how “physical” a horizon really is, i.e., does the black hole behave as a point particle, or does it behave as an extended object when put on a cylinder? Finally, we can learn much more about phase transitions in General Relativity. Is Cosmic Censorship upheld or not for a decaying light uniform string? Under which conditions can the topology of an event horizon change? We consider the ideas and results of Ref. [1] and this paper as providing new guiding principles and tools in the quest of understanding the full phase structure of neutral black holes and strings on cylinders so that all these important questions can find their answer.

Acknowledgements

We thank J. de Boer, H. Elvang, R. Emparan, G. Horowitz, V. Hubeny, M. Rangamani, S. Ross, K. Skenderis, M. Taylor, E. Verlinde and T. Wiseman for helpful discussions.

Appendix A. Equivalence of boundary conditions for ansatze in d = 5

In this appendix, we give a reﬁnement of the original proof [13] that for d = 5 the three- function conformal ansatz (3.3) of that paper can be mapped onto the two-function ansatz (3.15) proposed in [10]. Our starting point is the conformal ansatz of [13] r2 ds2 =− e2A dt2 + e2B dr2 + dz2 + e2C m + r2 dΩ2, (A.1) m + r2 3 where A,B,C are functions of (r, z), with asymptotic forms a b c c A 2 ,B 2 ,C 1 + 2 . (A.2) r2 r2 r r2 The new ingredient of the proof is that we ﬁrst show that by a suitable change of coordinates the leading term in the boundary condition of the function C can be eliminated, leaving 2 c2 + c /2 C 1 , (A.3) r2 while leaving the asymptotic forms of A and B unaffected. The relevance of this step will become clear once we perform the coordinate transformation to the two-function ansatz, as the boundary condition (A.3) will be essential in order to obtain the correct boundary condition for the transformed metric. 206 T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208

To prove this statement we perform the following sequence of coordinate transformations

ρ = m + r2 →ˆρ = ρ + c →ˆρ = m +ˆr2. (A.4) After some algebra the resulting metric in (r,z)ˆ coordinates becomes

rˆ2 ˆ ˆ ˆ ds2 =− e2A dt2 + e2B drˆ2 + g2(r)dzˆ 2 + e2C m +ˆr2 dΩ2, (A.5) 6 m +ˆr2 3 with asymptotic conditions a b c − c c + c (c − c) + c2/2 Aˆ 2 , Bˆ B 2 , Cˆ 1 + 2 1 1 , (A.6) rˆ2 rˆ2 rˆ rˆ2 and20 √ m 2B − m − m 1 − e 1 2 1 ˆ− 2 ( m+ˆr2−c)2 2mc g2(r)ˆ ≡ = ρ = (ρ c) = 1 − . (A.7) 2Bˆ 1 − m 1 − m 1 − m rˆ3 e ρˆ2 ρˆ2 m+ˆr2

As a result we see that for the specific choice c = c1 we can transform to a metric where the boundary condition on Cˆ becomes 2 c2 + c /2 Cˆ 1 . (A.8) r2 However, in the process we have lost the conformal form with one single function multiplying the factor (drˆ2 + dz2). We therefore first need to transform back to the original conformal form, which we achieve by changing variables rˆ = u(r)˜ with the function u satisfying the differential equation ∂r˜u(r)˜ = g u(r)˜ . (A.9) Then we find the form 2 u (r)˜ ˆ ˜ ˆ ˜ ds2 =− e2A(u(r)) dt2 + e2B(u(r)) dr˜2 + dz2 m + u2(r)˜ + 2C(u(ˆ r))˜ + 2 ˜ 2 e m u (r) dΩ3 . (A.10) The differential equation (A.9) is pretty hard to solve, but we only need to examine it asymptotically in order to make sure that no 1/r˜ terms in the asymptotic conditions are induced. We thus consider its asymptotic form mc ∂˜u(r)˜ = 1 − , (A.11) r u3(r)˜ obtained using (A.7). The solution is

mc u(r)˜ =˜r 1 + +··· , (A.12) 2r˜3

20 The expressions for Aˆ, Cˆ in terms of A,C are easily obtained but not important in the following. T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208 207 and obeys the desired property. After this sequence of transformations leading to (A.10), we change notation back to the symbols in (A.1) (but now we may drop the 1/r term in the asymptotic boundary condition of C). What remains is to show that we can change coordinates to the two-function ansatz (3.4). We do not repeat these steps here, as they can be obtained from the general treatment in Section 3.1 by substituting the particular value d = 5 in equations (3.5)–(3.14). For that value, these general expressions reduce to the steps that are equivalent to those originally performed in [13].

Appendix B. A useful identity for static perturbations

In this appendix we derive an identity which holds for static and Ricci-flat perturbations of a static metric. The identity is the starting point in Section 4 for obtaining an important relation on the variation of the mass δM which is independent from the generalized Smarr formula (2.5). The derivation and form of this identity follows the corresponding one in [26] (Eq. (12.5.42)) where the four-dimensional case was treated. µ Consider in a D-dimensional space–time two vector fields v and w satisfying Dµv = 0 µ and Dµw = 0. Let v and w be commuting, i.e., Lvw = Lwv = 0. It is not difficult to show [µ ν] [µ ν] µ ν ν µ that as a consequence one has Dµv w = 0where2v w = v w − v w .Using Gauss theorem we then get that [µ ν] [µ ν] dSµν v w = dSµν v w . (B.1)

S1 S2 = ∂ We now consider a static metric gµν and take w as the vector ﬁeld w ∂t and let v be ∂ µ = µ = µ = given so that ∂t v 0. First we see that Dµw 0sinceΓµ0 0 because gµν is static 0 µ µ α µ with respect to t = x . Secondly we see that (Lwv) = ∂0v − v ∂αw = 0so[v,w]=0. µ µ Thus, we only need to require that ∂0v = 0andDµv = 0 to get the desired properties of v and w Consider now a static perturbation of the metric δgµν which leaves the Ricci tensor = = α − α unchanged, i.e., δRµν 0. Using the Palatini identity δRµν Dν δΓµα DαδΓµν we can µν write g δRµν as µν µ µ µα νβ µν αβ g δRµν =−Dµv ,v= Dν g g δgαβ − g g δgαβ . (B.2)

Using now the requirement δRµν = 0aswellasthatδgµν is a static perturbation, we find µ µ that the v defined in (B.2) satisfies ∂0v = 0andDµv = 0 as desired. We are now ready to state the advertised identity which generalizes Wald’s identity (12.5.42) in [26]. Consider a static metric gµν and a static perturbation of the metric δgµν with δRµν = 0. Choose two oppositely oriented disjoint (D − 2)-dimensional surfaces S1 and S2 so that ∂V = S1 ∪ S2 for some (D − 1)-dimensional volume V . Then we have that µ ν µ ν dSµν v w = dSµν v w , (B.3)

S1 S2 208 T. Harmark, N.A. Obers / Nuclear Physics B 684 (2004) 183–208 with ∂ vµ = D gµαgνβ δg − gµν gαβ δg ,w= . (B.4) ν αβ αβ ∂t

References

[1] T. Harmark, N.A. Obers, New phase diagram for black holes and strings on cylinders, Class. Quantum Grav. 21 (2004) 1709–1724, hep-th/0309116. [2] B. Kol, E. Sorkin, T. Piran, Caged black holes: Black holes in compactified spacetimes I—theory, hep- th/0309190. [3] R. Gregory, R. Laflamme, Black strings and p-branes are unstable, Phys. Rev. Lett. 70 (1993) 2837–2840, hep-th/9301052. [4] R. Gregory, R. Laflamme, The instability of charged black strings and p-branes, Nucl. Phys. B 428 (1994) 399–434, hep-th/9404071. [5] G.T. Horowitz, K. Maeda, Fate of the black string instability, Phys. Rev. Lett. 87 (2001) 131301, hep- th/0105111. [6] M.W. Choptuik, et al., Towards the final fate of an unstable black string, Phys. Rev. D 68 (2003) 044001, gr-qc/0304085. [7] S.S. Gubser, On non-uniform black branes, Class. Quantum Grav. 19 (2002) 4825–4844, hep-th/0110193. [8] T. Wiseman, Static axisymmetric vacuum solutions and non-uniform black strings, Class. Quantum Grav. 20 (2003) 1137–1176, hep-th/0209051. [9] R. Gregory, R. Laflamme, Hypercylindrical black holes, Phys. Rev. D 37 (1988) 305. [10] T. Harmark, N.A. Obers, Black holes on cylinders, JHEP 0205 (2002) 032, hep-th/0204047. [11] G.T. Horowitz, Playing with black strings, hep-th/0205069. [12] B. Kol, Topology change in general relativity and the black-hole black-string transition, hep-th/0206220. [13] T. Wiseman, From black strings to black holes, Class. Quantum Grav. 20 (2003) 1177–1186, hep- th/0211028. [14] T. Harmark, N.A. Obers, Black holes and black strings on cylinders, Fortschr. Phys. 51 (2003) 793–798, hep-th/0301020. [15] B. Kol, T. Wiseman, Evidence that highly non-uniform black strings have a conical waist, Class. Quantum Grav. 20 (2003) 3493–3504, hep-th/0304070. [16] R. Casadio, B. Harms, Black hole evaporation and large extra dimensions, Phys. Lett. B 487 (2000) 209–214, hep-th/0004004. [17] R. Casadio, B. Harms, Black hole evaporation and compact extra dimensions, Phys. Rev. D 64 (2001) 024016, hep-th/0101154. [18] G.T. Horowitz, K. Maeda, Inhomogeneous near-extremal black branes, Phys. Rev. D 65 (2002) 104028, hep-th/0201241. [19] P.-J. De Smet, Black holes on cylinders are not algebraically special, Class. Quantum Grav. 19 (2002) 4877– 4896, hep-th/0206106. [20] B. Kol, Speculative generalization of black hole uniqueness to higher dimensions, hep-th/0208056. [21] E. Sorkin, T. Piran, Initial data for black holes and black strings in 5d, Phys. Rev. Lett. 90 (2003) 171301, hep-th/0211210. [22] A.V. Frolov, V.P. Frolov, Black holes in a compactified spacetime, Phys. Rev. D 67 (2003) 124025, hep- th/0302085. [23] R. Emparan, R.C. Myers, Instability of ultra-spinning black holes, JHEP 0309 (2003) 025, hep-th/0308056. [24] J. Traschen, D. Fox, Tension perturbations of black brane spacetimes, gr-qc/0103106. [25] P.K. Townsend, M. Zamaklar, The first law of black brane mechanics, Class. Quantum Grav. 18 (2001) 5269–5286, hep-th/0107228. [26] R.M. Wald, General Relativity, University of Chicago Press, 1984. Nuclear Physics B 684 (2004) 209–234 www.elsevier.com/locate/npe

Quantum gravity at a large number of dimensions

N.E.J. Bjerrum-Bohr

The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark Received 5 November 2003; accepted 11 February 2004

Abstract We consider the large-D limit of Einstein gravity. It is observed that a consistent leading large- D graph limit exists, and that it is built up by a subclass of planar diagrams. The graphs in the effective field theory extension of Einstein gravity are investigated in the same context, and it is seen that an effective field theory extension of the basic Einstein–Hilbert theory will not upset the latter leading large-D graph limit, i.e., the same subclass of planar diagrams will dominate at large-D in the effective field theory. The effective field theory description of large-D quantum gravity limit will be renormalizable, and the resulting theory will thus be completely well defined up to the Planck scale at ∼ 1019 GeV. The 1/D expansion in gravity is compared to the successful 1/N expansion in gauge theory (the planar diagram limit), and dissimilarities and parallels of the two expansions are discussed. We consider the expansion of the effective field theory terms and we make some remarks on explicit calculations of n-point functions.  2004 Elsevier B.V. All rights reserved.

PACS: 04.60.-m; 11.10.-z

1. Introduction

A traditional belief is that an adequate description of quantum gravity is necessarily complicated, and that we are very far from a satisfying explanation of its true physical content. This might be correct, however, progress on the problems in quantum gravity are being made, and gravitational interactions are today better understood than previously. The fate of gravity in the limit of Planckian scale energies is still unknown. At the physical transition to such energies ∼ 1019 GeV, fundamental issues are believed to arise. They concern the validity of the conceptual picture of space and time geometry. A completely

E-mail address: [email protected] (N.E.J. Bjerrum-Bohr).

0550-3213/$ – see front matter  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2004.02.012 210 N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234 new understanding of physics might be the consequence. String theory or the so called M-theory are candidates for such theories in high energy physics. Einstein’s theory of general relativity, on the other hand, presents us with an excellent classical model for the interactions of matter and space–time in the low energy regime of physics. Astronomical observations support the theoretical predictions, and there are no experimental reasons for a disbelief in the physical foundations of general relativity. The fact that general relativity is not a quantum theory is, however, intriguing. A fundamental description of nature should either be completely classical or quantum mechanical from a theoretical point of view, and so far experiments support quantum mechanics as the correct basis for interpreting physical measurements. It would be nice to unify Einstein’s theory in the classical low energy limit with a full quantum field theoretic description in the high energy limit. This seems not to be immediately possible, attempts of making a quantum field theoretic description of general relativity have normally proven to be incomplete in regard to the question of renormalizability of the fundamental action for the theory. The basic four-dimensional Einstein–Hilbert action: √ 4 2R LEH = d x −g + Lmatter , (1) κ2 is not renormalizable in the sense that the counter-terms which appear in the renormalized 2 µν 3 action, e.g., ∼ R , ∼ Rµν R and ∼ R ,...,cannot be absorbed in the original action [1– 2 3]. In the above equation κ is defined as 32πGD,whereGD is the D-dimensional Newton constant. The disappointing conclusion is usually that a field theoretical treatment of gravity is not possible, and that one should avoid attempting such a description. Effective field theory presents a solution to the renormalization problem of gravity. It initially originates from an idea put forward by Weinberg [4]. Including all possible invariants into the action, i.e., replacing the basic D-dimensional Einstein–Hilbert action with: √ D 2R 2 µν Leffective EH = d x −g + c1R + c2R Rµν +··· + Leff. matter , (2) κ2 solves the issue of renormalizability. No counter terms can be generated in a covariant renormalization scheme, which are not already present in the action. The effective field theory approach will always lead to consistent renormalizable theories in any dimension, e.g., for D = 4, D = 10 or even D = 127. It is important to note that in regard to its renormalizability, an effective field theory in the large-D limit of gravity is just as well defined as the planar diagram large-N limit, in e.g., a D = 4 Yang–Mills theory. A Yang– Mills theory, at D>4, will also need to be treated by means of effective field theory in order to be renormalizable. What we give up in the effective field treatment of gravity or in any other theory, is the finite number of terms in the action. The Einstein action is replaced with an action which has an infinite number of coupling constants. These have to be renormalized at each loop order. The coupling constants of the effective expansion of the action, i.e., c1,c2,..., will have to be determined by experiment and are running, i.e., they will depend on the energy scale. However, the advantage of the effective approach is quite clear. Quantum gravity is an effectively renormalizable theory up to the energies at the Planck scale! N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234 211

The effective field theory description of general relativity has directly shown its worth by explicit calculations of quantum corrections to the metric and to the Newtonian potential of large masses. This work was pioneered by Donoghue, Refs. [5–7], and as an effective field theory combined with QED, Refs. [8,9]. The conceptual picture of Einstein gravity is not affected by the effective field theory treatment of the gravitational interactions. The equations of motion will of course be slightly changed by the presence of higher derivative terms in the action, however, the contributions coming from such terms can at normal energies be shown to be negligible [10,11]. Effective field theory thus provides us with a very solid framework in the study of gravitational interactions and quantum effects at normal energies, and solves some of the essential problems in building field theoretical quantum gravity models. An intriguing paper by Strominger [12] deals with the perspectives of basic Einstein gravity at an infinite number of dimensions. The basic idea is to let the spatial dimension be a parameter in which one is allowed to expand the theory. Each graph of the theory is then associated with a dimensional factor. Formally, one can expand every Greens function of the theory as a series in 1/D and the gravitational coupling constant κ: 1 j G = (κ)i G . (3) D i,j i,j The contributions with highest-dimensional dependence will be the leading ones in the large-D limit. Concentrating on these graphs only simplifies the theory and makes explicit calculations easier. In the effective field theory we expect an expansion of the theory of the type: 1 j G = (κ)i (E)2kG , (4) D i,j,k i,j,k where (E) represents a parameter for the effective expansion of the theory in terms of the energy, i.e., each derivative acting on a massless field will correspond to a factor of E.The basic Einstein–Hilbert scale will correspond to the E2 ∼ ∂2g contribution, while higher order effective contributions will be of order E4 ∼ ∂4g, E6 ∼ ∂6g,...,etc. The idea of making a large-D expansion in gravity is somewhat similar to the large-N Yang–Mills planar diagram limit considered by ’t Hooft [13]. In large-N gauge theories, one expands in the internal symmetry index N of the gauge group, e.g., SU(N) or SO(N). The physical interpretations of the two expansions are, of course, completely dissimilar, e.g., the number of dimensions in a theory cannot really be compared to the internal symmetry index of a gauge theory. A comparison of the Yang–Mills large-N limit (the planar diagram limit), and the large-D expansion in gravity is, however, still interesting to perform, and as an example of a similarity between the two expansions the leading graphs in the large-D limit in gravity consists of a subset of planar diagrams. Higher-dimensional models for gravity are well known from string and supergravity theories. The mysterious M-theory has to exist in an 11-dimensional space–time. So there are many good reasons to believe that on fundamental scales, we might experience additional space–time dimensions. Additional dimensions could be treated as free or as compactified below the Planck scale, as in the case of a Kaluza–Klein mechanism. One 212 N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234 application of the large-D expansion in gravity could be to approximate Greens functions at finite dimension, e.g., D = 4. The successful and various uses of the planar diagram limit in Yang–Mills theory might suggest other possible scenarios for the applicability of the large-D expansion in gravity. For example, is effective quantum gravity renormalizable in its leading large-D limit, i.e., is it renormalizable in the same way as some non- renormalizable theories are renormalizable in their planar diagram limit? It could also be, that quantum gravity at large-D is a completely different theory, than Einstein gravity. A planar diagram limit is essentially a string theory at large distances, i.e., could gravity at large-D be interpreted as a large distance truncated string limit? We will here combine the successful effective field theory approach which holds in any dimension, with the expansion of gravity in the large-D limit. The treatment will be mostly conceptual, but we will also address some of the phenomenological issues of this theory. The structure of the paper will be as follows. First, we will discuss the basic quantization of the Einstein–Hilbert action, and then we will go on with the large-D behavior of gravity, i.e., we will show how to derive a consistent limit for the leading graphs. The effective extension of the theory will be taken up in the large-D framework, and we will especially focus on the implications of the effective extension of the theory in the large-D limit. The D-dimensional space–time integrals will then be briefly looked upon, and we will make a conceptual comparison of the large-D in gravity and the large-N limit in gauge theory. Here we will point out some similarities and some dissimilarities of the two theories. We will also discuss some issues in the original paper by Strominger, Ref. [12]. We will work in units: h¯ = c = 1, and metric: diag(ηµν ) = (1, −1, −1, −1,...,−1), i.e., with D − 1 minus signs.

2. Review of the large-D limit of Einstein gravity

In this section we will review the main idea. To begin we will formally describe how to quantize a gravitational theory. As is well known the D-dimensional Einstein–Hilbert Lagrangian has the form: √ 2R L = dDx −g . (5) κ2 If we neglect the renormalization problems of this action, it is possible to make a formal quantization using the path integral approach. Introducing a gauge breaking term in the action will fix the gauge, and because gravity is a non-Abelian theory we have to introduce a proper Faddeev–Popov ghost action as well. The action for the quantized theory will consequently be: √ 2R L = dDx −g + L + L , (6) κ2 gauge fixing ghosts where Lgauge fixing is the gauge fixing term, and Lghosts is the ghost contribution. In order to generate vertex rules for this theory an expansion of the action has to be carried out, and vertex rules have to be extracted from the gauge fixed action. The vertices will depend on the gauge choice and on how we define the gravitational field. N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234 213

It is conventional to deﬁne:

gµν ≡ ηµν + κhµν , (7) λ = 1 λ and work in harmonic gauge: ∂ hµλ 2 ∂µhλ. For this gauge choice the vertex rules for the 3- and 4-point Einstein vertices can be found in [14,15]. Yet another possibility is to use the background ﬁeld method, here we deﬁne:

gµν ≡˜gµν + κhµν , (8) and expand the quantum fluctuation: hµν around an external source, the background field: g˜µν ≡ ηµν + κHµν . In the background field approach we have to differ between vertices with internal lines (quantum lines: ∼ hµν ) and external lines (external sources: ∼ Hµν ). There are no real problems in using either method; it is mostly a matter of notation and conventions. Here we will concentrate our efforts on the conventional approach. In many loop calculations the background field method is the easiest to employ. See Refs. [2,5–7] for additional details about, e.g., the vertex rules in the background field method. Another possibility is to define: √ g˜µν ≡ −ggµν = ηµν + κhµν , (9) this field definition makes a transformation of the Einstein–Hilbert action into the following form: 1 1 L = dDx g˜ρσ g˜ g˜ g˜ακ g˜λτ − g˜ρσ g˜ g˜ g˜ακ g˜λτ 2 λα κτ ,ρ ,σ − ακ λτ ,ρ ,σ 2κ D 2 ακ ρτ − 2g˜ατ g˜ ,ρg˜ ,κ , (10) in D = 4 this form of the Einstein–Hilbert action is known as the Goldberg action [16]. In this description of the Einstein–Hilbert action its dimensional dependence is more obvious. It is, however, more cumbersome to work with in the course of practical large-D considerations. In the standard expansion of the action the propagator for gravitons take the form: + − 2 i ηαγ ηβδ ηαδηβγ D−2 ηαβ ηγδ Dαβ,γ δ(k) =− . (11) 2 k2 − i The 3- and 4-point vertices for the standard expansion of the Einstein–Hilbert action can be found in [14,15] and has the following form: V (3) (k ,k ,k ) µα,νβ,σ γ 1 2 3 1 1 = κ sym − P (k · k ηµαηνβ ησγ) − P (k νk βηµα ησγ) 2 3 1 2 2 6 1 1 1 + P (k · k ηµν ηαβ ησγ) + P (k · k ηµαηνσηβγ ) + 2P (k νk γ ηµα ηβσ) 2 3 1 2 6 1 2 3 1 1 − P (k k η η ) + P (k k η η ) + P (k k η η ) 3 1β 2µ αν σγ 3 1σ 2γ µν αβ 6 1σ 1γ µν αβ

+ 2P6(k1νk2γ ηβµηασ ) + 2P3(k1νk2µηβσηγα) − 2P3(k1 · k2ηανηβσηγµ) , (12) 214 N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234 and

(4) V (k1,k2,k3,k4) µα,νβ,σγ,ρλ 2 1 1 = κ sym − P (k · k ηµαηνβ ησγηρλ) − P (k νk β ηµαησγηρλ) 4 6 1 2 4 12 1 1 1 1 − P (k νk µηαβ ησγηρλ) + P (k · k ηµν ηαβ ησγηρλ) 2 6 1 2 4 6 1 2 1 1 + P (k · k ηµαηνβ ησρηγλ) + P (k νk βηµαησρηγλ) 2 6 1 2 2 12 1 1 1 + P (k νk µηαβ ησρηγλ) − P (k · k ηµν ηαβ ησρηγλ) 6 1 2 2 6 1 2 1 1 + P (k · k ηµαηνσηβγ ηρλ) + P (k νk βηµσ ηαγ ηρλ) 2 24 1 2 2 24 1 1 1 + P (k σ k γ ηµν ηαβ ηρλ) + P (k νk σ ηβµηαγ ηρλ) 2 12 1 2 24 1 2 − P12(k1 · k2ηανηβσηγµηρλ) + P12(k1νk2µηβσηγαηρλ)

+ P12(k1νk1σ ηβγ ηµαηρλ) − P24(k1 · k2ηµα ηβσηγρηλν)

− 2P12(k1νk1βηασ ηγρηλµ) − 2P12(k1σ k2γ ηαρ ηλνηβµ)

− 2P24(k1νk2σ ηβρ ηλµηαγ ) − 2P12(k1σ k2ρηγνηβµηαλ)

+ 2P6(k1 · k2ηασ ηγνηβρ ηλµ) − 2P12(k1νk1σ ηµαηβρηλγ )

− P12(k1 · k2ηµσ ηαγ ηνρ ηβλ) − 2P12(k1νk1σ ηβγ ηµρ ηαλ) − P (k k η η η ) − 2P (k k η η η ) 12 1σ 2ρ γλ µν αβ 24 1ν 2σ βµ αρ λγ 

− 2P12(k1νk2µηβσηγρηλα) + 4P6(k1 · k2ηανηβσ ηγρηλµ) , (13) in the above two expressions, ‘sym’, means that each pair of indices: µα,νβ,..., will have to be symmetrized. The momenta factors: k1,k2,..., are associated with the index pairs: µα,νβ,..., correspondingly. The symbol: P# means that a #-permutation of indices and corresponding momenta has to carried out for this particular term. As seen the algebraic structures of the 3- and 4-point vertices are already rather involved and complicated. 5- point vertices will not be considered explicitly in this paper. The explicit prefactors of the terms in the vertices will not be essential in this treatment. The various algebraic structures which constitutes the vertices will be more important. Different algebraic terms in the vertex factors will generate dissimilar traces in the ﬁnal diagrams. It is useful to adapt an index line notation for the vertex structure similar to that used in large-N gauge theory. In this notation we can represent the different algebraic terms of the 3-point vertex (see Fig. 1). For the 4-point vertex we can use a similar diagrammatic notation (see Fig. 2). There are only two sources of factors of dimension in a diagram. A dimension factor can arise either from the algebraic contractions in that particular diagram or from the space– time integrals. No other possibilities exists. In order to arrive at the large-D limit in gravity N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234 215

∼ − 1 · sym 2 P3(k1 k2ηµαηνβ ησγ) , 3A ∼ 1 · + · sym 2 P3(k1 k2ηµνηαβ ησγ) P6(k1 k2ηµαηνσ ηβγ ) , 3B ∼ sym P3(k1σ k2γ ηµνηαβ ) , 3C ∼ sym 2P6(k1ν k2γ ηβµηασ ) + 2P3(k1ν k2µηβσηγα) , 3D ∼ sym −2P3(k1 · k2ηανηβσηγµ) , 3E ∼ sym 2P3(k1ν k1γ ηµαηβσ) − P3(k1β k2µηανησγ) , 3F ∼ − 1 sym 2 P6(k1ν k1β ηµαησγ) . 3G

Fig. 1. A graphical representation of the various terms in the 3-point vertex factor. A dashed line represents a contraction of a index with a momentum line. A full line means a contraction of two index lines. The above vertex notation for the indices and momenta also apply here. both sources of dimension factors have to be considered and the leading contributions will have to be derived in a systematic way. Comparing to the case of the large-N planar diagram expansion, only the algebraic structure of the diagrams has to be considered. The symmetry index N of the gauge group does not go into the evaluation of the integrals. Concerns with the D-dimensional integrals only arises in gravity. This is a great difference between the two expansions. A dimension factor will arise whenever there is a trace over a tensor index in a diagram, αµ αβ γµ e.g., ηµαη ,ηµαη ηβγ η ,...∼ D. The diagrams with the most traces will dominate the large-D limit in gravity. Traces of momentum lines will never generate a factor of D. It is easy to be assured that some diagrams naturally will generate more traces than other diagrams. The key result of Ref. [12] is that only a particular class of diagrams will constitute the large-D limit, and that only certain contributions from these diagrams will be important there. The large-D limit of gravity will correspond to a truncation of the Einstein theory of gravity, where in fact only a subpart, namely, the leading contributions of the graphs will dominate. We will use the conventional expansion of the gravitational ﬁeld, and we will not separate conformal and traceless parts of the metric tensor. To justify this we begin by looking on how traces occur from contractions of the propagator. We can graphically write the propagator in the way shown in Fig. 3. The only way we can generate index loops is through a propagator. We can graphically represent a propagator contraction of two indices in a particular amplitude in the following way (see Fig. 4). Essentially (disregarding symmetrizations of indices etc.) only the following contractions of indices in the conventional graviton propagator can occur in an given amplitude, we can contract, e.g., (α and β)orboth(α and β)and(γ and δ). The results of such 216 N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234

∼ − 1 · sym 4 P6(k1 k2ηµαηνβ ησγηρλ) , 4A ∼ − 1 sym 4 P12(k1ν k1β ηµαησγηρλ) , 4B ∼ − 1 sym 2 P6(k1ν k2µηαβ ησγηρλ) , 4C ∼ 1 P (k · k η η η η ) + 1 P (k · k η η η η ) sym 4 6 1 2 µν αβ σγ ρλ 2 6 1 2 µα νβ σρ γλ 4D + 1 · 2 P24(k1 k2ηµαηνσ ηβγ ηρλ) , ∼ sym 1 P (k k η η η ) + 1 P (k k η η η ) 2 12 1ν 1β µα σρ γλ 2 24 1ν 1β µσ αγ ρλ 4E + 1 2 P12(k1σ k2γ ηµνηαβ ηρλ) , ∼ sym −2P12(k1ν k1β ηασ ηγρηλµ) − 2P12(k1σ k2γ ηαρ ηλνηβµ) , 4F ∼ sym 2P6(k1 · k2ηασ ηγνηβρ ηλµ) + 4P6(k1 · k2ηανηβσηγρηλµ) , 4G ∼ sym −P12(k1 · k2ηανηβσηγµηρλ) − P24(k1 · k2ηµαηβσηγρηλν) , 4H ∼ sym −2P24(k1ν k2σ ηβρ ηλµηαγ ) − 2P12(k1σ k2ρ ηγνηβµηαλ) 4I − P12(k1σ k2ρ ηγληµνηαβ ) − 2P24(k1ν k2σ ηβµηαρ ηλγ ) − 2P12(k1ν k2µηβσηγρηλα) , ∼ − 1 · − · sym 2 P6(k1 k2ηµνηαβ ησρηγλ) P12(k1 k2ηµσ ηαγ ηνρ ηβλ) , 4J ∼ + sym P24(k1ν k2σ ηβµηαγ ηρλ) P12(k1ν k2µηβσηγαηρλ) 4K − 2P12(k1ν k1σ ηµαηβρ ηλγ ) , ∼ − sym P6(k1ν k2µηαβ ησρηγλ 2P12(k1ν k1σ ηβγ ηµρ ηαλ) 4L − P12(k1σ k2ρ ηγληµνηαβ ) .

Fig. 2. A graphical representation of the terms in the 4-point vertex factor. The notation here is the same as the above for the 3-point vertex factor.

Fig. 3. A graphical representation of the graviton propagator, where a full line corresponds to a contraction of two indices. contractions are shown below and also graphically depicted in Fig. 5. 1 2 αβ D ηαγ ηβδ + ηαδηβγ − ηαβ ηγδ η = ηγδ − ηγδ, 2 D − 2 D − 2 1 2 αβ γδ 2D ηαγ ηβδ + ηαδηβγ − ηαβ ηγδ η η =− . (14) 2 D − 2 D − 2 N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234 217

Fig. 4. A graphical representation of a propagator contraction in an amplitude. The black half dots represent an arbitrary internal structure for the amplitude, the full lines between the two half dots are internal contractions of indices, the outgoing lines represents index contractions with external sources.

Fig. 5. Two possible types of contractions for the propagator. Whenever we have an index loop we have a contraction of indices. It is seen that none of the above contractions will generate something with a positive power of D.

Fig. 6. Two other types of index contractions for the propagator. Again it is seen that only the ηαγ ηβδ + ηαδηβγ part of the propagator is important in large-D considerations.

We can also contract, e.g., (α and γ ) or both (α and γ )and(β and δ) again the results are shown below and depicted graphically in Fig. 6. 1 2 αγ 1 1 1 ηαγ ηβδ + ηαδηβγ − ηαβ ηγδ η = ηβδD + ηβδ + , 2 D − 2 2 2 D − 2 1 2 αγ βδ 2 D ηαγ ηβδ + ηαδηβγ − ηαβ ηγδ η η = D − . (15) 2 D − 2 D − 2

As it is seen, only the ηαγ ηβδ + ηαδηβγ part of the propagator will have the possibility ηµν ηαβ to contribute with a positive power of D. Whenever the D−2 term in the propagator is ∼ D in play, e.g., what remains is something that goes as D−2 and which consequently will have no support to the large-D leading loop contributions. We have seen that different index structures go into the same vertex factor. In Fig. 7 we depict the same 2-loop diagram, but for different index structures of the external 3-point vertex factor (i.e., 3C and 3E) for the external lines. Different types of trace structures will hence occur in the same loop diagram. Every diagram will consist of a sum of contributions with a varying number of traces. It is seen that the graphical depiction of the index structure 218 N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234

Fig. 7. Different trace structures in the same diagram, generated by different index structures in the vertex (i.e., 3C and 3E, respectively), the ﬁrst diagram have three traces, ∼ D3, the other diagram have only two traces, ∼ D2.

Fig. 8. A diagrammatic representation of some leading contributions to the 2-point 1-loop graph in the large-D limit.

Fig. 9. Some non-leading contributions to the 2-point 1-loop graph in the large-D limit. in the vertex factor is a very useful tool in determining the trace structure of various diagrams. The following type of two-point diagram (see Fig. 8) build from the 3B or 3C parts of the 3-point vertex will generate a contribution which carries a maximal number of traces compared to the number of vertices in the diagram. The other parts of the 3-point vertex factor, e.g., the 3A or 3E parts, respectively, will not generate a leading contribution in the large-D limit, however, such contributions will go into the non-leading contributions of the theory (see Fig. 9). The diagram built from the 3A index structure will, e.g., not carry a leading contribution due to our previous considerations regarding contractions of the indices in the propagator (see Fig. 5). It is crucial that we obtain a consistent limit for the graphs, when we take D →∞.In order to cancel the factor of D2 from the above type of diagrams, i.e., to put it on equal footing in powers of D as the simple propagator diagram, it is seen that we need to rescale κ → κ/D.Inthelarge-D limit κ/D will define the ‘new’ redefined gravitational coupling. A finite limit for the 2-point function will be obtained by this choice, i.e., the leading D-contributions will not diverge, when the limit D →∞is imposed. By this choice all leading n-point diagrams will be well defined and go as ∼ (κ/D)(n−2) in a 1/D expansion. Each external line will carry a gravitational coupling.1 However, it is important to note that our rescaling of κ is solely determined by the requirement of creating a finite consistent large-D limit, where all n-point tree diagrams and their loop corrections are on an equal footing. The above rescaling seems to be the obvious to do from the viewpoint of creating a physical consistent theory. However, we note that other limits for the graphs are, in fact, possible by other redefinitions of κ but such rescalings will, e.g., scale away the loop corrections to the tree diagrams, thus such rescalings must be seen as rather unphysical from the viewpoint of traditional quantum field theory.

1 A tadpole diagram will thus not be well deﬁned by this choice but instead go as ∼ D, however, because we can dimensionally regularize any tadpole contribution away, any idea to make the tadpoles well deﬁned in the large-D limit would occur to be strange. N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234 219

Fig. 10. Two multi-loop diagrams with many traces. The rescaling of κ will, however, render both diagrams as non-leading contributions compared to diagrams with separate loops. Here and in the next coming ﬁgures we will employ the following notation for the index structure of the lines originating from loop contributions. Whenever an line originates from a loop contribution it should be understood that there are only two possibilities for its index structure, i.e., the lines has to be of the same type as in the contracted index lines in the index structure 3B or as the dashed lines in 3C. No assumptions on the index structures for tree external lines are made, here any viable index structures are present, but we still represent such lines with a single full line for simplicity.

For any loop with two propagators the maximal number of traces are obviously two. However, we do not know which diagrams will generate the maximal number of powers of D in total. Some diagrams will have more traces than others, however, because of the rescaling of κ they might not carry a leading contribution after all. For example, if we 4 look at the two graphs depicted in Fig. 10, they will, in fact, go as D3 × κ ∼ 1 and D4 D 8 D5 × κ ∼ 1 respectively, and thus not be leading contributions. D8 D3 In order to arrive at the result of Ref. [12], we have to consider the rescaling of κ which will generate negative powers of D as well. Let us now discuss the main result of Ref. [12]. Let Pmax be the maximal power of D associated with a graph. Next we can look at the function Π: ∞ Π = 2L − (m − 2)Vm. (16) m=3

Here L is the number of loops in a given diagram, and Vm counts the number of m-point vertices. It is obviously true that Pmax must be less than or equal to Π. The above function counts the maximal number of positive factors of D occurring for a given graph. The ﬁrst part of the expression simply counts the maximal number of traces ∼ D2 from a loop, the part which is subtracted counts the powers of κ ∼ 1/D arising from the rescaling of κ. Next, we employ the two well-known formulas for the topology of diagrams: L = I − V + 1, (17) ∞ mVm = 2I + E, (18) m=3

= ∞ where I is the number of internal lines, V m=3 Vm is total number of vertices and E is the number of external lines. Putting these three formulas together one arrives at: ∞ Pmax 2 − E − 2 Vm. (19) m=3 Hence, the maximal power of D is obtained in the case of graphs with a minimal number of external lines, and two traces per diagram loop. In order to generate the maximum of two traces per loop, separated loops are necessary, because whenever a propagator index 220 N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234

Fig. 11. Examples of multi-loop diagrams with leading contributions in the large-D limit.

Fig. 12. Of the following two diagrams with double trace loops the ﬁrst one will be leading.

Fig. 13. Leading bubble 2-point diagrams.

Fig. 14. Another type of 2-point diagram build from the index structure 4J, which will be leading in the large-D limit. line is shared in a diagram we will not generate a maximal number of traces. Considering only separated bubble graphs, thus the following type of n-point (see Fig. 11) diagrams will be preferred in the large-D limit. Vertices, which generate unnecessary additional powers of 1/D, are suppressed in the large-D limit; multi-loop contributions with a minimal number of vertex lines will dominate over diagrams with more vertex lines. That is, in Fig. 12 the ﬁrst diagram will dominate over the second one. Thus, separated bubble diagrams will contribute to the large-D limit in quantum gravity. The 2-point bubble diagram limit is shown in Fig. 13. Another type of diagrams which can make leading contributions, is the diagrams depicted in Fig. 14. These contributions are of a type where a propagator directly combines different legs in a vertex factor with a double trace loop. Because of the index structure 4J in the 4-point vertex such contributions are possible. These types of diagrams will have the 4 same dimensional dependence: κ × D4 ∼ 1, as the 2-loop separated bubble diagrams. It D4 is claimed in [12], that such diagrams will be non-leading. We have, however, found no evidence of this in our investigations of the large-D limit. Higher-order vertices with more external legs can generate multi-loop contributions of the vertex-loop type, e.g., a 6-point vertex can in principle generate a 3-loop contribution to the 3-point function, a 8-point vertex a 4-loop contribution the 4-point function, etc. (see Fig. 15). N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234 221

Fig. 15. Particular examples of leading vertex-loop diagrams build from 6-point and 8-point vertex factors, respectively.

The 2-point large-D limit will hence consist of the sum of the propagator and all types of leading 2-point diagrams which one considered above, i.e., bubble diagrams and vertex-loops diagrams and combinations of the two types of diagrams. Ghost insertions will be suppressed in the large-D limit, because such diagrams will not carry two traces, however, they will go into the theory at non-leading order. The general n-point function will consist of diagrams built from separated bubbles or vertex-loop diagrams. We have devoted Appendix B to a study of the generic n-point functions.

3. The large-D limit and the effective ﬁeld theory extension of gravity

So far we have avoided the renormalization difficulties of Einstein gravity. In order to obtain a effective renormalizable theory up to the Planck scale we will have to introduce the effective field theory description. The most general effective Lagrangian of a D-dimensional gravitational theory has the form: √ 2 L = dDx −g R + c R2 + c R Rµν +··· + L , (20) κ2 1 2 µν eff. matter α where R µνβ is the curvature tensor g =−det(gµν ) and the gravitational coupling 2 is defined as before, i.e., κ = 32πGD. The matter Lagrangian includes in principle everything which couples to a gravitational field [3], i.e., any effective or higher derivative couplings of gravity to bosonic and or fermionic matter. In this paper we will, however, solely look at pure gravitational interactions. The effective Lagrangian for the theory is then reduced to invariants built from the Riemann tensors: √ D 2R 2 µν L = d x −g + c1R + c2Rµν R +··· . (21) κ2 In an effective field theory higher derivative couplings of the fields are allowed for, while the underlying physical symmetries of the theory are kept manifestly intact. In gravity the general action of the theory has to be covariant with respect to the external gravitational fields. The renormalization problems of traditional Einstein gravity are trivially solved by the effective field theory approach. The effective treatment of gravity includes all possible invariants and hence all divergences occurring in the loop diagrams can be absorbed in a renormalized effective action. The effective expansion of the theory will be an expansion of the Lagrangian in powers of momentum, and the minimal powers of momentum will dominate the effective field 222 N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234

eff eff eff eff eff eff = 3 + 3 + 3 + 3 + 3 V3 κ κ κ κ κ 3A 3B 3C 3D 3E eff eff eff eff + κ3 + κ3 + κ3 + κ3 . 3F 3G 3H 3I

2 2 Fig. 16. A graphical representation of the R and the Rµν index structures in the effective 3-point vertex factor. A dashed line represents a contraction of a index with a momentum line. A full line symbolizes a contraction of two index lines.

eff ∼ sym −P η η η 3c k2(k · k ) + c 1 (k · k )(k · k ) − 1 k2k2 3 µα νσ βγ 1 1 2 3 2 2 1 2 1 3 4 2 3 3B − 2 2 + 1 · 2 − 1 2 · P6 ηµαηνσ ηβγ 2c1k1k3 c2 2 (k1 k3) 4 k2(k1 k3) ,

eff ∼ sym −P k k η η [3c k · k ] − 2P k k η η c k2 3 1 µ 1α νσ βγ 1 2 3 6 1µ 1α νσ βγ 1 3 3C + 1 P k k η η [c k · k ] + P k k η η [c k · k ] 2 6 1µ 2α νσ βγ 2 1 3 6 1µ 3α νσ βγ 2 1 3 − 1 P k k η η [c k2] − 1 P k k η η c k2 2 3 2µ 3α νσ βγ 2 1 2 6 3µ 3α νσ βγ 2 1 − 1 2 − 1 2 2 P6 k1β k3ν ηµσ ηαγ c2k2 2 P6 k3β k3ν ηµσ ηαγ c2k2 .

Fig. 17. Explicit index structure terms in the effective vertex 3-point factor, which can contribute with leading contributions in the large-D limit. theory at low energy scales. In gravity this means that the R ∼ (momentum)2 term will be dominant at normal energies, i.e., gravity as an effective field theory at normal energies will essentially still be general relativity. At higher energy scales the higher derivative terms R2 ∼ (momentum)4,..., R3 ∼ (momentum)6 ..., corresponding to higher powers of momentum will mix in and become increasingly important. The large-D limit in Einstein gravity is arrived at by expanding the theory in D and subsequently taking D →∞. This can be done in its effective extension too. Treating gravity as an effective theory and deriving the large-D limit results in a double expansion of the theory, i.e., we expand the theory both in powers of momentum and in powers of 1/D. In order to understand the large-D limit in the case of gravity as an effective field theory, we have to examine the vertex structure for the additional effective field theory terms. Clearly, any vertex contribution from, e.g., the effective field theory terms, R2, and, 2 Rµν , will have four-momentum factors and hence new index structures are possible for the effective vertices. 2 2 We find the following vertex structure for the R and the Rµν terms of the effective action (see Fig. 16 and Appendix A). We are particularly interested in the terms which will give leading trace contributions. For the 3Beff and 3Ceff index structures of we have shown some explicit results (see Fig. 17). It is seen that also in the effective theory there are terms which will generate double trace structures. Hence, an effective 1-loop bubble diagram with two traces is possible. 4 Such diagrams will go as κ c × D2 ∼ c1 . In order to arrive at the same limit for the D4 1 D2 N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234 223 effective tree diagrams as the tree diagrams in Einstein gravity we need to rescale c1 and 2 2 2 c2 by D , i.e., c1 → c1D and c2 → c2D . Thereby, every n-point function involving the effective vertices will go into the n-point functions of Einstein gravity. In fact, this will hold for any ci provided that the index structures giving double traces also exists for the higher- order effective terms. Other rescalings of the ci ’s are possible. But such other rescalings will give a different tree limit for the effective theory at large-D. The effective tree-level graphs will be thus be scaled away from the Einstein tree graph limit. 2 The above rescaling with ci → ciD corresponds to the maximal rescaling possible. n Any rescaling with, e.g., ci → ci D , n>2, will not be possible, if we still want to obtain a finite limit for the loop graphs at D →∞. A rescaling with D2 will give maximal support to the effective terms at large-D. The effective terms in the action are needed in order to renormalize infinities from the loop diagrams away. The renormalization of the effective field theory can be carried out at any particular D. Of course, the exact cancellation of pole terms from the loops will depend on the integrals, and the algebra will change with the dimension, but one can take this into account for any particular order of D = Dp by an explicit calculation of the counter terms at dimension Dp followed by an adjustment of the coefficients in the effective action. The exact renormalization will in this way take place order by order in 1/D. For any rescaling of the ci ’s it is thus always possible carry out a renormalization of the theory. Depending on how the rescaling of the coefficients ci are done, there will be different large-D limits for the effective field theory extension of the theory. The effective field theory treatment does, in fact, not change the large-D of gravity as much as one could expect. The effective enlargement of gravity does not introduce new exciting leading diagrams. Most parts of our analysis of the large-D limit depends only on the index structure for the various vertices, and the new index structures provided by the effective field theory terms only add new momentum lines, which do not give additional traces over index loops. Therefore, the effective extension of Einstein gravity is not a very radical change of its large-D limit. In the effective extension of gravity the action is trivially renormalizable up to a cut-off scale at MPlanck and the large-D expansion just as well defined as an large-N expansion of a renormalizable planar expansion of a Yang–Mills action. As a subject for further investigations, it might be useful to employ more general considerations about the index structures. Seemingly all possible types of index-structures are present in the effective field theory extension of gravity in the conventional approach. As only certain index structures go into the leading large-D limit, the ‘interesting’ index structures can be classified at any given loop order. In fact, one could consider loop amplitudes with only ‘interesting’ index structures present. Such applications might useful in very complicated quantum gravity calculations, where only the leading large-D behavior is interesting. A comparison of the index structures present in other field expansions and the conventional expansion might also be a working area for further investigations. Every index structure in the conventional expansion of the field can be build up from the following types of index structures, see Fig. 18. Thus, it is not possible to consider any index structures in a 3- or 4-point vertex, which cannot be decomposed into some product of the above types of index contractions. It might 224 N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234

Fig. 18. The ﬁrst 8 basic index structures which builds up any conventional vertex factor. be possible to extend this to something useful in the analysis of the large-D limit, e.g., a simpler description of the large-D diagrammatic truncation of the theory. This is another interesting working area for further research.

4. Considerations about the spaceÐtime integrals

The space–time integrals pose certain fundamental restrictions to our analysis of the large-D limit in gravity. So far we have considered only the algebraic trace structures of the n-point diagrams. This analysis lead us to a consistent large-D limit of gravity, where we saw that only bubble and vertex-loop diagrams will carry the leading contributions in 1/D. In a large-N limit of a gauge theory this would have been adequate, however, the space– time dimension is much more that just a symmetry index for the gauge group, and has a deeper significance for the physical theory than just that of a symmetry index. Space–time integrations in a D-dimensional world will contribute extra-dimensional dependencies to the graphs, and it thus has to be investigated if the D-dimensional integrals could upset the algebraic large-D limit. The issue about the D-dimensional integrals is not resolved in Ref. [12], only explicit examples are discussed and some conjectures about the contributions from certain integrals in the graphs at large-D are stated. It is clear that in the ∼ D−1 case of the 1-loop separated integrals the dimensional dependence will go as ( 2 ). Such a dimensional dependence can always be rescaled into an additional redefinition of κ, i.e., we can redefine κ → √κ where C ∼ (D−1 ). The dimensional dependence from both C 2 bubble graphs and vertex-loop graphs will be possible to scale away in this manner. Graphs which have a nested structure, i.e., which have shared propagator lines, will, however, be suppressed in D compared to the separated loop diagrams. Thus the rescaling of κ will make the dimensional dependence of such diagrams even worse. In the lack of rigorously proven mathematical statements, it is hard to be completely certain. But it is seen, that in the large-D limit the dimensional dependencies from the integrations in D dimensions will favor the same graphs as the algebraic graphs limit. The effective extension of the gravitational action does not pose any problems in these considerations, concerning the dimensional dependencies of the integrals and the extra rescalings of κ. The conjectures made for the Einstein–Hilbert types of leading loop diagrams hold equally well in the effective field theory case. A rescaling of the effective coupling constants have of course to be considered in light of the additional rescaling of κ. We will have ci → Cci in this case. Nested loop diagrams, which algebraically will be down in 1/D compared to the leading graphs, will also be down in 1/D in the case of an effective field theory. To directly resolve the complications posed by the space–time integrals, in principle all classes of integrals going into the n-point functions of gravity have to be investigated. Such N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234 225 an investigation would indeed be a very ambitious task and it is outside the scope of this investigation. However, as a point for further investigations, this would be an interesting place to begin a more rigorous mathematical justification of the large-D limit. To avoid any complications with extra factors of D arising from the evaluation of the integrals, one possibility is to treat the extra dimensions in the theory as compactified Kaluza–Klein space–time dimensions. The extra dimensions are then only excited above the Planck scale. The space–time integration will then only have to be preformed over a finite number of space–time dimensions, e.g., the traditional first four. In such a theory the large-D behavior is solely determined by the algebraic traces in the graphs and the integrals cannot not contribute with any additional factors of D to the graphs. Having a cutoff of the theory at the Planck scale is fully consistent with effective treatment of the gravitational action. The effective field theory description is anyway only valid up until the Planck scale, so an effective action having a Planckian Kaluza–Klein cutoff of the momentum integrations, is a possibility.

5. Comparison of the large-N limit and the large-D limit

The large-N limit of a gauge theory and the large-D limit in gravity has some similarities and some dissimilarities which we will look upon here. The large-N limit in a gauge theory is the limit where the symmetry indices of the gauge theory can go to inﬁnity. The idea behind the gravity large-D limit is to do the same, where in this case D will play the role of N, we thus treat the spatial dimension D as if it is only a symmetry index for the theory, i.e., a physical parameter, in which we are allowed to make an expansion. As it was shown by ’t Hooft in Ref. [13] the large-N limit in a gauge theory will be a planar diagram limit, consisting of all diagrams which can be pictured in a plane. The leading diagrams will be of the type depicted in Fig. 19. The arguments for this diagrammatic limit follow similar arguments to those we applied in the large-D limit of gravity. However, the index loops in the gauge theory are index loops of the internal symmetry group, and for each type of vertex there will only be one index

Fig. 19. A particular planar diagram in the large-N limit. A ‘×’ symbolizes an external source, the full lines are index lines, an full internal index loop gives a factor of N. 226 N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234

Fig. 20. The index structure for the 3-point in vertex in a gauge theory. structure. In Fig. 20 we show the particular index structure for a 3-point vertex in a gauge theory. This index structure is also present in the gravity case, i.e., in the 3E index structure, but various other index structure are present too, e.g., 3A, 3B, ..., and each index structure will generically give dissimilar traces, i.e., dissimilar factors of D for the loops. In gravity only certain diagrams in the large-D limit carry leading contributions, but only particular parts of the graph’s amplitudes will survive when D →∞because of the different index structures in the vertex factor. The gravity large-D limit is hence more like a truncated graph limit, where only some parts of the graph amplitude will be important, i.e., every leading graph will also carry contributions which are non-leading. In the Yang–Mills large-N limit the planar graph’s amplitudes will be equally important as a whole. No truncation of the graph’s amplitudes occur. Thus, in a Yang–Mills theory the large-N amplitude is found by summing the full set of planar graphs. This is a major dissimilarity between the two expansions The leading diagram limits in the two expansions are not identical either. The leading graphs in gravity at large-D are given by the diagrams which have a possibility for having a closed double trace structure, hence, the bubble and vertex-loop diagrams are favored in the large-D limit. The diagrams contributing to the gravity large-D limit will thus only be a subset of the full planar diagram limit.

6. Other expansions of the ﬁelds

In gravity it is a well-known fact that different definitions of the fields will lead to different results for otherwise similar diagrams. Only the full amplitudes will be gauge-invariant and identical for different expansions of the gravitational field. Therefore, comparing the large-D diagrammatical expansion for dissimilar definitions of the gravitational field, diagram for diagram, will have no meaning. The analysis of the large-D limit carried out in this paper is based on the conventional definition of the gravitational field, and the large-D diagrammatic expansion should hence be discussed in this light. Other field choices may be considered too, and their diagrammatic limits should be investigated as well. It may be worth some efforts and additional investigations to see, if the diagrammatic large-D limit considered here is an unique limit for every definition of the field, i.e., to see if the same diagrammatic limit always will occur for any definition of the gravitational field. A scenario for further investigations of this could, e.g., be to look N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234 227 upon if the field expansions can be dressed in such a way that we reach a less complicated diagrammatic expansion at large-D, at the cost of having a more complicated expansion of the Lagrangian. For example, it could be that vertex-loop corrections are suppressed for some definitions of the gravitational field, but that more complicated trace structures and cancellations between diagrams occurs in the calculations. In the case of the Goldberg definition of the gravitational field the vertex index structure is less complicated [12], however, the trace contractions of diagrams appears to much more complicated than in the conventional method. It should be investigated more carefully if the Goldberg definition of the field may be easier to employ in calculations. Indeed, the diagrammatical limit considered by [12] is easier, but it is not clear if results from two different expansions of the gravitational field are used to derive this. Further investigations should sheet more light on the large-D limit, and on how a field redefinition may or may not change its large-D diagrammatic expansion. Furthermore, one can consider the background field method in the context of the large- D limit as well. This is another working area for further investigations. The background field method is very efficient in complicated calculations and it may be useful to employ it in the large-D analysis of gravity as well. No problems in using background field methods together with a large-D quantum gravity expansion seems to be present.

7. Discussion

In this paper we have discussed the large-D limit of effective quantum gravity. In the large-D limit, a particular subset of planar diagrams will carry all leading 1/D contributions to the n-point functions. The large-D limit for any given n-point function will consist of the full tree n-point amplitude, together with a set of loop corrections which will consist of the leading bubble and vertex-loop graphs we have considered. The effective treatment of the gravitational action insures that a renormalization of the action is possible and that none of the n-point renormalized amplitudes will carry uncancelled divergent pole terms. An effective renormalization of the theory can be preformed at any particular dimensionality. The leading 1/D contributions to the theory will be algebraically less complex than the graphs in the full amplitude. Calculating only the 1/D leading contributions simplifies explicit calculations of graphs in the large-D limit. The large-D limit we have found is completely well defined as long as we do not extend the space–time integrations to the full D-dimensional space–time. That is, we will always have a consistent large-D theory as long as the integrations only gain support in a finite-dimensional space–time and the remaining extra dimensions in the space–time is left as compactified, e.g., below the Planck scale. We hence have a renormalizable, definite and consistent large-D limit for effective quantum gravity—good below the Planck scale! No solution to the problem of extending the space–time integrals to a full D- dimensional space–time has been presented in this paper, however, it is clear that the effective treatment of gravity do not impose any additional problems in such investigations. The considerations about the D-dimensional integrals discussed in Ref. [12] hold equally 228 N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234 well in a effective field theory, however, as well as we will have to make an additional rescaling of κ in order to account for the extra-dimensional dependence coming from the D-dimensional integrals, we will have make an extra rescaling of the coefficients ci in the effective Lagrangian too. Our large-D graph limit is not in agreement with the large-D limit of Ref. [12], the bubble graph are present in both limits, but vertex-loops are not allowed and claimed to be down in 1/D the latter. We do not agree on this point, and believe that this might be a consequence of comparing similar diagrams with different field definitions. It is a well-known fact in gravity that dissimilar field definitions lead to different results for the individual diagrams. Full amplitudes are of course unaffected by any particular definition of the field, but results for similar diagrams with different field choices cannot be immediately compared. Possible extensions of the large-D considerations discussed here would be to allow for external matter in the Lagrangian. This might present some new interesting aspects in the analysis, and would relate the theory more directly to external physical observables, e.g., scattering amplitudes, and corrections to the Newton potential at large-D or to geometrical objects such as a space–time metric. The investigations carried out in [6,7] could be discussed from a large-D point of view. Investigations in quantum gravitational cosmology may also present an interesting work area for applications of the large-D limit, e.g., quantum gravity large-D big bang models, etc. The analysis of the large-D limit in quantum gravity have prevailed that using the physical dimension as an expansion parameter for the theory, is not as uncomplicated as expanding a Yang–Mills theory at large N. The physical dimension goes into so many aspects of the theory, e.g., the integrals, the physical interpretations etc, that it is far more complicated to understand a large-D expansion in gravity than a large-N expansion in a Yang–Mills theory. In some sense it is hard to tell what the large-D limit of gravity really physically relates to, i.e., what describes a gravitational theory with infinitely many spatial dimensions? One way to look at the large-D limit in gravity, might be to see the large-D as a physical phase transition of the theory at the Planck scale. This suggests that the large-D expansion of gravity should be seen as a very high energy scale limit for a gravitational theory. Large-N expansions of gauge theories [13] have many interesting calculational and fundamental applications in, e.g., string theory, high energy, nuclear and condensed matter physics. A planar diagram limit for a gauge theory will be a string theory at large distance, this can be implemented in relating different physical limits of gauge theory and fundamental strings. In explicit calculations, large-N considerations have many important applications, e.g., in the approximation of QCD amplitudes from planar diagram considerations or in the theory of phase-transitions in condensed matter physics, e.g., in theories of superconductivity, where the number of particles will play the role of N. High energy physics and string theory are not the only places where large symmetry investigations can be employed to obtain useful information. As a calculational tool large- D considerations can be imposed in quantum mechanics to obtain useful information about eigenvalues and wave-functions of arbitrary quantum mechanical potentials with an extremely high precision [17]. N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234 229

Practical applications of the large-D limit in explicit calculations of n-point functions could be the scope for further investigations as well as certain limits of string theory might be related to the large-D limit in quantum gravity. The planar large-D limit found in our investigation might resemble that of a truncated string limit at large distances? Investigations of the (Kawai–Lewellen–Tye [18]) open/closed string, gauge theory/gravity relations could also be interesting to preform in the context of a large-D expansion. (See, e.g., [19–22] for recent work on this subject. See also [23,24].) Such questions are outside the scope of this investigation, however, knowing the huge importance of large- N considerations in modern physics, large-D considerations of gravity might pose very interesting tasks for further research.

Acknowledgements

I would like to thank Poul Henrik Damgaard for suggesting this investigation and for interesting discussions and advise.

Appendix A. Effective 3- and 4-point vertices

The effective Lagrangian takes the form: √ 2R L = dDx −g + c R2 + c R Rµν +··· . (A.1) κ2 1 2 µν √ √ − 2 − 2 In order to find the effective vertex factors we need to expand gR and gRµν . We are working in the conventional expansion of the field, so we define:

gµν ≡ ηµν + κhµν . (A.2) √ To second order we find the following expansion for −g: √ 2 2 1 κ α κ β α κ α 2 −g = exp log(ηµν + κhµν ) = 1 + h − h h + h ... . (A.3) 2 2 α 4 α β 8 α To first order in κ we have the following expansion for R: (1) = α β − α β R κ ∂α∂ hβ ∂ ∂ hαβ , (A.4) and to second order in κ2 we find: (2) 2 1 β α µ 1 β να α R = κ − ∂α h ∂ h + ∂β h 2∂αh − ∂ν h 2 µ β 2 ν α 1 ν ν ν α β βα β α + ∂αh + ∂β h − ∂ hβα ∂ h + ∂ν h − ∂ h 4 β α ν ν − 1 να − ν α β − 1 να β 2∂αh ∂ hα ∂ν hβ h ∂ν ∂αhβ 4 2 1 ν α β βα β α + h ∂β ∂ h + ∂ν h − ∂ h . (A.5) 2 α ν ν 230 N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234

InthesamewayweﬁndforRµν : (1) = κ β − β − β + 2 Rνα ∂ν ∂αhβ ∂β ∂αhν ∂β ∂ν hα ∂ hνα , (A.6) 2 (2) 2 1 βλ 1 βλ R = κ − ∂α h ∂ν hλβ + ∂β h (∂ν hλα + ∂αhνλ − ∂λhνα) να 2 2 + 1 λ + λ − λ λ β + β − β ∂β hν ∂ν hβ ∂ hνβ ∂ hα ∂αhλ ∂ hνα 4 1 − ∂ hλ + ∂ hλ − ∂λ ∂ hβ . (A.7) 4 α ν ν α αν λ β 2 2 From these equations we can expand to ﬁnd R and Rµν . Formally, we can write: √ 1 −gR2 = hαR(1)R(1) + 2R(1)R(2) 2 α 3 = κ γ α β − α β σ ρ − σ ρ hγ ∂α∂ hβ ∂ ∂ hαβ ∂σ ∂ hρ ∂ ∂ hσρ 2 3 σ ρ σ ρ 1 β α µ + 2κ ∂σ ∂ h − ∂ ∂ hσρ − ∂α h ∂ h ρ 2 µ β 1 β να α 1 ν ν ν + ∂β h 2∂αh − ∂ν h + ∂αh + ∂β h − ∂ hβα 2 ν α 4 β α × α β + βα − β α − 1 να − ν α β ∂ hν ∂ν h ∂ hν 2∂αh ∂ hα ∂ν hβ 4 1 να β 1 ν α β βα β α − h ∂ν ∂αh + h ∂β ∂ h + ∂ν h − ∂ h , (A.8) 2 β 2 α ν ν

√ 1 −gR2 = hγ R(1)R(1)µν − 2hαβ R(1)R(1)µ + 2R(1)R(2)µν µν 2 γ µν µα β µν κ3 = γ β − β − β + 2 hγ ∂ν ∂αhβ ∂β ∂αhν ∂β ∂νhα ∂ hνα 8 × ν α ρ − α νρ − ν αρ + 2 να ∂ ∂ hρ ∂ρ∂ h ∂ρ∂ h ∂ h 3 − κ ρσ γ − γ − γ + 2 h ∂µ∂σ hγ ∂γ ∂σ hµ ∂γ ∂µhσ ∂ hµσ 2 µ β µβ µ β 2 µ × ∂ ∂ρ h − ∂β ∂ρ h − ∂β ∂ h + ∂ h β ρ ρ + κ3 ∂ ∂ hγ − ∂ ∂ hγ − ∂ ∂ hγ + ∂2h ν α γ γ α ν γ ν α να 1 α βλ ν 1 βλ ν α α ν να × − ∂ h ∂ hλβ + ∂β h ∂ h + ∂ h − ∂λh 2 2 λ λ + 1 νλ + ν λ − λ ν βα + α β − β α ∂β h ∂ hβ ∂ hβ ∂λh ∂ hλ ∂ hλ 4 1 α νλ ν αλ λ αν β − ∂ h + ∂ h − ∂ h ∂λh . (A.9) 4 β N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234 231

Contracting only on terms which go into the 3Beff and 3Ceff index structures, and putting κ = 1 for simplicity, we have found for the R2 contribution: 2 = 2 ρ −3 β α µ − µ 2 β (R )(3B)eff ∂ hρ ∂αhµ∂ hβ 2hβ ∂ hµ , (A.10) 2 2 σ ρ 3 β α µ µ 2 β (R ) eff =−∂ ∂ hσρ − ∂αh ∂ h − 2h ∂ h . (A.11) (3C) 2 µ β β µ 2 For the Rµν contribution we have found:

2 1 γ 2 2 να 1 γ β να λ R eff = h ∂ hνα∂ h − ∂λ∂β h ∂ h ∂ hνα µν (3B) 8 γ 4 γ 1 γ να β λ 1 γ 2 λ να − ∂λ∂β h h ∂ ∂ hνα + ∂λh ∂ hνα∂ h , (A.12) 2 γ 4 γ and 1 R2 = ∂ ∂ hλ ∂σ hνα∂βh + ∂ ∂ hλ hνα∂σ ∂β h µν (3C)eff 2 λ β σ να λ β σ να 1 2 λ να β 1 2 να λ β − ∂ hλβ ∂ h ∂ hνα − ∂ hλβ h ∂ ∂ hνα 4 2 1 βλ 2 να 1 βλ 2 να − ∂β h ∂ hνα∂λh − h ∂ hνα∂λ∂β h . (A.13) 2 2 This leads to the following results for the 3B and 3C terms:

eff ∼ sym −P k k η η [3c k · k ] − 2P k k η η c k2 3 1 µ 1α νσ βγ 1 2 3 6 1µ 1α νσ βγ 1 3 3C + 1 P k k η η [c k · k ] + P k k η η [c k · k ] 2 6 1µ 2α νσ βγ 2 1 3 6 1µ 3α νσ βγ 2 1 3 − 1 P k k η η c k2 − 1 P k k η η c k2 2 3 2µ 3α νσ βγ 2 1 2 6 3µ 3α νσ βγ 2 1 − 1 2 − 1 2 2 P6 k1β k3ν ηµσ ηαγ c2k2 2 P6 k3β k3ν ηµσ ηαγ c2k2 .

Using the same procedure, we can ﬁnd expression for non-leading effective contributions, however, this will have no implications for the loop contributions. We have only calculated 3-point effective vertex index structures in this appendix, 4-point index structures are tractable too by the same methods, but the algebra gets much more complicated in this case.

Appendix B. Comments on general n-point functions

In the large-D limit only certain graphs will give leading contributions to the n-point functions. The diagrams favored in the large-D will have a simpler structure than the full 232 N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234

Fig. 21. The leading 1-loop n-point diagram, the index structure for the external lines is not decided, they can have the momentum structure, (i.e., as in 3C), or they can be of the contracted type, (i.e., as in 3B). External lines can of course also originate from 4-point vertex or a higher vertex index structure such as, e.g., the index structures 4D or 4E. This possibility will, however, not affect the arguments in this appendix, so we will leave this as a technical issue to be dealt with in explicit calculations of n-point functions.

Fig. 22. A diagrammatic expression for the generic n-point 1-loop correction.

Fig. 23. Some generic examples of graphs with leading contributions in the gravity large-D limit. The external lines originating from a loop can only have two leading index structures (i.e., they can be of the same type as the external lines originating from a 3B or a 3C) loop contribution. Tree external lines are not restricted to any particular structure, here the full vertex factor will contribute. n-point graphs. Calculations which would be too complicated to do in a full graph limit, might be tractable in the large-D regime. The n-point contributions in the large-D limit are possible to employ in approximations of n-point functions at any ﬁnite dimension, exactly as the n-point planar diagrams in gauge theories can be used, e.g., at N = 3, to approximate n-point functions. Knowing the full n-point limit is hence very useful in the course of practical approximations of generic n-point functions at ﬁnite D. This appendix will be devoted to the study of what is needed in order calculate n-point functions in the large-D limit. In the large-D limit the leading 1-loop n-point diagrams will be of the type, as displayed in Fig. 21. The completely generic n-point 1-loop correction will have the diagrammatic expression, as shown in Fig. 22. More generically the large-D, N-loop corrections to the n-point function will consist of combinations of leading bubble and vertex-loop contributions, such as shown in Fig. 23. N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234 233

For the bubble contributions it is seen that all loops are completely separated. Therefore, everything is known once the 1-loop contributions are calculated. That is, in order to derive the full n-point sole bubble contribution it is seen that if we know all j-point tree graphs, as well as all i-point 1-loop graphs (where i, j n) are known, the derivation the full bubble contribution is a matter of contractions of diagrams and combinatorics. For the vertex-loop combinations the structure of the contributions are a bit more complicated to analyze. The loops will in this case be connected through the vertices, not via propagators, and the integrals can therefore be joined together by contractions of the their loop momenta. However, this does, in fact, not matter; we can still do the diagrams if we know all integrals which occurs in the generic 1-loop n-point function. The integrals in the vertex-loop diagrams will namely be of completely the same type as the bubble diagram integrals. There will be no shared propagator lines which will combine the denominators of the integrals. The algebraic contractions of indices will, however, be more complicated to do for vertex-loop diagrams than for sole bubble loop diagrams. Multiple index contractions from products of various integrals will have to be carried out in order to do these types of diagrams. Computer algebraic manipulations can be employed to do the algebra in the diagrams, so we will not focus on that here. The real problem lays in the mathematical problem calculating generic n-point integrals. In the Einstein–Hilbert case, where each vertex can add only two powers of momentum, we see that problem of this diagram is about doing integrals such as: µ ν µ ν µ ν ··· l 1 l 1 l 2 l 2 ···l n l n (µ1ν1µ2ν2 µnνn) = D In d l 2 2 2 2 , l (l + p1) (l + p1 + p2) ···(l + p1 + p2 +···+pn) (B.1) ν µ ν µ ν ··· l 1 l 2 l 2 ···l n l n (ν1µ2ν2 µnνn) = D In d l 2 2 2 2 , l (l + p1) (l + p1 + p2) ···(l + p1 + p2 +···+pn) (B.2) . . = D 1 In d l 2 2 2 2 . (B.3) l (l + p1) (l + p1 + p2) ···(l + p1 + p2 +···+pn) Because the graviton is a massless particle such integrals will be rather badly defined. The denominators in the integrals are close to zero, and this makes the integrals divergent. One way to deal with these integrals is to use the following parametrization of the propagator: ∞ 1 = dxexp −xq2 , for q2 > 0, (B.4) q2 0 and define the momentum integral over a complex 2w-dimensional Euclidean space–time. Thus, what is left to do is Gaussian integrals in a 2w-dimensional complex space–time. Methods to do such types of integrals are investigated in [16,25–27]. The results for: I2, (µ) (µν) I2 , I2 and I3 are, in fact, explicitly stated there. It is outside our scope to actually delve into technicalities about explicit mathematical manipulations of integrals, but it should be 234 N.E.J. Bjerrum-Bohr / Nuclear Physics B 684 (2004) 209–234 clear that mathematical methods to deal with the 1-loop n-point integral types exist and that this could an working area for further research. Effective field theory will add more derivatives to the loops, thus the nominator will carry additional momentum contributions. Additional integrals will hence have to be carried out in order to do explicit diagrams in an effective field theory framework.

References

[1] G. ’t Hooft, M.J. Veltman, Ann. Inst. H. Poincaré Phys. Theor. A 20 (1974) 69. [2] M.H. Goroff, A. Sagnotti, Nucl. Phys. B 266 (1986) 709; A.E. van de Ven, Nucl. Phys. B 378 (1992) 309. [3] S. Deser, P. van Nieuwenhuizen, Phys. Rev. D 10 (1974) 401; S. Deser, P. van Nieuwenhuizen, Phys. Rev. D 10 (1974) 411; S. Deser, H.S. Tsao, P. van Nieuwenhuizen, Phys. Rev. D 10 (1974) 3337. [4] S. Weinberg, Physica A 96 (1979) 327. [5] J.F. Donoghue, Phys. Rev. D 50 (1994) 3874, gr-qc/9405057. [6] N.E.J. Bjerrum-Bohr, J.F. Donoghue, B.R. Holstein, Phys. Rev. D 68 (2003) 084005, hep-th/0211071. [7] N.E.J. Bjerrum-Bohr, J.F. Donoghue, B.R. Holstein, Phys. Rev. D 67 (2003) 084033, hep-th/0211072. [8] J.F. Donoghue, B.R. Holstein, B. Garbrecht, T. Konstandin, Phys. Lett. B 529 (2002) 132, hep-th/0112237. [9] N.E.J. Bjerrum-Bohr, Phys. Rev. D 66 (2002) 084023, hep-th/0206236. [10] K.S. Stelle, Gen. Relativ. Gravit. 9 (1978) 353. [11] J.Z. Simon, Phys. Rev. D 41 (1990) 3720. [12] A. Strominger, Phys. Rev. D 24 (1981) 3082. [13] G. ’t Hooft, Nucl. Phys. B 72 (1974) 461. [14] B.S. De Witt, Phys. Rev. 160 (5) (1967) 1113; B.S. De Witt, Phys. Rev. 162 (5) (1967) 1195; B.S. De Witt, Phys. Rev. 162 (5) (1967) 1239. [15] S. Sannan, Phys. Rev. D 34 (1986) 1749. [16] D.M. Capper, G. Leibbrandt, M. Ramon Medrano, Phys. Rev. D 8 (1973) 4320. [17] N.E.J. Bjerrum-Bohr, J. Math. Phys. 41 (2000) 2515, quant-ph/0302107. [18] H. Kawai, D.C. Lewellen, S.H. Tye, Nucl. Phys. B 269 (1986) 1. [19] Z. Bern, A.K. Grant, Phys. Lett. B 457 (1999) 23, hep-th/9904026. [20] Z. Bern, Living Rev. Rel. 5 (2002) 5, gr-qc/0206071. [21] N.E.J. Bjerrum-Bohr, Phys. Lett. B 560 (2003) 98, hep-th/0302131. [22] N.E.J. Bjerrum-Bohr, Nucl. Phys. B 673 (2003) 41, hep-th/0305062. [23] S. Deser, D. Seminara, Phys. Rev. Lett. 82 (1999) 2435, hep-th/9812136. [24] S. Deser, D. Seminara, Phys. Rev. D 62 (2000) 084010, hep-th/0002241. [25] D.M. Capper, M. Ramon Medrano, Phys. Rev. D 9 (1974) 1641. [26] G. Leibbrandt, Rev. Mod. Phys. 47 (1975) 849. [27] D.M. Capper, G. Leibbrandt, J. Math. Phys. 15 (1974) 795. Nuclear Physics B 684 (2004) 235–263 www.elsevier.com/locate/npe

Gödel spacetimes, Abelian instantons, the graviphoton background and other ﬂacuum solutions

Patrick Meessen, Tomás Ortín

Instituto de Física Teórica, C-XVI, Universidad Autónoma de Madrid, E-28049 Madrid, Spain Received 20 January 2004; accepted 19 February 2004

Abstract We study the relations between all the vacua of Lorentzian and Euclidean d = 4, 5, 6 SUGRAs with 8 supercharges, finding a new limiting procedure that takes us from the over-rotating near- horizon BMPV black hole to the Gödel spacetime. The timelike compactification of the maximally supersymmetric Gödel solution of N = 1, d = 5 SUGRA gives a maximally supersymmetric solution of pure Euclidean N = 2, d = 4 with flat space but non-trivial anti-selfdual vector field flux (flacuum) that, on the one hand, can be interpreted as an U(1) instanton on T 4 and that, on the other hand, coincides with the graviphoton background shown by Berkovits and Seiberg to produce the C-deformation introduced recently by Ooguri and Vafa. We construct flacuum solutions in other theories such as Euclidean type IIA supergravity.  2004 Elsevier B.V. All rights reserved.

1. Introduction

There has been much progress recently in the classiﬁcation of supersymmetric vacua in d = 4, 5, 6 Lorentzian supergravities (SUGRAs) with eight supercharges [1–4] and in the study of their relations via dimensional reduction and oxidation [5,9]. The study of these theories and their supersymmetric vacua is interesting not just as warm-up exercise for the more complicated cases of interest in string theory but also because these theories arise in dimensional reductions of 10-dimensional superstring effective theories and the solutions found can be uplifted to full 11- or 10-dimensional M/string theory vacua still preserving a large amount of supersymmetry.

E-mail addresses: [email protected] (P. Meessen), [email protected] (T. Ortín).

It is in this way that the supersymmetric Gödel spacetime solutions of M/string theory have been found: as a maximally supersymmetric vacuum of N = 1, d = 5 SUGRA which could be uplifted to 10 and 11 dimensions [2]. Many higher- [10] and also lower-dimensional [11] examples of supersymmetric supergravity solutions with a spacetime geometry similar to the original 4-dimensional cosmological solution by Gödel [12] (henceforth referred to as Gödel spacetimes) have been found after it was observed that Gödel spacetimes are T-duals of Hpp-wave spacetimes.1 This fact has raised many questions about the consistency of string theory in such spacetimes which, being supersymmetric vacua on which (for instance) black holes can be placed [16,17] and on which strings may be quantized [10,18,19], have generically closed timelike curves (CTCs). This is one of the various pathologies of General Relativity that supersymmetry or string theory are supposed to cure, forbid or explain in a consistent way and a great deal of interesting work has been done in this direction [9,20–23]. The supersymmetric vacua of M/string theory provide an interesting arena on which to study the general problems of quantization on curved spacetimes, due to the high degree of (super) symmetry they have, which, through the definition of conserved (super) charges and symmetry (super) algebra, gives a good control and understanding of the kinematics of the field theories defined on them. The fact that the aDS/CFT correspondence “works” is largely owed to the coincidence of the symmetry superalgebras of the theories involved. The symmetry (super) algebras of the M/string theory vacua (and of the field theories defined on them) arise deformations of the symmetry (super) algebra of the original theory of which the vacua are classical solutions, which is in general the Poincaré or anti-de Sitter (super) algebra. This suggests that quantization on curved backgrounds can be, at least in some cases, equivalent to quantization using a deformation of the standard Heisenberg algebra. In other words: quantum field theories in non-trivial backgrounds can be equivalent to certain non-commutative field theories. An example of this more general relation has been provided recently by Berkovits and Seiberg [8,24] who have found a background of Euclidean N = 2, d = 4 SUGRA whose symmetry superalgebra corresponds to the C-deformation introduced recently by Ooguri and Vafa [25]. We have rediscovered this background in our study of maximally supersymmetric vacua of d = 4, 5, 6 SUGRAs with 8 supercharges, in relation with the Gödel solution discussed above, as we are going to explain. Our goal in this paper is to complete our knowledge of the relations between the maximally supersymmetric vacua of d = 4, 5, 6 SUGRAs with 8 supercharges, extending our previous work [5] to Euclidean SUGRAs which arise in timelike dimensional reductions (see Fig. 1). Our main results are summarized in Fig. 2 and include the identification of the above mentioned maximally supersymmetric graviphoton background of Euclidean N = 2, d = 4 SUGRA that cannot be obtained by Wick-rotating any known solution of the Lorentzian theory. This solution can be obtained by timelike dimensional reduction of the 5-dimensional maximally supersymmetric Gödel vacuum and has a flat Euclidean space geometry in presence of a non-trivial anti-selfdual 2-form flux which has

1 Gödel spacetimes had previously been found as solutions of compactiﬁed string theory in [13,14]. More recent solutions of supergravities in various dimensions with Gödel spacetimes have been constructed in [15]. P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263 237

Fig. 1. Relations between the different Euclidean (E) and Lorentzian (L) d = 4, 5, 6 SUGRA theories with 8 supercharges by timelike (t) or spacelike (s) dimensional reductions. There is no Euclidean continuation of the Lorentzian N = (1, 0), d = 6 theory, due to the anti-selfduality of its 3-form ﬁeld strength, and there are two different Euclidean N = 2, d = 4 SUGRAs related by analytical continuation of the vector ﬁeld.

Fig. 2. Relations between the d = 4, 5, 6 vacua with 8 supercharges. Dimensional reduction in spacelike direction is represented by arrows with black arrowheads and in timelike directions with white arrowheads. The ellipse and hyperbolas represent the families of near-horizon limits of rotating and over-rotating extreme BMPV black holes. 3 2 In particular, the elements α and β of these families have the geometries of aDS2 × S and aDS3 × S . Dotted lines indicate that the corresponding solutions are related by a Penrose–Güven-like limiting procedure. suggested the name flacuum solution for it. Solutions of this kind are just as generic as those of Gödel type since they can be obtained by timelike compactifications of these and we study some examples and, in more detail, the 4-dimensional one which can also be interpreted as a U(1) instanton over a T 4. The existence of these solutions is due to the fact that the Euclidean “energy– momentum” tensor (the variation of the matter Lagrangian with respect to the metric) is not positive- or negative-definite and can vanish for non-trivial fields.2 This vanishing can

2 The roles and definiteness of the energy and action are interchanged when we go from Lorentzian to Euclidean signature. 238 P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263 be due to a cancellation between “energy–momentum” tensors of different fields (as in the case of the Einstein-frame D-instanton solution [26,27]) or to a cancellation between “electric” and “magnetic” components of a single field. In the case of a vector field in d = 4 Euclidean space, the “energy–momentum” tensor vanishes whenever the field strength is selfdual or anti-selfdual, as in the graviphoton background [8,24]. A standard procedure to construct Euclidean theories is to dimensionally reduce a Lorentzian theory in a timelike direction. This procedure avoids the difficulties of the Wick rotation of fermions and provides a mechanism to generate flacuum solutions from higher- dimensional Lorentzian solutions with metrics of the form ˆ2 = + 2 − 2 ds (dt ω) dxd , (1.1) which is precisely the general form of the supersymmetric Gödel solutions studied in [10]. This observation will allow us to construct flacuum solutions in d = 10 dimensions. This paper is organized as follows: in Section 2 we summarize the dimensional reduction of standard (Lorentzian)3 N = (1, 0), dˆ = 6 SUGRA in an arbitrary spacelike or timelike direction and its truncation to pure Lorentzian or Euclidean N = 1, d = 5 SUGRA. In Section 3 we summarize the dimensional reduction of the Lorentzian and Euclidean N = 1, dˆ = 5 SUGRAs obtained in the previous section in an arbitrary spacelike or timelike direction and its truncation to pure Lorentzian or Euclidean N = 2, d = 4 SUGRAs. The details of these reductions are explained in Appendix A, but the all the main formulae necessary for oxidation and reduction of solutions are contained in the first two sections. Observe that, as shown in Fig. 1, we obtain two different Euclidean N = 2, d = 4 SUGRAs. These two theories are related by an analytical continuation of the vector field. In the next two Sections 4 and 5 we apply the results of the first two to maximally supersymmetric vacua of N = (1, 0), d = 6 SUGRA to find maximally supersymmetric vacua of pure Euclidean and Lorentzian N = 1, d = 5andN = 2, d = 4 SUGRA. We only find one not previously known from the reduction of the Gödel solution, which coincides with the graviphoton background of Seiberg and Berkovits [8,24]. This solution, with flat space and constant anti-selfdual field strength, has interesting properties that we study in detail. In Section 6 we study more general flacuum solutions in higher dimensions. Section 8 contains our conclusions.

2. Dimensional reduction of N = (1, 0), d = 6 SUGRA in the timelike direction

N = (1, 0), d = 6 SUGRA,4 the minimal 6-dimensional SUGRA, consists of the âˆ ˆ − ˆ − = ˆ − metric e µˆ , 2-form field Bµˆ νˆ with anti-selfdual field strength H 3∂B and positive-

3 There is no Euclidean continuation of the N = (1, 0), dˆ = 6 theory because the anti-selfduality condition of the 3-form field strength cannot be consistently defined in Euclidean signature. The minimal Euclidean SUGRA in six dimensions has not yet been constructed. It would be interesting to construct this theory since it must be related by spacelike dimensional reduction to the same Euclidean N = 1, d = 5 SUGRA that we are going to obtain from timelike dimensional reduction. 4 Our conventions are essentially those of [5] which are those of [28] with some changes in the normalizations of the fields. In particular, we use mostly minus signature and Latin (respectively, Greek or underlined) ˆ =ˆ··· ˆ ˆ 2 =+ ˆ012345 =+ ˆ aˆ1···ân = letters for Lorentz (respectively, curved) indices. Further, γ7 γ0 γ5, γ7 1, 1, γ P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263 239 chirality symplectic Majorana–Weyl gravitino that can be described as a single, complex ˆ + Dirac spinor satisfying a Weyl condition ψµˆ . The bosonic equations of motion can be derived from the action 1 − Sˆ = d6xˆ |ˆg| Rˆ + (Hˆ )2 , (2.1) 24 imposing afterwards the anti-selfduality constraint Hˆ − =−Hˆ −. The gravitino supersymmetry transformation rule is (for zero fermions) + 1 ˆ − + + ˆ = ∇ˆ − √ ˆ ˆ δˆ ψˆ aˆ H/ γaˆ . (2.2) a 48 2 We can perform the dimensional reduction5 of N = (1, 0), dˆ = 6 SUGRA in one dimension can be performed simultaneously for the cases in which that dimension is ˆ ≡ =+ ˆ ≡ =− = timelike (η αdˆ 1) or spacelike (η αdˆ 1). The reduction gives N 1, a d = 5 SUGRA (the minimal 5-dimensional SUGRA) (metric e µ, graviphoton Vµ and symplectic-Majorana gravitino that can be described as a single complex, unconstrained Dirac spinor ψµ [29]) coupled to a vector multiplet consisting of a gaugino λ (the 6th component of the 6-dimensional gravitino, a real scalar (the KK one k) and a vector field Wµ. Vµ and Wµ are combinations of scalars, the KK vector field Aµ and the vector field that comes from the 6-dimensional 2-form Bµ. The identification of the right combinations is made by imposing consistency of the truncation to pure N = 1, d = 5 SUGRA.6 The result can be stated as follows: any solution of N = (1, 0), d = 6 SUGRA satisfying the truncation constraints √ ˆ = ˆ − =− ˆ ˆ = g α6, Bµ 2 gµ ,∂ 0, (2.3) can be consistently reduced in the direction x to a solution of pure (Lorentzian α6 =−1 or Euclidean α6 =+1) N = 1, d = 5 SUGRA whose bosonic action and gravitino supersymmetry transformation rule are given by 5 1 2 1 S = d x |g| R + α6F − α6 √ √ FFV (2.4) 4 12 3 |g| and i 3/2 bc b c δψa = ∇a − √ α γ γa + 2γ g a Fbc (2.5) 8 3 6

[n/2] ˆ ˆ (−1) aˆ ···ânb ···b − ˆ ± ˆ ± − ! ˆ 1 1 6 n γˆˆ ˆ γˆ7. Positive and negative chiralities are defined by γˆ7ψ =±ψ .Weuse (6 n) b1···b6−n b ab abc the notation F/a ≡ γ Fba , F/ ≡ γ Fab and H/ ≡ γ Habc. 5 In this section we use hats for 6-dimensional objects and no hats for 5-dimensional objects. In general, we will use hats for higher-dimensional (dˆ-dimensional) indices and no hats for d = (dˆ − 1)-dimensional indices. The index corresponding to the dimension which is being reduced is which will be t in the timelike case or z spacelike case. Observe that, with our mostly minus convention we get a negative-definite d-dimensional metric in the timelike case ηab =ˆηab =−δab. 6 The details of this dimensional reduction and truncation are explained in Appendix A. 240 P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263 with the same supersymmetries and with the same Killing spinor. The 5-dimensional fields are related to the 6-dimensional ones by √ gµν =ˆgµν − α6gˆµ gˆν ,Vµ =− 3 α6gˆµ . (2.6) Conversely, any solution of Lorentzian or Euclidean N = 1, d = 5 SUGRA can be uplifted to a solution of Lorentzian N = (1, 0), dˆ = 6 SUGRA whose fields are given by ˆ 2 gˆ = α , Bµ =− α Vµ, 6 3 6 1 1 gˆµ = √ α6Vµ, gˆµν = gµν + α6VµVν. (2.7) 3 3

3. Dimensional reduction of N = 1, dˆ = 5 SUGRA in the timelike direction

The dimensional reduction of N = 1, dˆ = 5 SUGRA in one dimension gives pure a N = 2, d = 4 SUGRA (metric e µ, graviphoton Vµ and a pair gravitinos that can be combined into a complex Dirac spinor ψµ) coupled to a vector multiplet (a vector Wµ, a pair of scalars k and l and a complex gaugino λ) [30]. As in the previous case, the matter vector field Wµ is a combination of the scalars KK vector field and of the 5-dimensional vector that has to be identified studying the consistency of truncation in the action and supersymmetry transformation rules. We can reduce simultaneously both the Euclidean (α6 =+1) and Lorentzian (α6 =−1) ˆ versions of N = 1, d = 5 SUGRA and, further, we can do it in timelike (α5 =+1) and 7 spacelike (α5 =−1) directions in one shot. The result will be a Euclidean theory when α6α5 =−1 and a Lorentzian theory when α6α5 =+1. The reduction, identification of Wµ and consistent truncation of the vector multiplet are performed in detail in Appendix A and we obtain the following action and gravitino supersymmetry transformation rule 4 1 2 i 3/2 S = d x |g| R + α F (V ) ,δψa = ∇a + α F(V)γ/ α . (3.1) 4 6 8 6

These expressions exhibit a curious dependence on α6 that indicates that we get two different Euclidean theories depending on the order in which we have reduced in the spacelike and timelike directions: if we compactify first in a spacelike direction α6 =−1 to Lorentzian N = 1, d = 5 SUGRA and then in a timelike direction to Euclidean N = 2, d = 4 SUGRA we get the same theory that we would obtain by Wick-rotating the Lorentzian theory considering the vector field Vµ as a tensor. If we compactify first in a timelike direction α6 =+1 to Euclidean N = 1, d = 5 SUGRA and then in a spacelike direction to Lorentzian N = 2, d = 4 SUGRA we get the same theory that we would obtain by Wick-rotating the Lorentzian theory considering Vµ as a pseudotensor. These = = two Euclidean theories that we denote by N 2α6 , d 4 are then related by an analytical continuation of the vector field Vµ → iVµ.

7 Only the case α6 = α5 =+1 is not possible, since we only have one time. P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263 241

This apparent contradiction in the results of the two compactifications in one timelike and one spacelike direction is related to the well-known fact that Wick rotations and Hodge duality (used in the reduction from d = 5tod = 4) do not commute. The detailed calculations and results of the appendix imply that any solution of ˆ Euclidean (α6 =+1) or Lorentzian (α6 =−1) N = 1, d = 5 SUGRA satisfying the truncation constraints √ ˆ gˆ = α5, V = 0, 3F(A)− F(B)= 0,∂ ˆ = 0, (3.2) can be reduced, preserving all the unbroken supersymmetries, in the direction x to a =− + = = solution of pure Euclidean or Lorentzian (α5α6 1, 1) N 2α6 , d 4 SUGRA with metric and vector field given by 2 ˆ gµν =ˆgµν − αgˆµ gˆν ,Vµ =−√ Vµ. (3.3) 3 = = Conversely, any solution of pure Euclidean or Lorentzian N 2α6 , d 4 SUGRA can always be uplifted, preserving all unbroken supersymmetries, to a solution of Euclidean ˆ (α6 =+1) or Lorentzian (α6 =−1) N = 1, d = 5 SUGRA whose fields are related to the 4-dimensional ones by ˆ gˆ = α5, V = 0, √ ˆ 3 gˆµ = α Aµ, Vµ =− Vµ, 5 2 gˆµν = gµν + α5AµAν, (3.4) where Aµ is defined in terms of Vµ by 1 F(A)=− F(V). (3.5) 2 These are the essential formulae that we have to use for oxidation and reduction of solutions and generalize those obtained in the Lorentzian cases in [5].

4. Maximally supersymmetric vacua of Euclidean N = 1, d = 5 SUGRA

The most interesting solutions of N = (1, 0), dˆ = 6 SUGRA to which the results of the previous sections may be applied are the maximally supersymmetric vacua since their reduction should give all the maximally supersymmetric vacua of Euclidean and Lorentzian N = 1, d = 5 SUGRA. The only maximally supersymmetric vacua of N = (1, 0), dˆ = 6 SUGRA are [2,3]

1. Minkowski spacetime. 2. The 1-parameter family of Kowalski-Glikman (KG6) Hpp-wave solutions found in Ref. [31]. 3 3. The 1-parameter family of solutions with aDS3 × S geometry found in Ref. [32] as the near-horizon limit of the selfdual string solution. 242 P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263

From these three solutions, all the maximally supersymmetric vacua of Lorentzian N = 1, d = 5 SUGRA, classiﬁed in [1] can be found by dimensional reduction in a spacelike direction [2,3,5,9] (see Fig. 2). They are

1. Minkowski spacetime. 2. The 1-parameter family of Kowalski-Glikman KG5 Hpp-wave solutions found in Ref. [31]. 2 3. The 1-parameter family of solutions with aDS3 × S geometry found in Ref. [32] as near-horizon limit of the extreme string solution. It coincides with the near-horizon limit of the critically-rotating ( = 1) BMPV black hole [33]. 3 4. The 1-parameter family of solutions with aDS2 × S geometry found in Ref. [34] as near-horizon limit of the extreme (non-rotating) black hole solution. 5. The 2-parameter family of N = 2, d = 5 solutions found in Ref. [35] as the near- horizon limit of the supersymmetric rotating BMPV black hole solution. 6. The 1-parameter family of Gödel-like solutions found in [1]. 7. A 2-parameter family of solutions found in [1] that has just been identiﬁed as the over-rotating (>1) BMPV black holes [9].

Now we want to ﬁnd non-trivial maximally supersymmetric solutions of Euclidean N = 1, d = 5 SUGRA by dimensional reduction in a spacelike direction. As we are going to see, all of them can be obtained by Wick rotation of maximally supersymmetric vacua of Lorentzian N = 1, d = 5 SUGRA. Let us consider the spacelike dimensional reductions of

3 4.1. The aDS3 × S vacuum

aDS3 can be seen as a spacelike U(1) fibration over aDS2 with metric (for unit radius) 1 dΠ2 = dΠ2 − (dη + sinh χdφ)2 , (4.1) (3) 4 (2) where 2 = 2 2 − 2 dΠ(2) cosh χdφ dχ , (4.2) 3 is the metric of aDS2 with unit radius, just as the round S can be seen as a Hopf U(1) fibration over S2 with metric (for unit radius) 1 dΩ2 = dΩ2 + (dψ + cosθdϕ)2 , (4.3) (3) 4 (2) where 2 = 2 + 2 2 dΩ(2) dθ sin θdϕ , (4.4) is the standard metric of S2 with unit radius. It is possible to perform spacelike dimensional reductions on linear combinations of these two fibers (i.e., of the coordinates η and ψ), obtaining the near-horizon limit of the extreme rotating BMPV black hole [5] (see Fig. 2). The metrics of all these 5-dimensional P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263 243

2 solutions are U(1) fibrations over aDS2 × S ,theU(1) fiber being orthogonal to the one we have reduced over. It is also possible to see aDS3 as a timelike U(1) fibration over the hyperbolic plane H2 with metric (for unit radius) 1 dΠ2 = (dt + coshχdφ)2 − dH2 , (4.5) (3) 4 (2) where (2) = 2 2 + 2 dH2 sinh χdφ dχ , (4.6) is the standard metric of the hyperbolic plane of unit radius. We can perform further spacelike and also timelike dimensional reductions on linear combinations of these two fibers. The spacelike reductions give the near-horizon limit of the extreme over-rotating BMPV black hole [9] and here we are going to see that the timelike reductions give their analytical continuations to Euclidean signature. 3 The maximally supersymmetric solution with aDS3 × S geometry is given by dsˆ2 = R2 dΠ2 − R2 dΩ2 , √3 (3) 3 (3) (4.7) ˆ − = [ − ] H 4 2R3 ωaDS3 ω(3) , 3 where ωaDS3 and ω(3) are the volume forms of the unit radius aDS3 and S metrics written above. Using the timelike U(1) fibration Eq. (4.5) we find that a convenient gauge for the 2-form potential is

− R Bˆ =−√3 [coshχdφ∧ dt − cosθdϕ∧ dψ]. (4.8) 2

Rescaling t and ψ by 2/R3 and boosting the metric in the direction ψ

2 t = (cosh ξt + sinh ξψ), R3 2 ψ = (sinh ξt + cosh ξψ), (4.9) R3 the solution takes the form 2 (t) 2 (ψ) 2 2 2 2 dsˆ = dt + A − dψ + A − (R3/2) dH + dΩ , √ √ (2) (2) (4.10) Bˆ − =− 2A(t) ∧ dt + 2A(ψ) ∧ dψ, where A(t,ψ) are the 1-forms R A(t) = 3 (cosh ξ cosh χdφ− sinh ξ cosθdϕ), 2 R A(ψ) = 3 (cosh ξ cosθdϕ− sinh ξ cosh χdφ). (4.11) 2 The truncation constraints (2.3) are satisfied for x = t and x = ψ and, thus, we can dimensionally reduce this solution to a maximally supersymmetric solution of Euclidean 244 P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263 or Lorentzian N = 1, d = 5 SUGRA. In the spacelike case we get   2 (t) 2 2 2 2  ds = dt + A − (R3/2) dH + dΩ , √ (2) (2)  √ 3 (4.12)  V =− 3A(ψ),F=− R (sinh ξω + cosh ξω ), 2 3 aDS2 (2) which is the near-horizon limit of the extreme critically rotating and over-rotating BMPV black hole [9].  = cosh ξ is the rotation parameter. The critically rotating case has  = 1, i.e., sinh ξ = 0 and the fibration takes place only over the hyperbolic plane, giving aDS3. The total metric describes the solution aDS3 × S2. For any other value of ξ, >1, and we are in the over-rotating case. In the timelike case, we get   2 (ψ) 2 2 2 2  ds =− dψ + A − (R3/2) dH + dΩ , √ (2) (2) √ (4.13)  (t) 3 V =− 3A ,F=− R (cosh ξωH + sinh ξω ). 2 3 2 (2) These two solutions are related by a Wick rotation t → iψ, V → iV which has to be accompanied of ξ → ξ − iπ/2tomakeV real again. The Euclidean solution can also be obtained by a Wick rotation t → iφ, V → iV from the near-horizon limit of the extreme BMPV black hole accompanied of ξ → iξ to make V real again. On the other hand, these solutions can be seen, respectively, as a timelike R or spacelike 2 U(1) fibration over H2 × S . Let us now consider the timelike dimensional reductions of

4.2. The KG6 Hpp-wave vacuum

As shown in [5], the KG6 solution can be conveniently written substituting guu completely by a Sagnac connection 1-form ω = λ(x1 dx2 − x3 dx4) whose main property is that dω is anti-selfdual in the 4-dimensional Euclidean space spanned by x4 = (x1,x2,x3,x4): √ dsˆ2 = 2 du dv + 2 ω − dx2, 4 (4.14) Bˆ − =−2ω ∧ du, √ and, using u = (t + z)/ 2, we can rewrite it in the form dsˆ2 = (dt + ω)2 − (dz − ω)2 − dx2, √ √ 4 (4.15) Bˆ − =− 2 ω ∧ dt − 2 ω ∧ dz, that shows that the truncation constraints (2.3) are satisfied for x = t and x = z and, thus, we can dimensionally reduce this solution to maximally supersymmetric solutions of Euclidean or Lorentzian N = 1, d = 5 SUGRA. The spacelike dimensional reduction gives the Gödel solution ds2 = (dt + ω)2 − dx2, √ 4 (4.16) V =− 3 ω, P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263 245 and the timelike reduction gives its Euclidean version ds2 =−(dz − ω)2 − dx2, √ 4 (4.17) V =− 3 ω, that can be obtained by Wick-rotating t → iz, V → iV plus λ →−iλ to make V real. These two solutions can also be seen as timelike R or spacelike U(1) fibrations over the 4-dimensional flat Euclidean space. No more timelike dimensional reductions satisfying the truncation constraints seem to be possible, and, thus, we do not expect any more maximally supersymmetric solutions of Euclidean N = 1, d = 5 SUGRA.

5. Maximally supersymmetric vacua of Euclidean N = 2±, d = 4 SUGRA

Next, we are going to dimensionally reduce the maximally supersymmetric vacua of Euclidean and Lorentzian N = 1, d = 5 SUGRA to ﬁnd maximally supersymmetric vacua of Euclidean N = 2±, d = 4 SUGRAs. The maximally supersymmetric vacua of Lorentzian N = 2, d = 4 SUGRA8 have been obtained by spacelike dimensional reduction of the vacua of Lorentzian N = 2, d = 5 SUGRA listed in Section 4 [5]. The only Lorentzian solutions that satisfy the truncation constraints for timelike reduction seem to be Minkowski spacetime, the near-horizon (NH) limit of the critically rotating and over-rotating BMPV black holes (BH) Eq. (4.12) and the Gödel solution Eq. (4.16). The only Euclidean solutions that satisfy the analogous truncation constraints are the Euclidean NH limit of the rotating BMPV BH9 Eq. (4.13) and the Euclidean Gödel solution Eq. (4.17). Let us study the non-trivial cases.

5.1. NH limits of the extreme critically rotating and over-rotating BMPV BHs

Using the formulae of Section 3 on the solution Eq. (4.12) we immediately get a maximally supersymmetric solution of Euclidean N = 2−, d = 4 SUGRA 2 2 2 2 ds =−(R3/2) dH + dΩ , 2 2 (5.1) = + F R3(sinh ξωH2 cosh ξω(2)), which is the Wick-rotated (φ =−it) version of the well-known dyonic Robinson–Bertotti solution [7]. Observe that, in order to have a real Euclidean solution, the Lorentzian electric–magnetic rotation angle has also been Wick-rotated ξ → iξ.

8 All the solutions of Lorentzian N = 2, d = 4 SUGRA that preserve some supersymmetry were found in [36]. The maximally supersymmetric vacua of the theory are just three: Minkowski spacetime, the dyonic Robinson– Bertotti solution [7] and the KG4 Hpp-wave solution [6]. 9 As we have seen, the Wick-rotated NH limits of the extreme over-rotating and “under-rotating” BMPV BHs are identical. 246 P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263

5.2. The Euclidean NH limits of extreme rotating BMPV BHs

Using the formulae of Section 3 on the solution Eq. (4.13) we immediately get a maximally supersymmetric solution of Euclidean N = 2+, d = 4 SUGRA 2 2 2 2 ds =−(R3/2) dH + dΩ , 2 2 (5.2) = + F R3(cosh ξωH2 sinh ξω(2)). This solution is related to the Euclidean dyonic Robinson–Bertotti solution by analytical continuation of the vector ﬁeld V → iV together with ξ → ξ − iπ/2.

5.3. The Euclidean and Lorentzian Gödel solutions and the ﬂacuum solution

The dˆ = 5 Gödel solutions are given in Eqs. (4.16) and (4.17). The formulae of Section 3 give, in both cases, the remarkable maximally supersymmetric 4-dimensional vacuum solution with constant anti-selfdual flux (flacuum) found in [8] ds2 =−dx2, 4 (5.3) V = 2ω. The triviality of the metric in presence of non-trivial matter fields is surprising at first sight, but it is not the first example of non-trivial Euclidean SUGRA solution with flat spacetime: the Einstein-frame D-instanton metric [27] (and that of its 4-dimensional analogue [26]) is also flat. In all these cases, the Euclidean version of the energy– momentum tensor vanishes identically, but for different reasons. In the D-instanton case the dilaton and RR 0-form energy–momentum tensors cancel each other on-shell. In the present case, anti-selfduality of the vector field strength makes the energy–momentum tensor vanish identically. This can be easily seen using Eq. (A.28) which tells us that, actually, the Euclidean version of energy–momentum tensor for any (anti-) selfdual Abelian or non-Abelian vector field vanishes identically and, thus, decouples from the metric.10 This implies that solutions such as the BPST instanton [37] are not just solutions of the Euclidean Yang–Mills equations on S4 with the standard metric, but they are also solutions of the full supergravity equations, provided that the metric satisfies the vacuum Einstein equations. We will discuss this and similar solutions in the next section. The above solution is guaranteed to be maximally supersymmetric in both Euclidean N = 2±, d = 4 SUGRA. It is worth studying how the Killing spinors integrability conditions are satisfied. They take the form11 1 ab 1 ρ 1 − Rµν γab + T[µ|ρ|γ ν] + ∇/ Fµν + Fµν γ = 0. (5.4) 4 4 4 In the case at hand the first term vanishes identically because the space is flat, the second because of the anti-selfduality of the vector field strength and the third and fourth because the field strength is covariantly constant (constant in Cartesian coordinates).

10 But not the other way around since (anti-) selfduality is established with respect to a given metric. 11 For simplicity we consider the α6 =−1 case. P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263 247

(Anti-) selfduality implies ﬂatness and, then, covariant constancy determines a unique solution (up to SO(4) rotations). Suppressing this last requirement allows for more solutions12 which would only preserve a half of the supersymmetries. This solution preserves all supersymmetries and the Killing spinors depend on all coordinates: 1 (x) = exp − F//x , (5.5) 8 0 µ where 0 is a constant spinor and x/ ≡ x γµ. Using the anti-selfduality of F , we can rewrite the exponent in this form

1 1 µ − F//x =− F/ µx (1 − γ ), (5.6) 8 4 where (γ )2 = 1 in this case. This implies that (F//x)2 = 0 and, thus, the Killing spinors can be written in the form 1 µ (x) = 1 − F/ µx (1 − γ ) . (5.7) 4 0 If we split our complex Dirac spinors into chiral halves, we see that the positive chirality Killing spinor is simply constant, as in empty Euclidean space, and the negative chirality Killing spinor has a x-dependent deformation with respect to that of empty Euclidean space which induces a deformation of the supersymmetry algebra [8,24] that affects only to negative-chirality objects. Using the methods of [38] and the notation of [39,40], and using 4-component complex (Dirac) spinors, we ﬁnd the following non-vanishing (anti-) commutators Q† Q = 1 a − 1 1 − (α), (β) (γ γ )αβ P(a) γ (1 γ ) M, 2 αβ β β [Q(α),P(a)]=−Q(β)Γs(P(a)) α, [Q(α),M]=−Q(β)Γs(M) α, b [P(a),M]=−P(b)Γv(M) a. (5.8)

The bosonic generators P(a) of this algebra are associated to the translational Killing 1 ab vectors ∂a. The bosonic generator M is a Lorentz (SO(4)) rotation 2 F Mab where the Lorentz generators Mab are associated the Killing vectors −2x[a∂b]. Both the PasandM act on spinors through their spinorial representations β β 1 Γs(P ) α = F/ a(1 − γ ) , (a) 4 α β β β 1 1 Γs(M) α = F/ = F(/ 1 + γ ) , (5.9) 2 α 4 α and on vectors through the vector representation b b Γv(M) a =−F a. (5.10)

12 For instance, A = dx /r + cos θdϕin spherical coordinates. 248 P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263

This superalgebra is a deformation of the standard Poincaré N = 2, d = 4 superalgebra. This is a common property of all the maximally supersymmetric vacua of N = 2, d = 4 SUGRA. However, this deformation is particularly interesting because the bosonic translations algebra is not modiﬁed and, actually, only the anticommutator of the negative chirality part of the supercharge is deformed and this deformation induces a deformation of the anticommutator of the fermionic superspace coordinates of negative chirality [8,24]. The relation between the ﬂacuum and Gödel solutions (whose Killing spinors are, according to our results, absolutely identical) implies a relation between their superalgebras which is important to understand. The only difference between them is the occurrence in the Gödel superalgebra of the bosonic generator P(0) associated to time translations in the anticommutator of the supercharges and, more importantly, in the commutator of the generators associated to space translations P(a). If we write the Gödel solution Eq. (4.16) in the gauge in which the 1-form ω takes the form

1 a b ω = Fabx dx ,a,b= 1, 2, 3, 4,F=−F = λ, (5.11) 2 12 34 we ﬁnd the Killing vectors13

1 b 1 ab ∂t ,∂a − Fabx ∂t , − F xa∂b, (5.12) 2 2 associated, respectively, to the generators of time and space translations P(0) and P(a) and to the rotation M. The non-vanishing (anti-) commutators of the Gödel superalgebra are Q† Q = 1 a + 1 − 1 1 − (α), (β) (γ γ )αβ P(a) (γ γ )αβ P(0) γ (1 γ ) M, 2 αβ β β [Q(α),P(a)]=−Q(β)Γs(P(a)) α, [Q(α),M]=−Q(β)Γs(M) α, b [P(a),M]=−P(b)Γv(M) a, [P(a),P(b)]=FabP(0). (5.13) In this background space translations commute only up to a time translation, because a = 1 ab space translations δx have to be compensated by a time translation δt 2 F xb to leave the metric invariant. This deformation is interesting because is suggests that string theory quantized in the Gödel background will give a non-commutative theory in which, instead of modifying the commutator of positions, it is the commutator of momenta which are deformed as above. Supersymmetry requires a corresponding modification of the anticommutator f the supercharges as in the flacuum solution. In the dimensional reduction to the flacuum solution the generator P(0) becomes a central charge that generates gauge transformations of the Kaluza–Klein vector field. Actually, it is easy to check that space translations only leave the vector field of the 1 ab flacuum solution up to gauge transformations with gauge parameter 2 F xb.Inthis sense, the flacuum superalgebra is not complete and one should add P(0)sasinthe Gödel superalgebra. However, if we are considering only t-independent 5-dimensional configurations which correspond to solutions with zero P(0) charge, the difference is

13 There are more Killing vectors, but they are not relevant insofar they do not appear in the anticommutator of the supercharges. P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263 249 immaterial. In N = 2, d = 4 SUGRA there are no charged perturbative states in N = 2, d = 4 SUGRA, but non-perturbative states that would be associated to 5-dimensional KK modes with non-trivial time dependence are charged with respect to P(0) and would feel the non-commutativity of the momenta.

5.4. The ﬂacuum solution as an Abelian instanton

Another interesting aspect of the flacuum solution is its possible interpretation as an instanton. This is only possible if the space is compactified into a T 4 to make the action integral finite. This is, also, a base space on which the U(1) gauge group can be nontrivially fibered. Euclidean Yang–Mills solutions on T 4 have been thoroughly studied in the literature14 and we are simply going to apply the known results to the present case. To compactify the solution on T 4 we take the quotient of R4 by the Z4 Abelian group of discrete translations along the four coordinates xa with periods la . The vector field of our solution, which we rewrite for convenience in a new gauge 1 2 2 1 3 4 4 3 a b V = λ x dx − x dx − x dx + x dx ≡ Fabx dx , (5.14) is not strictly periodic on T 4: when we move around the ath period from x to x +â it changes by a gauge transformation

(a) b V(x+â) = V(x)+ dΛa(x), Λa(x) = l F(a)bx , (5.15) where Λa(x) are the U(1) parameters, defined modulo 2π. Consistency requires that V(x+â + b)ˆ = V(x+ bˆ +â),thatis ˆ Λa(x + b) + Λb(x) = Λb(x +â) + Λa(x) mod(2π), (5.16) which in our case implies

λl1l2 = πn, λl3l4 = πm, (5.17) for two integers n, m that label the possible non-trivial bundles, the topological number being given by their product nm. The Euclidean action of these SUGRA solutions is, therefore, in our conventions

S =−4π2|nm|. (5.18) Let us consider now taking the quotient on the spinor bundle, which requires making the identiﬁcations l(a) (x +ˆa) = exp − Fγ/ (x) ∼ (x), (5.19) 8 (a) consistently. These transformations are not SO(4) rotations but are still elements of the bigger holonomy group of the supercovariant derivative of N = 2, d = 4 SUGRA, which,

14 See, for instance, the pedagogical review [41] whose notation we will loosely follow. In particular, we use c c (a) the period vectors (a)ˆ ≡ δ(a) l , where summation is not to be made over indices within parenthesis. 250 P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263 by arguments similar to those in [42,43] can be shown to be SL(4, R), and, in this sense, from this more general point of view, they should be admissible. On the other hand, the consistency of these identifications is ensured by the fact that the above transformations are indeed a representation of the commutative Z4 group of discrete a translations: the generators of translations in the x directions, P(a) are represented on the spinors by the mutually commuting operators Γs(P(a)) given in Eq. (5.9). The consistency of this construction has interesting implications: the compactness of the gauge group (U(1) instead of R), reflected by the periodicity of the gauge parameters ˆ Λa(x) implies the compactness of the d = 5 time coordinate. All the vacua of N = 1, dˆ = 5 SUGRA (except the KG5 Hpp-wave) seem to be non-trivial spacelike or timelike 2 fibrations over 4-dimensional Euclidean space or over H2 × S . In this respect, it is remarkable that the only solutions that can be reduced consistently in the timelike direction (preserving all supersymmetries) have closed timelike curves or in the  = 1 closed lightlike curves.

6. (Anti-) selfdual vacua in Euclidean SUGRA

It is not difficult to construct generalizations of the flacuum solution. Let us consider a (p + 1) form potential A(p+1) with field strength F(p+2) = dA(p+1) coupled to d-dimensional gravity: − p d ( 1) 2 S = d x |g| R − F + . (6.1) 2 · (p + 2)! (p 2) The Einstein equation is − p 1 ( 1) (p+1) Rµν − gµν R = T , (6.2) 2 2 · (p + 1)! µν where the energy–momentum tensor is given by 1 (p+1) ρ1···ρp+1 2 T = F + F + ··· + − g F + , (6.3) µν (p 2)µ (p 2)νρ1 ρp 1 2 · (p + 2) µν (p 2) or, equivalently, by the more useful expression (p+1) 1 ρ ···ρ + ρ ···ρ ˜+ T = F + 1 p 1 F + ··· − α F + 1 p 1 F + ··· , µν 2 (p 2)µ (p 2)νρ1 ρp+1 (p 2)µ (p 2)νρ1 ρp˜+1 (6.4) where p˜ = d − p − 4. We are interested in solutions with vanishing energy–momentum tensor. A necessary condition for the energy–momentum tensor to vanish is that its trace does. Using (6.3) we find that (p+1)µ d 2 T ∼ 1 − F + , (6.5) µ 2(p + 2) (p 2) P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263 251

2 = + = which vanishes when F(p+2) 0orwhenp 2 d/2. For non-trivial potentials, the ﬁrst case is only possible in Lorentzian signature, but we know that the Lorentzian energy– momentum tensor only vanishes for vanishing ﬁeld strengths. Thus, a vanishing energy– momentum tensor is only possible in Euclidean signature for d = 2(p +2) and, using (6.4), it requires

··· ··· ρ1 ρp+1 = ρ1 ρp+1 F(p+2)µ F(p+2)νρ1···ρp+1 F(p+2)µ F(p+2)νρ1···ρp+1 , (6.6) which can be solved in any even dimension: in d = 4n using (anti-) selfdual field strengths (aswehaveseenind = 4) and in d = 4n + 2 using configurations which would be (anti-) selfdual in Lorentzian signature. For example, in d = 6 we could use F(3)123 = F(3)456 = λ which has F(3)123 =− F(3)123 but F(3)123 =+ F(3)123 (i.e., not definite selfduality properties) but clearly satisfies the above condition. Unfortunately, these simple ansatzes for flacuum solutions do not work in presence of 2 scalars such as the string dilaton that couple to F(p+2). Still, since the sign of the Euclidean “kinetic” terms and energy–momentum tensors is not uniquely defined,15 it is possible that the energy–momentum tensors and their contributions to the source for the dilaton of two fields cancel each other. Examples of these solutions are provided by the dimensional reduction in the time direction of the supersymmetric Gödel solutions of N = 1, d = 11 SUGRA found in [10]. These solutions take the form   ˆ2 = + i 2 − 2  ds (dt ciω ) dx10, (6.7)  ˆ 1 i j G = aij dω ∧ dω , 4 where the ωi s are 1-forms defined by

+ + ωi = x(i) dx5 (i) − x5 (i) dx(i), (6.8) and where the constants ci and aij (with aii = 0) have to satisfy the following two conditions:

2 1 ij klm a a = 4c + cj cj ,aij cj =− η ajkalm, (6.9) (i)j j(i) i 8 where the ηij klm symbol is one is all its indices take different values and zero otherwise. It is convenient to label the solutions of this family by the two vectors of constants

(c1,c2,c3,c4,c5), (a12,a13,a14,a15,a23,a24,a25,a34,a35,a45). (6.10) The supersymmetry of some explicit Gödel solutions was studied in [10] and some examples are given in Table 1.

15 They depend on whether the Lorentzian ﬁeld is a tensor or a pseudotensor, as the KK vector generically turns out to be. 252 P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263

Table 1 Some Gödel solutions of N = 1, d = 11 SUGRA. The ﬁrst example gives, after dimensional reduction, the maximally supersymmetric Gödel solution of N = 1, d = 5 SUGRA Eq. (4.16) and, therefore, it also gives rise to the ﬂacuum solution Eq. (5.3)

(ci )(aij ) #Susies (i) λ(1, 1, 0, 0, 0)λ(0, −2, −2, −2, −2, −2, −2, 0, 0, 0) 20 (ii) λ(1, 1, 1, 1, 0)λ(4, 0, 0, −2, 0, 0, −2, 4, −2, −2, −2) 0 (iii) λ(1, 1, 1, −1, 0)λ(4, 0, 0, −2, 0, 0, −2, 4, −2, −2, +2) 20 (iv) λ(1, 0, 0, 0, 0)λ(2, 0, 0, −2, 0, 0, 0, 2, 0, 0) 8 (v) λ(2, 1, 1, 1, 1)λ(−6, 2, 2, 2, 0, 0, 0, 4, 4, 4) 18

It is clear that the timelike reduction of any of these Gödel spacetimes gives a ﬂacuum solutions of Euclidean type IIA SUGRA with ﬂat metric and RR 4-form and 1-form given by  2 =− 2  ds dx10,  (2) i G = ci dω ,  (6.11)   (4) 1 i j G = aij dω ∧ dω . 4 It is important to understand how the vanishing of the energy–momentum tensor takes place and why the dilaton is constant in this case. The action of the Euclidean type II SUGRA obtained by dimensional reduction from standard, Lorentzian 11-dimensional SUGRA is given by 1 1 S = d10x |g| e−2φ R − (∂φ)2 − α H 2 + α G(2) 2 4 · ! 11 11 2 3 4 1 1 1 − G(4) 2 − √ ∂C(3)∂C(3)B . (6.12) 2 · 4! 144 |g|

This action differs from the type IIA action by the occurrences of α11’s, the definition of the various field-strengths staying the same.16 Observe that, to obtain this bosonic action by a standard Wick rotation one has to consider both the RR 1-form and the NSNS 2-form pseudotensors. The RR 1- and 3-form kinetic terms (G(2))2 and (G(4))2 have opposite signs when α11 =+1 and the dilaton does not couple directly to them in the string frame, but only through the Ricci scalar. Thus, the exact cancellation between the respective “energy– momentum” tensors which makes the metric flat is enough to render the dilaton constant. To end this section, let us observe that the BPST instanton [37] also provides an example of flacuum solution since its energy–momentum tensor is identically zero because of the (anti-) selfduality of its field strength on the round S4. Thus, it is a solution of the Euclidean Einstein–Yang–Millstheory with positive cosmological constant proportional to the square inverse radius of the sphere. The embedding of this solution in a supergravity theory is more problematic because a DS-type gauged SUGRA would have to be used. It should be

16 Our conventions here are the same as in Ref. [44]. P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263 253 possible to uplift the solution to Euclidean purely gravitational d = 7 solution with the S7 metric [45].

6.1. pp-waves, lightlike reductions and ﬂacuum solutions

The fact that the N = 2, d = 4 ﬂacuum solution originates in a timelike and spacelike dimensional reduction of the KG6 Hpp wave suggests that there must exist a relation between this kind of solutions and pp-waves. pp-wave metrics can always be put in the form

ˆ 2 i i j ds = 2 du(dv + Kdu+ Ai dx ) + gij dx dx , (6.13) where all the functions in the metric are independent of v. K or Ai can be removed by coordinate transformations that preserve the above form of the metric, but we will keep both. Furthermore, we are going to assume independence on the second light-cone coordinate u. The components of the dˆ-dimensional Ricci tensor are

ˆ ˆ j ˆ i 1 2 Rij = Rij , 2Ri u =∇ Fji, Ru u =∇i ∂ K − F , (6.14) 4 where all the objects without hats are calculated with the d = (dˆ − 2)-dimensional Euclidean wavefront metric gij and where Fij = 2∂[iAj]. The Ricci scalar is just ˆ ˆ ˆ R = R and, therefore, d-dimensional vacuum Einstein equations Gµˆ νˆ = 0 imply the d- dimensional vacuum Einstein equations Gij = 0, the vacuum Maxwell equation in the ∇ ji = ∇2 = 1 2 background metric j F 0 and the scalar equation of motion K 4 F . This is similar to the result of a Kaluza–Klein reduction, except for the fact that neither the scalar nor the vector field contribute to the Euclidean “energy–momentum” tensor17 as in flacuum solutions. Actually, the flacuum solution can be embedded in a purely ˆ 2 gravitational d = 6 pp-wave with K ∼ λ|x4| . Another wide class of similar solutions is given by

1 2 2 Fi = 2∇i H, Fij = ij k √ ∇kH, K = H , ∇ H = 0. (6.15) |g| It is possible to perform dimensional reductions in one lightlike dimension [46], but it does not very useful as the resulting theories have singular metric. The above observation suggests that lightlike reductions should be made in the two light-cone coordinates to give a Euclidean (dˆ − 2)-dimensional theory that has essentially a metric an a vector ﬁeld constrained to have vanishing energy–momentum tensor and to satisfy the constraint F 2 = 4∇2K for some scalar K.Ind = 4 it is possible to deﬁne Euclidean theories in which the selfdual or anti-selfdual parts of the spin connection are the only variables. It might be possible to construct a theory in which only the selfdual or anti-selfdual part of F occurs and we can speculate that this kind of theory is the one associated to lightlike dimensional reductions.

17 Since either of them can be removed by reparametrizations, only one of them may contribute. 254 P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263

7. Limiting solutions

Apart from the dimensional reduction from the KG6 solution, there is another way of obtaining the Gödel solution through a limiting procedure which is very similar to the m Penrose–Güven limit [47,48] that relates aDSn × S supergravity vacua to Kowalski- Glikman-type Hpp-wave solutions and in general any supergravity solution to a plane wave solution with equal or larger number of isometries and supersymmetries [49,50]. Let us consider the NH limit of the extreme over-rotating BMPV BH Eq. (4.12). Its 2 ξ → 0 limit is well defined and gives the vacuum with aDS3 ×S geometry. The naive limit ξ →∞is singular, though, but can be made regular by using the following observation: under the field rescalings a −1 a −1 e µ → ε e µ,V→ ε V, (7.1) the action of N = 1, d = 5 SUGRA Eq. (2.4) scales homogeneously S → ε−3S. Then, we can combine this rescaling that transforms solutions into solutions with the reparametrization R χ → εχ, θ → εθ, t → εt + 3 (coshξφ− sinh ξϕ), (7.2) 2 and the gauge transformation of the vector field √ 3 δV = R (coshξdϕ− sinh ξdφ), (7.3) 2 3 and take the double limit ξ →∞, ε → 0, whilst keeping the product ε sinh ξ = 1. The result is the solution   R 2  ds2 = dt + 3 χ2 dφ − θ 2 dϕ   4  2 R3 − dχ2 + χ2 dφ2 + dθ2 + θ 2 dϕ2 , (7.4)  2   √  3  V = R θ 2 dϕ + χ2 dφ , 4 3 which is the Gödel solution Eq. (4.16) in polar coordinates with λ = 1/R3. In this limit the number of supersymmetries and symmetries has been preserved. Observe that the gravitino supersymmetry rule Eq. (2.5) δψµ is invariant under the above field rescalings. Actually, the arguments of [49,50] for the Penrose–Güven limit apply equally to this kind of limit. In the remaining of this section we are going to apply it to two cases: the Euclidean 5 Robinson–Bertotti solutions of N = 2±, d = 4 SUGRA Eqs. (5.1), (5.2) and the aDS5 ×S solution of d = 10 type IIB SUGRA.

7.1. The limit of the Euclidean dyonic Robinson–Bertotti solutions

The action and gravitino supersymmetry rule of N = 2, d = 4 SUGRA also transform homogeneously under the rescalings Eq. (7.1) and we can use them to take a smooth limit P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263 255 of the Euclidean dyonic Robinson–Bertotti solutions Eqs. (5.1), (5.2), exactly as in the previous case. The calculations are clearly identical and the result is, not surprisingly the ﬂacuum solution Eq. (5.3).

5 7.2. Boosted Hopf oxidation of aDS5 × S

It is well known that S5 can be seen as an S1 fibration over CP2. In [51], it was observed 1 that the aDS2n+1 spacetimes can also be seen as a timelike S fibration over a non-compact version of CPn, which the authors called CPn. This construction was, then, used to derive solution in the Euclidean IIA theory via timelike T-duality. The relation of the NH limit of the extreme BMPV BH solutions to the N = (1, 0), d = 6 3 vacuum aDS3 × S , however, shows that in general we before performing dimensional reduction or a T-duality transformation one can mix the S1 fibers of the sphere and the aDS spacetime by a rotation, if both fibers are spacelike, or by a boost if one is spacelike and the other is timelike. 5 In the case of aDS5 × S the only mixing that we can do is a boost on the timelike 5 fiber in the aDS5 and the spacelike fiber for the S . After the boost, we have basically 2 options: T-dualize over the new timelike direction, obtaining a generalization of the solutions discussed in [51], or T-dualize over the new spacelike direction, which will lead to a solution similar to the NH limit of the extreme over-rotating BMPV BH, and that will 2 contain the aDS5 × CP × R solution as a special limit. 5 Let us write the aDS5 × S solution as   2 = − ¯ 2 − − ¯ 2 − CP2 − CP2 dsIIB dt B(ξ,ξ) dy A(z, z) d [ξ] d [z], (7.5)  (5) G = 4 dt ∧ ωCP2 + 4 dy ∧ ωCP2 , where we have introduced (i, j = 1, 2) the metrics i ¯i ¯i i j ¯j ¯i i − i ¯i 2 dz dz z dz z dz i z dz z dz d CP[ ] = − ,A= , z 1 +|z|2 (1 +|z|2)2 2 1 +|z|2 i ¯ i i ¯ i ¯j j i ¯ i − ¯ i i 2 dξ dξ ξ dξ ξ dξ i ξ dξ ξ dξ d CP[ ] = + ,B= , (7.6) ξ 1 −|ξ|2 (1 −|ξ|2)2 2 1 −|ξ|2

ωCP2 and ωCP2 being their corresponding volume 4-forms. Boosting on the (t, y)-plane, with boost parameter β, T-dualizing over y and lifting the solution up to M-theory, we find the solution 2 = − − 2 − CP2 − CP2 − R2 ds11 (dt cosh βB sinhβA) d [ξ] d [z] d [x], (7.7) 1 2 G = 4coshβωCP2 − 4sinhβωCP2 − sinhβdB+ coshβdA ∧ dx ∧ dx . Now we can take the β →∞limit using the fact that the action for the bosonic fields of M-theory scales homogeneously under the field rescalings a −1 a −3 e µ → ε e µ,Cµνρ → ε Cµνρ . (7.8) We perform the above rescaling and a change of coordinates t → εt, ξi → εξi ,zi → εzi,xi → εxi, (7.9) 256 P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263 and now take the double limit β →∞ε → 0 with sinh βε = 1. It is readily seen that the limiting solution is the n = 4 Gödel spacetime that preserves 20 supersymmetries [10].

8. Conclusions

In this paper we have explored the relations between maximally supersymmetric vacua of d = 4, 5, 6 Euclidean and Lorentzian SUGRA theories with 8 supercharges and we have found that the maximally supersymmetric graviphoton background of [8,24] which we have called flacuum solution is related to the maximally supersymmetric Gödel solution by timelike dimensional reduction, a situation that can be generalized to higher-dimensional Gödel solutions of M-theory, for instance, and higher-dimensional flacuum solutions, of Euclidean type II theory, for instance. We have studied other instances in which Euclidean solutions with vanishing energy– momentum tensor appear, as in the (“double”) lightlike dimensional reduction. Apart from the relations between solutions via spacelike and timelike dimensional reduction and oxidation we have found a limiting procedure similar in may respects to the well-known Penrose–Güven limiting procedure that, instead of giving plane-wave spacetimes gives, at least in the examples considered, Gödel solutions. The main interest in flacuum solutions, for the moment, seems to be due to the fact that their superalgebras are small deformations of the superalgebra that “defines” the SUGRA theory in question. All the maximally supersymmetric vacua if a given SUGRA have, in fact, a symmetry superalgebra that can be seen as a deformation of the SUGRA superalgebra in which some central charges are activated and these, together with some momenta, are given non-trivial commutation relations between them and with the supercharges. These new non-vanishing commutators and the central charges vanish for some value of the deformation parameter. For instance, the (Lorentzian) N = 2, d = 4 Poincaré superalgebra is given by the only on-vanishing anticommutator (in a Majorana basis)

a {Q(αi), Q(βj)}=−iδij (Cγ )αβ Pa + iCαβ ij Q + (Cγ5)αβ ij P, (8.1) where Q and P are the central charges. All the bosonic charges commute. Now, let us consider different maximally supersymmetric vacua of this theory. Minkowski spacetime has the same superalgebra, with vanishing central charges. The superalgebra of the 2 Robinson–Bertotti solution (aDS2 × S ) has the same anticommutator but, now, the six bosonic charges generate an SO(2, 1) × SO(3) Lie algebra 1 1 1 [Q, P1]= P3, [Q, P0]= P1, [P1,P0]=− Q, R2 R2 R2 1 1 1 [P,P2]= P3, [P,P3]=− P2, [P2,P3]= P, (8.2) R2 R2 R2 2 that become again Abelian when the aDS2 and S radius R2 goes to inﬁnity. The commutators of the supercharges and these bosonic generators also become nontrivial and have a similar behavior. P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263 257

Another example is provided by the KG4 Hpp-wave solution [6]. The supercharges of its symmetry superalgebra also have the same anticommutator as above, but now the six bosonic charges generate the Heisenberg algebra with non-vanishing commutators λ λ [Q, P ]=− Pv, [P,P ]=− Pv, [Pu,P ]=−λQ, 1 4 2 4 1 λ λ [P ,P ]=λP, [P ,q]=− P , [P ,P]=− P , (8.3) u 2 u 4 1 u 4 2 that also becomes Abelian in the λ → 0 limit. m The maximally supersymmetric aDSn × S vacua of 11- and 10-dimensional supergravities can also be seen as deformations of the Poincaré superalgebra and the same is valid for Hpp-wave backgrounds. It is natural to expect that quantum theories deﬁned in these backgrounds satisfy some non-commutative generalization of the Heisenberg algebra.

Acknowledgements

We thank E. Lozano-Tellechea for interesting conversations. T.O. would like to thank M.M. Fernández for her support. The work of T.O. has been supported in part by the Spanish grant BFM2003-01090.

Appendix A. The detailed dimensional reductions

The dimensional reduction of the Einstein–Hilbert term is identical in the two cases N = (1, 0), dˆ = 6andN = 1, dˆ = 5 SUGRA. Further, in both the timelike and spacelike cases we can make the ansatz a µ aˆ eµ kAµ µˆ ea −Aa eˆ ˆ = , eˆˆ = , (A.1) µ 0 k a 0 k−1 µ where Aa = ea Aµ.Alld-dimensional ﬁelds with Lorentz indices will be assumed to have been contracted with the d-dimensional Vielbeins. This ansatz gives the following non-vanishing components of the Ricci rotation coefﬁcients ˆ ˆ 1 ˆ 1 Ωabc = Ωabc, Ωab = α ˆFab, Ωa = α ˆ∂a logk, (A.2) 2 d 2 d ˆ and of the spin connection ωaˆbˆcˆ 1 ωˆ abc = ωabc, ωˆ ab =− α ˆkFab, 2 d 1 ωˆ bc = α ˆkFbc, ωˆ b = α∂b logk, (A.3) 2 d where µ ν Fab = ea eb Fµν ,Fµν = 2∂[µAν] (A.4) 258 P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263

(which we will also denote by F(A) in presence of other vector fields) is the KK vector field strength. The standard procedure gives the KK-frame action ˆ dˆ ˆ dˆ−1 1 2 2 S = d xˆ |ˆg| R = d x |g| k R + α ˆk F . (A.5) 4 d The sign of the Maxwell term in the timelike case is unconventional because we would get the opposite sign by Wick-rotating the standard Lorentzian Maxwell action. However, it cannot be deemed as “wrong” since there is no propagation, no concepts of motion or energy in Euclidean space. On the other hand, if we Wick-rotated a pseudovector field we would get precisely the above sign.

A.1. N = (1, 0), dˆ = 6 SUGRA

In the reduction of the action of N = (1, 0), dˆ = 6 SUGRA Eq. (2.1) to d = 5wehave to take into account the anti-selfduality constraint, which can be imposed on the action after dimensional reduction, just as in the reduction of N = 2B, d = 10 supergravity on a circle [44]. Thus, we ﬁrst perform the naive dimensional reduction of Eq. (2.1). We can use our previous results for the reduction of the Einstein–Hilbert term and we only need to reduce the 2-form kinetic term. The reduction of Hˆ − gives a 3- and a 2-form ﬁeld strengths ˆ − 1 1 Habc = H ,H= 3 ∂B − AF (B) − BF(A) , abc 2 2 = −1 ˆ − = Fab(B) k Hab ,F(B)2∂B, (A.6) where ˆ ˆ Bµν = Bµν − A[µBν],Bµ = Bµ . (A.7) The anti-selfduality constraint becomes −1 −1 H =−k F(B) ⇐⇒ H = α6k F(B). (A.8) We immediately get 1 1 − 1 S = d5x |g| k R + α k2F 2(A) + α k 2F 2(B) + H 2 . (A.9) 4 6 8 6 24

As in Ref. [44], we Poincaré-dualize the 2-form into a third vector field Cµ andthenwe identify it with Bµ in the action, that is left with the metric and the unconstrained vector fields Aµ and Bµ and takes the form 5 1 2 2 1 −2 2 S = d x |g| k R + α6k F (A) + α6k F (B) 4 4 −1 − α6 √ k F(A)F(B)B . (A.10) 8 |g| The truncation to pure supergravity involves setting to zero the scalar k = 1 consistently, i.e., in such a way that its equation of motion is always satisfied. The k equation of motion P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263 259 with k = 1 (upon use of Einstein’s equation) implies the constraint

F 2(B) = 2F 2(A). (A.11)

We also have to set to zero the gaugino λ and the matter vector ﬁeld Wµ, which must be a combination of Aµ and Bµ related to λ by supersymmetry. Thus, to identify Wµ,wehave to analyze the reduction and truncation of the gravitino supersymmetry transformation rule Eq. (2.2). The 6- and 5-dimensional gamma matrices are related by

ˆ a = a ⊗ 1 ˆ = I ⊗ 1/2 2 ≡ˆ γ γ σ , γ α6 σ γ , 3 γˆ7 =ˆγ0 ···γ ˆ5 = I ⊗ σ , (A.12) a where the γ s Lorentzian or Euclidean 5-dimensional gamma matrices satisfying γ0 ···γ4 = I and γ1 ···γ5 = iI. Using this relation, the decompositions of the spin connection and 3-form field strength, the anti-selfduality constraint Eq. (A.8), the chirality of ˆ+ and assuming that the supersymmetry parameter ˆ+ is independent of x and setting k = 1, we find + i 3/2 i 1/2 + ˆ = ∇ − + √ δˆ ψa a α F/a(A) α F(B)γ/ a , 4 6 8 2 6 − √ ˆ + 1 δˆ+ ψ = √ 2 α6F(A)/ +F(B)/ , (A.13) 8 2 1 + 3 ˆ+ = where 2 (1 σ ) . ˆ + 1 + 3 ˆ + = ˆ + ψa is to be identified with the 5-dimensional gravitino 2 (1 σ )ψa ψa. ψ only 1 + 3 ˆ + ≡ transforms into vectors and it is to be identified with the gaugino 2 (1 σ )ψ λ.This, in turn, implies that the combination of vector field strengths in the r.h.s. is proportional to F(W) (for k = 1). The field strengths of V and W must be an SO(2) rotation of those of A and B to preserve the canonical normalization and diagonal form of the kinetic terms in the action. The right combination is that in which λ only transforms into W and not into V : 1 3 F(V)≡ √ F(A)− α6F(B), 3 2 2 1 F(W)≡ α6F(A)+ √ F(B). (A.14) 3 3

Setting Wµ = 0 (to make consistent λ = 0) implies the constraint √ 2 α6F(A)=−F(B), (A.15) which implies the constraint Eq. (A.11). Therefore, we can truncate the vector multiplet setting consistently k = 1, λ = 0, Wµ = 0. The result is the action of pure Lorentzian or Euclidean N = 1, d = 5 SUGRA Eq. (2.4) and its gravitino supersymmetry transformation law Eq. (2.5). 260 P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263

A.2. N = 1, dˆ = 5 SUGRA

The reduction of the Einstein–Hilbert term in the action Eq. (2.4) goes as in the previous ˆ case. The reduction of the Maxwell field Vµˆ gives rise to a d-dimensional vector field that we denote by Bµ and a scalar l: ˆ ˆ V = l, Vµ = Bµ + lAµ. (A.16) ˆ As for the field strength, we define Fµν (B) and the combination Gab = Fab

Fµν (B) = 2∂[µBν], G = F(B)+ lF (A), (A.17) where F(A)is the KK vector ﬁeld strength. Taking into account

ˆ −1 Fa = k ∂al (A.18) we ﬁnd that the reduction of the Einstein–Hilbert and Maxwell terms of the action Eq. (2.4) gives 1 − 1 1 d4x |g| k R + α α k 2(∂l)2 + α k2F 2(A) + α G2 . (A.19) 2 5 6 4 5 4 6 The reduction of the Chern–Simons term is identical for both the timelike and spacelike cases, using ˆabcd = abcd, and gives after integration by parts the d = 4 action 1 1 1 S = d4x |g| k R + α α k−2(∂l)2 + α k2F 2(A) + α G2 2 5 6 4 5 4 6 1 −1 2 − √ α6k l √ [G + 2A∂l] . (A.20) 4 3 |g|

This theory is pure (Euclidean α6α5 =−1 or Lorentzian α6α5 =+1) N = 2, d = 4 SUGRA theory coupled to a “matter” vector supermultiplet that contains a vector and the two scalars k and l [30]. The matter and supergravity vector fields are combinations of the two vectors A, V . To identify the mater vector field we can use the fact that eliminating a matter supermultiplet is always a consistent truncation. The equations of motion of k and l after setting k = 1andl = 0 give the constraints 2 2 3α5F (A) + α6F (B) = 0, √ (A.21) 3F(A)− F(B) F(B)= 0. The second constraint is solved by √ 3F(A)− F(B)= 0, (A.22) 18 which also solves the first. This implies√ that the matter vector field that we denote by Wµ is proportional to the combination 3F(B)− F(B) or to its Hodge dual, at least for the

18 2 2 Observethatwehave( F) =−α6α5F and F =−α6α5F because α5α6 =+1 corresponds to Lorentzian signature and α5α6 =−1 to Euclidean signature. P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263 261 k = 1, l = 0 case. The orthogonal combination will be the field strength of the supergravity vector that we will denote by Vµ. To fully identify the supergravity and matter vector field strengths we have to check the consistency of the truncation from the supersymmetric point of view, that is, in the gravitino supersymmetry transformation rule Eq. (2.5) (adding hats everywhere). Assuming that the supersymmetry parameter ˆ is independent of the internal coordinate so it can be identified with the 4-dimensional one we get and setting from now on k = 1andl = 0 ˆ 1 i 3/2 δψa = ∇a − α5F/a(A)γ − √ α F(B)γ/ a + 2F/a(B) , 4 8 3 6 √ ˆ 1 δψ =− √ α5 3F(A)/ +F(B)γ/ , (A.23) 8 3 a a where γ =ˆγ with a = 0, 1, 2, 3 in the Lorentzian case α5α6 =+1, and a =

1, 2, 3, 4 in the Euclidean case α5α6 =−1, and γ =ˆγ is a matrix proportional to the 4-dimensional γ5. It can be shown that 1/2 1 1/2 F/ aγ =−iα Fγ/ a − F/ a , Fγ/ = iα F,/ (A.24) 6 2 6 and using these identities we can rewrite the gravitino supersymmetry transformation rule as follows: ˆ i 3/2 1 δψa = ∇a − α −α α F(A)/ + √ F(B)/ γa 8 6 5 6 3 1 3/2 1 + α −α α F/ a(A) − √ F/a(B) , 4 6 5 6 3 ˆ 1 1 δψ =− α5 F(A)/ − √ F(B)/ . (A.25) 8 3 ˆ The components ψa clearly become the 4-dimensional gravitino ψa. The internal ˆ component of the gravitino ψ only transforms into a combination vector field strengths and, thus, it can be identified with the gaugino λ of the matter vector multiplet. The combination of vector field strengths is proportional to the matter vector field strength F(W) (for k = 1andl = 0 only), in agreement with the constraint Eq. (A.22) found in the truncation of the action. The normalization factor and the definition of the supergravity vector field strength F(V) are found by imposing the diagonalization and correct normalization of the energy–momentum tensors of the vectors in the Einstein equation: the r.h.s. of the Einstein equation contains 1 1 − α Tµν (A) − α Tµν (B), (A.26) 2 5 2 6 where Tµν , the energy–momentum tensor of a vector field in four Euclidean or Lorentzian dimensions, is usually written in the form 1 T = F ρ F − g F 2, (A.27) µν µ νρ 2 µν 262 P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263 but can be rewritten in the more useful form 1 ρ ρ Tµν = Fµ Fνρ + α α Fµ Fνρ . (A.28) 2 5 6 This still leaves some ambiguities in the identification of the supergravity vector field, but they are irrelevant when W = 0 and lead to a unique relation

F(V)= 2α5α6 F(A). (A.29)

Now we can eliminate consistently the gaugino λ, the matter vector ﬁeld Wµ and scalars k and l, getting the bosonic action19 and gravitino supersymmetry transformation rule which are given in Eq. (3.1).

References

[1] J.P. Gauntlett, J.B. Gutowski, C.M. Hull, S. Pakis, H.S. Reall, Class. Quantum Grav. 20 (2003) 4587, hep- th/0209114. [2] J.B. Gutowski, D. Martelli, H.S. Reall, Class. Quantum Grav. 20 (2003) 5049, hep-th/0306235. [3] A. Chamseddine, J. Figueroa-O’Farrill, W. Sabra, hep-th/0306278. [4] M.M. Caldarelli, D. Klemm, JHEP 0309 (2003) 019, hep-th/0307022. [5] E. Lozano-Tellechea, P. Meessen, T. Ortín, Class. Quantum Grav. 19 (2002) 5921, hep-th/0206200. [6] J. Kowalski-Glikman, Phys. Lett. B 150 (1985) 125. [7] I. Robinson, Bull. Acad. Pol. Sci. 7 (1959) 351; B. Bertotti, Phys. Rev. 116 (1959) 1331. [8] N. Berkovits, N. Seiberg, JHEP 0307 (2003) 010, hep-th/0306226. [9] B. Fiol, C. Hofman, E. Lozano-Tellechea, hep-th/0312209. [10] T. Harmark, T. Takayanagi, Nucl. Phys. B 662 (2003) 3, hep-th/0301206. [11] M.M. Caldarelli, D. Klemm, hep-th/0310081. [12] K. Gödel, Rev. Mod. Phys. 21 (1949) 447. [13] G.T. Horowitz, A.A. Tseytlin, Phys. Rev. D 51 (1995) 2896, hep-th/9409021. [14] J.G. Russo, A.A. Tseytlin, Nucl. Phys. B 448 (1995) 293, hep-th/9411099. [15] M. Gurses, A. Karasu, O. Sarioglu, hep-th/0312290. [16] E.G. Gimon, A. Hashimoto, Phys. Rev. Lett. 91 (2003) 021601, hep-th/0304181. [17] D. Brecher, U.H. Danielsson, J.P. Gregory, M.E. Olsson, JHEP 0311 (2003) 033, hep-th/0309058. [18] M. Blau, P. Meessen, M. O’Loughlin, JHEP 0309 (2003) 072, hep-th/0306161. [19] D. Brace, C.A.R. Herdeiro, S. Hirano, hep-th/0307265. [20] E.K. Boyda, S. Ganguli, P. Horava, U. Varadarajan, Phys. Rev. D 67 (2003) 106003, hep-th/0212087. [21] N. Drukker, B. Fiol, J. Simon, Phys. Rev. Lett. 91 (2003) 231601, hep-th/0306057. [22] Y. Hikida, S.J. Rey, Nucl. Phys. B 669 (2003) 57, hep-th/0306148. [23] D. Brecher, P.A. DeBoer, D.C. Page, M. Rozali, JHEP 0310 (2003) 031, hep-th/0306190. [24] N. Seiberg, JHEP 0306 (2003) 010, hep-th/0305248. [25] H. Ooguri, C. Vafa, Adv. Theor. Math. Phys. 7 (2003) 53, hep-th/0302109. [26] S.B. Giddings, A. Strominger, Phys. Lett. B 230 (1989) 46. [27] G.W. Gibbons, M.B. Green, M.J. Perry, Phys. Lett. B 370 (1996) 37, hep-th/9511080. [28] H. Nishino, E. Sezgin, Nucl. Phys. B 278 (1986) 353. [29] E. Cremmer, LPTENS 80/17, Invited paper at the Nufﬁeld Gravity Workshop, Cambridge, England, June 22–July 12, 1980.

19 The bosonic action cannot be found by performing the above ﬁeld redeﬁnitions and has to bes constructed so as to reproduce the truncated equations of motion. P. Meessen, T. Ortín / Nuclear Physics B 684 (2004) 235–263 263

[30] A.H. Chamseddine, H. Nicolai, Phys. Lett. B 96 (1980) 89. [31] P. Meessen, Phys. Rev. D 65 (2002) 087501, hep-th/0111031. [32] G.W. Gibbons, G.T. Horowitz, P.K. Townsend, Class. Quantum Grav. 12 (1995) 297, hep-th/9410073. [33] J.C. Breckenridge, R.C. Myers, A.W. Peet, C. Vafa, Phys. Lett. B 391 (1997) 93, hep-th/9602065. [34] R. Kallosh, A. Rajaraman, W.K. Wong, Phys. Rev. D 55 (1997) 3246, hep-th/9611094. [35] J.P. Gauntlett, R.C. Myers, P.K. Townsend, Class. Quantum Grav. 16 (1999) 1, hep-th/9810204. [36] K.P. Tod, Phys. Lett. B 121 (1983) 241. [37] A.A. Belavin, A.M. Polyakov, A.S. Shvarts, Y.S. Tyupkin, Phys. Lett. B 59 (1975) 85. [38] J.M. Figueroa-O’Farrill, Class. Quantum Grav. 16 (1999) 2043, hep-th/9902066. [39] T. Ortín, Class. Quantum Grav. 19 (2002) L143, hep-th/0206159. [40] N. Alonso-Alberca, E. Lozano-Tellechea, T. Ortín, hep-th/0208158. [41] A. González-Arroyo, hep-th/9807108. [42] C. Hull, hep-th/0305039. [43] G. Papadopoulos, D. Tsimpis, Class. Quantum Grav. 20 (2003) L253, hep-th/0307127. [44] P. Meessen, T. Ortín, Nucl. Phys. B 541 (1999) 195, hep-th/9806120. [45] A. Trautman, Int. J. Theor. Phys. 16 (1977) 561. [46] B. Julia, H. Nicolai, Nucl. Phys. B 439 (1995) 291, hep-th/9412002. [47] R. Penrose, in: Differential Geometry and Relativity, Reidel, Dordrecht, 1976, pp. 271–275. [48] R. Gueven, Phys. Lett. B 482 (2000) 255, hep-th/0005061. [49] M. Blau, J. Figueroa-O’Farrill, C. Hull, G. Papadopoulos, hep-th/0201081. [50] M. Blau, J. Figueroa-O’Farrill, G. Papadopoulos, hep-th/0202111. [51] C.N. Pope, A. Sadrzadeh, S.R. Scuro, Timelike Hopf duality and type IIA* string solutions, Class. Quantum Grav. 17 (2000) 623, hep-th/9905161. Nuclear Physics B 684 (2004) 264–280 www.elsevier.com/locate/npe

Is Θ+(1540) a kaon–skyrmion resonance?

Nissan Itzhaki, Igor R. Klebanov, Peter Ouyang, Leonardo Rastelli

Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA Received 28 November 2003; accepted 3 February 2004

Abstract We reconsider the relationship between the bound state and the SU(3) rigid rotator approaches to strangeness in the Skyrme model. For non-exotic S =−1 baryons the bound state approach matches for small mK onto the rigid rotator approach, and the bound state mode turns into the rotator zero- mode. However, for small mK , we find no S =+1 kaon bound states or resonances in the spectrum, confirming previous work. This suggests that, at least for large N and small mK , the exotic state may be an artifact of the rigid rotator approach to the Skyrme model. An S =+1 near-threshold state comes into existence only for sufficiently large SU(3) breaking. If such a state exists, then it + = = 1 has the expected quantum numbers of Θ : I 0, J 2 and positive parity. Other exotic states + + + + = P = 3 = P = 1 = P = 5 = P = 3 with (I 1,J 2 ), (I 1,J 2 ), (I 2,J 2 ) and (I 2,J 2 ) appear as its SU(2) rotator excitations. As a test of our methods, we also identify a D-wave S =−1 near-threshold resonance that, upon SU(2) collective coordinate quantization, reproduces the mass splittings of the observed states Λ(1520), Σ(1670) and Σ(1775) with good accuracy.  2004 Published by Elsevier B.V.

1. Introduction

A remarkable recent event in hadronic physics is the discovery of a S =+1 baryon (dubbed the Z+ or Θ+) with a mass of 1540 MeV and width less than 25 MeV [1]. This discovery was promptly conﬁrmed in [2–5]. At present the spin, parity and magnetic moment of this state have not been determined; one group, the SAPHIR Collaboration [4], found that the isospin of the Θ+ is zero. Because it appears as a resonance in the system K+n, the minimal possibility for its quark content is uudds¯ which is manifestly exotic, i.e., it cannot be made out of three non-relativistic quarks. Early speculations on this kind of exotic baryons were made in [6,7]. Remarkably, a state with these quantum numbers

E-mail address: [email protected] (L. Rastelli).

0550-3213/$ – see front matter  2004 Published by Elsevier B.V. doi:10.1016/j.nuclphysb.2004.02.004 N. Itzhaki et al. / Nuclear Physics B 684 (2004) 264–280 265 appears naturally [8–10] in the rigid rotator quantization of the three-flavor Skyrme model [11–13], and detailed predictions for its mass and width were made by Diakonov, Petrov and Polyakov [14]. Their results provided motivation for the experimental searches that led to the discovery of Θ+ very close to the predicted parameters.1 The rigid rotator quantization of the Skyrme model that was used in [14] relied on working directly with N = 3(N is the number of colors). Then the model predicts the well-known 8 and 10 of SU(3), followed by an exotic 10 multiplet whose S =+1member is the Θ+. The approach of [14] began by postulating that the established N(1710) and Σ(1880) states are members of the antidecuplet. Then, using group theory techniques, and constraints from a rigid rotator treatment of chiral solitons, they estimated the mass and width of the other states in this multiplet. They predicted that the lowest member of the antidecuplet has a mass of 1530 MeV and width of about 9 MeV. These results appear to be confirmed strikingly by experiment.2 In this paper we will follow a somewhat different strategy, trying to develop a systematic 1/N expansion for Θ+. In the two-flavor Skyrme model, the quantum numbers of the low- = = 1 3 lying states do not depend on N, as long as it is odd: I J 2 , 2 ,... (I is the isospin and J is the spin). These states are identified with the nucleon and the ∆ [25,26]. The three-flavor case is rather different, since even the lowest SU(3) multiplets depend on N = 2n + 1 and become large as N →∞. The allowed multiplets must contain states of hypercharge N/3, i.e., of strangeness S = 0. In the notation where SU(3) multiplets are = 1 − labeled by (p, q), the lowest multiplets one finds are (1,n)with J 2 and (3,n 1) with = 3 J 2 [27–31]. These are the large N analogues of the octet and the decuplet. The rigid rotator mass formula, valid in the limit of unbroken SU(3),is 1 1 N2 M(p,q) = M + J(J + 1) + C(p,q) − J(J + 1) − , (1.1) cl 2Ω 2Φ 12 where Ω and Φ are moments of inertia, which are of order N. Using the formula for the quadratic Casimir, 1 C(p,q) = p2 + q2 + 3(p + q)+ pq , (1.2) 3 + 1 − = one notes [27–31] that the lowest lying SU(3) multiplets (2J,n 2 J),ofspinJ 1 3 2 , 2 ,...obey the mass formula N 1 M(J)= M + + J(J + 1). (1.3) cl 4Φ 2Ω Exactly the same multiplets appear when we construct baryon states out of N quarks. The splittings among them are of order 1/N, as is usual for soliton rotation excitations. + = 1 The large N analogue of the exotic antidecuplet is (0,n 2) with J 2 . Its splitting N + 0 from the lowest multiplets is 4Φ O(1/N). The fact that it is of order N raises questions

+ 1 For other recent theoretical models of Θ , see, e.g., [15–20]. 2 Note that the authors of [21–24] have argued that the experimental data actually indicate an even smaller width. 266 N. Itzhaki et al. / Nuclear Physics B 684 (2004) 264–280 about the validity of the rigid rotator approach to these states [29,30,32]. Indeed, we will argue that a better approximation to these states is provided by the bound state approach [33]. In the bound state approach one departs from the rigid rotator ansatz, and adopts more general kaon fluctuation profiles. This has O(N0) effect on energies of states even in the low-lying non-exotic multiplets, after SU(3) breaking is turned on. In the limit mK → 0 the bound state description of non-exotic baryons smoothly approaches the rigid rotator description, and the bound state wave function approaches the zero-mode sin(F (r)/2) [33,34], where F(r)is the radial profile function of the skyrmion. Our logic leads us to believe that, at least from the point of view of the large N expansion, Θ+ should be described by a near-threshold kaon–skyrmion resonance or bound state of S =+1, rather than by a rotator state (a similar suggestion was made independently in [30,35]). This is akin to the bound state description of the S =−1 baryons in [33] where a possibility of such a description of an exotic S =+1 state was mentioned as well. This leads us to a puzzle, however, since, in contrast with the situation for S =−1, for S =+1 there is no fluctuation mode that in the mK → 0 limit approaches the rigid rotator N mode of energy 4Φ (this will be shown explicitly in Section 3). An essential difficulty is that, for small mK , this is not a near-threshold state; hence, it is not too surprising that it does not show up in the more general fluctuation analysis. Thus, for large N and small mK the rigid rotator state with S =+1 appears to be an artifact of the rigid rotator approximation (we believe this to be a general statement that does not depend on the details of the chiral Lagrangian). Next, we ask what happens as we increase mK keeping other parameters in the kaon– skyrmion Lagrangian fixed. For mK = 495 MeV, and with the standard fit values of fπ and e, neither bound states [34] nor resonances [36] exist for S =+1.3 If, however, we increase mK to ≈ 1 GeV then a near-threshold state appears. Thus, we reach a surprising conclusion that, at least for large N, the exotic S =+1 state exists only due to the SU(3) breaking and disappears when the breaking is too weak. While this certainly contradicts the philosophy of [31], it is actually in line with some of the earlier literature (see, for example, [38] and the end of [39]). An intuitive way to see the necessity of the SU(3) breaking for the existence of the exotic is that the breaking keeps it a near-threshold state. One of the purposes of this paper is to examine how sensitive the existence of this state is to parameter choices. If we set mK = 495 MeV, then minor adjustments of fπ and e do not make the S =+1 resonance appear. We will find, however, that if the strength of the Wess–Zumino term is reduced by roughly a factor of 0.4 compared to its SU(3) value, then a near-threshold state corresponding to Θ+ is indeed found. Although we do not have a good a priori explanation for this reduction, it could be caused by unexpectedly large SU(3) breaking effects on this particular term. This issue clearly requires further investigation. The structure of the paper is as follows. In the next section we will review the rigid rotator approach to Skyrme model at large N, and recall a method of large N expansion introduced in [32], which involves expanding in rigid motions around the SU(2) subgroup

3 Such states seem to appear in [37]. However, in the normalization of the Wess–Zumino term, which is repulsive for S =+1 states, a large factor of e2 ∼ 30 was apparently omitted there (compare Eq. (7) of [37] with [34]). When this factor is reinstated, both bound states and resonances disappear for standard parameter choices, as claimed in [34,36] and conﬁrmed in this work. N. Itzhaki et al. / Nuclear Physics B 684 (2004) 264–280 267 of SU(3). In Section 3 we proceed to the bound state approach that, by introducing extra degrees of freedom, has O(N0) effect on baryon spectra. In Section 4 we review the status of S =−1 baryons based on kaon–skyrmion bound states, and also study a near-threshold D-wave resonance [36] that, upon SU(2) collective coordinate quantization, reproduces the observed states Λ(1520), Σ(1670) and Σ(1775) with good accuracy. In Section 5 we carry out the search for S =+1 kaon–skyrmion bound states and resonances. In Section 6 we attempt a different ﬁt with a non-zero pion mass. We offer some concluding remarks in Section 7.

2. Three-ﬂavor Skyrme model at large N

The Skyrme approach to baryons begins with the Lagrangian [25] 2 = fπ † µ + 1 † † 2 LSkyrme Tr ∂µU ∂ U 2 Tr ∂µUU ,∂νUU 16 32 e + Tr M(U + U † − 2) , (2.1) where U(xµ) is a matrix in SU(3) and M is proportional to the matrix of quark masses. Later on, it will be convenient to choose units where efπ = 1. There is an additional term in the action, called the Wess–Zumino term: iN 5 µναβγ † † † † † S =− d x Tr ∂µUU ∂ν UU ∂αUU ∂β UU ∂γ UU . (2.2) WZ 240π2 In the limit of unbroken SU(3) flavor symmetry, its normalization is fixed by anomaly considerations [40]. The Skyrme Lagrangian is a theory of mesons but it describes baryons as well. The simplest baryons in the Skyrme model are the nucleons. Classically, they have no strange quarks, so we may set the kaon fluctuations to zero and consider only the SU(2)-isospin subgroup of SU(3). Skyrme showed that there are topologically stabilized static solutions of hedgehog form: eiτ·ˆrF(r) 0 U = U = (2.3) 0 π,0 01 in which the radial profile function F(r) satisfies the boundary conditions F(0) = π,F(∞) = 0. By substituting the hedgehog ansatz (2.3) into the Skyrme Lagrangian (2.1), and considering the corresponding equations of motion one obtains an equation for F(r) which is straightforward to solve numerically. The non-strange low-lying excitations of this soliton are given by rigid rotations of the pion field A(t) ∈ SU(2): −1 U(x,t)= A(t)U0A (t). (2.4) For such an ansatz the Wess–Zumino term does not contribute. By expanding the Lagrangian about U0 and canonically quantizing the rotations, one finds that the Hamiltonian is 1 H = M + J(J + 1), (2.5) cl 2Ω 268 N. Itzhaki et al. / Nuclear Physics B 684 (2004) 264–280 where J is the spin and the c-numbers Mcl and Ω are complicated integrals of functions of the soliton profile. Numerically, for vanishing pion mass, one finds that f M 36.5 π , (2.6) cl e 107 Ω 3 . (2.7) e fπ For N = 2n + 1, the low-lying quantum numbers are independent of the integer n.The = = 1 = = 3 lowest states, with I J 2 and√I J 2 , are√ identified with the nucleon and ∆ particles respectively. Since fπ ∼ N,ande ∼ 1/ N, the soliton mass is ∼ N, while the rotational splittings are ∼ 1/N. Adkins, Nappi and Witten [26] found that they could fit the N and ∆ masses with the parameter values e = 5.45,fπ = 129 MeV. In comparison, the physical value of fπ = 186 MeV. A generalization of this rigid rotator treatment that produces SU(3) multiplets of baryons is obtained by making the collective coordinate A(t) an element of SU(3).Then the WZ term makes a crucial constraint on allowed multiplets [8,9,11–13]. As discussed in the introduction, large N treatment of this 3-flavor Skyrme model is more subtle than for its 2-flavor counterpart. When N = 2n + 1 is large, even the lowest lying (1,n) SU(3) multiplet contains (n + 1)(n + 3) states with strangeness ranging from S = 0to S =−n − 1 [29]. When the strange quark mass is turned on, it will introduce a splitting of order N between the lowest and highest strangeness baryons in the same multiplet. Thus, SU(3) is badly broken in the large N limit, no matter how small ms is [29]. We will find it helpful to think in terms of SU(2)×U(1) flavor quantum numbers, which do have a smooth large N limit. In other words, we focus on low strangeness members of these multiplets, whose I, J quantum numbers do have a smooth large N limit, and to try identify them with observable baryons. Since the multiplets contain baryons with up to ∼ N strange quarks, the wave functions of baryon with fixed strangeness deviate only an amount ∼ 1/N into the strange directions of the collective coordinate space. Thus, to describe them, one may expand the SU(3) rigid rotator treatment around the SU(2) collective coordinate. The small deviations from SU(2) may be assembled into a complex SU(2) doublet K(t). This method of 1/N expansion was implemented in [32], and reviewed in [29]. From the point of view of the Skyrme model the ability to expand in small fluctuations is due to the Wess–Zumino term which acts as a large magnetic field of order N. The method 0 works for arbitrary kaon mass, and has the correct limit as mK → 0. To order O(N ) the Lagrangian has the form [32] N L = 4ΦK˙ †K˙ + i K†K˙ − K˙ †K − ΓK†K. (2.8) 2 The Hamiltonian may be diagonalized:

† † N H = ω−a a + ω+b b + , (2.9) 4Φ where 2 N 2 2 N ω± = 1 + (mK /M ) ± 1 ,M= . (2.10) 8Φ 0 0 16ΦΓ N. Itzhaki et al. / Nuclear Physics B 684 (2004) 264–280 269

The strangeness operator is S = b†b − a†a. All the non-exotic multiplets contain a† N − → + → ∼ 0 excitations only. In the SU(3) limit, ω 0, but ω 4Φ N . Thus, the “exoticness” quantum number mentioned in [31] is simply E = b†b here, and the splitting between N multiplets of different “exoticness” is 4Φ , in agreement with results found from the exact rigid rotator mass formula (1.1). The rigid rotator prediction for O(N0) splittings are not exact, however, for reasons explained long ago [29,32,33] and reviewed in the introduction. Even for non-exotic states, as the soliton rotates into strange directions, it experiences deformation which grows 0 with mK . The bound state approach allows it to deform, which has a signiﬁcant O(N ) effect on energy levels. We now turn to review the bound state approach.

3. Review of the bound state approach

Another approach to strange baryons, which proves to be quite successful in describing the light hyperons, is the so-called bound state method [33]. The basic strategy involved is to expand the action to second order in kaon fluctuations about the classical hedgehog soliton. Then one can obtain a linear differential equation for the kaon field, incorporating the effect of the kaon mass, which one can solve exactly. The eigenenergies of the kaon field are then precisely the differences between the Skyrmion mass and the strange baryons. In order to implement this strategy, it is convenient to write U in the form

U = Uπ UK Uπ , (3.1) j a where Uπ = exp[2iλj π /fπ ] and UK = exp[2iλaK /fπ ] with j running from 1 to 3 and a 4 running from 4 to 7. The λa are the standard SU(3) Gell-Mann matrices. We will collect the Ka into a complex isodoublet K: 4 − 5 + = √1 K iK = K K 6 7 0 . (3.2) 2 K − iK K Though the Wess–Zumino term can only be written as an action term, if we expand it to second order in K, we obtain an ordinary Lagrangian term: iN = µ † − † LWZ 2 B K DµK (DµK) K , (3.3) fπ where 1 † † DµK = ∂µK + U ∂µ Uπ + Uπ ∂µ U K, (3.4) 2 π π and Bµ is the baryon number current. Now we decompose the kaon ﬁeld into a set of partial waves. Because the background soliton ﬁeld is invariant under combined spatial

4 There is actually a second coupling constant fK which replaces fπ in the deﬁnition of UK ; experimentally, fK ∼ 1.22fπ . To incorporate the difference between these coupling constants, one simply replaces fπ by fK when expanding in powers of the kaon ﬁeld, but does not rescale the kinetic and kaon mass terms, which are required to have standard normalization. Then those terms that follow from the four-derivative term, the Wess– 2 Zumino term, and the pion mass in the Skyrme Lagrangian change by a factor of (fπ /f K ) . 270 N. Itzhaki et al. / Nuclear Physics B 684 (2004) 264–280 and isospin rotations T = I + L, a good set of quantum numbers is T, Land Tz,andsowe write the kaon eigenmodes as = K k(r,t)YTLTz . (3.5)

Substituting this expression into LSkyrme + LWZ we obtain an effective Lagrangian for the radial kaon ﬁeld k(r,t): L = 4π r2 dr f(r)k˙†k˙ + iλ(r) k†k˙ − k˙†k d d − h(r) k† k − k†k m2 + V (r) , dr dr K eff with 2 Ne λ(r) =− F sin2 F, f(r)= 1 + 2s(r) + d(r), h(r)= 1 + 2s(r), 2π2r2 F d(r)= F 2,s(r)= (sin F/r)2,c(r)= sin2 , 2 and + + 1 1 d s 2 Veff =− (d + 2s) − 2s(s + 2d)+ L(L + 1) + 2c + 4cI · L 4 r2 6 d + s c2 + 2cI · L − I · L + (c + I · L)F sin F r2 dr m2 − π (1 − cosF). (3.6) 2 The resulting equation of motion for k is −f(r)k¨ + 2iλ(r)k˙ + Ok = 0, 1 O ≡ ∂ h(r)r2∂ − m2 − V (r). (3.7) r2 r r K eff Expanding k in terms of its eigenmodes gives ˜ = ˜ iωnt † + iωnt k(r,t) kn(r)e bn kn(r)e an , (3.8) n>0 with ωn, ω˜ n positive. The eigenvalue equations are thus f(r)ω2 + 2λ(r)ω + O k = 0 (S =−1), n n n ˜ 2 − ˜ + O ˜ = =+ f(r)ωn 2λ(r)ωn kn 0 (S 1). (3.9) Crucially, the sign in front of λ, which is the contribution of the WZ term, depends on whether the relevant eigenmodes have positive or negative strangeness. The important result here is the set of equations (3.9) which we will now solve and whose solutions we will match with the spectrum of baryons. It is possible to examine these equations analytically for mK = 0. The S =−1 equation has an exact solution with ω = 0andk(r) ∼ sin(F (r)/2). This is how the rigid rotator zero N. Itzhaki et al. / Nuclear Physics B 684 (2004) 264–280 271 mode is recovered in the bound state treatment [34]. As mK is turned on, this solution turns into an actual bound state [33,34]. One the other hand, the S =+1 equation does not have ˜ = N ∼ a solution with ω 4Φ and k(r) sin(F (r)/2). This is why the exotic rigid rotator state is not reproduced by the more precise bound state approach to strangeness. In Section 5 we further check that, for small mK , there is no resonance that would turn into the rotator N state of energy 4Φ in the SU(3) limit.

4. Baryons with S =−1

In this section we recall the description of S =−1 baryons as antikaon–skyrmion bound states or resonances. We will set the kaon mass equal to its physical value, mK = 495 MeV, but set the pion mass equal to zero. If we wish to fit both the nucleon and delta masses to their physical values using the SU(2) rotator approximation, then we must take e = 5.45 and fπ = 129 MeV; let us begin with these values as they are somewhat traditional in analyses based on the Skyrme model. = = 1 The lightest strange excitation is in the channel L 1, T 2 , and its mass is Mcl + 0.218 efπ 1019 MeV. As the lightest state with S =−1, it is natural to identify it with the Λ(1115), Σ(1190),andΣ(1385) states, where the additional splitting arises from SU(2) rotator corrections. Let us compute these corrections. The relevant formula for S =−1 [34] is 1 3 M = M + ω + cJ(J + 1) + (1 − c)I(I + 1) + (c2 − c) , (4.1) cl 1 2Ω 4 where ω1 is the kaon eigenenergy and c is a number defined in terms of the bound state eigenfunction k1 by ∗ 4 2 2 F − d 2 − 4 2 2 F dr k1 k1 3 fr cos 2 2 dr (r F sin F) 3 sin F cos 2 c = = 1 − ω . l 1 1 2 ∗ + r dr k1k1(f ω1 λ) (4.2) = = 1 = In this L 1, T 2 channel, we find from numerical integration that c 0.617. The masses, including SU(2) corrections, appear in columns (a) of Table 1. The two features to note here are that first, these states are all somewhat overbound, and second, the SU(2) splittings match rather closely with experiment (one of the successes of the bound state approach [34]). = = 1 The next group of strange excitations is in the channel L 0, T 2 , and we have determined its mass before including rotator corrections to be Mcl + 0.523 efπ 1233 MeV. In this channel the formula for c is given by ∗ 4 2 2 F + d 2 + 4 2 2 F dr k2 k2 3 fr sin 2 2 dr (r F sin F) 3 sin F sin 2 c = = 1 − ω , l 0 2 2 ∗ + r dr k2 k2(f ω2 λ) (4.3) and for the relevant bound state this gives c ∼ 0.806. The SU(2) corrections (4.1) raise the mass of the lightest state in this channel to 1281 MeV. From the quantum numbers, it is natural to identify this state with the Λ(1405) state, but as we see it is rather overbound. 272 N. Itzhaki et al. / Nuclear Physics B 684 (2004) 264–280

Table 1 Masses (in MeV) of the light S =−1 hyperons as calculated from the bound state approach, with (a) e = 5.45, fπ = fK = 129 MeV, (b) e = 4.82, fπ = fK = 186 MeV, with an overall constant added to ﬁt the N and ∆ masses, and (c) the same parameters as (a) but with the WZ term artiﬁcially decreased by a factor of 0.4. In all cases mπ = 0 Particle JILMass (expt) Mass (a) Mass (b) Mass (c) 1 Λ 2 0 1 1115 1048 1059 1121 1 Σ 2 1 1 1190 1122 1143 1289 3 Σ 2 1 1 1385 1303 1309 1330 1 Λ 2 0 0 1405 1281 1346 1366

We have seen that with the traditional values e = 5.45 and fπ = 129 MeV the Skyrme model successfully captures qualitative features of the baryon spectrum such as the presence of the Λ(1405) state, but that the bound states are all too light. It is possible that the zero-point energy of kaon fluctuations, which is hard to calculate explicitly, has to be added to all masses. Thus it is easiest to focus on mass splittings. Then from the SU(2) rotator quantization, we would obtain only one constraint (2.7), from the nucleon–∆ splitting, and be able to adjust e and fπ to improve the fit to known masses. As we increase fπ , we find that the particle masses increase, improving agreement with experiment. For definiteness, let us try to set fπ to its experimental value, fπ = 186 MeV, which then requires e = 4.82. We report the results for the masses in column (b) of Table 1. More dramatic increases in the particle masses may be obtained by distinguishing between the pion decay constant fπ and the kaon decay constant fK , as shown by Rho, Riska and Scoccola [43], who worked in a modified Skyrme model with explicit vector mesons [44]. For fK = 1.22 fπ they were able to essentially eliminate the over-binding = = 1 problem for the L 1, T 2 states, though they still found the analogue of the Λ(1405) state to be overbound by about 100 MeV. We should add that the natural appearance of this Λ(1405) with negative parity is a major success of the bound state approach [34,45]. In quark or bag models such a baryon is described by a p-wave quark excitation, which typically turns out to be too heavy (for a discussion, see the introduction of [45]). A third way to raise the masses of the Λ and Σ states is to modify the Wess–Zumino term. For the S =−1 states, the WZ term results in an attractive force between the Skyrmion and the kaon, so if we reduce this term by hand, the hyperons will become less tightly bound and their masses should increase.5 We will address this approach in the next section, and we will see that such a reduction helps to produce an S =+1 near-threshold state. However, too great a reduction of the WZ term will spoil the hyperfine splittings governed by the parameter c. Finally, let us note that the philosophy of the bound state approach can be applied successfully to states above threshold. Such states will appear as resonances in kaon– nucleon scattering, which we may identify by the standard procedure of solving the

5 Note that for unbroken SU(3) the WZ term is quantized and cannot be changed by hand. However, SU(3) breaking is likely to change the WZ term. N. Itzhaki et al. / Nuclear Physics B 684 (2004) 264–280 273

= = 3 =− Fig. 1. Phase shift as a function of energy in the L 2, T 2 , S 1 channel. The energy ω is measured in units of ef π (with the kaon mass subtracted, so that ω = 0 at threshold), and the phase shift δ is measured in radians. Here e = 5.45 and fπ = 129 MeV.

Table 2 Masses (in MeV) of the S =−1 D-wave resonances calculated from the bound state approach, with fπ = 129 MeV, e = 5.45 Particle JILMass (expt) Mass (th) 3 Λ(D03) 2 0 2 1520 1462 3 Σ(D13) 2 1 2 1670 1613 5 Σ(D15) 2 1 2 1775 1723 appropriate kaon wave equation and studying the phase shifts of the corresponding = = 3 solutions as a function of the kaon energy. In the L 2, T 2 channel there is a 6 resonance at Mcl + 0.7484 efπ = 1392 MeV (see Fig. 1). In this channel, the SU(2) splitting parameter c is given by the formula [41] ∗ 2 + 4 2 F 2 − 4 d 2 + 4 2 2 F dr k3 k3 3 1 5 cos 2 fr 5 dr (r F sin F) 3 sin F sin 2 c = = 1 − ω . l 2 3 2 ∗ + r dr k3 k3(f ω3 λ) Numerically, we evaluate this coefficient by cutting off the radial integral around the point where k(r) begins to oscillate. We find c ∼ 0.23. The states are split into the channels 3 3 5 with (I, J ) given by (0, 2 ), (1, 2 ), (1, 2 ) [36], with masses 1462 MeV, 1613 MeV, and 1723 MeV, respectively (see Table 2). We see that these correspond nicely to the known negative parity resonances Λ(1520) (which is D03 in standard notation), Σ(1670) (which is D13)andΣ(1775) (which is D15) [42]. As with the bound states, we find that the resonances are somewhat overbound (the overbinding of all states is presumably related to the necessity of adding an overall zero-point energy of kaon fluctuations), but that the mass splittings within this multiplet are accurate to within a few percent. In fact, we find

6 The existence of this resonance was noted long ago in a vector-meson stabilized Skyrme model [36]. 274 N. Itzhaki et al. / Nuclear Physics B 684 (2004) 264–280 that the ratio M 1, 5 − M 0, 3 2 2 ≈ 1.73, (4.4) 3 − 3 M 1, 2 M 0, 2 while its empirical value is 1.70. This very good agreement with the states observed above the K–N threshold is an additional success of the kaon ﬂuctuation approach to strange baryons.

5. A baryon with S =+1?

For states with positive strangeness, the eigenvalue equation for the kaon field is the same except for a change of sign in the contribution of the WZ term. This sign change makes the WZ term repulsive for states with s¯ quarks and introduces a splitting between ordinary and exotic baryons [33]. In fact, with standard values of the parameters (such as those in the previous section) the repulsion is strong enough to remove all bound states and resonances with S = 1, including the newly-observed Θ+. It is natural to ask how much we must modify the Skyrme model to accommodate the pentaquark. The simplest modification we can make is to introduce a coefficient a multiplying the WZ term. Qualitatively, we expect that reducing the WZ term will make the S =+1 baryons more bound, while the opposite should happen to the ordinary baryons. Another modification we will attempt is to vary the mass of the kaon; we will find that for sufficiently large kaon mass the Θ+ becomes stable. In all cases, we have found empirically that raising fK relative to fπ makes the pentaquark less bound, so for this section we will take fK = fπ . The most likely channel in which we might find an exotic has the quantum numbers = = 1 L 1, T 2 , as in this case the effective potential is least repulsive near the origin. For 3 fπ = 129, 186, and 225 MeV, with e fπ fixed, we have studied the effect of lowering the WZ term by hand. Interestingly, in all three cases we have to set a 0.39 to have a bound state at threshold. If we raise a slightly, this bound state moves above the threshold, but does not survive far above threshold; it ceases to be a sharp state for a 0.46. We have 7 plotted phase shifts for various values of a in Fig. 2. With a = 0.39 and fπ = 129 MeV, this state (with mass essentially at threshold) has SU(2) splitting parameter c ∼−0.48. The SU(2) collective coordinate quantization of the state proceeds analogously to that of the S =−1 bound states, and the mass formula is again of the form (4.1). Thus the lightest =+ = = 1 =+ S 1 state we find has I 0, J 2 and positive parity, i.e., it is an S 1 counterpart of the Λ. This is our candidate Θ+ state. Its first SU(2) rotator excitations have I = 1, + + J P = 3 and I = 1, J P = 1 (a relation of these states to Θ+ also follows from general 2 2 + + P = 1 3 large N relations among baryons [35]). The counterparts of these J 2 , 2 states in the rigid rotator quantization lie in the 27-plets of SU(3). These states were recently discussed in [46,47].

7 When the state is above the threshold, we do not ﬁnd a full π variation of the phase. Furthermore, the variation and slope of the phase shift decrease rapidly as the state moves higher, so it gets too broad to be identiﬁable. So, the state can only exist as a bound state or a near-threshold state. N. Itzhaki et al. / Nuclear Physics B 684 (2004) 264–280 275

From the mass formula (4.1) we deduce that 1 1 c M 1, 1 − M 0, 1 = (1 − c), M 1, 3 − M 0, 1 = 1 + . (5.1) 2 2 Ω 2 2 Ω 2 Since c<0, the J = 3 state is lighter than J = 1 .Usingc ∼−0.48, we ﬁnd that the + 2 2 + = P = 3 ∼ + 8 = P = 1 I 1, J 2 state is 148 MeV heavier than the Θ , while the I 1, J 2 ∼ + state is 289 MeV heavier than the Θ . + + = P = 3 5 We may further consider I 2 rotator excitations which have J 2 , 2 . Such states are allowed for N = 3 (in the quark language the charge +3 state, for example, is given by + + ¯ P = 3 5 uuuus). The counterparts of these J 2 , 2 states in the rigid rotator quantization lie in the 35-plets of SU(3) [46,47]. From the mass formula (4.1) we deduce that 5 − 1 = 1 + ∼ M 2, 2 M 0, 2 (3 c) 494 MeV, Ω 3 c M 2, 3 − M 0, 1 = 1 − ∼ 729 MeV. (5.2) 2 2 Ω 2 While the value of c certainly depends on the details of the chiral Lagrangian, we may form certain combinations of masses of the exotics from which it cancels. In this way we ﬁnd “model-independent relations” which rely only the existence of the SU(2) collective coordinate: 3 + 1 − 1 = − = 2M 1, 2 M 1, 2 3M 0, 2 2(M∆ MN ) 586 MeV, 3 5 + 3 − 5 1 = − = M 2, 2 M 2, 2 M 0, 2 5(M∆ MN ) 1465 MeV, 2 2 5 M 2, 3 − M 2, 5 = M 1, 1 − M 1, 3 , (5.3) 2 2 3 2 2 − = 3 where we used M∆ MN 2Ω . These relations are analogous to the sum rule [33]

2MΣ∗ + MΣ − 3MΛ = 2(M∆ − MN ), (5.4) which is obeyed with good accuracy. If the I = 1, 2 exotic baryons are discovered, it will be very interesting to compare the relations (5.3) with experiment. Suppose we tried to require the existence of a pentaquark state by setting a = 0.4. How would this change affect the spectrum of negative strangeness baryons? The results are = = = 1 somewhat mixed. Let us take fπ 129 MeV as an example. In the L 1, T 2 channel, the bound state has a mass of 1209 MeV before including SU(2) rotator corrections. The parameter c which characterizes the SU(2) splittings falls to c ∼ 0.14. Including the splittings we ﬁnd the masses given in column (c) of Table 1. Notice that the Σ is far above its experimental mass of 1190 MeV, signaling drastic disagreement with the Gell-Mann– = = 1 Okubo relations, as we would expect for this small value of c.IntheL 0, T 2 channel the Λ resonance is at 1366 MeV (including rotator corrections), still somewhat overbound.

8 This value is close to those predicted in [46] but is signiﬁcantly higher than the 55 MeV reported in [47]. For a comparison of these possibilities with available data, see [19]. 276 N. Itzhaki et al. / Nuclear Physics B 684 (2004) 264–280

=+ = = 1 Fig. 2. Phase shifts δ as a function of energy in the S 1, L 1, T 2 channel, for various choices of the parameter a (strength of the WZ term). The energy ω is measured in units of ef π (e = 5.45, fπ = fK = 129 MeV) and the phase shift δ is measured in radians. ω = 0 corresponds to the K–N threshold.

=+ = = 1 Fig. 3. Phase shifts δ as a function of energy in the S 1, L 1, T 2 channel, for various values of mK . Here e = 5.45 and fπ = fK = 129 MeV.

As another probe of the parameter space of our Skyrme model, let us vary the mass of the kaon and see how this affects the pentaquark. As observed in Section 3, in the limit of =+ = = 1 infinitesimal kaon mass, there is no resonance in the S 1, L 1, T 2 channel. We 9 find that to obtain a bound state in this channel, we must raise mK to about 1100 MeV. Plots of the phase shift vs. energy for different values of mK are presented in Fig. 3. This figure clearly shows that increasing SU(3) breaking leads to increasing variation of the scattering phase.

9 Since both D and B mesons are much heavier than this, we may infer following [38] that there exist exotic bound state baryons which, in the quark model language, are pentaquarks containing an anti-charm or an anti- bottom quark (provided that the associated meson decay constants fD and fB are not too large). This prediction is rather insensitive to the details of the chiral Lagrangian [38]. See also [48] for a more careful analysis of these exotics that incorporates heavy quark symmetry. N. Itzhaki et al. / Nuclear Physics B 684 (2004) 264–280 277

Table 3 Masses (in MeV) of the light S =−1 hyperons as calculated from the bound state approach, with e = 4.84, fπ = fK = 108 MeV, and mπ = 138 MeV. Column (d) reports the masses with the usual WZ term; in column (d) the WZ term has been artiﬁcially reduced by a factor 0.75 Particle JILMass (expt) Mass (d) Mass (e) 1 Λ 2 0 1 1115 1031 1050 1 Σ 2 1 1 1190 1126 1177 3 Σ 2 1 1 1385 1276 1279 1 Λ 2 0 0 1405 1253 1283

6. Fits with massive pion

One interesting way to extend the model is to include the mass of the pion, as first explored by Adkins and Nappi [49]. We find that this actually improves matters for the pentaquark. The basic procedure is simply to take mπ = 138 MeV; there will be a modification in the kaon effective potential and also a change in the variational equation for the Skyrmion profile function F(r). As a result, the constraints on e and fπ change: f M 38.7 π , (6.1) cl e . 62 9 Ω 3 . (6.2) e fπ Adkins and Nappi found that the best fit to the nucleon and delta masses was given by e = 4.84 and fπ = 108 MeV. In dimensionless units, mπ = 0.263. =− = = 1 For the S 1 baryons (Table 3), we find a bound state in the L 1, T 2 channel with mass 1012 MeV. The SU(2) rotator parameter is cl=1 ∼ 0.51, giving a mass spectrum = = 1 of 1031, 1126, and 1276 MeV. In the L 0, T 2 sector there is a bound state with mass 1204 MeV which is increased by rotator corrections (c ∼ 0.82) to give a state with mass = = 3 1253 MeV corresponding to Λ(1405).Furthermore,intheL 2, T 2 channel there is a bound state slightly below the threshold [41] (in the massless pion fit this state was a resonance slightly above the threshold). While these states are all overbound, one can presumably improve the fit by adjusting parameters as in the previous sections. Let us first note that with the massive pion fit, the pentaquark appears with a smaller = = 1 =+ adjustment in parameters. In the L 1, T 2 , S 1 channel, there is a bound state for a ∼ 0.69. Let us study the effect of setting a = 0.75 on the bound states. With this change, the bound state energies for negative strangeness baryons rise. In the L = 1, = 1 ∼ T 2 channel, the mass is 1114 MeV, and cl=1 0.35, giving masses of 1050, 1177 and 1279 MeV for the Λ,Σ,Σ states. cl=1 is smaller than its experimental value of 0.62 but is not disastrously small, and the overall masses have increased towards their = = 1 experimental values. In the L 0, T 2 channel, the Λ state also increases to 1283 MeV (including rotator corrections). One further interesting result is that the pentaquark seems more sensitive to the mass of the kaon with mπ = 138 MeV; the pentaquark actually becomes bound for mK = 700 MeV. 278 N. Itzhaki et al. / Nuclear Physics B 684 (2004) 264–280

It is also possible to vary the pion mass; for masses around 200 MeV the bound state can appear for S = 1 with a ∼ 0.8.

7. Discussion

There are several implications of the analysis carried out in the preceding sections. First of all, we have to admit that the bound state approach to the Skyrme model could not have been used to predict the existence of an exotic S =+1 baryon. Indeed, for typical parameter choices we find neither kaon–skyrmion bound states nor resonances with S =+1, confirming earlier results from the 80s [29,34,36]. At the time these results appeared consistent with the apparent absence of such exotic resonances in kaon scattering data.10 We have found, however, that by a relatively large adjustment of parameters in the minimal bound state Lagrangian, such as reduction of the WZ term to 0.4 of its SU(3) value, the near-threshold S =+1 kaon state can be made to appear. In this case, however, the agreement of the model with the conventional strange baryons is worsened somewhat. A better strategy may be to vary more parameters in the Lagrangian, and perhaps to include other terms; then there is hope that properties of both exotic and conventional baryons will be reproduced nicely. This would be a good project for the future. Finally, our work sheds new light on connections between the bound state and the rigid rotator approaches to strange baryons. These connections were explored in the 80s, and it was shown that the bound state approach matches nicely to 3-flavor rigid rotator quantization carried out for large N [29,32,34]. The key observation is that, in both approaches, the deviations into strange directions become small in the large N limit due to the WZ term acting as a large magnetic field. Thus, for baryons whose strangeness is of order 1, the harmonic approximation is good for any value of mK .TheS =−1 bound state mode smoothly turns into the rotator zero-mode in the limit mK → 0, which shows explicitly that rotator modes can be found in the small fluctuation analysis around the SU(2) skyrmion. However, for small mK there is no fluctuation mode corresponding to exotic S =+1 rigid rotator excitations. In our opinion, this confirms the seriousness of questions raised about such rotator states for large N [29,30,32]. If the large N expansion is valid, then we conclude that the exotic baryon appears in the spectrum only for sufficiently large SU(3) breaking. The simplest way to parametrize this breaking is to increase mK while keeping coefficients of all other terms fixed at their SU(3) values. Then we find that the resonance appears at a value of mK ∼ 1 GeV.However, in reality SU(3) breaking will also affect other coefficients, in particular it may reduce somewhat the strength of the WZ term, thus helping the formation of the resonance. In principle, the coefficients in the chiral Lagrangian should be fitted from experiment, and also higher derivative terms may need to be included. What does our work imply about the status of the Θ+ baryon in the real world? As usual, this is the most difficult question. If N = 3 is large enough for the semiclassical approach

10 In fact, the experimental situation is still somewhat confused (see [21–24] for discussions of remaining puzzles). N. Itzhaki et al. / Nuclear Physics B 684 (2004) 264–280 279 to skyrmions to be valid, then we believe that our picture of Θ+ as a kaon–skyrmion near-threshold state is a good one. It is possible, however, that the rigid rotator approach carried out directly for N = 3, as in [14], is a better approximation to the real world, as suggested by its successful prediction of the pentaquark. It is also possible that quark model approaches, such as those in [16,17], or lattice calculations [20], will eventually prove to be more successful. Clearly, further work, both experimental and theoretical, is needed to resolve these issues.

Acknowledgements

We are grateful to C. Callan, D. Diakonov, G. Farrar, C. Nappi, V. Petrov and E. Witten for useful discussions. This material is based upon work supported by the National Science Foundation Grants No. PHY-0243680 and PHY-0140311. Any opinions, ﬁndings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reﬂect the views of the National Science Foundation.

References

[1] T. Nakano, et al., LEPS Collaboration, Evidence for a narrow S =+1 baryon resonance in photoproduction from the neutron, Phys. Rev. Lett. 91 (2003) 012002, hep-ex/0301020. [2] V.V. Barmin, et al., DIANA Collaboration, Observation of a baryon resonance with positive strangeness in + K collisions with Xe nuclei, hep-ex/0304040. [3] S. Stepanyan, et al., CLAS Collaboration, Observation of an exotic S =+1 baryon in exclusive photoproduction from the deuteron, hep-ex/0307018. [4] J. Barth, et al., SAPHIR Collaboration, Observation of the positive-strangeness pentaquark Theta+ in photoproduction with the SAPHIR detector at ELSA, hep-ex/0307083. [5] A.E. Asratyan, A.G. Dolgolenko, M.A. Kubantsev, Evidence for formation of a narrow K0(S)p resonance with mass near 1533 MeV in neutrino interactions, hep-ex/0309042. [6] R.L. Jaffe, Baryon Excitations In The Bag Model, SLAC-PUB-1774, Talk presented at the Topical Conf. on Baryon Resonances, Oxford, England, July 5–9, 1976. [7] D. Strottman, Multi-quark baryons and the mit bag model, Phys. Rev. D 20 (1979) 748. [8] A.V. Manohar, Equivalence of the chiral soliton and quark models in large N, Nucl. Phys. B 248 (1984) 19. [9] M. Chemtob, Skyrme model of baryon octet and decuplet, Nucl. Phys. B 256 (1985) 600. [10] M. Praszalowicz, SU(3) Skyrmion, TPJU-5-87, Talk presented at the Cracow Workshop on Skyrmions and Anomalies, Mogilany, Poland, February 20–24, 1987. [11] E. Witten, Global aspects of current algebra, Nucl. Phys. B 223 (1983) 422; E. Witten, Current algebra, baryons, and quark conﬁnement, Nucl. Phys. B 223 (1983) 433. [12] E. Guadagnini, Baryons as solitons and mass formulae, Nucl. Phys. B 236 (1984) 35. [13] P.O. Mazur, M.A. Nowak, M. Praszalowicz, SU(3) extension of the Skyrme model, Phys. Lett. B 147 (1984) 137. [14] D. Diakonov, V. Petrov, M.V. Polyakov, Exotic anti-decuplet of baryons: prediction from chiral solitons, Z. Phys. A 359 (1997) 305, hep-ph/9703373. [15] F. Stancu, D.O. Riska, Stable uudds¯ pentaquarks in the constituent quark model, hep-ph/0307010. [16] M. Karliner, H.J. Lipkin, The constituent quark model revisited: quark masses, new predictions for hadron masses and KN pentaquark, hep-ph/0307243. [17] R.L. Jaffe, F. Wilczek, Diquarks and exotic spectroscopy, hep-ph/0307341. + [18] K. Cheung, Pentaquark Theta , constituent quark structures, and prediction of charmed Theta/c0 and bottomed Theta/b+, hep-ph/0308176. 280 N. Itzhaki et al. / Nuclear Physics B 684 (2004) 264–280

[19] B.K. Jennings, K. Maltman, hep-ph/0308286. [20] F. Csikor, Z. Fodor, S.D. Katz, T.G. Kovacs, Pentaquark hadrons from lattice QCD, hep-lat/0309090. + [21] S. Nussinov, Some comments on the putative Theta (1543) exotic state, hep-ph/0307357. + [22] R.A. Arndt, I.I. Strakovsky, R.L. Workman, K nucleon scattering and exotic S =+1 baryons, nucl- th/0308012. [23] A. Casher, S. Nussinov, The narrow Theta(1543)—a QCD dilemma: tube or not tube?, hep-ph/0309208. + [24] J. Haidenbauer, G. Krein, Influence of a Z (1540) resonance on K + N scattering, hep-ph/0309243. [25] T.H. Skyrme, A nonlinear field theory, Proc. R. Soc. London A 260 (1961) 127; T.H. Skyrme, A unified field theory of mesons and baryons, Nucl. Phys. 31 (1962) 556. [26] G.S. Adkins, C.R. Nappi, E. Witten, Static properties of nucleons in the Skyrme model, Nucl. Phys. B 228 (1983) 552. [27] V. Kaplunovsky, unpublished. [28] Z. Dulinski, M. Praszalowicz, Large N(c) limit of the Skyrme model, Acta Phys. Pol. B 18 (1988) 1157. [29] I.R. Klebanov, Strangeness In The Skyrme Model, PUPT-1158, Lectures given at NATO ASI on Hadron and Hadronic Matter, Cargese, France, August 8–18, 1989, http://curie.princeton.edu/klebanov/ strangeness_paper.pdf. + [30] T.D. Cohen, Chiral soliton models, large N(c) consistency and the Theta exotic baryon, hep-ph/0309111. [31] D. Diakonov, V. Petrov, Exotic baryon multiplets at large number of colours, hep-ph/0309203. [32] D.B. Kaplan, I.R. Klebanov, The role of a massive strange quark in the large N Skyrme model, Nucl. Phys. B 335 (1990) 45. [33] C.G. Callan, I.R. Klebanov, Bound state approach to strangeness in the Skyrme model, Nucl. Phys. B 262 (1985) 365. [34] C.G. Callan, K. Hornbostel, I.R. Klebanov, Baryon masses in the bound state approach to strangeness in the Skyrme model, Phys. Lett. B 202 (1988) 269. + [35] T.D. Cohen, R.F. Lebed, Partners of the Theta in large N(c) QCD, hep-ph/0309150. [36] N.N. Scoccola, Hyperon resonances in SU(3) soliton models, Phys. Lett. B 236 (1990) 245. [37] M. Karliner, M.P. Mattis, Hadron dynamics in the three flavor Skyrme model, Phys. Rev. Lett. 56 (1986) 428; M. Karliner, M.P. Mattis, Pi N, K N and anti-K N scattering: Skyrme model versus experiment, Phys. Rev. D 34 (1986) 1991. [38] D.O. Riska, N.N. Scoccola, Anti-charm and anti-bottom hyperons, Phys. Lett. B 299 (1993) 338. + [39] H. Weigel, Radial excitations of low-lying baryons and the Z penta-quark, Eur. Phys. J. A 2 (1998) 391, hep-ph/9804260. [40] E. Witten, Global aspects of current algebra, Nucl. Phys. B 223 (1983) 422. [41] U. Blom, K. Dannbom, D.O. Riska, The hyperons as bound states in the Skyrme model, Nucl. Phys. A 493 (1989) 384. [42] K. Hagiwara, et al., The review of particle physics, Phys. Rev. D 66 (2002) 010001. [43] M. Rho, D.O. Riska, N.N. Scoccola, The energy levels of the heavy flavor baryons in the topological soliton model, Z. Phys. A 341 (1992) 343. [44] N.N. Scoccola, D.P. Min, H. Nadeau, M. Rho, The strangeness problem: an SU(3) skyrmion with vector mesons, Nucl. Phys. A 505 (1989) 497. [45] C.L. Schat, N.N. Scoccola, C. Gobbi, Lambda (1405) in the bound state soliton model, Nucl. Phys. A 585 (1995) 627, hep-ph/9408360. [46] H. Walliser, V.B. Kopeliovich, Exotic baryon states in topological soliton models, hep-ph/0304058. [47] D. Borisyuk, M. Faber, A. Kobushkin, New family of exotic Theta baryons, hep-ph/0307370. [48] Y. Oh, B.Y. Park, D.P. Min, Heavy quark symmetry and the Skyrme model, Phys. Rev. D 50 (1994) 3350, hep-ph/9407214; Y. Oh, B.Y. Park, D.P. Min, Pentaquark exotic baryons in the Skyrme model, Phys. Lett. B 331 (1994) 362, hep-ph/9405297. [49] G.S. Adkins, C.R. Nappi, The Skyrme model with pion masses, Nucl. Phys. B 233 (1984) 109. Nuclear Physics B 684 (2004) 281–317 www.elsevier.com/locate/npe

Axial-vector form factors for Kl2γ and πl2γ at O(p6) in chiral perturbation theory

C.Q. Geng a,b, I-Lin Ho a,T.H.Wua,1

a Department of Physics, National Tsing Hua University, Hsinchu, Taiwan b Theory Group, TRIUMF, Vancouver, BC V3T 2A3, Canada Received 20 June 2003; received in revised form 9 December 2003; accepted 31 December 2003

Abstract

We present two-loop calculations on the axial-vector form factors FA of semileptonic radiative kaon and pion decays in chiral perturbation theory. The relevant dimension-6 terms of the Lagrangian are evaluated from the resonance contribution and the results of the irreducible two-loop graphs of the sunset topology are given in detail. We also explicitly show that the divergent parts in FA are cancelled exactly as required.  2004 Elsevier B.V. All rights reserved.

PACS: 13.20.Cz; 12.39.Fe; 13.40.Gp; 11.30.Rd

1. Introduction

Chiral perturbation theory (ChPT) [1,2] has established itself as a powerful effective theory of low energy interactions. While it works for the strong interaction, it also includes the electroweak one whose dynamics can be completely ﬁxed by introducing the corresponding gauge bosons through the usual covariant derivative. Since the external momenta and quark masses are the expansion parameters for the generating function in ChPT [3–5], they need to be small compared to the physical scale of the chiral symmetry breaking, i.e., about 1 GeV. Therefore, one expects that the semileptonic radiative kaon + + + + (pion) decays of K → l νlγ (π → l νlγ ) can be well described in ChPT [6,7]. It is known that these radiative decays [8–11] could provide us with information on new physics

E-mail address: [email protected] (C.Q. Geng). 1 Deceased.

[12,13] by searching for the lepton polarization effects, which depend on the vector and axial-vector form factors FV,A of the structure dependent parts. In this paper we deal mainly with the SU(3) ⊗ SU(3) chiral symmetry. We will + + present two-loop calculations on the axial-vector form factors in K → l νlγ(Kl2γ ) and + + 6 π → l νlγ(πl2γ ) by virtue of the recent progresses in the p -Lagrangian [14–16] and the massive two-loop integrals [17,18]. Some remarks related to the form factors in Kl2γ and πl2γ follow:

• The usual one-loop ChPT for the timelike form factor does not satisfy the unitarity or the final-state theorem and makes poor approximations [19–21]. Substantial corrections are expected at a higher order, especially at the two-loop level. • The form factors in the decays receive the first non-vanishing contributions at O(p4) but the first sizable ones, as well as the estimates of the accuracy, arise at O(p6). • The question of convergence in ChPT needs to be clearly addressed [22]. 6 • In ChPT, the O(p ) contributions to the vector form factors FV in the decays have been studied in Ref. [23], but those to the axial-vector ones FA have been done only for π → lνl γ basedontheSU(2) × SU(2) symmetry [24].

In this paper, by using the relevant dimension-6 terms of the Lagrangian [18,20] from the resonance contribution [23–25], we perform a detailed calculation for the irreducible two-loop graphs of the sunset topology [21], which give the dominant contributions to FA 6 in both π and K decays at O(p ). For completeness, we will also evaluate FV and compare our results with those in Ref. [23]. The paper is organized as follows. In Section 2, we give the matrix elements for the decays. We review the Lagrangians of ChPT to O(p6) in Section 3. In Section 4, we display the two-loop calculations for both vector and axial-vector form factors with the detailed formulas placed in Appendices A–C. In Section 5, we show the analytical results. We present our numerical values and conclusions in Section 6.

2. The matrix elements

+ + + We consider the decay of P(p)→ l (l )νl(s )γ (k) with P = K or π ,whereγ is a real photon with k2 = 0. The matrix element M for the decay [6,26,27] can be written as

eGF µ∗ ν MP →l+ν γ =−i √ θP Mµν (p, k) (k)u(s¯ )γ (1 − γ5)v(l ), (1) l 2 µ where is the photon polarization and θK+(π+) = cosθ(sin θ) with θ being the Cabibbo angle. In Eq. (1), the hadronic part of the quantity Mµν is given by = 4 iq·x | em wk Mµν (p, k) d xe 0 T Jµ (x)Jν (0) P(p) , (2) which has the general structure √ (p − k) √ M (p, k) =− 2F ν P(p− k)J emP(p) + 2F g µν P − 2 − 2 µ P µν (p k) MP C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 283 2 − FA (p − k)µkν − gµν k · (p − k) − rA kµkν − gµν k α β + iFV εµναβ k p , (3) where the ﬁrst line represents the Born diagram, in which the photon couples to hadrons through the known KKγ(ππγ) coupling, with FP being the P meson decay constant, and the subsequent lines correspond to axial-vector and vector portions of the weak currents with FV(A) being the vector (axial-vector) form factor. In Eq. (3), rA is non-zero only for − + + − those processes with virtual photons, such as P → l l l ν¯l . In terms of the form factors FA and FV , we can write the vector and axial-vector parts in Eqs. (1) and (3) as + eGF θP µ∗ ν α β MV P → l νlγ = √ FV εµναβ l k p , (4) 2 + eGF θP ν µ∗ MA P → l νlγ = i √ FAl kν(p · ) − gµν (p · k) , (5) 2 respectively, where

ν ν l =¯u(s )γ (1 − γ5)v(l ). (6) 2 2 Both FA and FV are real functions for q ≡ (p − k) below the physical threshold, which is the region of interest here, based on time-reversal invariance, and they are analytic functions of q2 with cuts on the positive real axis. One of the reasons to perform the present calculation is that the q2 dependence of the form factors starts at O(p6).

3. The Lagrangians of chiral perturbation theory

In the usual formulation of ChPT [1,2,28] with the chiral symmetry SU(3)L × SU(3)R, the pseudoscalar fields are collected in a unitary 3 × 3matrix Φ(x) U(x)= exp i , (7) F where F absorbs the dimensional dependence of the fields and, in the chiral limit, is equal to the pion decay constant, Fπ = 92.4MeV.TheΦ is given by the 3 × 3matrix  √ √  π0 + √η 2π+ 2K+  √ 3 √  = =  2π− −π0 + √η 2K0  Φ λaϕa  3  , (8) √ √ − 2K− 2K¯ 0 √2η 3 where λa (a = 1, 2,...,8) are the Gell-Mann matrices. An explicit breaking of the chiral symmetry is introduced via the mass matrix   2 mπ 00 =  2  χ 0 mπ 0 , (9) 2 − 2 002mK mπ 284 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 where mπ(K) is the unrenormalized π(K)mass. We note that the mass of η to this order is given by the Gell-Mann–Okubo relation 4 1 m2 = m2 − m2 . (10) η 3 K 3 π The mass term in Eq. (9) is related to the quark masses by χ = const × diag(mu,md ,ms ) with mu = md . To calculate the form factors, we have to include the interaction with external boson fields. As previously stated, the electroweak gauge fields Aµ and Wµ are introduced via the covariant derivative D U = ∂ U + iU − ir U, µ µ µ µ  0cosθW+ sin θW+ g µ µ = − √  −  µ eAµQ cosθWµ 00, 2 − sin θWµ 00 rµ = eAµQ, (11) where Q is the quark charge√ matrix in units of e = g sin θW , with θW standing for the 2 2 Weinberg angles and GF / 2 = g /(8M ). The final lepton pair (for clarity we will + + W always refer to l νl coming from W ) appears in the leptonic current when substituting g g W → √ l = √ u(s)γ¯ (1 − γ )v(l). (12) µ 2 µ 2 µ 5 2 2MW 2 2MW The Lagrangians of ChPT contain both normal (or non-anomalous) and anomalous parts. Since the form factor FV is related by an isospin rotation to the amplitude for π0 → γγ, it can be absolutely predicted from the axial anomaly. For this reason, we must also include the effect of the axial anomaly. At the two lowest orders, the full non- anomalous Lagrangian is given by [1,2] 2 2 L(2) = F µ † + F † + † n Tr DµUD U Tr χU Uχ , (13) 4 4 L(4) = L Tr D UDµU † 2 + L Tr D UD U † Tr DµUDν U † n 1 µ 2 µ ν + L Tr D UDµU †D UDν U † 3 µ ν µ † † † + L4 Tr DµUD U Tr χU + Uχ µ † † † † † 2 + L5 Tr DµUD U χU + Uχ + L6 Tr χU + Uχ + L Tr χ†U − U †χ 2 + L Tr χU†χU† + Uχ†Uχ† 7 8 µ ν † µ † ν † + iL9 Tr Lµν D UD U + Rµν D U D U + L10 Tr Lµν URµν U , (14) where Lµν and Rµν are the field-strength tensors of external sources, defined by

Lµν = ∂µν − ∂ν µ − i[µ,ν ],

Rµν = ∂µrν − ∂ν rµ − i[rµ,rν ], (15) 6 and {Li} are unrenormalized coupling constants. At O(p ), the non-anomalous Chiral Lagrangian contains 90 independent terms plus four contact terms for SU(3) [18]. The C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 285 terms relevant to Kl2γ (πl2γ ) decays are found to be (6) µν µν µν L = y χ+h h + y χ+ h h + y χ+f+ f n 17 µν 18 µν 81 µν + µν µ ν + y χ+ f+ f + iy f+ χ+,u u 82 µν + 83 µν µ ν µ ν + iy χ+ f+ u u + iy f+ u χ+u 84 µν 85 µν νρ µ µν + iy f+ f ,h + y χ+f− f 100 µν − ρ 102 µν − µν µν + y χ+ f− f + y f+ f ,χ− 103 µν − 104 µν − ρ µν µρ ν + y109 ∇ρf−µν ∇ f− + iy110 ∇ρ f+µν h ,u +···, (16) where † † uµ = i U (∂µ − irµ)U − U(∂µ − iµ)U , † † † χ± = U χU + Uχ U, µν µν † † µν f± = UL U ± U R U,

hµν =∇µuν +∇ν uµ, † † † i χ±µ = U DµχU ± UDµχ U =∇µχ± − {χ∓,uµ}. (17) 2 The covariant derivative ∇µX = ∂µX +[Γµ,X] is deﬁned in terms of the chiral condition 1 Γ = U †(∂ − ir )U − U(∂ − i )U † . (18) µ 2 µ µ µ µ 4 The anomalous part begins at O(p ) with the Wess–Zumino (WZ) term LWZ [29] containing pieces with zero, one and two gauge boson ﬁelds. The terms with one and two gauge bosons as well as the anomalous p6-Lagrangian [20,29] are given by (4) 1 µναβ † † † † L =− ε tr U∂µU ∂ν U∂αU β − U ∂µU∂ν U ∂αUrβ , (19) WZ,1 16π2 (4) i µναβ † † L =− ε tr ∂µU ∂ν αUrβ − ∂µU∂ν rαU β WZ,2 16π2 + U∂ U †( ∂ + ∂ ), (20) µ ν α β ν α β (6) µναβ µναβ L = iC ε χ−f+ f+ +iC ε χ−[f+ ,f− ] (21) a 7 µν αβ 11 µν αβ µναβ + C22ε uµ{∇γ f+γν,f+αβ } +···. From the above expressions, the Feynman rules can be derived by expanding U = exp(iΦ/F ) everywhere in L = L(2) + L(4) + L(6) and identifying the relevant vertex monomials. In the next section we display the main result of the two-loop calculations for the form factors.

4. The form factors

To O(p6), the finite matrix elements in ChPT are obtained by multiplying the unrenormalized Feynman diagrams obtained from L = L(2) + L(4) + L(6) with a factor 286 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 √ Z per external meson, where Z is the wave function renormalization constant. To get these results, we start to calculate the mass and wave function renormalizations as well as that of the pion decay constant. Since the form factors at O(p4) are related to the finite counterterm contribution, we 2 4 only need the wave function renormalization, mK,π and FK,π to O(p ) in our calculations. Explicitly, we have −1 1 2 2 2 2 2 Z = 1 − I m + 2I m − 24 2m + m L4 − 24m L5 , (22) π 3F 2 K π K π π − 1 Z 1 = 1 − I m2 + 2I m2 + I m2 − 32 2m2 + m2 L K 4F 2 η K π K π 4 − 2 32mKL5 , (23) 1 2 2 2 2 2 2 2 4 δmπ = −m I m + 3m I m − 48m 2m + m L − 48L m 6F 2 π η π π π K π 4 5 π + 2 2 + 2 + 4 96L6mπ 2mK mπ 96L8mπ , (24) 1 2 2 2 2 2 4 4 δmK = 4m I m − 96L m 2m + m − 96L m + 192L m 12F 2 K η 4 K K π 5 K 8 K 2 2 2 + 192L6m 2m + m , (25) K K π 1 1 2 2 2 2 2 Fπ = F 1 + − I m − I m + 4 2m + m L + 4m L , (26) F 2 2 K π K π 4 π 5 1 3 2 3 2 3 2 2 2 FK = F 1 + − I m − I m − I m + 4 2m + m L F 2 8 η 4 K 8 π K π 4 + 2 4mK L5 , (27) where D − d q i I m2 ≡ µ4 D (2π)D q2 − m2 m2 1 m2 =− + 1 + ln(4π)− γ − ln (28) 16π2 ε µ2 is the standard tadpole integral. It should be noted that the renormalization constants δm(≡ mphys − m) and δFK,π(≡ FK,π − F) defined above are finite. The divergences and (4) scale dependences of the loop integrals are cancelled by similar factors for Li in L .

4.1. The vector form factors FV

Using the chiral Lagrangians mentioned above, one immediately obtains the tree-level contribution for the anomalous parts of our processes. For the semileptonic radiative K decays, we have C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 287

(a) (b) (c)

Fig. 1. One-loop diagrams that contribute to FV in P2γ 1 256 2 2 W 2 2 2 W FV,K,tree = √ 1 − π m C + 256π m − m C 2 K 7 K π 11 4 2π F 3 64 + π2 q2 + k2 CW . (29) 3 22 Loop corrections to the above tree-level contribution proceed through diagrams involving at least one vertex given by the WZ Lagrangian. As shown in Fig. 1 with P = K, due to 6 the initial order, most one-loop diagrams can contribute to FV to O(p ) constructed with one vertex coming from the WZ Lagrangian and another one from the lowest order chiral Lagrangian. Performing the calculation in the three-ﬂavor case, we get = √ 1 1√ 2 + 2 + 2 FV,K,loop(a) λ mη 2mK 3mπ 4 2π2F 16π2( 2F)2 m2 m2 2 − 2 η − 2 K − 2 mπ mη ln 2 2mK ln 2 3mπ ln 2 , µ µ µ = √ 1 1√ − 2 + 2 2 FV,K,loop(b) 4mπ k λ 4 2π2F 16π2( 2F)2 3 m2 − x( − x)k2 + 2 − − 2 π 1 4 mπ x(1 x)k ln 2 , µ 1 1 2 2 2 2 2 FV,K,loop(c) = √ √ −m − m − 2m + q λ 2 2 2 π η K 4 2π F 16π ( 2F) 3 + 2 + − 2 − − 2 2 xmη (1 x)mK x(1 x)q xm2 + (1 − x)m2 − x(1 − x)q2 × η K ln 2 dx µ + 2 + − 2 − − 2 2 xmπ (1 x)mK x(1 x)q xm2 + (1 − x)m2 − x(1 − x)q2 × ln π K dx , (30) µ2 from the three loops in Fig. 1, respectively, where 1 λ = + 1 + ln(4π)− γ. (31) ε 288 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317

Putting all of the contributions together as well as the usual renormalizations of the pseudoscalar wave-functions and decay constants, we obtain the following results: 1 256 2 2 2 2 2 FV,K = √ 1 − π m C7 + 256π m − m C11 2 K K π 4 2π FK 3 64 + π2 q2 + k2 C 3 22 m2 + √1 5 2 − 2 + 2 2 + 2 − 3 2 η mK mπ λ k q λ mη ln 16π2( 2F )2 3 3 2 µ2 K m2 7 m2 − 3m2 ln K − m2 ln π K µ2 2 π µ2 m2 − x( − x)k2 + 2 − − 2 π 1 4 mπ x(1 x)k ln 2 µ + 2 + − 2 − − 2 2 xmπ (1 x)mK x(1 x)q xm2 + (1 − x)m2 − x(1 − x)q2 × ln π K 2 µ + 2 + − 2 − − 2 2 xmη (1 x)mK x(1 x)q xm2 + (1 − x)m2 − x(1 − x)q2 × ln η K . (32) µ2

Similarly, for πl2γ we derive = √ 1 − 256 2 2 + 64 2 2 + 2 FV,π 1 π mπ C7 π q k C22 4 2π2F 3 3 π m2 m2 + √1 − 2 K − 2 π 4mK ln 4mπ ln 16π2( 2F )2 µ2 µ2 π m2 − x(1 − x)k2 + 4 m2 − x(1 − x)k2 ln K K µ2 m2 − x(1 − x)q2 2 + 4 m2 − x(1 − x)q2 ln π + k2 + q2 λ . (33) π µ2 3 We note that the numerical results of Eqs. (32) and (33) agree with those in Ref. [23] as will be shown in Section 6. The one-loop corrections contain divergent terms proportional = 1 + + − to λ ε 1 ln(4π) γ coming from the dimensional regularization scheme. Obviously, the presence of these divergent terms requires the introduction of the corresponding counterterms in the anomalous section of the Lagrangian at O(p6), which were already given in Refs. [20,30,31]. Their infinite parts cancel the λ-terms in Eqs. (32) and W (33) and the coefficients Ci are simply substituted for by the remaining finite parts, Wr the renormalized coefficients Ci . The values of these finite contributions from the counterterms to our processes are not pinned down in ChPT and they have to be deduced C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 289 from data fitting [29] or, alternatively, from the hypothesis of resonance saturation (RS) of the counterterms [23]. The estimations of using RS are described in Section 6.

4.2. The axial-vector form factors FA

In this subsection we aim at the extraction of the form factor FA, which is the only one that has a contribution proportional to gµν (p · k). The presence of gµν requires that the axial-vector and vector insertions are in the same one-particle irreducible subdiagrams. This immediately removes a large part of the diagrams. Furthermore, the (p ·k) kinematical factor guarantees that it is not part of the internal Bremsstrahlung contribution. We now discuss the contributions from the diagrams to O(p6).

4.2.1. Tree-level diagrams With the chiral Lagrangians in Eqs. (13), (14) and (16), one obtains tree-level contributions for the processes as follows: √ (4) = 4 2 + FA,π,tree (L9 L10), (34) F√ (4) = 4 2 + FA,K,tree (L9 L10), (35) F √ √ √ 2 48 2(2m2 + m2 ) 2 (6) =− 48 2mπ − K π + 16 2mπ FA,π,tree y17 y18 y81 √F F √ √F 2 + 2 2 2 + 2 16 2(2mK mπ ) 16 2mπ 8 2(2mK mπ ) + y82 − y83 − y84 √ F √ F √ F 2 2 − · 2 8 2mπ 8 2(pπ pπ k) 16 2mπ − y85 + y100 − y102 F√ F √ F√ 2 + 2 2 · − 2 16 2(2mK mπ ) 16 2mπ 8 2(2pπ k pπ ) − y103 + y104 + y109 √ F F F 4 2pπ · k − y110 , (36) √ F √ √ 48 2m2 48 2(2m2 + m2 ) 16 2(2m2 + m2 ) (6) =− K − K π + K π FA,K,tree y17 y18 y81 √F F √ 3F 2 + 2 2 + 2 16 2(2mK mπ ) 16 2(2mK mπ ) + y82 − y83 √ F √ 3F 2 + 2 2 − 2 8 2(2mK mπ ) 8 2(4mK mπ ) − y84 − y85 √ F √3F √ 2 − · 2 2 + 2 8 2(pK pK k) 16 2mK 16 2(2mK mπ ) + y100 − y102 − y103 √ F √ F √ F 2 2 16 2m 8 2(2pK · k − p ) 4 2p · k + y K + y K − y K . (37) 104 F 109 F 110 F We note that for the tree-level contributions in Eqs. (36) and (37) at O(p6) one needs to perform renormalization with the ﬁnite parts, which will be discussed in Section 5. 290 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317

4.2.2. One-loop diagrams We now consider the one-loop diagrams shown in Fig. 1. Since it contains only photon- even-meson vertices in the non-anomalous chiral Lagrangian to O(p4), Fig. 1(c) does 4 not contribute to FA. Moreover, one-loop diagrams with an O(p ) vertex insertion on a propagator in the loop never produce the factor (p · k) and hence do not contribute to FA. From Figs. 1(a) and (b), we get √ √ L9 2 2 FA,π, ( ) =− 14 2I m + 28 2I m loop a 3F 3 K π 2L √ √ − 10 10 2I m2 + 20 2I m2 , 3F 3 K π √ √ √ L9 2 2 2 FA,K, ( ) =− 6 2I m + 24 2I m + 12 2I m loop a 3F 3 η K π L √ √ √ − 2 10 2 + 2 + 2 3 3 2I mη 18 2I mK 9 2I mπ , 3√F √ √ −16 2L 8 2L 8 2L 1 F = 1 I m2 + 2 I m2 − 3 I m2 + I m2 A,π,loop(b) F 3 π F 3 π F 3 π 2 K √ L − 2 2 9 2 + 2 3 I mK 2I mπ , √F √ √ −16 2L 8 2L 8 2L 1 F = 1 I m2 + 2 I m2 − 3 I m2 + I m2 A,K,loop(b) F 3 K F 3 K F 3 K 2 π √ 2 2L − 9 I m2 + 2I m2 . (38) F 3 π K 4.2.3. Two-loop diagrams The two-loop diagrams which may contribute to FA are shown in Fig. 2. The last six diagrams with non-overlapping loops in Fig. 2, which can be written as the products of one-loop integrals and produce no (p · k) factor, do not contribute to FA. The only possible non-vanishing diagrams are the first three irreducible ones in Fig. 2. Since there are three different mass scales of (mπ ,mK ,mη) with the same order of magnitude in the SU(3) ChPT, the irreducible integrals can no longer be expressed by elementary analytical functions. We will quote only the numerical results in Sections 5 and 6. We now give the detailed calculations for the gµν terms of MA inEq.(5).Itisclear that the first irreducible diagram in Fig. 2 does not contribute to FA since there is no (p · k) term. The second and third irreducible diagrams with genuine massive two-loop integrals [21] are depicted in Figs. 3 and 4, respectively. Our calculations on these diagrams are summarized in Appendix A. We point out that the two-point irreducible diagrams in Fig. 3 vanished in Ref. [24] for π2γ in the linear sigma model parametrization in which there is neither a photon nor a four-pion vertex [32]. As shown in Section 5, these diagrams play a very important role in the cancellation of divergent parts with 1/2 in the final results. We note that the two-loop amplitudes shown in Appendix A have been classified by the functions {I,II,...,A,B,...,L} related to each diagram. For clarity, we always refer to all inward particles for vertices. The Feynman diagrams which contain tensor structures at C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 291

Fig. 2. Two-loop diagrams for the form factors in P2γ .

Fig. 3. Two-point irreducible diagrams.

ab the numerator can be reduced to the calculation of scalar integrals Pα1,α2,α3 by introducing µ µ 2 µ the transverse components of the loop momenta s and q as s⊥ = s − (s · l/l )l and µ µ 2 µ q⊥ = q − (q · l/l )l , respectively, for the two-point diagrams [21]. For the three-point = diagrams, we ﬁrst combine the denominators by using the Feynman formula 1/ab 1 [ + − ]2 = + µ = µ − · 2 µ 0 dx/ ax b(1 x) and then replace l by l l kx,i.e.,s⊥ s (s l /l )l µ µ 2 µ and q⊥ = q − (q · l /l )l . The corresponding relations among {I,II,...,A,B,...,L} ab ab and Pα1,α2,α3 are displayed in Appendices B and C, where the scalar integrals Pα1,α2,α3, deﬁned in the Euclidean space, are given by ab ; 2 Pα1,α2,α3 m1,m2,m3 l (s · l)a(q . l)b = dnsdnq . (39) 2 + 2 α1 2 + 2 α2 + 2 + 2 α1 (s m1) (q m2) ((s q) m3) 292 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317

Fig. 4. Three-point irreducible diagrams. C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 293

The detailed contributions from all diagrams for FA in π2γ and K2γ are summarized in the next section.

5. Analytical results

5.1. Renormalization scheme

In our calculations, we use the following dimensional regularization and renormalization scheme [1,2,22,28]. Each diagram of O(p2n) is multiplied by a factor (cµ)(n−1)(D−4), where D = 4 − 2 is the dimension of space–time and c is given by 1 π2 1 ln c =− 1 − γ + ln(4π) − + + O 2 . (40) 2 2 12 2

(n−1)(D−4) (4) Using the renormalization factor of (cµ) , the low energy constants Li of Ln in Eq. (14) are deﬁned by

(D−4) Li = (cµ) Li(µ, D). (41) (6) Similarly, for the Ln parameters yi in Eq. (16),

2(D−4) yi = (cµ) yi(µ, D). (42)

We note that Li (µ, D) have the same µ-dependences as the one-loop integrals, whereas yi(µ, D) behave like the two-loop ones. Their values at the two different scales (D−4) of µ1 and µ2 are related by Li (µ1,D) = (µ2/µ1) Li (µ2,D) and yi(µ1,D) = 2(D−4) (µ2/µ1) yi(µ2,D), respectively.

5.2. Analytical forms of FA

6 We now try to obtain the analytical forms of FA from each diagram to O(p ) at the scale of mρ . From Section 5.1, for the unrenormalized coefﬁcients in the chiral Lagrangian (4) Ln , we use [22,28] (D−4) γi r Li = (cµ) − + L (µ) 32π2 i 2 (D−4) γi 1 mρ r () = µ − − γs − ln + L (mρ) + L (mρ ,µ) , (43) 32π2 µ2 i i which is a Laurent series expanded around = 0. Of the values in Eq. (43), γi are shown in =− − + r Table 1 [22], γs 1 ln(4π) γ , Li correspond to measurable low energy constants and () () =− r + 2 2 + · Li are given by Li (mρ,µ) Li (mρ)(γs ln(mρ/µ )) γi f(mρ,µ). Similarly, for (6) the chiral Lagrangian Ln ,wehave 294 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317

Table 1 Coefﬁcients of γi in the Minkowski space [22] i 12345678910 3 3 1 3 11 5 1 −1 γi 32 16 0 8 8 144 0 48 4 4

Table 2 (2) (1,L) Coefﬁcients of Γi and Γi with the double-pole and single-pole divergences, respectively, in the Minkowski space [22] (2) 2 (1) (L) yi Γi 16π Γi Γi 19 − 13 2 r + 4 r + 8 r + 3 r 17 64 768 3 L1 3 L2 9 L3 4 L5 67 − 1 8 r + 8 r + 23 r + 3 r + 1 r 18 192 2304 3 L1 9 L2 27 L3 2 L4 4 L5 − 3 r − 3 r 81 0 0 4 L9 4 L10 − 1 r − 1 r 82 0 0 4 L9 4 L10 − 1 5 1 r − 11 r − 5 r − 3 r + 3 r 83 4 384 3 L1 6 L2 36 L3 4 L5 8 L9 − 4 − 5 − 32 r − 32 r − 157 r − r − r + 1 r 84 3 144 3 L1 9 L2 54 L3 6L4 L5 2 L9 − 1 5 2 r − 11 r − 16 r − 3 r + 3 r 85 2 192 3 L1 3 L2 9 L3 2 L5 4 L9 − 5 49 1 r − 1 r + 1 r − 1 r − 1 r − 9 r 100 24 1152 3 L1 6 L2 4 L3 3 L4 4 L5 8 L9 − 25 − 17 − 2 r − 4 r − 8 r − 3 r + 3 r 102 64 768 3 L1 3 L2 9 L3 4 L5 4 L10 − 73 − 101 − 8 r − 8 r − 23 r − 3 r − 1 r + 1 r 103 192 2304 3 L1 9 L2 27 L3 2 L4 4 L5 4 L10 7 5 − 1 r + 1 r − 1 r + 1 r + 1 r + 5 r 104 96 144 6 L1 12 L2 8 L3 6 L4 8 L5 16 L9 − 1 3 − 1 r 109 16 128 2 L9 1 − 31 − 2 r + 1 r − 1 r + 2 r + 1 r + 1 r 110 6 576 3 L1 3 L2 2 L3 3 L4 2 L5 4 L9

2(D−4) (cµ) r (1) (L) 1 (2) 1 yi = y (µ) + Γ + Γ (µ) − Γ F 2 i i i 32π2 i 322π42 2(D−4) (1) + (L) 2 µ r (Γi Γi (mρ )) 1 mρ = y (mρ) + − 2γs − 2ln F 2 i 32π2 µ2 g(m ,µ,) − Γ (2) ρ ; (44) i 322π42

(1) (2) (L) the relevant constants of Γi , Γi and Γi are shown in Table 2 [22]. We note that f(mρ,µ) and g(mρ ,µ,), which receive contributions from the (/2) term in Eq. (40), will not contribute to FA. 4 6 By writing the unrenormalized contributions to FA of O(p ) and O(p ) diagrams 4 6 6 6 as FA,tree(p ), FA,tree(p ), FA,1-loop(p ) and FA,2-loop(p ), respectively, using the wave function renormalizations, we find 4 6 FA = ZP FA,tree p (1 + δFP ) + FA,tree p 6 6 + FA,1-loop p + FA,2-loop p , (45) C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 295 which leads to 4 1 6 FA = FA, p 1 + δFP + δZP + FA, p tree 2 tree 6 6 + FA,1-loop p + FA,2-loop p , (46) where P = π or K. In Eq. (46), for P = π,wehave 1 FA, 1 + δFπ + δZπ tree 2 √ 4 2 1 I(m2 ) 2I(m2 ) = Lr + Lr 1 − K + π 9 10 2 Fπ F 3 3 √ 2 2 2 4 2 r r 1 1 mρ mK 2mπ = L + L 1 + − 2γs − 2ln + F 9 10 16π2F 2 ε µ2 3 3 π π 1 m2 m2 2m2 m2 − K ln K + π ln π , (47) 16π2F 2 3 m2 3 m2 π ρ ρ 2 1 1 mρ FA,1-loop = √ − 2γs − 2ln F 3π2 µ2 6 2 π × 6 2Lr − Lr m2 + m2 + 2m2 3Lr + 5Lr + 5Lr 1 2 π K π 3 9 10 m2 + √ 1 − r + r 2 π 12L 6L mπ ln 6 2F 3π2 1 2 m2 π ρ m2 m2 − 3Lr + 5Lr + 5Lr m2 ln K + 2m2 ln π , (48) 3 9 10 K m2 π m2 ρ ρ 4 2 + 2 + 2 2 1 π (116mK 184mπ 21mη) 1 mρ FA,2-loop = − √ − 2γs − 2ln 6F 3(2π)8 12 2 µ2 π 3π4 1 m2 − √ − 2γ − 2ln ρ (p · k) s 2 2 2 µ − 1421.4 − 1167.7(p · k) + 1228.0 + 1123.2(p · k) , (49) 1 m2 1 3π4 F p6 =− − 2γ − 2ln ρ − √ (p · k) A,tree s 2 3 8 µ 6Fπ (2π) 2 2 π4(432m2 + 531m2 ) − 1 K√ π 3 8 6Fπ (2π) 36 2 1 + √ 3m2 Lr + 2Lr + 2Lr 3 2 K 3 9 10 6 2Fπ π + 2 r − r + r + r + r 6mπ 2L1 L2 L3 2L9 2L10 √ 4 2 − 4m2 6yr − 2yr + yr + 2yr 3 K 18 82 84 103 Fπ 296 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 + 2m2 6yr + 6yr − 2yr − 2yr + 2yr + yr + yr π 17 18 81 82 83 84 85 − yr + 2yr + 2yr − 2yr + yr 100 102 103 104 109 + r − r + r pk 2y100 4y109 y110 . (50) For P = K,weget 1 FA, 1 + δFK + δZK tree 2 √ I(m2) I(m2 ) I(m2 ) = 4 2 r + r − 1 η + K + π L9 L10 1 2 FK F 4 2 4 √ 4 2 1 1 m2 = Lr + Lr 1 + − 2γ − 2ln ρ 9 10 2 2 s 2 FK 16π FK ε µ m2 m2 m2 × η + K + π 4 2 4 2 2 2 2 2 2 1 mη mη m m m m − ln + K ln K + π ln π , (51) 2 2 2 2 2 16π FK 4 mρ 2 mρ 4 mρ 2 1 1 mρ FA,1-loop = √ − 2γs − 2ln 4 2F 3 π2 µ2 K × m2 8Lr − 4Lr + 4Lr + 6Lr + 6Lr K 1 2 3 9 10 + 2 r + r + r + 2 r + r mπ 2L3 3L9 3L10 mη L9 L10 1 m2 + √ − 2Lr + 3Lr + 3Lr m2 ln π 3 2 3 9 10 π m2 4 2FK π ρ m2 − Lr + Lr m2 ln η 9 10 η 2 mρ m2 − 8Lr − 4Lr + 4Lr + 6Lr + 6Lr m2 ln K , (52) 1 2 3 9 10 K 2 mρ 1 π4(99m2 + 20m2 − 12m2) F = − K √ π η A,2-loop 3 8 6FK (2π) 4 2 2 1 mρ × − 2γs − 2ln µ2 4 2 3π 1 mρ − √ − 2γs − 2ln (p · k) 2 2 µ2 − 1465.2 − 923.0(p · k) + 1266.9 + 1208.3(p · k) , (53) C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 297 1 m2 1 3π4 F p6 =− − 2γ − 2ln ρ − √ (p · k) A,tree s 2 3 8 µ 6FK (2π) 2 2 π4(747m2 + 216m2 ) − 1 K√ π 3 8 6FK (2π) 36 2 + √ 1 2 r + r + r mπ 2L 3L 3L 4 2F 3 π2 3 9 10 K + 2 r − r + r + r + r mK 8L1 4L2 4L3 9L9 9L10 √ 4 2 − 2m2 18yr − 2yr − 6yr + 2yr + 3yr − yr + 6yr 3F 3 π 18 81 82 83 84 85 103 K + 2m2 18yr + 36yr − 4yr − 12yr + 4yr + 6yr + 4yr K 17 18 81 82 83 84 85 − 3yr + 6yr + 12yr − 6yr + 3yr 100 102 103 104 109 + r − r + r 3pk 2y100 4y109 y110 . (54) We note that, in the above expressions, one special case must be treated separately. For the finite part of FA,2-loop, the functions of g(y) and fi (y) in Appendix B seem to introduce additional singularities in the integrals, such as at y = 0 and 1. However, this problem can be resolved by noticing that at y = 0 (the situation is the same at y = 1), the function, e.g., g(y), behaves like ln(x2), which is integrable. Consequently, the main question of evaluating the finite part of FA,2-loop is how to implement the formula in a computer program by correct and numerically stable forms. As seen from the last two terms for FA,2-loop in Eqs. (49) and (53), we have found the reliable and stable numerical results due to the contributions of functions hi , i.e., the integrals of g and fi , by fitting the numerical data with recursive analytic methods. We remark that in Eqs. (47)–(54) we have explicitly shown the single poles, subtracted 6 2 via FA,tree(p ). We emphasize that in our results there are no divergent parts with 1/ and all the terms related to 1/ are cancelled explicitly by the renormalization of the coupling (6) constants in Ln as well as the Gell-Mann–Okubo relation in Eq. (10). It is clear that the disappearance of 1/2 terms relies on the two-point irreducible diagrams in Fig. 3. Moreover, our results are scale independent since the scale terms with ln µ2 can be grouped into the ones with 1/, i.e., they are always associated with 1/ terms. These also serve as checks of our calculations. Now we can extract the axial-vector form factor by placing the related physical quantities into the Eq. (46). Explicitly, we may write Eq. (46) into more transparent forms r r FA,π = 66.86 L + L 9 10 yr yr yr yr + 2.41 − 122.96(pk) 100 − 4.82 102 − 125.35 103 + 4.82 104 F 2 F 2 F 2 F 2 yr yr yr yr + −2.41 + 245.95(pk) 109 − 61.49(pk) 110 − 14.46 17 − 376.05 18 F 2 F 2 F 2 F 2 298 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 yr yr yr yr yr + 81 + 82 − 83 − 84 − 85 4.82 2 125.35 2 4.82 2 62.67 2 2.41 2 F F F F F + 12.30Lr − 6.15Lr + 16.11Lr + 26.85Lr + 26.85Lr 1 2 3 9 10 − − + −1.70 × 10 2 − 3.92 × 10 3(pk) (55) and F = 54.99 Lr + Lr A,K 9 10 yr yr yr yr + 24.75 − 101.01(pk) 100 − 49.50 102 − 102.97 103 + 49.50 104 F 2 F 2 F 2 F 2 yr yr yr + −24.75 + 202.03(pk) 109 − 50.50(pk) 110 − 148.50 17 F 2 F 2 F 2 yr yr yr yr yr − 308.90 18 + 34.32 81 + 102.97 82 − 34.32 83 − 51.48 84 F 2 F 2 F 2 F 2 F 2 yr − 85 32.34 2 F + 22.08Lr − 11.04Lr + 12.75Lr + 21.71Lr + 21.71Lr 1 2 3 9 10 − − + −0.97 × 10 2 + 13.93 × 10 3(pk) , (56) where the four {···} terms in Eqs. (55) and (56) correspond to those in Eq. (46), respectively.

6. Numerical values and conclusions

As shown in Section 5, the divergent terms for FA in loop-diagrams are cancelled by the corresponding counterterms in the Lagrangian at O(p6). The infinite parts cancel each other and thus they can be simply substituted by the remaining finite part of r the counterterms, yi . We now study the finite parts which contain the actual physical information. We will present the results in numerical forms, with the scale at mρ = r L(4) 0.77 GeV.In Table 3, we show the standard values for the couplings Li in n [6] and those in the two-loop calculation of ChPT; we chose the Main fit in Ref. [33] as an illustration. Other two-loop studies in ChPT can be found in Refs. [34,35]. We note that in the table r =− the central value of α10 5.5 is kept, and our numerical results for FA are sensitive to this value. Several O(p6) low-energy constants of the normal chiral Lagrangian have been evaluated from the RS. In Table 4 we illustrate the values of yi in the lowest meson dominance (LMD) approximation [25] and the resonance Lagrangian (RL) [23,25,36].

Table 3 r L(4) Values of Li in n 3 r 10 Li 12 3910 O(p4) [6] 0.4 ± 0.31.35 ± 0.30 −3.5 ± 1.16.9 ± 0.7 −5.5 ± 0.7 Main ﬁt [33] 0.53 ± 0.25 0.71 ± 0.27 −2.72 ± 1.12 6.9 ± 0.7 −5.5 ± 0.7 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 299

Table 4 (6) Values of yi in Ln −4 2 yi (in units of 10 /F ) y100 y104 y109 y110 LMD 1.09 −0.36 0.40 −0.52 RL I 1.09 −0.29 0.47 −0.16 RL II 1.49 −0.39 0.65 −0.14

Table 5 Wr L(6) Values of Ci in a in various models [29] Wr −3 −2 Ci (10 GeV ) 71122 ChPT 0.013 ± 1.17 −6.37 ± 4.54 6.52 ± 0.78 20.3 ± 18.75.07 ± 0.71 3 VMD 2 2 8.01 64Mρ π CQM 0.51 ± 0.06 −0.00143 ± 0.03 3.94 ± 0.43

Table 6 2 fA at q = 0forP = K and π 2 4 6 6 fA(q = 0)O(p) [6] O(p )|SU(2) O(p )|SU(3) Experiment P = K 0.041 − 0.034 0.035 ± 0.020 [37,38] P = π 0.0102 0.0117 [24] 0.0112 0.0116 ± 0.0016 [39]

To study the vector form factors, we need to consider the anomalous chiral Lagrangian. The set of anomalous coefficients is treated by phenomenological fitting in ChPT as well as by the two main alternative models of vector meson dominance (VMD) method and constituent chiral quark model (CQM) [29]. The relevant terms for our purposes are shown in Table 5. Other physical inputs are mK = 0.495, mπ = 0.14, mη = 0.55, FK = 0.112, F = 0.0871 and Fπ = 0.092 GeV. We note that some of the actual values of Fπ and the masses in our calculations may be different from those in the literature. In particular Fπ could differ around 1% from one paper to another; however, we expect that the changes on our O(p6) results due to the different sets of parameters are less than 5% since O(p6) 3 contributions are at least proportional to Fπ . To compare our results with those in the literature, we use dimensionless form factors of fV,A,definedby

fi = mP Fi (i = V,A) (57) to replace FV,A. In Figs. 5 and 6, we plot the dimensionless vector and axial-vector form factors fV,A 2 as functions of q with the photon on mass-shell for πe2γ and Ke2γ , respectively. Similar ﬁgures can also be drawn for the µ modes. In Table 6, we show the form factors of fA at q2 = 0atO(p4) and O(p6) with SU(2) and SU(3) symmetries as well as experimental values for P = K and π. 4 In Fig. 5, the dot, solid and dashed curves stand for the contributions to fV at O(p ), 4 6 4 6 O(p ) + O(p ) in VMD and O(p ) + O(p ) in CQM, respectively. We note that for Kl2γ 300 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317

2 Fig. 5. The vector form factors fV as functions of the momentum transfer q for π2γ and K2γ with = e.The dot, solid and dashed curves stand for the contributions of O(p4), O(p4) + O(p6) in VMD and O(p4) + O(p6) in CQM, respectively.

Fig. 6. Same as Fig. 5 but for the axial-vector form factors fA. The dashed and solid curves represent the contributions at O(p4) in Table 3 and the ﬁtting O(p4) + O(p6), respectively.

4 in Fig. 5 F has been set to Fπ in Eq. (29) for the curve of O(p ) as in the literature and 4 6 6 FK in Eq. (32) for those of O(p ) + O(p ). As shown in Fig. 5, the O(p ) contribution obtained for the π radiative decays is very small (< 5%) for all the kinematical allowed 6 values. However, it is interesting to see that the O(p ) correction for K2γ is much larger. 4 In Fig. 6, for fA, the dashed and solid curves represent the contributions at O(p ) in Table 3 plus the two-loop calculations using either the LMD or the RL determinations, respectively. It is easy to see that, as shown in the ﬁgures, the two-loop contributions to 4 fA are sizable and destructive compared with those at the pure O(p ) for both π and K modes, but their q2-dependences, which are dominated by the irreducible diagrams, are 2 6 small. At q = 0, the contributions to fA from fA,tree(p ) are vanishingly small, which r implies that the ﬁnal results of fA are insensitive to the known values of yi .However, those from the irreducible two-loop diagrams and the one-loop diagrams with one vertex (4) 4 6 of Ln give the dominant corrections to fA at O(p ). All together the O(p ) corrections keep around 25% for both decays. We remark that the uncertainties in Eqs. (55) and (56) r due to the errors of yi [25] are less than 1%. Moreover, our results of tree contributions at O(p6) are also insensitive to the choice of the scale, as expected. C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 301

6 We note that, as shown in Fig. 5, the numerical result for π → eνeγ at O(p ),usingthe SU(3) ⊗ SU(3) chiral symmetry, is found to be comparable to the one in Ref. [24] from resonance estimates of O(p6) low-energy constants based on SU(2)⊗SU(2).Furthermore, 6 our result of the O(p ) correction for fA in Kl2γ also confirms the speculation in Ref. [24]. In summary, we have studied the O(p6) corrections to the vector and axial-vector form factors in π2γ and K2γ decays. These include the contributions from loop diagrams and the ones from higher dimension terms in the Lagrangians. The former can be exactly calculated in terms of the known parameters of the chiral Lagrangians. The latter is mainly evaluated from the resonance contributions. For the axial-vector form factors of fA,we have found that the divergent parts cancel order by order in ChPT, while the finite numerical results in both K and π modes contain considerable corrections from loops diagrams; they also agree with the recent experimental determination [37,38]. This demonstrates numerically the statement about the final-state theorem mentioned in Refs. [19–21]. 2 Finally, we remark that our result of fA at q = 0 for the kaon case is consistent with that found in the light front QCD model [40,41] as well.

Acknowledgements

We would like to thank Drs. Bijnens and Talavera for encouragement and comments. We would also like to thank Elizabeth Walter for reading the manuscript. This work was supported in part by the National Science Council of the Republic of China under contract numbers NSC-91-2112-M-007-043.

Appendix A

From Fig. 3, we have √ µ ν √ 2eGF l ε M = cosθ −40 2I(m ,m ,m ) A,π,2-point × 3 K K π 12 √24F √ + 40 2I(m ,m ,m ) + 24 2I(m ,m ,m ) √ π K K √ K K η − 24 2I(m ,m ,m ) − 240 2II(m ,m ,m ) √ η K K √ K π K − 40 2II(m ,m ,m ) − 320 2II(m ,m ,m ) √ π K K √ π π π − 48 2II(m ,m ,m ) − 72 2II(m ,m ,m ) √ K K η √ η K K + 240 2III(m ,m ,m ) + 40 2III(m ,m ,m ) √ K K π √ K π K + 320 2III(mπ ,mπ ,mπ ) + 96 2III(mη,mK ,mK ) √ √ + 72 2III(m ,m ,m ) − 48 2III(m ,m ,m ) , (A.1) √ K η K K K η 2eG lµεν √ √ M = F sin θ −36 2I(m ,m ,m ) + 36 2I(m ,m ,m ) A,K,2-point × 3 η η K K η η 12 √24F √ − 36 2I(mπ ,mπ ,mK ) + 36 2I(mK ,mπ ,mπ ) 302 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 √ √ − 36 2II(m ,m ,m ) − 336 2II(m ,m ,m ) √ K η η √ K K K − (m ,m ,m ) − (m ,m ,m ) 36√2II K π π 168√ 2II π π K − 36 2II(m ,m ,m ) − 96 2II(m ,m ,m ) √ η K π √ K π η − 12 2II(m ,m ,m ) + 36 2III(m ,m ,m ) √ π K η √ η K η + (m ,m ,m ) + (m ,m ,m ) 336√ 2III K K K 168√ 2III π π K + 36 2III(m ,m ,m ) − 36 2III(m ,m ,m ) √ π K π √ π η K + 72 2III(m ,m ,m ) + 48 2III(m ,m ,m ) √ K η π √ η K π + 48 2III(m ,m ,m ) + 24 2III(m ,m ,m ) √ π K η K π η − 12 2III(mη,mπ ,mK ) . (A.2) Form Fig. 4, we obtain

MA,K,3-point √ √ i 2 µ ν 2 = eGF l ε sin θ 6 2m A(mK ,mK ,mK ) 12F 3 K √ √ + 2 + 2 + 2 2 + 2 2 mK mπ A(mπ ,mπ ,mK ) mK mπ A(mK ,mπ ,mπ ) √ 2√ + 2 2 − 2 + 2 2 − 2 3mK mπ A(mK ,mη,mη) mK mπ A(mπ ,mη,mK ) √2 3 − 2 2 − 2 mK mπ A(mK ,mη,mπ ) √3 √ − 2 m2 + m2 B(m ,m ,m ) − 6 2m2 B(m ,m ,m ) √ K π π π K √ K K K K − 2 2 + 2 − 2 2 − 2 mK mπ B(mπ ,mK ,mπ ) 3mK mπ B(mη,mK ,mη) √2 √2 + 2 2 − 2 − 2 2 2 − 2 mK mπ B(mπ ,mK ,mη) mK mπ B(mK ,mπ ,mη) √6 √3 + 2 2 − 2 + 2 2 − 2 mK mπ B(mη,mK ,mπ ) mK mπ B(mη,mπ ,mK ) 6 √ 3 √ √ 2 2 + 2C(mK ,mK ,mK ) − C(mπ ,mπ ,mK ) − C(mK ,mπ ,mπ ) √ √2 2 3 2 2 √ − C(mK ,mη,mη) − C(mπ ,mη,mK ) − 2C(mK ,mη,mπ ) 2 2√ √ √ 5 2 7 2 − 7 2D(mK ,mK ,mK ) − D(mπ ,mπ ,mK ) − D(mπ ,mK ,mπ ) √ √ 2 4 2 2 √ − D(mη,mπ ,mK ) − D(mπ ,mK ,mη) − 2 2D(mK ,mπ ,mη) √2 4√ 2 3 2 − D(mη,mK ,mπ ) − D(mη,mK ,mη) 4 4 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 303 √ √ √ 5 2 7 2 − 7 2E(mK ,mK ,mK ) − E(mπ ,mK ,mπ ) − E(mK ,mπ ,mπ ) √ √ 2 √ 4 2 2 3 2 − E(mπ ,mK ,mη) − E(mK ,mη,mπ ) − E(mK ,mη,mη) √2 4 4 2 √ − E(mK ,mπ ,mη) − 2 2E(mπ ,mη,mK ) 4 √ √ √ 2 2 + 2F(mK ,mK ,mK ) − F(mπ ,mK ,mπ ) − F(mπ ,mπ ,mK ) √ √2 2 3 2 2 √ − F(mη,mη,mK ) − F(mη,mK ,mπ ) − 2F(mη,mπ ,mK ) 2 2 √ √ √ 2 − 2G(mK ,mK ,mK ) − 2G(mπ ,mK ,mπ ) + G(mK ,mπ ,mπ ) √ √ √2 3 2 2 2 + G(mK ,mη,mη) + G(mK ,mη,mπ ) + G(mK ,mπ ,mη) √2 2 √2 3 2 √ 2 + G(mπ ,mπ ,mK ) + 2G(mπ ,mK ,mη) − G(mπ ,mη,mK ) 2 √ 2 √ 3 2 √ − 2H(mK ,mK ,mK ) + H(mπ ,mπ ,mK ) − 2H(mK ,mπ ,mπ ) √ 2 √ 2 √ 2 + H(mπ ,mK ,mπ ) + 2H(mK ,mπ ,mη) + H(mπ ,mK ,mη) 2√ √ 2√ 3 2 2 2 + H(mη,mK ,mη) + H(mη,mK ,mπ ) − H(mη,mπ ,mK ) √2 √2 √ 2 + 8 2I(m ,m ,m ) + 2I(m ,m ,m ) + 2I(m ,m ,m ) √ K K K √ K π π π K π 5 2 3 2 √ + I(mπ ,mπ ,mK ) − I(mη,mη,mK ) + 2I(mK ,mη,mπ ) 4 √ 4 √ √ 2 2 + 2I(mη,mK ,mπ ) − I(mπ ,mη,mK ) − I(mη,mπ ,mK ) 4 4√ √ √ 2 − 2J(mK ,mK ,mK ) − 2J(mπ ,mπ ,mK ) + J(mK ,mπ ,mπ ) √ √ 2√ 2 3 2 2 − J(mπ ,mK ,mη) + J(mπ ,mK ,mπ ) + J(mK ,mπ ,mη) √2 √2 2 2 3 2 √ + J(mK ,mη,mπ ) + J(mK ,mη,mη) + 2J(mπ ,mη,mK ) √2 2√ √ + 8 2K(m ,m ,m ) + 2K(m ,m ,m ) + 2K(m ,m ,m ) √ K K K √ K π π √ π π K 5 2 3 2 2 + K(mπ ,mK ,mπ ) − K(mη,mK ,mη) − K(mπ ,mK ,mη) √4 4 4 2 √ √ − K(mη,mK ,mπ ) + 2K(mη,mπ ,mK ) + 2K(mK ,mπ ,mη) 4 304 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 √ √ √ 2 − 2L(mK ,mK ,mK ) − 2L(mK ,mπ ,mπ ) + L(mπ ,mπ ,mK ) √ √ 2 3 2 2 √ + L(mη,mη,mK ) + L(mη,mπ ,mK ) + 2L(mK ,mη,mπ ) 2 2 √ √ √ 3 2 2 2 + L(mπ ,mK ,mπ ) + L(mπ ,mη,mK ) − L(mη,mK ,mπ ) , 2 2 2 (A.3) MA,π,3-point √ √ i 2 µ ν 2 2 = eGF l ε cosθ 2 m + m A(mK ,mπ ,mK ) 12F 3 K π √ √ + 20 2 2 + 2 2 2 + 2 mπ A(mπ ,mπ ,mπ ) mK mπ A(mπ ,mK ,mK ) √3 3 − 2 2 − 2 mπ mK A(mK ,mK ,mη) 3 √ √ − 2 + 2 − 2 2 2 + 2 2 mK mπ B(mK ,mK ,mπ ) mK mπ B(mK ,mπ ,mK ) √ √ 3 − 20 2 2 + 2 2 2 − 2 mπ B(mπ ,mπ ,mπ ) mπ mK B(mη,mK ,mK ) √3 3 − 2 2 − 2 mπ mK B(mK ,mK ,mη) 3 √ √ √ 4 2 2 2 + 2C(mK ,mπ ,mK ) − C(mπ ,mπ ,mπ ) − C(mπ ,mK ,mK ) √ 3√ 3 3 2 2 − C(mK ,mK ,mπ ) − C(mK ,mK ,mη) √2 2 √ √ 5 2 20 2 10 2 − D(mK ,mK ,mπ ) − D(mπ ,mπ ,mπ ) − D(mK ,mπ ,mK ) √2 3 3 2 √ − D(mK ,mK ,mη) − 2 2D(mη,mK ,mK ) 2√ √ 5 2 20 2 − E(mK ,mπ ,mK ) − E(mπ ,mπ ,mπ ) 2√ 3 √ 10 2 √ 2 − E(mπ ,mK ,mK ) − 2 2E(mK ,mK ,mη) − E(mK ,mη,mK ) √3 √ 2 2 4 2 − F(mπ ,mK ,mK ) − F(mπ ,mπ ,mπ ) 2√ √3 2 2 2 − F(mK ,mK ,mπ ) − F(mη,mK ,mK ) 3 √2 √ √ 4 2 2 2 − 2G(mK ,mπ ,mK ) + G(mπ ,mπ ,mπ ) + G(mπ ,mK ,mK ) 3 3 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 305 √ √ 3 2 2 √ + G(mK ,mK ,mπ ) − G(mK ,mK ,mη) + 2G(mK ,mη,mK ) 2 √2 √ √ 2 2 4 2 − 2H(mπ ,mK ,mK ) + H(mK ,mπ ,mK ) + H(mπ ,mπ ,mπ ) √ 3 √3 3 2 √ 2 + H(mK ,mK ,mπ ) + 2H(mη,mK ,mK ) − H(mK ,mK ,mη) 2 √2 √ √ 16 2 + 2I(mπ ,mK ,mK ) + 2I(mK ,mπ ,mK ) + I(mπ ,mπ ,mπ ) √ 3 8 2 √ √ + I(mK ,mK ,mπ ) + 2I(mK ,mη,mK ) + 2I(mη,mK ,mK ) 3 √ √ √ 4 2 2 2 − 2J(mK ,mK ,mπ ) + J(mπ ,mπ ,mπ ) + J(mπ ,mK ,mK ) √ 3 √3 3 2 √ 2 + J(mK ,mπ ,mK ) + 2J(mK ,mK ,mη) − J(mK ,mη,mK ) 2 2√ √ √ 16 2 + 2K(mK ,mK ,mπ ) + 2K(mπ ,mK ,mK ) + K(mπ ,mπ ,mπ ) √ 3 8 2 √ √ + K(mK ,mπ ,mK ) + 2K(mη,mK ,mK ) + 2K(mK ,mK ,mη) 3 √ √ √ 4 2 2 2 − 2L(mπ ,mK ,mK ) + L(mπ ,mπ ,mπ ) + L(mK ,mK ,mπ ) 3 3 √ √ 3 2 2 √ + L(mK ,mπ ,mK ) − L(mK ,mη,mK ) + 2L(mη,mK ,mK ) . 2 2 (A.4)

Appendix B

We list the functions {II, III,...,A,B,...,L} in Figs. 3 and 4 by scalar integrals of ab Pα1,α2,α3 and one-loop tadpole integrals T1, T2 in the Euclidean space. We note that the µν function I does not contain g and thus it has no contribution to FA. For simplicity, we only give the formulas related to terms with gµν due to the deﬁnition of FA.Wehave

(m ,m ,m ) II 1 2 3 − µν ig 2 2 2 00 = T1 m T1 m − m P (m1,m2,m3) (2π)2n(n − 1) 2 3 1 111 1 − P 20 (m ,m ,m ) +···, (B.1) 2 111 1 2 3 III(m1,m2,m3) −igµν 1 1 1 = T m2 T m2 − T m2 T m2 − T m2 T m2 (2π)2n(n − 1) 2 1 1 1 2 2 1 2 1 3 2 1 1 1 3 306 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317

+ 1 2 + 2 − 2 − 2 00 − 10 m1 m2 m3 P111(m1,m2,m3) P111(m1,m2,m3) 2 01 1 11 − P (m1,m2,m3) − P (m1,m2,m3) +···, (B.2) 111 2 111 A(m1,m2,m3) −2gµν = dx(−1) P 00 (m ,m ,m ) − m2P 00 (m ,m ,m ) 2n − 111 1 2 3 1 211 1 2 3 (2π) (n 1) 1 20 − P (m1,m2,m3) +···, (B.3) 2 211 B(m1,m2,m3) − µν g 2 2 2 2 = dx(−1) T2 m T1 m − T2 m T1 m (2π)2n(n − 1) 2 1 2 3 − 00 + 2 + 2 − 2 − 2 00 P111(m2,m1,m3) m1 m2 m3 P211(m2,m1,m3) − 2P 10 (m ,m ,m ) − 2P 01 (m ,m ,m ) 211 2 1 3 211 2 1 3 2 11 − P (m2,m1,m3) +···, (B.4) 2 211 + C(m1,m2,m3) F(m 3,m1,m2) µν g 00 = dx 2(1 − x)(pM · k)P (m1,m2,m3) (2π)2n(n − 1) 111 2(p · ) − M P 10 (m ,m ,m ) − T m2 T m2 + T m2 T m2 2 111 1 2 3 1 1 1 2 1 2 1 3 − T m2 T m2 1 1 1 3 + 10 + − 2 + 2 + 2 + 2 00 2P111(m1,m2,m3) m1 m2 m3 P111(m1,m2,m3) (p · ) − 2m2(1 − x)(p · k)P00 (m ,m ,m ) + 2m2 M P 10 (m ,m ,m ) 1 M 211 1 2 3 1 2 211 1 2 3 + m2T m2 T m2 − m2P 00 (m ,m ,m ) + m2T m2 T m2 1 2 1 1 2 1 111 1 2 3 1 2 1 1 3 − 2 10 − 2 − 2 + 2 + 2 + 2 00 2m1P211(m1,m2,m3) m1 m1 m2 m3 P211(m1,m2,m3) 2(1 − x)(p · k) 2(p · ) − M P 20 (m ,m ,m ) + M P 30 (m ,m ,m ) 2 211 1 2 3 4 211 1 2 3 − 2 30 − 1 − 2 + 2 + 2 + 2 20 2 P211(m1,m2,m3) 2 m1 m2 m3 P211(m1,m2,m3) n · 2 1 2 d p(p ) 1 20 + T1 m − P (m1,m2,m3) 2 2 (p2 + m2)2 2 111 1 1 dnp(p· )2 + T m2 +···, (B.5) 2 1 3 2 + 2 2 (p m1) D(m ,m ,m ) + E(m ,m ,m ) 2 1 3 1 3 2 −gµν p2 = dx T m2 dnp 2n − 1 1 2 + 2 2 (2π) (n 1) (p m2) C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 307 − − · 2 2 (1 x)( k)T2 m2 T1 m1 · (pM ) 01 00 − P (m2,m1,m3) + (1 − x)( · k)P (m2,m1,m3) 2 111 111 − · − · (1 x)( k) 01 (1 x)( k) 10 + P (m2,m1,m3) + P (m2,m1,m3) 2 111 2 111 1 1 − T m2 T m2 + m2P 00 (m ,m ,m ) 2 1 1 1 3 2 2 111 2 1 3 1 1 − T m2 T m2 + T m2 T m2 2 1 2 1 1 2 1 2 1 3 − 1 2 − 2 − 2 00 + 01 m1 m3 P111(m2,m1,m3) P111(m2,m1,m3) 2 + (p · )T m2 T m2 M 2 2 1 3 (p · ) + m2 + m2 − 2 − m2 M P 01 (m ,m ,m ) 1 2 3 2 211 2 1 3 − − · 00 (1 x)( k)P211(m2,m1,m3) − · − · (1 x)( k) 01 (1 x)( k) 10 − P (m2,m1,m3) − P (m2,m1,m3) 2 211 2 211 1 1 + P 00 (m ,m ,m ) − m2P 00 (m ,m ,m ) 2 111 2 1 3 2 2 211 2 1 3 + 1 2 2 − 1 2 2 T2 m2 T1 m1 T2 m2 T1 m3 2 2 1 + m2 − 2 − m2 P 00 (m ,m ,m ) − P 01 (m ,m ,m ) 2 1 3 211 2 1 3 211 2 1 3 · (pM ) 02 01 − 2 P (m2,m1,m3) + 2(1 − x)( · k)P (m2,m1,m3) 2 211 211 2(1 − x)( · k) 2(1 − x)( · k) + P 02 (m ,m ,m ) + P 11 (m ,m ,m ) 2 211 2 1 3 2 211 2 1 3 − P 01 (m ,m ,m ) + m2P 01 (m ,m ,m ) − 2T m2 T m2 111 2 1 3 2 211 2 1 3 2 2 1 3 − 2 − 2 − 2 01 + 02 m1 m3 P211(m2,m1,m3) 2P211(m2,m1,m3) · (pM ) 11 10 − 2 P (m2,m1,m3) + 2(1 − x)( · k)P (m2,m1,m3) 2 211 211 2(1 − x)( · k) 2(1 − x)( · k) + P 11 (m ,m ,m ) + P 20 (m ,m ,m ) 2 211 2 1 3 2 211 2 1 3 − P 10 (m ,m ,m ) + m2P 10 (m ,m ,m ) 111 2 1 3 2 211 2 1 3 − 2 − 2 − 2 10 m1 m3 P211(m2,m1,m3) 2(pM · ) + 2P 11 (m ,m ,m ) − P 12 (m ,m ,m ) 211 2 1 3 4 211 2 1 3 − · − · 2(1 x)( k) 11 (1 x)( k) 12 + P (m2,m1,m3) + 2 P (m2,m1,m3) 2 211 4 211 308 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317

− · 2(1 x)( k) 21 1 11 + P (m2,m1,m3) − P (m2,m1,m3) 4 211 2 111 1 (m2 − 2 − m2) + m2P 11 (m ,m ,m ) − 1 3 P 11 (m ,m ,m ) 2 2 211 2 1 3 2 211 2 1 3 2 1 dnp(p· )2 + P 12 (m ,m ,m ) − T m2 +···, (B.6) 2 211 2 1 3 2 1 3 2 + 2 2 (p m2) H(m ,m ,m ) + J(m ,m ,m ) 2 1 3 1 3 2 −gµν 2(p · ) = dx M P 01 (m ,m ,m ) + 2P 10 (m ,m ,m ) (2π)2n(n − 1) 2 111 2 1 3 111 2 1 3 − · 00 2(1 x)( k) 10 − 2(1 − x)( · k)P (m2,m1,m3) − P (m2,m1,m3) 111 2 111 2(1 − x)( · k) − P 01 (m ,m ,m ) + T m2 T m2 2 111 2 1 3 1 2 1 1 + −m2 + m2 − 2 − m2 P 00 (m ,m ,m ) − 2P 10 (m ,m ,m ) 2 1 3 111 2 1 3 111 2 1 3 − 01 + 2 2 − 2 2 2P111(m2,m1,m3) T1 m1 T1 m3 T1 m2 T1 m3 2 2m (pM · ) − 2 P 01 (m ,m ,m ) − 2m2P 10 (m ,m ,m ) 2 211 2 1 3 2 211 2 1 3 2 2m (1 − x)( · k) + 2m2(1 − x)( · k)P00 (m ,m ,m ) + 2 P 10 (m ,m ,m ) 2 211 2 1 3 2 211 2 1 3 2m2(1 − x)( · k) + 2 P 01 (m ,m ,m ) − m2T m2 T m2 2 211 2 1 3 2 1 2 1 1 − m2 −m2 + m2 − 2 − m2 P 00 (m ,m ,m ) + 2m2P 10 (m ,m ,m ) 2 2 1 3 211 2 1 3 2 211 2 1 3 + 2 01 + 2 2 2 − 2 00 2m2P211(m2,m1,m3) m2T2 m2 T1 m3 m2P111(m2,m1,m3) 2(p · ) 2 − M P 21 (m ,m ,m ) − P 30 (m ,m ,m ) 4 211 2 1 3 2 211 2 1 3 2(1 − x)( · k) (1 − x)( · k) + P 20 (m ,m ,m ) + 2 P 30 (m ,m ,m ) 2 211 2 1 3 4 211 2 1 3 2 n (p· ) 2(1 − x)( · k) d p 2 + P 21 (m ,m ,m ) − T m2 4 211 2 1 3 1 1 2 + 2 2 (p m2) (−m2 + m2 − 2 − m2) 2 − 2 1 3 P 20 (m ,m ,m ) + P 30 (m ,m ,m ) 2 211 2 1 3 2 211 2 1 3 2 n (p· ) 2 1 d p 2 + P 21 (m ,m ,m ) − P 20 (m ,m ,m ) + T m2 2 211 2 1 3 2 111 2 1 3 1 3 2 + 2 2 (p m2) +···, (B.7) + I(m3,m2,m1) K(m 2,m1,m3) µν g 2 2 = dx (pM · )T m T m (2π)2n(n − 1) 2 3 1 2 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 309

· 00 (pM ) 10 − (pM · )P (m3,m2,m1) − P (m3,m2,m1) 111 2 111 (p · ) (1 − x)( · k) − M P 01 (m ,m ,m ) + P 01 (m ,m ,m ) 2 111 3 2 1 2 111 3 2 1 1 1 1 − T m2 T m2 + T m2 T m2 + T m2 T m2 2 1 3 1 2 2 1 2 1 1 2 1 3 1 1 − 1 2 − 2 − 2 + 2 00 + 10 m2 m1 m3 P111(m3,m2,m1) P111(m3,m2,m1) 2 n 2 01 2 2 2 d p(p) + P (m3,m2,m1) − (1 − x)( · k)T2 m T1 m + T1 m 111 3 1 1 (p2 + m2)2 3 + 2 + 2 − 2 − 2 · 00 m3 m2 m1 (pM )P211(m3,m2,m1) · · (pM ) 10 (pM ) 01 + P (m3,m2,m1) + P (m3,m2,m1) 2 211 2 211 − · (1 x)( k) 01 1 2 2 1 00 − P (m3,m2,m1) + T2 m T1 m − P (m3,m2,m1) 2 211 2 3 2 2 111 − 1 2 2 + 1 2 + 2 − 2 − 2 00 T2 m3 T1 m1 m3 m2 m1 P211(m3,m2,m1) 2 2 − 10 − 01 P211(m3,m2,m1) P211(m3,m2,m1) · 10 2(pM ) 20 − 2(pM · )P (m3,m2,m1) − P (m3,m2,m1) 211 2 211 · − · − 2(pM ) 11 + (1 x)( k) 11 2 P211(m3,m2,m1) 2 2 P211(m3,m2,m1) + 10 − 2 + 2 − 2 − 2 10 P111(m3,m2,m1) m3 m2 m1 P211(m3,m2,m1) + 20 + 11 2P211(m3,m2,m1) 2P211(m3,m2,m1) 2(p · ) − 2(p · )P 01 (m ,m ,m ) − M P 11 (m ,m ,m ) M 211 3 2 1 2 211 3 2 1 · − · − 2(pM ) 02 + (1 x)( k) 02 2 P211(m3,m2,m1) 2 2 P211(m3,m2,m1) + P 01 (m ,m ,m ) − 2T m2 T m2 111 3 2 1 2 3 1 1 − 2 + 2 − 2 − 2 01 m3 m2 m1 P211(m3,m2,m1) · 11 02 2(pM ) 11 + 2P (m3,m2,m1) + 2P (m3,m2,m1) − P (m3,m2,m1) 211 211 2 211 (p · ) (p · ) − 2 M P 21 (m ,m ,m ) − 2 M P 12 (m ,m ,m ) 4 211 3 2 1 4 211 3 2 1 (1 − x)( · k) 1 + 2 P 12 (m ,m ,m ) + P 11 (m ,m ,m ) 4 211 3 2 1 2 111 3 2 1 2 + 2 − 2 − 2 (m3 m2 m1) 11 2 21 − P (m3,m2,m1) + P (m3,m2,m1) 2 211 2 211 310 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 2 1 dnp(p· )2 + P 12 (m ,m ,m ) − T m2 +···, (B.8) 2 211 3 2 1 2 1 1 2 + 2 2 (p m3) G(m ,m ,m ) + L(m ,m ,m ) 1 2 3 3 1 2 gµν dnpp2 = dx (p · k)(1 − x)T m2 T m2 − T m2 2n − M 2 1 1 2 2 1 2 + 2 (2π) (n 1) (p m2) (p · ) + M P 10 (m ,m ,m ) − (p · k)(1 − x)P00 (m ,m ,m ) 2 111 1 2 3 M 111 1 2 3 1 + P 01 (m ,m ,m ) + T m2 T m2 111 1 2 3 2 1 1 1 2 1 1 − T m2 T m2 + T m2 T m2 2 1 2 1 3 2 1 1 1 3 + 1 2 − 2 − 2 − 2 00 − 10 m1 m2 m3 P111(m1,m2,m3) P111(m1,m2,m3) 2 − P 01 (m ,m ,m ) − (1 − x)(p · k)T m2 T m2 111 1 2 3 M 2 1 1 3 (p · ) − m2 + m2 − 2 − m2 M P 10 (m ,m ,m ) 1 2 3 2 211 1 2 3 − · − 00 + 01 (pM k)(1 x)P211(m1,m2,m3) P211(m1,m2,m3) 1 1 1 + T m2 T m2 − P 00 (m ,m ,m ) + T m2 T m2 2 2 1 1 2 2 111 1 2 3 2 2 1 1 3 + 1 2 − 2 − 2 − 2 00 − 10 m1 m2 m3 P211(m1,m2,m3) P211(m1,m2,m3) 2 (p · ) − P 01 (m ,m ,m ) + 2 M P 20 (m ,m ,m ) 211 1 2 3 2 211 1 2 3 − 2(p · k)(1 − x)P10 (m ,m ,m ) + 2P 11 (m ,m ,m ) M 211 1 2 3 211 1 2 3 − 10 + 2 − 2 − 2 − 2 10 P111(m1,m2,m3) m1 m2 m3 P211(m1,m2,m3) − 20 − 11 2P211(m1,m2,m3) 2P211(m1,m2,m3) · + (pM ) 11 − · − 01 2 2 P211(m1,m2,m3) 2(pM k)(1 x)P211(m1,m2,m3) + 2P 02 (m ,m ,m ) − P 01 (m ,m ,m ) − 2T m2 T m2 211 1 2 3 111 1 2 3 2 1 1 3 + 2 − 2 − 2 − 2 01 − 11 m1 m2 m3 P211(m1,m2,m3) 2P211(m1,m2,m3) − 02 2P211(m1,m2,m3) 2(p · ) (p · k)(1 − x) + M P 21 (m ,m ,m ) − 2 M P 11 (m ,m ,m ) 4 211 1 2 3 2 211 1 2 3 2 12 1 11 + P (m1,m2,m3) − P (m1,m2,m3) 2 211 2 111 (m2 − m2 − 2 − m2) + 1 2 3 11 − 2 21 2 P211(m1,m2,m3) 2 P211(m1,m2,m3) 2 1 dnp(p· )2 − P 12 (m ,m ,m ) − T m2 +···, (B.9) 2 211 1 2 3 2 1 3 2 + 2 2 (p m1) C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 311

where = + kx, n = 4 − 2, PM represents the meson momentum, and {···}correspond to the terms without gµν .

Appendix C

ab ; 2 In this appendix, we will replace the set of ten functions P211(m1,m2,m3 ) by 2 the following equivalent one of Hi(m1,m2,m3; ) [17]. The functions Hi are free of quadratic divergences and for this reason they have simpler integral representations. For the well-known one-loop integrals with the deﬁnition γs = γ − 1 − ln(4π),wehave D − d q i I m2 ≡ µ4 D (2π)D q2 − m2 − m2 m2 = Γ(−1 + ) 16π2 4πµ2 −m2 1 m2 = − γs − ln 16π2 ε µ2 π2 m2 1 m2 2 + − γs − ln + γs + 1 + ln , (C.1) 12 µ2 2 µ2 − (2π)4 nT m2 1 − 1 = (2πµ)4 n dnp p2 + m2 1 m2 =−m2π2 − γ − ln s µ2 π2 m2 1 m2 2 + − γs − ln + γs + 1 + ln , (C.2) 12 µ2 2 µ2 − (2π)4 nT m2 2 − 1 = (2πµ)4 n dnp (p2 + m2)2 1 m2 = π2 − 1 − γ − ln s µ2 2 2 2 2 π 1 γs m m + + + γs + + ln + γs ln 12 2 2 µ2 µ2 1 m2 m2 + ln ln . (C.3) 2 µ2 µ2 We note that the expressions of Eq. (C.1) and Eqs. (C.2) and (C.3) are deﬁned in Minkowski and Euclidean spaces, respectively. 312 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317

ab For the two-point functions, the relations between P111 and Hi are given by 00 ; 2 P111 m1,m2,m3 −1 = m2 + 2 H (m ,m ,m ) + H (m ,m ,m ) − 1 1 1 2 3 2 1 2 3 n 3 2 2 + m H1(m2,m1,m3) + m H1(m3,m1,m2) , (C.4) 2 3 10 2 P m1,m2,m3; 111 01 2 = P m2,m1,m3; 111 −1 2 = P 00 (m ,m ,m ) − P 20 (m ,m ,m ) − 5 2 111 1 2 3 211 1 2 3 n 2 − m2H (m ,m ,m ) − m22H (m ,m ,m ) − m2H (m ,m ,m ) 1 2 1 2 3 1 1 1 2 3 2 3 2 1 3 + 2 + 2 m3H2(m3,m2,m1) m3H3(m3,m2,m1) , (C.5) P 11 m ,m ,m ; 2 111 1 2 3 −1 = m2P 11 (m ,m ,m ) − m2P 11 (m ,m ,m ) n − 2 2 211 2 1 3 3 211 3 2 1 2 02 2 2 01 − m P (m3,m2,m1) − m P (m3,m2,m1) 3 211 3 211 2 + m2P 11 (m ,m ,m ) − P 21 (m ,m ,m ) + P 01 (m ,m ,m ) , (C.6) 1 211 1 2 3 211 1 2 3 2 111 1 2 3 02 2 P m1,m2,m3; 111 = 20 ; 2 P111 m2,m1,m3 −1 = m2P 20 (m ,m ,m ) + m2P 02 (m ,m ,m ) − 2 211 2 1 3 3 211 3 2 1 n 2 + 2 02 − 12 m1P211(m1,m2,m3) P211(m1,m2,m3) . (C.7) For the three-point functions, we use [17] 00 2 P m1,m2,m3; = H1(m1,m2,m3), (C.8) 211 10 2 2 P m1,m2,m3; =−H2(m1,m2,m3) − H1(m1,m2,m3), (C.9) 211 01 2 P m1,m2,m3; =−H3(m1,m2,m3), (C.10) 211 20 ; 2 P211 m1,m2,m3 2 = + − 2 − 2 H4(m1,m2,m3) (n 1) m1 H1(m1,m2,m3) n 00 + 2(n − 1)H2(m1,m2,m3) + P (m1,m2,m3) , (C.11) 111 11 ; 2 P211 m1,m2,m3 2 = H5(m1,m2,m3) + H3(m1,m2,m3) C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 313

2 + m2 + m2 − m2 + 2 H (m ,m ,m ) + 2H (m ,m ,m ) 2n 1 2 3 1 1 2 3 2 1 2 3 00 2 2 2 2 − P (m1,m2,m3) + T2 m T1 m − T2 m T1 m , (C.12) 111 1 2 1 3 02 ; 2 P211 m1,m2,m3 2 = H (m ,m ,m ) + −m2H (m ,m ,m ) + T m2 T m2 , (C.13) 6 1 2 3 n 2 1 1 2 3 2 1 1 3 30 ; 2 P211 m1,m2,m3 10 =−H7(m1,m2,m3) − P (m1,m2,m3) 111 32 n − 1 − 2 − m2 2H (m ,m ,m ) n + 2 3 1 1 1 2 3 + − 2 − 2 + (n 1) m1 H2(m1,m2,m3) nH4(m1,m2,m3) , (C.14) 21 ; 2 P211 m1,m2,m3 32 2 =−H (m ,m ,m ) − (n − 1)H (m ,m ,m ) 8 1 2 3 n + 2 3 5 1 2 3 n − 1 + 2 − m2 H (m ,m ,m ) − P 01 (m ,m ,m ) 3 1 3 1 2 3 111 1 2 3 n − 1 − 4 m2 + m2 − m2 + 2 H (m ,m ,m ) + 2H (m ,m ,m ) n(n + 2) 1 2 3 1 1 2 3 2 1 2 3 00 2 2 2 2 − P (m1,m2,m3) + T2 m T1 m − T2 m T1 m , (C.15) 111 1 2 1 3 12 ; 2 P211 m1,m2,m3 2 =−H9(m1,m2,m3) − H6(m1,m2,m3) 2 22 − 2H (m ,m ,m ) + − m2 H (m ,m ,m ) n + 2 5 1 2 3 n 2 2 1 2 3 + 2 + 2 − 2 + 2 + 01 m1 m2 m3 H3(m1,m2,m3) P111(m1,m2,m3)

4 − −P 00 (m ,m ,m ) n(n + 2) 111 1 2 3 2 2 2 2 × m − (n + 1)m − m + H1(m1,m2,m3) 1 2 3 2 2 2 2 + T2 m T1 m − (n − 1)T2 m T1 m , (C.16) 1 2 1 3 03 ; 2 P211 m1,m2,m3 32 =−H (m ,m ,m ) − −m2H (m ,m ,m ) + 2T m2 T m2 . 10 1 2 3 n + 2 2 3 1 2 3 2 1 1 3 (C.17) 314 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317

The functions of Hi are expressed as follows H m ,m ,m ; 2 1 1 2 3 2 4 2 1 1 π 2 = π − (1 − 2γm ) − + − γm + γ + h (m ,m ,m ) , (C.18) ∆2 ∆ 1 2 12 1 m1 1 1 2 3 H m ,m ,m ; 2 2 1 2 3 2 1 1 13 π2 γ = π42 − + − 2γ + − + m1 − γ 2 2 m1 m1 ∆ ∆ 2 8 12 2

− h2(m1,m2,m3) , (C.19) 2 H3 m1,m2,m3; 1 1 1 13 π2 γ γ 2 = 4 2 − − − + − m1 + m1 π 2 γm1 ∆ ∆ 4 16 24 4 2

+ h3(m1,m2,m3) , (C.20) 2 H4 m1,m2,m3; 2 3γ 2 4 4 3 1 3γm1 175 π m1 3 = π + − + + + h4(m1,m2,m3) , (C.21) 2∆2 ∆ 2 96 16 4 4 2 H5 m1,m2,m3; 3 1 3γ 175 π2 3γ 2 3 = π44 − − m1 + − − m1 . − h (m ,m ,m ) , (C.22) 4∆2 ∆ 4 192 32 8 4 5 1 2 3 H m ,m ,m ; 2 6 1 2 3 1 1 1 γ 19 π2 γ = π44 − − m1 − + − m1 2∆2 ∆ 24 2 32 48 24 γ 2 3 + m1 + h (m ,m ,m ) , (C.23) 4 4 6 1 2 3 H m ,m ,m ; 2 7 1 2 3 2 4 6 1 1 5 287 π 5γm1 = π − − + γm + − − ∆2 ∆ 24 1 192 24 24 γ 2 1 − m1 − h (m ,m ,m ) , (C.24) 2 2 7 1 2 3 H m ,m ,m ; 2 8 1 2 3 1 1 5 γ 287 π2 5γ = π46 + + m1 − + + m1 2∆2 ∆ 48 2 384 48 48 γ 2 1 + m1 + h (m ,m ,m ) , (C.25) 4 2 8 1 2 3 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 315 H m ,m ,m ; 2 9 1 2 3 1 1 1 γ 95 π2 γ = π46 − − + m1 + − − m1 3∆2 ∆ 24 3 192 72 24 γ 2 1 − m1 − h (m ,m ,m ) , (C.26) 6 2 9 1 2 3 H m ,m ,m ; 2 10 1 2 3 1 1 1 γ 283 π2 γ = π46 + + m1 − + + m1 4∆2 ∆ 96 4 768 96 96 γ 2 1 + m1 + h (m ,m ,m ) , (C.27) 8 2 10 1 2 3 2 2 where ∆ =−2 and γm = γ + ln(πm /µ ). 2 The ultraviolet ﬁnite parts hi (m1,m2,m3) of the function Hi(m1,m2,m3; ) have the following one-dimensional integral representations: 1 h1(m1,m2,m3) = dy g(y) , 0 1 h2(m1,m2,m3) = dy g(y) + f1(y) , 0 1 h3(m1,m2,m3) = dy g(y) + f1(y) (1 − y), 0 1 h4(m1,m2,m3) = dy g(y) + f1(y) + f2(y) , 0 1 h5(m1,m2,m3) = dy g(y) + f1(y) + f2(y) (1 − y), 0 1 2 h6(m1,m2,m3) = dy g(y) + f1(y) + f2(y) (1 − y) , 0 1 h7(m1,m2,m3) = dy g(y) + f1(y) + f2(y) + f3(y) , 0 1 h8(m1,m2,m3) = dy g(y) + f1(y) + f2(y) + f3(y) (1 − y), 0 316 C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317

1 2 h9(m1,m2,m3) = dy g(y) + f1(y) + f2(y) + f3(y) (1 − y) , 0 1 3 h10(m1,m2,m3) = dy g(y) + f1(y) + f2(y) + f3(y) (1 − y) . (C.28) 0 All ten integral representations are built up by the following four basic functions: 1 1 y1 y2 g(y) = Sp + Sp + y1 ln + y2 ln , 1 − y1 1 − y2 y1 − 1 y2 − 1 1 1 − ν2 y y f (y) = − + y2 ln 1 + y2 ln 2 , 1 2 1 2 2 κ y1 − 1 y2 − 1 − 2 − 2 2 = 1 − 2 − 1 ν − 1 ν + 3 y1 + 3 y2 f2(y) 2 2 2 y1 ln y2 ln , 3 κ 2κ κ y1 − 1 y2 − 1 1 4 1 3 1 − ν2 1 1 − ν2 2 1 − ν2 3 f3(y) = − − + − − 4 κ2 3 κ2 2κ2 2 κ2 κ2 + 4 y1 + 4 y2 y1 ln y2 ln , (C.29) y1 − 1 y2 − 1 where z ln(1 − t) Sp(z) = − dt, t 0 1 + κ2 − ν2 ± (1 + κ2 − ν2)2 + 4ν2κ2 − 4iκ2η y , = , 1 2 2κ2 ay + b(1 − y) m2 m2 2 ν2 = ,a= 2 ,b= 3 ,κ2 = . (C.30) − 2 2 2 y(1 y) m1 m1 m1 Finally, we must transform the parameters back into the Minkowski space and change the inward directions of the momenta l, k for the ﬁnal particles by outward ones: 2 →− 2 − · mK,π 2p k , 2 →− 2 − · − mK,π 2p k(1 x) , p · k → p · k, · → 2 − · − p mK,π p k(1 x), k · →−p · k, (C.31) · 2 − 2 2 − 2 where 0 p k (mK,π m)/2, and we have replaced ∂ by ∂ as calculated in the Euclidean space. C.Q. Geng et al. / Nuclear Physics B 684 (2004) 281–317 317

References

[1] J. Gasser, H. Leutwyler, Ann. Phys. 158 (1984) 142. [2] J. Gasser, H. Leutwyler, Nucl. Phys. B 250 (1985) 465. [3] A. Pich, Rep. Prog. Phys. 58 (1995) 563. [4] V. Bernard, N. Kaiser, U.-G. Meissner, J. Mod. Phys. E 4 (1995) 193. [5] G. Ecker, Prog. Part. Nucl. Phys. 35 (1995) 1. [6] J. Bijnens, G. Ecker, J. Gasser, Chiral perturbation theory, hep-ph/9411232; J. Bijnens, G. Colangelo, G. Ecker, J. Gasser, Semileptonic kaon decays, hep-ph/9411311; J. Bijnens, G. Colangelo, G. Ecker, J. Gasser, in: L. Maiani, G. Pancheri, N. Paver (Eds.), The Second Daphne Physics Handbook, INFN, Frascati, 1995. [7] J. Bijnens, G. Ecker, J. Gasser, Nucl. Phys. B 396 (1993) 81, hep-ph/9209261. [8] V.N. Bolotov, Phys. Lett. B 243 (1990) 308. [9] A.A. Poblaguev, Phys. Lett. B 238 (1990) 109. [10] E787 Collaboration, S.C. Adler, et al., Phys. Rev. Lett. 85 (2000) 2256, hep-ex/0003019. [11] KEK E246 Collaboration, V.V. Anisimovsky, et al., hep-ex/0304027. [12] C.Q. Geng, S.K. Lee, Phys. Rev. D 51 (1995) 99, hep-ph/9410347. [13] C.H. Chen, C.Q. Geng, C.C. Lih, Phys. Rev. D 56 (1997) 6856, hep-ph/9709447. [14] J. Bijnens, G. Colangelo, G. Ecker, JHEP 9902 (1999) 020, hep-ph/9902437. [15] H.W. Fearing, S. Scherer, Phys. Rev. D 53 (1996) 315. [16] T. Ebertshauser, H.W. Fearing, S. Scherer, Phys. Rev. D 65 (2002) 054033, hep-ph/0110261. [17] A. Ghinculov, Y.-P. Yao, Nucl. Phys B 516 (1998) 385, hep-ph/9702266. [18] P. Post, J.B. Tausk, Mod. Phys. Lett. A 11 (1996) 2115. [19] T.N. Truong, Phys. Rev. Lett. 61 (1988) 2526. [20] A. Dobado, M.J. Herrero, T.N. Truong, Phys. Lett. B 235 (1990) 134. [21] T.N. Truong, EFI-90-26-CHICAGO, lectures given at Ettore Majoranava International School on Low Energy Antiproton Physics, Erice, Italy, January 25–31, 1990. [22] J. Bijnens, G. Colangelo, G. Ecker, JHEP 9902 (1999) 020; J. Bijnens, G. Colangelo, G. Ecker, Ann. Phys. B 280 (2000) 100, hep-ph/9907333. [23] L. Ametller, J. Bijnens, A. Bramon, F. Cornet, Phys. Lett. B 303 (1993) 140, hep-ph/9302219. [24] J. Bijnens, P. Talavera, Nucl. Phys. B 489 (1997) 387, hep-ph/9610269. [25] M. Knecht, A. Nyffeler, Eur. Phys. J. C 21 (2001) 659, hep-ph/0106034. [26] John F. Donoghue, Barry R. Holstein, Phys. Rev. D 40 (1989) 2378. [27] John F. Donoghue, Barry R. Holstein, Phys. Rev. D 40 (1989) 3700. [28] P. Post, K. Schilcher, Eur. Phys. J. C 25 (2002) 427, hep-ph/0112352. [29] O. Strandberg, hep-ph/0302064. [30] J. Bijnens, A. Bramon, F. Cornet, Phys. Rev. Lett. 61 (1988) 1453. [31] J. Bijnens, A. Bramon, F. Cornet, Phys. Lett. B 237 (1990) 488. [32] J. Bijnens, private communication. [33] G. Amoros, J. Bijnens, P. Talavera, Nucl. Phys. B 602 (2001) 87, hep-ph/0101127. [34] J. Bijnens, P. Talavera, hep-ph/0303103; J. Bijnens, P. Talavera, JHEP 0203 (2002) 046, hep-ph/0203049; G. Amoros, J. Bijnens, P. Talavera, Nucl. Phys. B 585 (2000) 293, hep-ph/0003258; G. Amoros, J. Bijnens, P. Talavera, Nucl. Phys. B 598 (2001) 665, Erratum; G. Amoros, J. Bijnens, P. Talavera, Nucl. Phys. B 568 (2000) 319, hep-ph/9907264; E. Golowich, J. Kambor, Nucl. Phys. B 447 (1995) 373, hep-ph/9501318; E. Golowich, J. Kambor, Phys. Rev. D 58 (1998) 036004, hep-ph/9710214. [35] S. Scherer, hep-ph/0210398. [36] J. Prades, Z. Phys. C 63 (1994) 491; J. Prades, Eur. Phys. J. C 11 (1999) 571. [37] A.A. Poblaguev, et al., Phys. Rev. Lett. 89 (2002) 061803, hep-ex/0204006. [38] F.L. Bezrukov, D.S. Gorbunov, Y.G. Kudenko, Phys. Rev. D 67 (2003) 091503, hep-ph/0302106. [39] Particle Data Group, K. Hagiwara, et al., Phys. Rev. D 66 (2002) 010001. [40] C.Q. Geng, C.C. Lih, W.M. Zhang, Phys. Rev. D 57 (1998) 5697, hep-ph/9710323. [41] C.Q. Geng, C.C. Lih, C.C. Liu, Phys. Rev. D 62 (2000) 034019, hep-ph/0004164. Nuclear Physics B 684 [PM] (2004) 321–350 www.elsevier.com/locate/npe

N = 8 superconformal mechanics

S. Bellucci a,E.Ivanovb, S. Krivonos b, O. Lechtenfeld c

a INFN-Laboratori Nazionali di Frascati, C.P. 13, 00044 Frascati, Italy b Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia c Institut für Theoretische Physik, Universität Hannover, Appelstraße 2, 30167 Hannover, Germany Received 19 January 2004; accepted 19 February 2004

Abstract We construct new models of N = 8 superconformal mechanics associated with the off-shell N = 8, d = 1 supermultiplets (3, 8, 5) and (5, 8, 3). These two multiplets are derived as N = 8 Goldstone superfields and correspond to nonlinear realizations of the N = 8, d = 1 superconformal group | | OSp(4|4) in its supercosets OSp(4 4) and OSp(4 4) , respectively. The irreducibility con- U(1)R⊗SO(5) SU(2)R⊗SO(4) straints for these superfields automatically follow from appropriate superconformal covariant conditions on the Cartan superforms. The N = 8 superconformal transformations of the superspace coordinates and the Goldstone superfields are explicitly given. Interestingly, each N = 8 supermultiplet admits two different off-shell N = 4 decompositions, with different N = 4 superconformal subgroups SU(1, 1|2) and OSp(4|2) of OSp(4|4) being manifest as superconformal symmetries of the corresponding N = 4, d = 1 superspaces. We present the actions for all such N = 4 splittings of the N = 8 multiplets considered.  2004 Elsevier B.V. All rights reserved.

1. Introduction

Supersymmetric quantum mechanics (SQM) [1–26]1 has plenty of applications ranging from the phenomenon of spontaneous breaking of supersymmetry [1,6,7] to the description of the moduli of supersymmetric monopoles and black holes [12,13,20,21]. The latter topic is closely related to the AdS2/CFT1 pattern of the general AdS/CFT correspondence [28], and it is the superconformal versions of SQM [3–5,13–15,17,22,25] which are relevant in this context. Most attention has been paid to SQM models with extended N = 2n

E-mail addresses: [email protected] (S. Bellucci), [email protected] (E. Ivanov), [email protected] (S. Krivonos), [email protected] (O. Lechtenfeld). 1 See [27] for a more exhaustive list of references.

0550-3213/$ – see front matter  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2004.02.023 322 S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350 supersymmetries (where n = 1, 2,...)2 because the latter are related to supersymmetries in d>1 dimensions via some variant of dimensional reduction. So far, the mostly explored example is SQM with N 4 supersymmetry. However, the geometries of d = 1 sigma models having N 4 supercharges are discussed in [12,18]. Up to now, concrete SQM models with N = 8, d = 1 supersymmetry have been of particular use. In [12] such sigma models were employed to describe the moduli of certain solitonic black holes. In [11], a superconformal N = 8, d = 1 action was constructed for the low energy effective dynamics of a D0-brane moving in D4-brane and/or orientifold plane backgrounds (see also [26,29]). In [21], N = 8 SQM yielded the low energy description of half-BPS monopoles in N = 4SYMtheory. It appears to be desirable to put the construction and study of N = 8 (and perhaps N>8) SQM models on a systematic basis by working out the appropriate off-shell superfield techniques. One way to build such models is to perform a direct reduction of four-dimensional superfield theories. For instance, one may start from a general off-shell action containing k copies of the d = 4 hypermultiplet, naturally written in analytic harmonic superspace [30,31], and reduce it to d = 1 simply by suppressing the dependence on the spatial coordinates. The resulting action will describe the most general N = 8 extension of a d = 1 sigma model with a 4k-dimensional hyper-Kähler target manifold. Each reduced hypermultiplet yields a N = 8, d = 1 off-shell multiplet (4, 8, ∞) [32] containing four physical bosons, eight physical fermions and an infinite number of auxiliary fields just like its d = 4 prototype. A wider class of N = 8 SQM models can be constructed reducing two-dimensional N = (4, 4) or heterotic N = (8, 0) sigma models. In this way one recovers the off-shell N = 8, d = 1 multiplets (4, 8, 4) and (8, 8, 0) [33]. The relevant superfield actions describe N = 8, d = 1 sigma models with strong torsionful hyper-Kähler or octonionic-Kähler bosonic target geometries, respectively [12]. Although any d = 1 super-Poincaré algebra can be obtained from a higher-dimensional one via dimensional reduction, this is generally not true for d = 1 superconformal algebras [34,35] and off-shell d = 1 multiplets. For instance, no d = 4 analog exists for the N = 4, d = 1 multiplet with off-shell content (1, 4, 3) [5] or (3, 4, 1) [9,10,20] (while the multiplet (2, 4, 2) is actually a reduction of the chiral d = 4 multiplet). Moreover, there exist off-shell d = 1 supermultiplets containing no auxiliary fields at all, something impossible for d 3 supersymmetry. Examples are the multiplets (4, 4, 0) [16,19] and (8, 8, 0) [33] of N = 4 and N = 8 supersymmetries in d = 1. This zoo of d = 1 supermultiplets greatly enlarges the class of admissible target geometries for supersymmetric d = 1 sigma models and the related SQM models. Besides those obtainable by direct reduction from higher dimensions, there exist special geometries particular to d = 1 models and associated with specific d = 1 supermultiplets. In fact, dimensional reduction is not too useful for obtaining superconformally invariant d = 1 superfield actions which are important for the study of the AdS2/CFT1 correspondence. One of the reasons is that the integration measures of superspaces having

2 By N in SQM models we always understand the number of real d = 1 supercharges. S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350 323 the same Grassmann-odd but different Grassmann-even dimensions possess different dilatation weights and hence different superconformal transformation properties. As a result, some superconformal invariant superfield Lagrangians in d = 1 differ in structure from their d = 4 counterparts. Another reason is the already mentioned property that most d = 1 superconformal groups do not descend from superconformal groups in higher dimensions. The most general N = 4, d = 1 superconformal group is the exceptional one- parameter (α) family of supergroups D(2, 1; α) [35] which only at the special values α = 0 and α =−1 (and at values equivalent to these two) reduces to the supergroup SU(1, 1|2) obtainable from superconformal groups SU(2, 2|N) in d = 4. Taking into account these circumstances, it is advantageous to have a convenient superfield approach to d = 1 models which does not resort to dimensional reduction and is self-contained in d = 1. Such a framework exists and is based on superfield nonlinear realizations of d = 1 superconformal groups. It was pioneered in [5] and recently advanced in [22,36]. Its basic merits are, firstly, that in most cases it automatically yields the irreducibility conditions for d = 1 superfields and, secondly, that it directly specifies the superconformal transformation properties of these superfields. The physical bosons and fermions, together with the d = 1 superspace coordinates, prove to be coset parameters associated with the appropriate generators of the superconformal group. Thus, the differences in the field content of various supermultiplets are attributed to different choices of the coset supermanifold inside the given superconformal group. Using the nonlinear realizations approach, in [36] all known off-shell multiplets of N = 4, d = 1 Poincaré supersymmetry were recovered and a few novel ones were found, including examples of nontrivial off-shell superfield actions. With the present paper we begin a study of N = 8, d = 1 supermultiplets along the same line. We will demonstrate that the (5, 8, 3) multiplet of Ref. [11] comes out as a Goldstone one, parametrizing a specific coset of the supergroup OSp(4|4) such that four physical bosons parametrize the coset SO(5)/SO(4) while the fifth one is the dilaton. The appropriate irreducibility constraints in N = 8, d = 1 superspace immediately follow as a consequence of covariant inverse Higgs [37] constraints on the relevant Cartan forms. As an example of a different N = 8 mechanics, we construct a new model associated with the N = 8, d = 1 supermultiplet (3, 8, 5). This supermultiplet parametrizes another coset of OSp(4|4) such that SO(5) ⊂ OSp(4|4) belongs to the stability subgroup while one out of three physical bosons is the coset parameter associated with the dilatation generator and the remaining two parametrize the R-symmetry coset SU(2)R/U(1)R. This model is a direct N = 8 extension of some particular case of the N = 4, d = 1 superconformal mechanics considered in [22] and the corresponding bosonic sectors are in fact identical. For both N = 8 supermultiplets considered, we construct invariant actions in N = 4, d = 1 superspace. We find an interesting peculiarity of the N = 4, d = 1 superfield representation of the model associated with a given N = 8 multiplet. Depending on the N = 4 subgroup of the N = 8 super-Poincaré group, with respect to which we decompose the given N = 8 superfield, we obtain different splittings of the latter into irreducible N = 4off- shell supermultiplets and, consequently, different N = 4 off-shell actions, which however produce the same component actions. There exist two distinct N = 4 splittings of the 324 S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350 considered multiplets, namely

(5, 8, 3) = (3, 4, 1) ⊕ (2, 4, 2) or (5, 8, 3) = (1, 4, 3) ⊕ (4, 4, 0), (1.1) (3, 8, 5) = (1, 4, 3) ⊕ (2, 4, 2) or (3, 8, 5) = (3, 4, 1) ⊕ (0, 4, 4). (1.2) The first splitting in (1.1) is just what has been employed in [11]. The second splitting is new. For both of them we write down N = 8 superconformal actions in standard N = 4, d = 1 superspace. The latter is also suited for setting up the off-shell superconformal action corresponding to the first option in (1.2). As for the second splitting in (1.2), the equivalent off-shell superconformal action can be written only by employing N = 4, d = 1 harmonic superspace [23], since the kinetic term of the multiplet (0, 4, 4) is naturally defined just in this superspace. The paper is organized as follows. In Section 2 we give a N = 8 superfield formulation of the multiplet (3, 8, 5), based on a nonlinear realization of the superconformal supergroup OSp(4|4). In Sections 3 and 4 we present two alternative N = 4 superfield formulations of this multiplet and the relevant off-shell superconformal actions. Section 5 is devoted to treating the multiplet (5, 8, 3) along the same lines. A summary of our results and an outlook are the contents of the concluding Section 6.

2. N = 8, d = 1 superspace and tensor multiplet

2.1. N = 8, d = 1 superspace as a reduction of N = 2, d = 4

It will be more convenient for us to start from the N = 8, d = 1 superfield description of the off-shell multiplet (3, 8, 5), since it is tightly related to the N = 4, d = 1 model studied in [22]. We shall first recover it within the dimensional reduction from N = 2, d = 4 superspace. The maximal automorphism group of N = 8, d = 1 super-Poincaré algebra (without central charges) is SO(8) and so eight real Grassmann coordinates of N = 8, d = 1 superspace can be arranged into one of three 8-dimensional real irreps of SO(8). For our purpose in this paper we shall split these 8 coordinates in another way, namely into two real quartets on which three commuting automorphism SU(2) groups will be realized. = = m α ¯iα˙ A convenient point of departure is N 2, d 4 superspace (x ,θi , θ ) whose automorphism group is SL(2,C)× U(2)R. The corresponding covariant derivatives form the following algebra:3 Di D¯ j = ij α, α˙ 2i ∂αα˙ , (2.1) ¯ j where Djα˙ ≡−(Dα).TheN = 2, d = 4 tensor multiplet is described by a real isotriplet superfield V(ik) subjected to the off-shell constraints D(iVjk) = D¯ (iVjk) = α α˙ 0. (2.2)

3 jk = k = 21 = We use the following convention for the skew-symmetric tensor : ij δi , 12 1. S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350 325

These constraints can be reduced to d = 1 in two different ways, yielding multiplets of two different d = 1 supersymmetries. To start with, it has been recently realized [36] that many N = 4, d = 1 superconformal multiplets have their N = 2, d = 4 ancestors (though the standard dimensional reduction to N = 4, d = 1 superspace proceeds from N = 1, d = 4 superspace, see, e.g., [9]). The relation between such d = 4andd = 1 supermultiplets is provided by a special reduction procedure, whose key feature is the suppression of space–time indices in d = 4 spinor derivatives. The superﬁeld constraints describing the N = 4, d = 1 supermultiplets are obtained from the N = 2, d = 4 ones by discarding the spinorial SL(2,C) indices and keeping only the R-symmetry indices. For instance, the constraints (2.2) after such a reduction become

D(iV jk) = D¯ (iV jk) = 0. (2.3)

Here, the ‘reduced’ N = 4, d = 1 spinor derivatives Di , D¯ j are subject to i ¯ j ij D , D = 2i ∂t (2.4)

ij and ∂αα˙ → ∂t .TheN = 4, d = 1 superfield V obeying the constraints (2.3) contains four bosonic (three physical and one auxiliary) and four fermionic off-shell components, i.e., it defines the N = 4, d = 1 multiplet (3, 4, 1). It has been employed in [9] for constructing a general off-shell sigma model corresponding to the N = 4 SQM model of [2] and [8]. The same supermultiplet was independently considered in [10,20] and then has been used in [22] to construct a new version of N = 4 superconformal mechanics. It has been also treated in the framework of N = 4, d = 1 harmonic superspace [23]. Other N = 2, d = 4 supermultiplets can be also reduced in this way to yield their N = 4, d = 1 superspace analogs [36]. It should be emphasized that, though formally reduced constraints look similar to their N = 2, d = 4 ancestors, the irreducible component field contents of the relevant multiplets can radically differ from those in d = 4 due to the different structure of the algebra of the covariant derivatives. For example, (2.2) gives rise to a notoph type differential constraint for a vector component field, while (2.3) does not impose any restriction on the t-dependence of the corresponding component fields. As an extreme expression of such a relaxation of constraints, some N = 2, d = 4 supermultiplets which are on-shell in the standard d = 4 superspace in consequence of their superfield constraints, have off-shell N = 4, d = 1 counterparts. This refers, e.g., to the N = 2, d = 1 hypermultiplet without central charge [38] (leaving aside the formulations in N = 2, d = 4 harmonic superspace [30,31]). Another, more direct way to reduce some constrained N = 2, d = 4 superfield to d = 1 is to keep as well the SL(2,C) indices of spinor derivatives, thus preserving the total number of spinor coordinates and supersymmetries and yielding a N = 8, d = 1 supermultiplet. However, such a reduction clearly breaks SL(2,C) down to its SU(2) subgroup iα ¯ jβ ij αβ D , D = 2i ∂t . (2.5) 326 S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350

4 A closer inspection of the relation (2.5) shows that, besides the U(2)R automorphisms and the manifest SU(2) realized on the indices α, β and inherited from SL(2,C),italso possesses the hidden automorphisms Di = β D¯ i D¯ i =− ¯β Di α = δ α Λα β ,δα Λα β ,Λα 0, (2.6) which emerge as a by-product of the reduction. Together with the manifest SU(2) and overall phase U(1)R transformations, (2.6) can be shown to form USp(4) ∼ SO(5).The ¯α =− α transformations with Λβ Λβ close on the manifest SU(2) and, together with the latter, constitute the subgroup Spin(4) = SU(2) × SU(2) ⊂ USp(4). The manifest SU(2) forms a diagonal in this product. Passing to the new basis 1 i Dia = √ Dia + D¯ ia , ∇iα = √ Diα − D¯ iα , 2 2 ia iα D =−Dia, ∇ =−∇iα, (2.7) one can split (2.5) into two copies of N = 4, d = 1 anticommutation relations, such that the covariant derivatives from these sets anticommute with each other Dia,Djb = 2iij ab∂ , ∇iα, ∇jβ = 2iij αβ ∂ , t t Dia, ∇jα = 0. (2.8) These two mutually anticommuting algebras pick up the left and right SU(2) factors of Spin(4) as their automorphism symmetries. Correspondingly, the N = 8, d = 1 superspace is parametrized by the coordinates (t, θia,ϑkα), subjected to the reality conditions ia iα (θia) = θ , (ϑiα) = ϑ . (2.9) Such a representation of the algebra of N = 8, d = 1 spinor derivatives manifests three mutually commuting automorphism SU(2) symmetries which are realized, respectively, on the doublet indices i, a and α. The transformations from the coset USp(4)/Spin(4) ∼ i ∇i SO(5)/SO(4) rotate Da and α through each other (and the same for θia and ϑiα). In this basis, the N = 8, d = 1 reduced version of N = 2, d = 4 tensor multiplet (2.2) is defined by the constraints (i jk) =∇(i jk) = Da V α V 0. (2.10) This off-shell N = 8, d = 1 multiplet can be shown to comprise eight bosonic (three physical and five auxiliary) and eight fermionic components, i.e., it is (3, 8, 5).Weshall see that (2.10) implies a d = 1 version of the d = 4 notoph field strength constraint whose effect in d = 1 is to constrain some superfield component to be constant [5]. In the next section we shall show that the N = 8, d = 1 tensor multiplet V ij defined by (2.10) can support, besides the manifest N = 8, d = 1 Poincaré supersymmetry, also a realization of the N = 8, d = 1 superconformal algebra osp(4|4). While considered as a carrier of the latter, it can be called ‘N = 8, d = 1 improved tensor multiplet’. In fact, like

4 Hereafter, we reserve the term ‘R-symmetry’ for those automorphisms of d = 1 Poincaré superalgebra which originate from R-symmetries in d = 4. S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350 327 its N = 4, d = 1 counterpart (3, 4, 1), it can be derived from a nonlinear realization of the supergroup OSp(4|4) in the appropriate coset supermanifold, without any reference to the dimensional reduction from d = 4.

2.2. Superconformal properties of the N = 8, d = 1 tensor multiplet

The simplest way to ﬁnd the transformation properties of the N = 8, d = 1tensor multiplet V ij and prove the covariance of the basic constraints (2.10) with respect to the superconformal N = 8 superalgebra osp(4|4) is to use the coset realization technique. All steps in this construction are very similar to those employed in [22]. So we quote here the main results without detailed explanations. We use the standard deﬁnition of the superalgebra osp(4|4) [35]. It contains the following sixteen spinor generators: iaA iαA iaA = jbA = Q1 ,Q2 , Q ij abQ , i,a,α,A 1, 2, (2.11) and sixteen bosonic generators:

AB ij ab αβ aα T0 ,T,T1 ,T2 ,U. (2.12) The indices A,i,a and α refer to fundamental representations of the mutually commuting ∼ AB ∼ ij ab αβ aα sl(2,R) T0 and three su(2) T ,T1 ,T2 algebras. The four generators U belong ab αβ to the coset SO(5)/SO(4) with SO(4) generated by T1 and T2 . The bosonic generators form the full bosonic subalgebra sl(2,R)⊕ su(2)R ⊕ so(5) of osp(4 |4). The commutator of any T -generator with Q has the same form, and it is sufﬁcient iaA to write it for some particular sort of indices, e.g., for a,b (other indices of Q1,2 being suppressed): i T ab,Qc =− acQb + bcQa . (2.13) 2 The commutators with the coset SO(5)/SO(4) generators U aα have the following form aα ibA =− ab iαA aα iβA =− αβ iaA U ,Q1 i Q2 , U ,Q2 i Q1 . (2.14) At last, the anticommutators of the fermionic generators read QiaA,QjbB =−2 ij abT AB − 2ij ABT ab + abAB T ij , 1 1 0 1 QiαA,QjβB =−2 ij αβ T AB − 2ij ABT αβ + αβ AB T ij , 2 2 0 2 iaA jαB = ij AB aα Q1 ,Q2 2 U . (2.15) iaA iαA From (2.15) it follows that the generators Q1 and Q1 , together with the corresponding bosonic generators, span two osp(4|2) subalgebras in osp(4|4). For what follows it is convenient to pass to another notation, ≡ 22 ≡ 11 ≡− 12 P T0 ,KT0 ,DT0 , 22 ¯ 11 12 V ≡ T , V ≡ T ,V3 ≡ T , ia ≡− ia2 Qiα ≡− iα2 ia ≡ ia1 Siα ≡ iα1 Q Q1 , Q2 ,SQ1 , Q2 . (2.16) 328 S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350

One can check that P and Qia, Qiα constitute a N = 8, d = 1 Poincaré superalgebra. The generators D,K and Sia, Siα stand for the d = 1 dilatations, special conformal transformations and conformal supersymmetry, respectively. The full structure of the superalgebra osp(4|4) is given in Appendix A. Now we shall construct a nonlinear realization of the superconformal group OSp(4|4) in the coset superspace with an element parametrized as ia+ Qiα ia+ Siα + ¯ ¯ g = eitP eθiaQ ϑiα eψiaS ξiα eizKeiuDeiφV iφV . (2.17)

The coordinates t, θia, ϑiα parametrize the N = 8, d = 1 superspace. All other supercoset parameters are Goldstone N = 8 superfields. The stability subgroup contains a subgroup U(1)R of the group SU(2)R realized on the doublet indices i, hence the Goldstone ¯ superfields φ, φ parametrize the coset SU(2)R/U(1)R. The group SO(5) is placed in the stability subgroup. It linearly rotates the fermionic Goldstone superfields ψ and ξ through each other, equally as the N = 8, d = 1 Grassmann coordinates θ and ϑ’s. To summarize, | in the present case we are dealing with the supercoset OSp(4 4) . U(1)R⊗SO(5) The semi-covariant (fully covariant only under N = 8 Poincaré supersymmetry) spinor derivatives are defined by

ia ∂ ia iα ∂ iα D = + iθ ∂t , ∇ = + iϑ ∂t . (2.18) ∂θia ∂ϑiα Their anticommutators, by construction, coincide with (2.8). A natural way to find conformally covariant irreducibility conditions on the coset superfields is to impose the inverse Higgs constraints [37] on the left-covariant Cartan one-form Ω valued in the superalgebra osp(4|4). This form is defined by the standard relation − g 1 dg = Ω. (2.19) In analogy with [22,36], we impose the following constraints:

ωD = 0,ωV |=ω ¯V |=0, (2.20) where | denotes the spinor projection. These constraints are manifestly covariant under the whole supergroup. They allow one to express the Goldstone spinor superfields and the superfield z via the spinor and t-derivatives, respectively, of the remaining bosonic Goldstone superfields u, φ, φ¯ iDiau =−2ψia,i∇iαu =−2ξ iα, u˙ = 2z, iD1aΛ = 2Λ ψ1a + Λψ2a ,iD2aΛ =−2 ψ1a + Λψ2a , i∇1αΛ = 2Λ ξ 1α + Λξ 2α ,i∇2αΛ =−2 ξ 1α + Λξ 2α , (2.21) where tan φφ¯ tan φφ¯ Λ = φ, Λ¯ = φ.¯ (2.22) φφ¯ φφ¯ Simultaneously, Eq. (2.21) imply some irreducibility constraints for u, φ, φ¯.After ij ij ji ik i k introducing a new N = 8 vector superfield V such that V = V and V = ii kk V , S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350 329 via √ Λ √ Λ¯ i 1 − ΛΛ¯ V 11 =−i 2 eu ,V22 = i 2 eu ,V12 = √ eu , 1 + ΛΛ¯ 1 + ΛΛ¯ 2 1 + ΛΛ¯ 2 ik 2u V ≡ V Vik = e , (2.23) and eliminating the spinor superfields from (2.21), the differential constraints on the remaining Goldstone superfields can be brought in the manifestly SU(2)R-symmetric form (i jk) = ∇ (i jk) = Da V 0, α V 0, (2.24) which coincides with (2.10). For further use, we present one important consequence of (2.24) a ij +∇α∇ ij = ⇒ ∇ α∇ ij = − a ij ∂t Di DjaV i jαV 0 i jαV 6m Di DjaV , m = const. (2.25) The specific normalization of the arbitrary real constant m is chosen for convenience. Besides ensuring the covariance of the basic constraints (2.24) with respect to OSp(4|4) and clarifying their geometric meaning, the coset approach provides the easiest way to find the transformation properties of all coordinates and superfields. Indeed, all transformations are generated by acting on the coset element (2.17) from the left by the elements of osp(4|4). Since all bosonic transformations appear in the anticommutator of the conformal supersymmetry and Poincaré supersymmetry, it is sufficient to know how V ij is transformed under these supersymmetries. The N = 8, d = 1 Poincaré supersymmetry is realized on superspace coordinates in the standard way ia iα δt =−i ηiaθ + ηiαϑ ,δθia = ηia,δϑiα = ηiα (2.26) and V ik is a scalar with respect to these transformations. On the other hand, N = 8 superconformal transformations are nontrivially realized both on the coordinates and coset superfields =− ia + iα + i ja + i jα b + β δt it θia ε ϑiα aθ εαϑ θibθj ϑiβ ϑj , = − j b + j b − j α + α j δθia tia ia θjbθi 2ib θi θja ia ϑjαϑi 2iεj ϑiαθa , j β j β j a j a δϑiα = tεiα − iε ϑjβϑ + 2iε ϑ ϑjα − iε θjaθ + 2ia θ ϑjα, α i β i α i i ia iα 2 δu =−2i θia + ε ϑiα ,δΛ= a + ibΛ+¯aΛ , (2.27) where a = 2i aθ + εαϑ , a¯ = 2i aθ + εαϑ , 2 2a 2 2α 1 1a 1 1α =− a + a + α + α b 2 1 θ2a 2 θ1a ε1 ϑ2α ε2 ϑ1α . (2.28) ij The conformal supersymmetry transformation of V has the manifestly Spin(4) × SU(2)R covariant form ij = ka + kα ij + a(i + a (i + α(i + α (i j)k δV 2i θ ka ϑ εkα V θka k θa ε ϑkα εk ϑα V . (2.29) 330 S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350

Thus we know the transformation properties of the N = 8 ‘tensor’ multiplet V ij under the N = 8 superconformal group OSp(4|4). Before turning to the construction of superconformal invariant actions, in the next section we will study how this N = 8 multiplet is described in N = 4, d = 1 superspace.

3. N = 8, d = 1 tensor multiplet in N = 4 superspace

While being natural and very useful for deriving irreducibility constraints for the superfields and establishing their transformation properties, the N = 8, d = 1 superfield approach is not too suitable for constructing invariant actions. The basic difficulty is, of course, the large dimension of the integration measure, which makes it impossible to write the action in terms of this basic superfields without introducing the prepotential and/or passing to some invariant superspaces of lower Grassmann dimension, e.g., chiral or harmonic analytic superspaces. Another possibility is to formulate the N = 8tensor multiplet in N = 4, d = 1 superspace. In such a formulation half of N = 8 supersymmetries is hidden and only N = 4 supersymmetry is manifest. Nevertheless, it allows a rather straightforward construction of N = 8 supersymmetric and superconformal actions. Just this approach was used in [11]. In the next sections we shall reproduce the action of [11] and construct new N = 4 superfield actions with second hidden N = 4 supersymmetry, both for vector and tensor N = 8 multiplets. In the present section we consider two N = 4 superfield formulations of the N = 8 tensor multiplet.

3.1. (3, 8, 5) = (3, 4, 1) ⊕ (0, 4, 4)

In order to describe the N = 8 tensor multiplet in terms of N = 4 superﬁelds we should choose the appropriate N = 4 superspace. The ﬁrst (evident) possibility is to consider the N = 4 superspace with coordinates

(t, θia). (3.1) In this superspace the N = 4 conformal supergroup | ∼ ij ab ia ia OSp(4 2) P,K,D,T ,T1 ,Q ,S is naturally realized, while the rest of the osp(4|4) generators mixes two irreducible N = 4 superfields comprising the (3, 8, 5)N= 8 supermultiplet in question. Expanding ij ij the N = 8 superfields V in ϑiα, one finds that the constraints (2.24) leave in V the following four bosonic and four fermionic N = 4 projections: ij = ij i ≡∇ ij ≡∇α∇ ij v V ,ξα jαV ,Ai jαV , (3.2) where | means restriction to ϑiα = 0. Each N = 4 superfield is subjected, in virtue of (2.24), to an additional constraint (i jk) = (i j) = Da v 0,Da ξα 0, = − a ij = A 6m Di Daj v ,mconst, (3.3) where we used (2.25). S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350 331

Thus, we conclude that our N = 8 tensor multiplet V ij , when rewritten in terms of N = 4 superfields, amounts to a direct sum of the N = 4 ‘tensor’ multiplet vij with the = i (3, 4, 1) off-shell content and a fermionic analog of the N 4 hypermultiplet ξα with the (0, 4, 4) off-shell content,5 plus a constant m of dimension of mass (i.e., (cm−1)). The conservation-law type condition (2.25) yielding this constant is in fact a d = 1analogof m the ‘notoph’ condition ∂mA (x) = 0oftheN = 2, d = 4 tensor multiplet; the appearance of a similar constant in the case of the N = 4, d = 1 supermultiplet (1, 4, 3),whichis defined by the d = 1 reduction of the N = 1, d = 4 tensor multiplet constraints, was earlier observed in [5]. The transformations of the implicit N = 4 Poincaré supersymmetry completing the manifest one to the full N = 8 Poincaré supersymmetry have the following form in terms of N = 4 superfields:

∗ ij (i j) (i ¯j) ∗ i ij 1 i ¯ jk i δ v = η ξ −¯η ξ ,δξ =−2iη¯j v˙ − η¯ Dj Dkv + 6mη¯ , (3.4) 3 wherewepassedtostandardN = 4, d = 1 derivatives (2.4)

Di ≡ Di1, D¯ i ≡ Di2 (3.5) and the complex notation

ηi ≡ η1i, η¯i ≡ η2i,ξi ≡ ξ 1i , ξ¯ i ≡ ξ 2i. (3.6) Since any other osp(4|4) transformation appears in the anticommutator of N = 4 conformal supersymmetry (realized as in (2.27)–(2.29) with ε = ϑ = 0) and implicit N = 4 Poincaré supersymmetry transformations, it sufﬁces to require invariance under these two basic supersymmetries when constructing invariant actions for the considered system in the N = 4, d = 1 superspace. Note that we could equally choose the N = 4, d = 1 superspace ij (t, ϑiα) to deal with, and expand V with respect to θia.TheﬁnalN = 4, d = 1 splitting of the multiplet (3, 8, 5) will be the same modulo the interchange α ↔ a. These two ‘mirror’ choices manifestly preserve covariance under the R-symmetry group SU(2)R realized on the indices i, k.

3.2. (3, 8, 5) = (1, 4, 3) ⊕ (2, 4, 2)

Surprisingly, there is a more sophisticated choice of a N = 4 subspace in the N = 8, d = 1 superspace which gives rise to a different N = 4 splitting of the considered N = 8 supermultiplet, namely, that into the multiplets (1, 4, 3) and (2, 4, 2). It corresponds to dividing the N = 8, d = 1 Grassmann coordinates with respect to the SU(2)R doublet indices. As a result, the SU(2)R symmetry becomes implicit, as opposed to the previous options. The essential difference between two possible N = 4 splittings is closely related to the fact that they are covariant under different N = 4, d = 1 superconformal groups.

5 This supermultiplet has been introduced in [33]; its off-shell superﬁeld action was given in [23] in the framework of N = 4, d = 1 harmonic superspace. 332 S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350

The most general N = 4, d = 1 superconformal group is known to be D(2, 1; α) and its (anti)commutation relations involve an arbitrary parameter α.TheN = 4, d = 1 superconformal group OSp(4|2) ⊂ OSp(4|4) which is explicit in the N = 4, d = 1 superfield formulation discussed in the previous subsection is just D(2, 1; α = 1) ∼ OSp(4|2). One can wonder whether OSp(4|4) contains as a subgroup also some other special case of D(2, 1; α) corresponding to a different choice of α. Our approach immediately implies that OSp(4|4) contains another N = 4, d = 1 superconformal group SU(1, 1|2) ∼ D(2, 1; α =−1). Although this fact can be of course established by an inspection of the root diagrams for OSp(4|4), in our approach it is visualized in the possibility of the just mentioned alternative splitting of the multiplet (3, 8, 5) into the sum of the off-shell N = 4, d = 1 multiplets (1, 4, 3) and (2, 4, 2). The latter is a chiral N = 4 multiplet, while the only N = 4, d = 1 superconformal group which respects chirality is just SU(1, 1|2) [22,23]. Using the (anti)commutation relations (2.13)–(2.15) one can check that the following fermionic generators: a = 1a2 + 1a2 ¯ a = 1a2 − 1a2 Q Q1 iQ2 , Q Q1 iQ2 , a = 1a1 + 1a1 ¯a = 1a1 − 1a1 S Q1 iQ2 , S Q1 iQ2 , (3.7) together with the bosonic ones T ab ≡ ab + ab ≡ 12 − 21 − 12 P, D, K, T1 T2 ,ZU U 2iT , (3.8) indeed form a su(1, 1|2) superalgebra with central charge Z. Let us note that the only su(2) subalgebra which remains manifest in this basis is the diagonal su(2) ∼ T ab in ∼ ab αβ o(4) (T1 ,T2 ). The splitting of the N = 8 superfield V ij in terms of the N = 4 ones in the newly introduced basis can be performed as follows. Firstly, we define the new covariant derivatives with the indices of the fundamental representations of the diagonal su(2) ⊂ Spin(4) as a ≡ √1 1a + ∇1a ¯ ≡ √1 2 − ∇2 D D i , Da Da i a , 2 2 i i ∇a ≡ √ 2a + ∇2a ∇¯ ≡ √ 1 − ∇1 D i , a Da i a , (3.9) 2 2 with the only nonzero anticommutators being a ¯ =− a ∇a ∇¯ =− a D , Db 2iδb ∂t , , b 2iδb ∂t . (3.10) Now, as an alternative N = 4 superspace we choose the set of coordinates closed under a ¯ the action of D , Da ,i.e. 1a 1a t, θ1a − iϑ1a,θ + iϑ , (3.11) while the N = 8 superfields are expanded with respect to the orthogonal combinations a − a a + a a ¯ θ2 iϑ2 , θ1 iϑ1 annihilated by D , Da . S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350 333

The basic constraints (2.24), being rewritten through (3.9), read

Daϕ = 0,Dav −∇aϕ = 0, ∇av + Daϕ¯ = 0, ∇aϕ¯ = 0, ¯ ¯ ¯ ¯ ¯ ¯ ∇aϕ = 0, ∇av + Daϕ = 0, Dav − ∇aϕ¯ = 0, Daϕ¯ = 0, (3.12) where

v ≡−2iV 12,ϕ≡ V 11, ϕ¯ ≡ V 22. (3.13) a ¯ Due to the constraints (3.12), the derivatives ∇ and ∇a of every N = 8 superfield in the 12 11 22 a ¯ triplet (V ,V ,V ) can be expressed as D , Da of some other superfield. Therefore, a − a = a + a = = only the zeroth order (i.e., taken at θ2 iϑ2 θ1 iϑ1 0) components of each N 8 superfield are independent N = 4 superfields. Moreover, these N = 4 superfields prove to be subjected to the additional constraints (which also follow from (3.12)) a ¯ ¯ a a ¯ D Dav = DaD v = 0,Dϕ = 0, Daϕ¯ = 0. (3.14) The N = 4 superfields ϕ,ϕ¯ comprise the standard N = 4, d = 1 chiral multiplet (2, 4, 2), while the N = 4 superfield v subject to (3.14) and having the (1, 4, 3) off- shell content is recognized as the one employed in [5,6] for constructing the N = 4 superconformal mechanics and N = 4 supersymmetric quantum mechanics with partially broken supersymmetry. An immediate question is as to how the previously introduced constant m reappears in this new setting. The answer can be found in [5]. First of all, from the constraints (3.14) it follows that ∂ Da, D¯ v = 0 ⇒ Da, D¯ v = const. (3.15) ∂t a a Secondly, rewriting the constraint (2.25) in terms of the new variables we obtain ∇α∇ ij + a ij =− a ¯ = ⇒ a ¯ =− i jαV Di DjaV 3 D , Da v 6m D , Da v 2m. (3.16) Thus, the constant m now is recovered as a component of the N = 4 superfield v.Aswe already know from [5], the presence of this constant in v is crucial for generating the scalar potential term in the action. The realization of the implicit N = 4 supersymmetry on the N = 4 superfields v,ϕ,ϕ¯ is as follows:

∗ a a ¯ ∗ a ∗ a ¯ δ v = ηaD ϕ¯ +¯η Daϕ, δ ϕ =−ηaD v, δ ϕ¯ =−¯η Dav. (3.17) It is worth noting that in the considered N = 4 formulation, as distinct from the previous one, the constant m does not appear explicitly in the transformations of the implicit N = 4 supersymmetry (cf. (3.4)). It is hidden inside the superﬁeld v and shows up only at the level of the component action (see Section 4.1). To summarize, we observed a new interesting phenomenon: four different off-shell N = 4 supermultiplets with the component contents (3, 4, 1), (0, 4, 4), (1, 4, 3) and (2, 4, 4) can be paired in two different ways to yield the same N = 8 tensor supermultiplet (3, 8, 5). In a forthcoming paper [40] we show that in fact all N = 8, d = 1 supermultiplets 334 S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350 admitting OSp(4|4) as a superconformal group have two different N = 4 ‘faces’, where either OSp(4|2) or SU(1, 1|2)N= 4 superconformal symmetries are manifest. In Section 5 we demonstrate that the N = 8 supermultiplet with the off-shell content (5, 8, 3) employed in [11], besides the known splitting (3, 4, 1) ⊕ (2, 4, 2) in terms of N = 4 ‘tensor’ and chiral supermultiplets, also admits a representation in terms of the N = 4 superﬁeld v obeying (3.14) and the N = 4, d = 1 ‘hypermultiplet’. This yields an alternative (1, 4, 3) ⊕ (4, 4, 0) splitting of this N = 8 multiplet. Now we have all the necessary ingredients to construct invariant actions for the N = 8, d = 1 tensor multiplet in N = 4 superspace.

4. N = 4 superﬁeld actions for V ij

4.1. Actions for the (1, 4, 3) ⊕ (2, 4, 2) splitting

In order to construct an invariant action for the N = 8 tensor multiplet in N = 4 superspace, one can proceed from any of its two alternative N = 4 superfield representations described above. We shall consider first the actions in terms of the N = 4 scalar superfield v subjected to the constraints (3.14) and the chiral superfields ϕ,ϕ¯, i.e., for the splitting described in Section 3.2. The equivalent action in terms of the N = 4 tensor multiplet vij and the fermionic variant of the hypermultiplet ξ i , Eq. (3.3), is naturally constructed in the N = 4, d = 1 harmonic superspace of Ref. [23]. It will be considered in the next subsection. The N = 8 supersymmetric free action for the supermultiplet v,ϕ,ϕ¯ reads

1 Sfree =− dt d4θ v2 − 2ϕϕ¯ . (4.1) 4 It is easy to check that it is indeed invariant with respect to the hidden N = 4 Poincaré supersymmetry (3.17) which completes the manifest N = 4 one to N = 8. The free action (4.1) is not invariant with respect to (super)conformal transformations. In order to construct a N = 8 superconformal invariant action for N = 4 superﬁelds v,ϕ,ϕ¯, we follow a strategy which goes back to [39] and was applied in [22] for constructing a N = 4 superconformal action for the N = 4 tensor multiplet in N = 2, d = 1 superspace. The ‘step-by-step’ construction of [22] adapted to the given case works as follows. Let us start from the N = 4 superconformal invariant action for the superﬁeld v [5]6

1 S =− dt d4θL ,L= v log v. (4.2) 0 4 0 0

The variation of L0 with respect to (3.17), up to a total derivative, reads δ(ϕϕ)¯ δL =− . (4.3) 0 v

6 In (4.2) and the subsequent N = 4 actions we always assume that the N = 4 superﬁelds containing the dilaton (i.e., v in the present case) start with a constant, hence no actual singularities appear. S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350 335

Thus, in order to ensure N = 8 supersymmetry, we have to add to L0 in (4.2) a term which cancels (4.3), namely ϕϕ¯ L = . (4.4) 1 v Continuing this recursive procedure further, we ﬁnd the full Lagrangian yielding the N = 8 supersymmetric action ∞ (2(n − 1))! ϕϕ¯ n L = v log v + (−1)n ! ! 2 = n n v n1 = v log v + v2 + 4ϕϕ¯ + 1 − log 2 − v2 + 4ϕϕ.¯ (4.5)

Discarding terms which do not contribute to the action as a consequence of the basic constraints (3.14), we eventually obtain 1 S =− dt d4θ v log v + v2 + 4ϕϕ¯ − v2 + 4ϕϕ¯ . (4.6) 4 The last step is to check that the action (4.6) is invariant with respect to N = 4 superconformal transformations and so is N = 8 superconformal (since the closure of N = 4 superconformal and N = 8 Poincaré supersymmetries is N = 8 superconformal symmetry). The subgroup of (2.27)–(2.29) with respect to which the N = 4 superspace (3.11) is closed (i.e., SU(1, 1|2)) is singled out by the following identiﬁcation of the transformation parameters

a ¯a 1a =−iε1a ≡ √ ,2a = iε2a =−√ . (4.7) 2 2 The corresponding subset of N = 8 superconformal transformations is formed just by those a − a a + a transformations which leave intact the combinations θ2 iϑ2 and θ1 iϑ1 anticommuting a ¯ with both D and Da.TheN = 4 superconformal transformations of the superfields v,ϕ,ϕ¯ have the following form: ¯a a ¯a a δv =−2i aθ +¯ θa v, δϕ =−4iaθ ϕ, δϕ¯ =−4i¯ θaϕ,¯ (4.8) while the full integration measure transforms as 4 ¯a a 4 δ dt d θ = 2i aθ +¯ θa dt d θ . (4.9) The combination ϕϕ/v¯ 2 is invariant under (4.8). Thus, only the first multiplier in (4.5), i.e., the superfield v, is transformed. Therefore, the transformation of L (4.5) is ¯a a δL =−2i aθ +¯ θa L (4.10) and cancels out the transformation of the measure (4.9), ensuring the N = 4 superconformal invariance of the action (4.6). Hence the latter, being also invariant under N = 8 Poincaré supersymmetry, is invariant under the entire N = 8 superconformal group OSp(4|4). 336 S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350

The full component action is rather lengthy, but its pure bosonic core is remarkably simple. Integrating in (4.6) over Grassmann variables, discarding all fermions and (a ¯ b) ¯ ¯ a eliminating the auxiliary fields D D v| and (DaD )ϕ by their equations of motion, we end up with the bosonic action:7 ˙¯ 1 2 ˙ 2 4imϕϕ SB = dt v˙ + 4ϕ˙ϕ¯ − m − 2imv˙ − , (4.11) v2 + 4ϕϕ¯ v + v2 + ϕϕ¯ where the fields v,ϕ,ϕ¯ are the first, θ-independent terms in the N = 4 superfields v,ϕ,ϕ¯, respectively. The meaning of the action (4.11) can be clarified by passing to the new field variables q,λ,λ¯ defined as

1 − λλ¯ eq λ eqλ¯ v = eq ,ϕ= , ϕ¯ = . (4.12) 1 + λλ¯ 1 + λλ¯ 1 + λλ¯ In terms of these fields the action (4.11) becomes ˙ ¯˙ ¯˙ − ˙ ¯ q 2 2 −q q λλ λλ λλ SB = dt e q˙ − m e + 4e − 2im . (4.13) (1 + λλ)¯ 2 1 + λλ¯ This amounts to the sum of two conformally invariant d = 1 actions, i.e., that of conformal mechanics [41] and that of a charged particle moving in the field of a Dirac monopole. Thus, the bosonic part of the action (4.5) exactly coincides with the bosonic part of the action of N = 4, D(2, 1; α) superconformal mechanics [22] corresponding to the particular choice α = 1, for which D(2, 1; α) becomes OSp(4|2). The mass parameter m,which is present among the components of the original N = 8 superfield V ik, plays the role of the coupling constant, in analogy with [5]. Note that in [22] an analogous coupling constant appeared as a free parameter of the action, while in the considered case it arises as a consequence of the irreducibility constraints in N = 8, d = 1 superspace. We conclude that the action (4.6) describes a N = 8 supersymmetric extension of the conformally invariant action of the coupled system for the dilaton field and a charged particle in a Dirac monopole background. The fact that the bosonic sector of the action of this N = 8 superconformal mechanics coincides with that of the α = 1 case of the N = 4 superconformal mechanics action associated with the multiplet (3, 4, 1) finds its natural explanation in the existence of the alternative splitting (3, 8, 5) = (3, 4, 1) ⊕ (0, 4, 4), whose N = 4 actions are presented in the next subsection. The second multiplet contains no physical bosons and so does not contribute to the bosonic sector. The conformal N = 4 supergroup OSp(4|2) is hidden in the action (4.6), but it becomes a manifest symmetry of the alternative N = 4 action.

4.2. Superconformal action for the (3, 4, 1) ⊕ (0, 4, 4) splitting

The action for the splitting of Section 3.1 consists of two pieces. The ﬁrst one is written in the customary N = 4, d = 1 superspace and the second one in the analytic subspace

7 When passing to the component action we should take into account that the N = 4 superfield v contains the a ¯ constant m among its components, [D , Da]v =−2m. S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350 337 of N = 4, d = 1 harmonic superspace [23]. So, we first need to rewrite the defining = (ik) i relations of the N 4 superfields v and ξα (first two equations in (3.3)) in the harmonic superspace. We use the definitions and conventions of Ref. [23]. The harmonic variables parametrizing the coset SU(2)R/U(1)R are defined by the relations +i − = ⇐⇒ + − − + − = +i = − u ui 1 ui uj uj ui ij , u ui . (4.14) ik i The harmonic projections of v , ξα are defined by ++ = ij + + + = i + ¯ + = ¯ i + v v ui uj ,ξξ ui , ξ ξ ui , (4.15) +− 1 −− ++ −− −− +− v = D v ,v= D v (4.16) 2 and the relevant part of constraints (3.3) is rewritten as + ++ + ++ ++ ++ D v = D¯ v = 0,Dv = 0, + + + + ++ + ++ + D ξ = D¯ ξ = 0,Dξ = D ξ¯ = 0. (4.17) + = i + ¯ + = ¯ i + ±± = ±i ∓i Here D D ui , D D ui , D u ∂/∂u (in the central basis of the harmonic superspace), Di , D¯ i are given in (3.5) and we use the notation (ξ i , ξ¯ i) ≡ ξ αi. The relations (4.17) imply that v++ and ξ +, ξ¯ + are analytic harmonic N = 4, d = 1 superfields living ± ≡ + ¯+ ± on the invariant subspace (ζ, ui ) (tA,θ , θ ,ui ). In this setting, the transformations of the hidden N = 4 supersymmetry (3.4) are rewritten as ∗ + + + − +− + −− + δ ξ = D D¯ 2η¯ v −¯η v + 6mη¯ , ∗ + + + − +− + −− + δ ξ¯ = D D¯ 2η v − η v + 6mη , δ∗v++ = η+ξ + −¯η+ξ¯+, ∗ + −− − +− + + +− − ++ − v δ vij = η v − η v ξ − η v + η v ξ ij + −− − +− + + +− − ++ − − η¯ v −¯η v ξ¯ + η¯ v −¯η v ξ¯ . (4.18) The action is given by 1 S = dt d4θ v2 − √ dudζ−− 2 ξ +ξ¯ + vˆ++ × + √ √ −− ++ 12m , (4.19) (1 + c vˆ )3/2 1 + c−−vˆ++(1 + 1 + c−−vˆ++) −− + ¯+ where dudζ = dudtA dθ dθ is the measure of integration over the analytic superspace and ++ =ˆ++ + ++ ±± = ik ± ± ik = v v c ,c c ui uk ,c const. (4.20) The first term in (4.19) is the superconformal (i.e., OSp(4|2)) invariant kinetic term of vik. The second and third ones are the kinetic term of ξ + and the superconformal invariant potential term of vik [23]. The action (4.19) is manifestly N = 4 supersymmetric since it is written in terms of N = 4 superfields. However, its invariance with respect to the hidden 338 S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350

N = 4 supersymmetry (4.18) must be explicitly checked. It is enough to demonstrate the invariance of (4.19) with respect to the η part of (4.18). The variations of the three terms in (4.19) are

ij ∗ 4 v δ vij δS1 = dt d θ √ , 2 v + ¯ + − +− + −− + + + 1 −− D D [(2η v − η v )ξ ] 6mη ξ δS2 = √ dudζ √ + √ , 2 ( 1 + c−−vˆ++ )3 ( 1 + c−−vˆ++ )3 + + 1 −− 6mη ξ δS3 =−√ dudζ √ . (4.21) 2 ( 1 + c−−vˆ++ )3

It is seen that the m-dependent terms are cancelled among δS2 and δS3. The ﬁrst term in δS2 may be rewritten as an integral over the full N = 4 superspace using

dt dud4θ = dudζ−− D+D¯ +.

Thus, the variation of the action (4.19) reads ij ∗ − +− + −− + v δ vij 1 (2η v − η v )ξ δS = dt d4θ √ + √ du √ . (4.22) v2 2 ( 1 + c−−vˆ++ )3 Now, using (4.18), we identically rewrite the numerator in the second term in (4.22) as − +− + −− + ∗ − +− + + +− − − ++ − 2η v − η v ξ =−vij δ v + η v ξ − η v ξ + η v ξ ij ij ∗ ≡−v δ vij + (B1) + B2 . (4.23) The ﬁrst term in (4.23) does not depend on harmonics, so the corresponding part of the harmonic integral in (4.22) can be easily computed to yield8

ij ∗ 1 ij ∗ 1 v δ vij −√ v δ vij du √ =− √ . (4.24) 2 ( 1 + c−−vˆ++ )3 v2 This term is cancelled by the ﬁrst one in (4.22). Finally, we are left with

1 B + B δS = √ dt d4θdu √ 1 2 . (4.25) 2 ( 1 + c−−vˆ++ )3 Let us prove that the expression −3/2 −− ++ (B1 + B2)(1 + X) ,X≡ c vˆ (4.26) is reduced to a total harmonic derivative, modulo purely analytic terms which disappear under the Berezin integral over the full harmonic superspace. The proof for the terms ∼ B1 and ∼ B2 can be carried out independently, so we start with the first term. +− − +− + + +− − As a first step, let us note that v in B1 = η v ξ − η v ξ can be replaced ˆ+− = +− − +− +− = ik + − by v v c , c c ui uk .ForthefirstterminB1 this is evident, since the

8 This harmonic integral has been computed in [39]. S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350 339 difference is an analytic superfield. For the second term this follows from the identity + −− − + +− − − ++ η c ξ η c ξ (1 + X) 3/2 = D √ √ , (4.27) 1 + X(1 + 1 + X) which can be easily checked. Next, we need to show the existence of a function F˜ ++, such that ˆ −3/2 − +− + + +− − −3/2 −− ˜ ++ B1(1 + X) = η vˆ ξ − η vˆ ξ (1 + X) = D F . (4.28) Choosing for F˜ ++ the ansatz ˜ ++ − ++ + + ++ − F = f1(X) η vˆ ξ + f2(X) η vˆ ξ , (4.29) and comparing both sides of (4.28), we find 1 f1 =−f2 = √ √ . (4.30) 1 + X(1 + 1 + X) Proceeding in a similar way, it is straightforward to show that the term proportional to B2 in (4.26) is also reduced to a total harmonic derivative −3/2 ++ −− B2(1 + X) = D G + analytic terms. (4.31) Indeed, making the ansatz −− + −− − − +− − G = g1(X) η c ξ + g2(X) η c ξ , (4.32) it is easy to find 1 1 g2 = √ ,g1 =− √ . (4.33) 1 + X 1 + 1 + X In obtaining this result, we made use of the relation ++ −− +− c c − c 2 = 1. (4.34) Thus, we proved that the term (4.25) vanishes and so the action (4.19) possesses N = 8 supersymmetry. Finally, we should prove the N = 4 superconformal invariance of the entire action (4.19). The invariance of the first and third terms was shown in [22] and [23], so it remains for us to prove this property for the second term. As shown in [23], the measure of integration over the analytic superspace is invariant under the N = 4 superconformal group (OSp(4|2) in our case), while the involved superfields are transformed as + + + + ++ ++ ++ ++ +− δξ = Λξ ,δξ¯ = Λξ¯ ,δvˆ = 2Λ vˆ + c − 2Λ c , (4.35) where Λ++ = D++Λ, (D++)2Λ = 0. Here the analytic superfunction Λ contains all parameters of superconformal transformations (see [23] for details). The variation of the second term in (4.19) under these transformations is ++ −− ++ +− −− + + Λ Λ(X + c c ) D Λc c ξ ξ¯ 2 − 3 + 3 . (4.36) (1 + X)3/2 (1 + X)5/2 (1 + X)5/2 340 S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350

The last term in (4.36) is vanishing under the harmonic integral, because 2 +− −− − 1 ++ 3 + 3X + X −− c c (1 + X) 5/2 = D c 2 2 (1 + X)3/2[1 + (1 + X)3/2] and one can integrate by parts with respect to D++ and make use of the relations (D++)2Λ = 0andD++ξ + = 0. Then (4.36) is reduced to +− 2 + + 1 + X + 3(c ) −ξ ξ¯ Λ . (4.37) (1 + X)5/2 It can be checked that the expression in the square brackets is equal to ++ 2 −− 2 D f1(X) c , where the function f1(X) already appeared in (4.30). Integrating by parts with respect to (D++)2 we find that the variation (4.37) is vanishing under the harmonic integral, once again due to harmonic constraints on ξ + and Λ. We finish with two comments. First, we wish to point out that the actions (4.6) and (4.19), though describing the same system, are defined on N = 4 superspaces with drastically different superconformal properties and so cannot be related by any equivalence transformation. Indeed, the manifest superconformal group of (4.6) realized on the coordinates of the corresponding N = 4 superspace is SU(1, 1|2), and it is the only N = 4, d = 1 superconformal group compatible with chirality. On the contrary, the manifest superconformal group of (4.19) acting in the relevant N = 4 superspaces is OSp(4|2). Both actions have hidden N = 4 superconformal symmetries which close on OSp(4|4) together with the manifest ones; they are given by OSp(4|2) in the first case and SU(1, 1|2) in the second one. These hidden superconformal symmetries are not realized on the N = 4 superspace coordinates, but rather they transform the involved N = 4 superfields through each other. Thus, the actions (4.6) and (4.19) bring to light different symmetry aspects of the same N = 8 superconformal system. Of course, these two different N = 4 actions yield the same component action; also, they both can presumably be obtained from the single superconformal action of V ij in N = 8, d = 1 superspace, e.g., in N = 8, d = 1 harmonic superspace which is a straightforward reduction of the N = 2, d = 4 one [30,31]. We do not address the latter issue in this paper. The second comment concerns an interesting dual role of the parameter m. In the action (4.6) it is hidden inside the superfield v, does not appear in the transformations of the implicit N = 4 Poincaré supersymmetry (3.17) and is revealed only after passing to the component action, where it produces a scalar potential and magnetic monopole terms (it yet reappears, due to (3.16), in the anticommutators of the implicit Poincaré supersymmetry and the manifest SU(1, 1|2) superconformal symmetry). On the contrary, in the alternative splitting considered here this parameter appears already in the transformations of the second N = 4 Poincaré supersymmetry (3.4), (4.18) and is explicitly present in the superfield action (4.19) as the coefficient before the superconformal superfield potential term. As follows from (3.4) and (4.18), at m = 0 the implicit half of N = 8, d = 1 Poincaré supersymmetry is spontaneously broken, ξ i , ξ¯ i being the relevant Goldstone fermions with an inhomogeneous transformation law. Since, by construction, all fermions S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350 341 in the theory are the Goldstone ones for the conformal supersymmetry and so transform inhomogeneously under the latter, we conclude that there is a linear combination of the implicit N = 4 Poincaré supersymmetry generators and half of those of the N = 8 conformal supersymmetry which remains unbroken at m = 0. At m = 0 unbroken is the entire N = 8, d = 1 Poincaré supersymmetry. So the structure of the vacuum symmetries of the model essentially depends on the parameter m.

5. N = 8, d = 1 vector multiplet

In this section we shall analyse, along the same line, the off-shell N = 8 multiplet (5, 8, 3) considered in [11,26,29]. As new results, we shall study its superconformal properties (which has not been done before), show the existence of a second N = 4 splitting for it and construct the corresponding N = 8 superconformal invariant off-shell actions.

5.1. Superconformal properties in N = 8 superspace

In analogy with the N = 8 tensor multiplet, the N = 8 multiplet (5, 8, 3) employed in [11] can also be obtained from the proper nonlinear realization of the supergroup OSp(4|4). The corresponding supercoset contains the bosonic coset USp(4)/Spin(4) ∼ SO(5)/SO(4), four out of five physical bosonic fields of the multiplet in question being the corresponding coset parameters. The remaining field, as in the previous case, is the dilaton associated with the generator D. As opposed to the previous case in which the supercoset contained the coset SU(2)R/U(1)R of the R-symmetry group SU(2)R (with generators ik T ), in the present case this SU(2)R as a whole is placed into the stability subgroup. Thus we are going to construct a nonlinear realization of the superconformal group OSp(4|4) | in the coset superspace OSp(4 4) parametrized as SU(2)R⊗SO(4) ia+ Qiα ia+ Siα αa g = eitP eθiaQ ϑiα eψiaS ξiα eizKeiuDeivαaU . (5.1) In order to find superconformal covariant irreducibility conditions on the coset superfields, we must impose, once again, the inverse Higgs constraints [37] on the left- covariant osp(4|4)-valued Cartan one-form Ω. In full analogy with (2.20), we impose the following constraints = αa = ωD 0,ωU 0. (5.2) These constraints are manifestly covariant under the left action of the whole supergroup. They allow one to trade the Goldstone spinor superfields and the superfield z for the spinor and t-derivatives of the remaining bosonic Goldstone superfields u, vαa. Simultaneously, they imply the irreducibility constraints for the latter ibV + b∇i U = ∇iβ V + β i U = D αa δa α 0, αa δα Da 0, (5.3) where 2 2 v − 2V − 2 − V tan V = e u αa , U = e u ,V= 2 v . (5.4) αa 2 2 αa αa 2 + V 2 + V v2 2 342 S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350

Note that the superfields U, Vαa provide an example of how to construct a linear representation of SO(5) symmetry in terms of its nonlinear realization. Indeed, with respect to SO(4) ∼ SU(2) × SU(2) the superfield Vαa is a 4-vector, in accordance with its spinor indices, and U is a scalar. With respect to the SO(5)/SO(4) transformations generated by the left action of the group element αa iaαaU g1 = e , (5.5) these superfields transform as αa δVαa = aαaU,δU =−2aαaV , (5.6) thus constituting a vector representation of SO(5). Roughly speaking, the SO(5)/SO(4) −u superfield Vαa represents the angular part of this SO(5) vector, while e is its radial part. Indeed, − U2 + 2V2 = e 2u. (5.7) Under (5.5) the coordinates also transform, as = α = a δθia aαaϑi ,δϑiα aαaθi . (5.8) The transformation properties of the superfields u, V αa under the conformal supersymmetry generated by the left shifts with

ia iα iaS +εiαS g2 = e (5.9) can be easily found δu =−2i iaθ + εiαϑ , ia iα = β b + b β + b + β δVαa 2i δα δa Vα Va Aβb 2iAabVα 2iAαβ Va . (5.10) Here = i + i = i + i = i + i Aαa θiaεα ϑiαa,Aab θiab θiba,Aαβ ϑiαεβ ϑiβ εα. (5.11)

5.2. Two N = 4 superﬁeld formulations

As in the case of the N = 8 tensor multiplet, we may consider two different splittings of the N = 8 vector multiplet into irreducible N = 4 superfields. They correspond to two different choices of N = 4, d = 1 superspace as a subspace of N = 8, d = 1 superspace. The first N = 4 superspace is parametrized by the coordinates {t,θia}. It follows from the constraints (5.3) that the spinor derivatives of all involved superfields with respect to ϑiα are expressed in terms of spinor derivatives with respect to θia. This means that the only essential N = 4 superfield components of Vαa and U in their ϑ-expansion are the first ones ˆ ˆ Vαa ≡ Vαa|ϑ=0, U ≡ U|ϑ=0. (5.12) They accommodate the whole off-shell component content of the N = 8 vector multiplet. These five bosonic N = 4 superfields are subjected, in virtue of (5.3), to the irreducibility S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350 343 constraints in N = 4 superspace i(aVˆ b)α = i(a b)Uˆ = D 0,DDi 0. (5.13) Thus, from the N = 4 superspace perspective, the vector N = 8 supermultiplet amounts ˆ to the sum of the N = 4, d = 1 hypermultiplet Vαa with the (4, 4, 0) off-shell component content and the N = 4 ‘old’ tensor multiplet Uˆ with the (1, 4, 3) content. The transformations of the implicit N = 4 Poincaré supersymmetry, completing the manifest one to the full N = 8 supersymmetry, have the following form in terms of N = 4 superfields:

∗ ˆ i ˆ ∗ ˆ 1 ia ˆ α δ Vaα = ηiαD U,δU = ηiαD V . (5.14) a 2 a Another interesting N = 4 superfield splitting of the N = 8 vector multiplet can be achieved by passing to the complex parametrization of the N = 8 superspace as ¯ ia ia ia t, Θia = θia + iϑia, Θ = θ − iϑ , (5.15) with a manifest diagonal su(2) (this is just the parametrization corresponding to the direct reduction from N = 2, d = 4 superspace, with the spinor derivatives satisfying the algebra (2.5)). In this superspace we define the covariant derivatives Diα, D¯ jβ as 1 1 Diα ≡ √ Diα − i∇iα , D¯ iα ≡ √ Diα + i∇iα , 2 2 iα ¯ jβ ij αβ D , D = 2i ∂t (5.16) (the indices β and a are indistinguishable with respect to the diagonal SU(2)) and pass to the new notation αa αβ (αβ) 1 αβ βα V ≡−αaV , W ≡ V = V + V , 2 W ≡ V + iU, W¯ ≡ V − iU. (5.17) In this basis of N = 8 superspace, the original constraints (5.3) amount to9 1 DiαWβγ =− βαD¯ iγ W¯ + γαD¯ iβ W¯ , 4 1 D¯ iαWβγ =− βαDiγ W + γαDiβ W , 4 DiαW¯ = D¯ iαW = DkαDi W = D¯ k D¯ iα W¯ 0, 0, α α . (5.18) ¯a Next, we single out the N = 4, d = 1 superspace in (5.15) as (t, θa ≡ Θ1a, θ ) and split our N = 8 superfields into N = 4 ones in the standard way. As in the previous cases, ¯ 2a the spinor derivatives of each N = 8 superfield with respect to Θ and Θ2a, due to the constraints (5.18), are expressed as derivatives of other superfields with respect to θ¯a

9 This form of constraints is a direct d = 1 reduction of the N = 2 superﬁeld constraints of N = 2, d = 4 supersymmetric Abelian gauge theory. They can be solved through the d = 1 analytic harmonic gauge potential [29] like their d = 4 counterparts [30,31]. 344 S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350

¯ 2a and θa. Therefore, only the first (i.e., taken at Θ = 0andΘ2a = 0) N = 4 superfield components of N = 8 superfields are independent. They accommodate the entire off-shell field content of the multiplet. These N = 4 superfields are defined as Φ ≡ W|, Φ¯ ≡ W¯ |,Wαβ ≡ Wαβ (5.19) and satisfy the constraints following from (5.18) α ¯ ¯ (α βγ) ¯ (α βγ) D Φ = 0, DαΦ = 0, D W = D W = 0, α 1α ¯ ¯ D ≡ D , Dα ≡ D1α. (5.20) They tell us that the N = 4 superfields Φ and Φ¯ form the standard N = 4 chiral multiplet (2, 4, 2), while the N = 4 superfield W αβ is recognized as the N = 4 tensor multiplet (3, 4, 1). Just this N = 4 superfield set has been proposed in [11] to represent the vector N = 8, d = 1 supermultiplet in question. As we see, it provides another ‘face’ of the same N = 8 multiplet defined by the N = 8 superspace constraints (5.3). The first ‘face’ (unknown before) is the set (1, 4, 3) ⊕ (4, 4, 0) considered earlier. The implicit N = 4 supersymmetry is realized on W αβ , Φ and Φ¯ as ∗ αβ 1 (α ¯ β) ¯ (α β) ∗ 4 ¯ β α δ W = η D Φ −¯η D Φ ,δΦ = ηαD W , 2 3 β ∗ ¯ 4 α β δ Φ =− η¯ Dβ W . (5.21) 3 α 5.3. N = 8 supersymmetric actions in N = 4 superspace

As we showed, the N = 8 vector multiplet has two different descriptions in terms of N = 4 superfields, the ‘new’ and ‘old’ ones. The first (new) description involves the N = 4 ˆ ˆ ‘old tensor’ multiplet U and the ‘hypermultiplet’ Vaα obeying the constraints (5.13). For constructing the N = 4 superfield superconformal action for this case, we start with the following ansatz:

1 Vˆ 2 n 1 L = L0 + An = L0 + an ,L0 = , (5.22) U Uˆ 2 Uˆ n=1 n=1 where L0 is the OSp(4 |2) superconformal invariant action for U [36]. The complete action (5.22), by construction, possesses N = 4 superconformal OSp(4|2) invariance. Indeed, with respect to the particular subset of transformations (5.10) with

1a = a,2a =−¯a,εiα = 0, (5.23) ˆ ˆ which do not mix the superﬁelds U and Vaα and preserve the N = 4 superspace (t, θia) ≡ ¯ (t, θa, −θa), the measure transforms as 4 ¯a a 4 δ dt d θ = 2i aθ +¯ θa dt d θ , (5.24) while δ V 2 = 0 (5.25) S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350 345 and, therefore, ˆ 2 ¯a a ˆ 2 ˆ ¯a a ˆ δ V = 4i aθ +¯ θa (V) ,δU = 2i aθ +¯ θa U (5.26)

(recall the relations (5.4)). Since (Vˆ 2/Uˆ 2)n is invariant with respect to (5.26), it follows from the representation (5.22) that L transforms as ¯a a δL =−2i aθ +¯ θa L, (5.27) which cancels the variation of the measure (5.24). Now we should choose the coefﬁcients an so as to make the action (5.22) invariant with respect to the implicit N = 4 supersymmetry (5.14). The variation of the ﬁrst term under (5.14) can be represented as

δ(Vˆ 2) 1 δL0 = ⇒ a1 =− . (5.28) 2Uˆ 3 2

Going further, one ﬁnds the following recurrence relation between the coefﬁcients an which ensures the invariance of the action (5.22): (2n + 1) (−1)n(2n − 1)! an+1 =− an ⇒ an = . (5.29) (n + 2)(n + 1) 2n−1(n + 1)!(n − 1)!

Let us note that the same relations among the coefﬁcients an (5.29) appear if we require the invariance of the action (5.22) under the SO(5)/SO(4) transformations (5.6), (5.8) which, in terms of N = 4 superﬁelds, look as follows

∗ ˆ ˆ b i ˆ ∗ ˆ ˆ αa 1 a ib ˆ α δ Vαa = aαaU + aαbθ D U,δU =−2aαaV + aαaθ D V . (5.30) i a 2 i b

Indeed, the variation of L0 after integrating by parts can be cast in the form

1 Y ˆ αa a i ˆ ˆ αb δL0 = X + , where X ≡ aαaV ,Y≡ aαaθ D UV , (5.31) Uˆ 2 Uˆ i b while the variation of generic term An in (5.22) is as follows: (2n + 1)(n + 1) Vˆ 2 Vˆ 2(n−1) Y δAn = 2an n + X + . (5.32) n + 2 Uˆ 2 Uˆ 2n Uˆ

It is easy to see that, with an as in (5.29), all terms in the SO(5) variation of (5.22) are cancelled among themselves. Summing up all terms, we get the N = 8 supersymmetric Lagrangian in the closed form 2 L = . (5.33) Uˆ + Uˆ 2 + 2Vˆ 2 Thus, the action

1 S = 2 dt d4θ (5.34) Uˆ + Uˆ 2 + 2Vˆ 2 346 S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350 possesses both N = 4 superconformal symmetry and N = 8 supersymmetry. The closure of these two supersymmetries yields the whole N = 8 conformal supergroup OSp(4|4), which thus represents the full invariance group of (5.34). The second possibility includes the N = 4tensorW αβ and the chiral Φ, Φ¯ multiplets (5.20). The free action invariant under the implicit N = 4 supersymmetry (5.21) reads

3 S = dt d4θ W 2 − ΦΦ¯ . (5.35) free 4 The N = 4 superconformal invariant action which is invariant also under (5.21) has the very simple form √ 

log( W 2 + W 2 + 1 ΦΦ)¯ 4 2 Skin = 2 dt d θ √ . (5.36) W 2 The proof of N = 4 superconformal symmetry in this case (it is now SU(1, 1|2)) is similar to that performed in Section 4.1. Indeed, under the restriction (4.7) the transformation of the measure is the same as in (4.9) and 2 ¯a a 2 ¯ ¯a ¯ a δ W = 4i aθ +¯ θa W ,δΦ = 4iaθ Φ, δΦ = 4i¯ θaΦ. (5.37) Once again, one may be easily convinced that the variation of the Lagrangian in (5.36) cancels the variation of the measure. This proves the N = 4 (and hence N = 8) superconformal invariance of the action (5.36). The bosonic part of the action (5.36) has the very simple form

˙ ˙ αβ + 1 ˙ ˙¯ Wαβ W 2 ΦΦ SB = dt . (5.38) 2 + 1 ¯ 3/2 (W 2 ΦΦ) It is invariant under the SO(5) transformations (5.5) which are realized on the bosonic ﬁelds as αβ ¯ ¯ αβ δΦ =−iaΦ − 2iaαβ W ,δΦ = iaΦ + 2iaαβ W , i ¯ δWαβ =− aαβ (Φ − Φ), (5.39) 2 where αa αβ (αβ) 1 αβ βα a ≡−αaa ,a≡ a = a + a . (5.40) 2 This action supplies a particular case of the SO(5) invariant bosonic target metric found in [11]. The most general SO(5) invariant metric of [11] can be reproduced from the superﬁeld action constructed as the sum of (5.36) and the nonconformal (but SO(5) invariant) action (5.35). Let us recall that the N = 4, d = 1 tensor multiplet admits a N = 4 superconformal invariant potential term [22,23], but the latter can be written only in terms of the prepotential for W αβ , or as an integral over the analytic harmonic superspace [23]. This term can be promoted to a N = 8 superconformally invariant one. However, this issue is out of the scope of the present paper. We plan to consider it elsewhere. S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350 347

6. Conclusions

In this paper we constructed new models of N = 8 superconformal mechanics associated with the off-shell multiplets (3, 8, 5) and (5, 8, 3) of N = 8, d = 1 Poincaré supersymmetry. We showed that both multiplets can be described in a N = 8 superfield form as properly constrained Goldstone superfields associated with suitable cosets of the nonlinearly realized N = 8, d = 1 superconformal group OSp(4|4).TheN = 8 superfield irreducibility conditions were derived as a subset of superconformal covariant constraints on the Cartan super one-forms. The superconformal transformation properties of these N = 8, d = 1 Goldstone superfields were explicitly given, alongside with the transformation of the coordinates of N = 8, d = 1 superspace. Although these superfield irreducibility constraints can also be reproduced, via dimensional reduction, from the analogous constraints defining off-shell d = 4 tensor and vector multiplets in d = 4 superspace, our method, being self-contained in d = 1, is not tied to such a procedure. Also, the field contents of the considered d = 1 multiplets differ essentially from those of their d = 4 ancestors. Apart from the N = 8 superfield description, we presented several N = 4 superfield formulations of these d = 1 multiplets. As an interesting phenomenon, we revealed the existence of two different N = 4 ‘faces’ of each single N = 8 multiplet. These alternative N = 4 formulations deal with different pairs of off-shell N = 4 multiplets (see (1.1) and (1.2)) and generically manifest different N = 4 superconformal subgroups of OSp(4|4). More concretely for both (1.1) and (1.2), the first splitting features the superconformal N = 4 subgroup SU(1, 1|2) (with some central charges added) while the second one involves OSp(4|2). We constructed N = 8 superconformal invariant off-shell N = 4 superfield actions for all four splittings considered. To our knowledge, these actions have not yet appeared in the literature. As a subsymmetry of OSp(4|4) they all respect a hidden USp(4) ∼ SO(5) symmetry. Hence, for the multiplet (5, 8, 3) our superconformal actions provide a special case of the SO(5) invariant N = 4 superfield action of [11] (which presented the target metric but not the action). An obvious project for future study is to investigate the implications of our N = 8 superconformal mechanics models for the AdS2/CFT1 correspondence. Indeed, the N = 4 superconformal mechanics associated with the multiplet (3, 4, 1) [23] is believed to be equivalent, both classically and quantum-mechanically [42,43], to a superparticle on the SU(1,1|2) × 2 = supercoset SO(1,1)⊗U(1) which is a superextension of AdS2 S . One of the N 8 models considered here, namely the one associated with the multiplet (3, 8, 5),isa further superextension of the N = 4 superconformal mechanics just mentioned, with the 2 bosonic sector unchanged. It is therefore natural to expect a relation to some AdS2 × S superparticles defined on appropriate cosets of the supergroup OSp(4|4) since it contains both SU(1, 1|2) and OSp(4|2) as supersubgroups. The superconformal (5, 8, 3) SQM model can also have an interesting application in a similar context. As argued in [44], this 4 type of N = 8 superconformal mechanics may be relevant to the near-horizon AdS2 × S geometry of a D5-brane in an orthogonal D3-brane background. The supergroup OSp(4|4) just defines the corresponding superisometry. As proposed in [26], the models associated with the multiplet (5, 8, 3) can describe the dimensionally reduced Coulomb branch of d = 4 Seiberg–Witten theory, and the 348 S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350 corresponding actions may be connected with the low-energy quantum effective action of this theory. It would be interesting to elucidate the role of the superconformal invariant model as well as the existence of its two different but equivalent N = 4 descriptions from this point of view. Our main focus in this paper was on superconformally invariant actions of the N = 8 multiplets considered. One could equally well employ these off-shell multiplets for constructing N = 8 supersymmetric, but not superconformal, d = 1 models. It is desirable to learn what is the most general bosonic target geometry associated with such sigma models. Finally, let us recall that, besides OSp(4|4), there exist other N = 8, d = 1 superconformal groups, namely OSp(8|2), F(4) and SU(1, 1|4) (see, e.g., [34]). It is clearly of interest to set up nonlinear realizations of all these supergroups, deducing the corresponding N = 8 multiplets and constructing the associated N = 8 supersymmetric and superconformal mechanics models. Certain steps towards this goal will be presented in a forthcoming paper [40].

Acknowledgements

S.K. would like to thank B. Zupnik for many useful discussions. This work was partially supported by the European Community’s Human Potential Programme under contract HPRN-CT-2000-00131 Quantum Spacetime, the INTAS-00-00254 grant, the NATO Collaborative Linkage Grant PST.CLG.979389, RFBR-DFG grant No. 02-02- 04002, grant DFG No. 436 RUS 113/669, RFBR grant No. 03-02-17440 and a grant of the Heisenberg–Landau programme. E.I. and S.K. thank the Institute for Theoretical Physics of the University of Hannover and the INFN-Laboratori Nazionali di Frascati for the warm hospitality extended to them during the course of this work.

Appendix A. Superalgebra osp(4|4)

Here we present the explicit structure of the superalgebra osp(4|4) in the basis (2.16). Boson–boson sector:

i[D,P]=P, i[D,K]=−K, i[P,K]=−2D, ij kl = ik jl + jl ik ab cd = ac bd + bd ac i T ,T T T ,iT1 ,T1 T1 T1 , 1 i T αβ ,Tγρ = αγ T βρ + βρT αγ ,iT ab,Ucα = acU bα + bcU aα , 2 2 2 2 1 2 1 i T αβ ,Uaγ = αγ U aβ + βγ U aα ,iU aα,Ubβ = 2 αβ T ab + abT αβ . 2 2 1 2 (A.1) Mixed boson–fermion sector: i P,Qia = 0,iP,Qiα = 0,iP,Sia =−Qia, S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350 349

1 1 i P,Siα =−Qiα,iD,Qia = Qia,iD,Qiα = Qiα, 2 2 1 1 i D,Sia =− Sia,iD,Siα =− Siα,iK,Qia = Sia, 2 2 i K,Qiα = Siα,iK,Sia = 0,iK,Siα = 0, i U aα,Qib = abQiα,iU aα, Qiβ = αβ Qia,iU aα,Sib = abSiα, 1 i U aα, Siβ = αβ Sia,iT ab,Qc = acQb + bcQa . (A.2) 2 Fermion–fermion sector: Qia,Qjb =−2ij abP, Qiα, Qjβ =−2ij αβ P, Sia,Sjb =−2ij abK, Siα, Sjβ =−2ij αβ K, Qia,Sjb = 2 abT ij − ij abD − 2ij T ab , 1 Qia, Sjα =−2ij U aα, Qiα,Sja =−2ij U aα, Qiα Sjβ = αβ ij − ij αβ − ij αβ , 2 T D 2 T2 . (A.3)

References

[1] E. Witten, Nucl. Phys. B 188 (1981) 513. [2] M. De Crombrugghe, V. Rittenberg, Ann. Phys. 151 (1983) 99. [3] V. Akulov, A. Pashnev, Teor. Mat. Fiz. 56 (1983) 344. [4] S. Fubini, E. Rabinovici, Nucl. Phys. B 245 (1984) 17. [5] E. Ivanov, S. Krivonos, V. Leviant, J. Phys. A: Math. Gen. 22 (1989) 4201. [6] E. Ivanov, S. Krivonos, A. Pashnev, Class. Quantum Grav. 8 (1991) 19. [7] E.E. Donets, A. Pashnev, J.J. Rosales, M.M. Tsulaia, Phys. Rev. D 61 (2000) 043512, hep-th/9907224. [8] A.V. Smilga, Nucl. Phys. B 291 (1987) 241. [9] E.A. Ivanov, A.V. Smilga, Phys. Lett. B 257 (1991) 79. [10] V.P. Berezovoj, A.I. Pashnev, Class. Quantum Grav. 8 (1991) 2141. [11] D.-E. Diaconescu, R. Entin, Phys. Rev. D 56 (1997) 8045–8052, hep-th/9706059. [12] G.W. Gibbons, G. Papadopoulos, K.S. Stelle, Nucl. Phys. B 508 (1997) 623, hep-th/9706207. [13] P. Claus, M. Derix, R. Kallosh, J. Kumar, P.K. Townsend, A. Van Proeyen, Phys. Rev. Lett. 81 (1998) 4553, hep-th/9804177. [14] J.A. de Azcárraga, J.M. Izquierdo, J.C. Perez Bueno, P.K. Townsend, Phys. Rev. D 59 (1999) 084015, hep- th/9810230. [15] G.W. Gibbons, P.K. Townsend, Phys. Lett. B 454 (1999) 187, hep-th/9812034. [16] S. Hellerman, J. Polchinski, Supersymmetric quantum mechanics from light cone quantitation, in: M.A. Shifman (Ed.), The Many Faces of the Superworld, World Scientiﬁc, 2000, hep-th/9908202. [17] V. Akulov, M. Kudinov, Phys. Lett. B 460 (1999) 365, hep-th/9905070. [18] G. Papadopoulos, Class. Quantum Grav. 17 (2000) 3715, hep-th/0002007; J. Michelson, A. Strominger, Commun. Math. Phys. 213 (2000) 1, hep-th/9907191; J. Michelson, A. Strominger, JHEP 9909 (1999) 005, hep-th/9908044; R.A. Coles, G. Papadopoulos, Class. Quantum Grav. 7 (1990) 427. [19] C.M. Hull, The geometry of supersymmetric quantum mechanics, hep-th/9910028. [20] A. Maloney, M. Spradlin, A. Strominger, JHEP 0204 (2002) 003, hep-th/9911001. [21] D. Bak, K. Lee, P. Yi, Phys. Rev. D 62 (2000) 025009, hep-th/9912083. 350 S. Bellucci et al. / Nuclear Physics B 684 [PM] (2004) 321–350

[22] E. Ivanov, S. Krivonos, O. Lechtenfeld, JHEP 0303 (2003) 014, hep-th/0212303. [23] E. Ivanov, O. Lechtenfeld, JHEP 0309 (2003) 073, hep-th/0307111. [24] S. Bellucci, A. Nersessian, Phys. Rev. D 64 (2001) 021702, hep-th/0101065. [25] S. Bellucci, A. Galajinsky, S. Krivonos, Phys. Rev. D 68 (2003) 064010, hep-th/0304087. [26] A.V. Smilga, Nucl. Phys. B 652 (2003) 93, hep-th/0209187. [27] R. de Lima Rodrigues, The quantum mechanics SUSY algebra: an introductory review, hep-th/0205017. [28] J. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231, hep-th/9711200; S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Phys. Lett. B 428 (1998) 105, hep-th/9802109; E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253, hep-th/9802150. [29] B. Zupnik, Nucl. Phys. B 554 (1999) 365; B. Zupnik, Nucl. Phys. B 644 (2002) 405, Erratum, hep-th/9902038. [30] A. Galperin, E. Ivanov, V. Ogievetsky, E. Sokatchev, Pis’ma Zh. Eksp. Teor. Fiz. 40 (1984) 155, JETP Lett. 40 (1984) 912 (in English); A.S. Galperin, E.A. Ivanov, S. Kalitzin, V.I. Ogievetsky, E.S. Sokatchev, Class. Quantum Grav. 1 (1984) 469. [31] A.S. Galperin, E.A. Ivanov, V.I. Ogievetsky, E.S. Sokatchev, Harmonic Superspace, Cambridge Univ. Press, Cambridge, 2001, 306 p. [32] S. Bellucci, E. Ivanov, S. Krivonos, Phys. Lett. B 460 (1999) 348, hep-th/9811244. [33] S.J. Gates Jr., L. Rana, Phys. Lett. B 342 (1995) 132, hep-th/9410150. [34] A. Van Proeyen, Tools for supersymmetry, hep-th/9910030. [35] L. Frappat, P. Sorba, A. Sciarrino, Dictionary on Lie superalgebras, hep-th/9607161. [36] E. Ivanov, S. Krivonos, O. Lechtenfeld, Class. Quantum Grav. 21 (2004) 1, hep-th/0310299. [37] E.A. Ivanov, V.I. Ogievetsky, Teor. Mat. Fiz. 25 (1975) 164. [38] M.F. Sohnius, Nucl. Phys. B 138 (1978) 109. [39] A. Galperin, E. Ivanov, V. Ogievetsky, Yad. Fiz. 45 (1987) 245, Sov. J. Nucl. Phys. 45 (1987) 157 (in English); A. Galperin, E. Ivanov, V. Ogievetsky, Phys. Scr. T 15 (1987) 176. [40] S. Bellucci, E. Ivanov, S. Krivonos, O. Lechtenfeld, in preparation. [41] V. De Alfaro, S. Fubini, G. Furlan, Nuovo Cimento A 34 (1974) 569. [42] E. Ivanov, S. Krivonos, J. Niederle, Nucl. Phys. B 677 (2004) 485, hep-th/0210196. [43] S. Bellucci, A. Galajinsky, E. Ivanov, S. Krivonos, Phys. Lett. B 555 (2003) 99, hep-th/0212204. [44] J. Gomis, P.K. Townsend, M.N.R. Wohlfarth, JHEP 0212 (2002) 027, hep-th/0211020. Nuclear Physics B 684 [PM] (2004) 351–368 www.elsevier.com/locate/npe

Cohomological extension of Spin(7)-invariant super-Yang–Mills theory in eight dimensions

D. Mülsch a,B.Geyerb

a Wissenschaftszentrum Leipzig e.V., D-04103 Leipzig, Germany b Universität Leipzig, Naturwissenschaftlich-Theoretisches Zentrum and Institut für Theoretische Physik, D-04109 Leipzig, Germany Received 4 November 2003; accepted 9 February 2004

Abstract It is shown that the Spin(7)-invariant super-Yang–Mills theory in eight dimensions, which relies on the existence of the Cayley invariant, permits the construction of a cohomological extension, which relies on the existence of the eight-dimensional analogue of the Pontryagin invariant arising from a quartic chiral primary operator.  2004 Elsevier B.V. All rights reserved.

PACS: 11.15.-q; 11.30.Pb

1. Introduction

Topological quantum ﬁeld theory (TQFT) has attracted a lot of interest over the last years, both for its own sake and due to their connection with string theory (for a review, see, e.g., [1]). Particularly interesting are TQFT’s in D = 2 [2], because of their connection with N = 2 superconformal theories and with Calabi–Yau moduli spaces [3]. Considerable impact on physics and mathematics has had Witten’s construction of topological Yang– Mills theory in D = 4 and the discovery of its relation with the Donaldson map, which relates the de Rham cohomology groups on four-manifolds with those on the moduli space of quaternionic instantons, as well as its relation to the topologically twisted super-Yang– Mills theory [4]. Recently, the construction of cohomological gauge theories on manifolds of special holonomy in D>4 have received considerable attention, too [5–7]. These theories, which

E-mail addresses: [email protected] (D. Mülsch), [email protected] (B. Geyer).

0550-3213/$ – see front matter  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2004.02.011 352 D. Mülsch, B. Geyer / Nuclear Physics B 684 [PM] (2004) 351–368 arise without the necessity for a topological twist, acquire much of the characteristics of a TQFT. However, such theories are not fully topological, since they are only invariant under such metric variations which do not change the reduced holonomy structure. For D = 8 examples of cohomological gauge theories have been constructed for the cases when the holonomy group in SO(8) is either Spin(7) [5,6] (Joyce manifolds) or Spin(6) ∼ SU(4) [5] (Calabi–Yau four-folds) or Spin(5) ∼ Sp(4) [8] (hyper-Kähler eight-folds). At present, the physical content of these theories has not been entirely revealed, especially, since they are not renormalizable. But, recent developments of string theory have renewed the interest in super-Yang–Mills theories (SYM) in D>4, particulary because of their crucial role in the study of D-branes and in the matrix approach to M-theory. It is widely believed that the low-energy effective world volume theory of D-branes obtains through dimensional reduction of N = 1, D = 10 SYM [9]. Hence, the above mentioned cohomological theories in D = 8 provide effective field theories on the world volume of Euclidean Dirichlet 7-branes wrapping around manifolds of special holonomy. In order to improve their renormalizibility, counterterms are needed at high energies arising from string theory compactifications down to eight dimensions. Such terms were computed in [10]. In [11] it was argued that the complete string-corrected counterterms for the Spin(7)-invariant theory, whose construction relies on the existence of the Cayley invariant, should be still cohomological. In this paper we show—without going beyond the scope of a cohomological theory— that the Spin(7)-invariant theory, in fact, has a cohomological extension. It relies on the existence of the eight-dimensional analogue of the Pontryagin invariant which arises = 1 4 from the primary operator W0 4 tr φ . This extension, which in flat space is uniquely determined by the shift and vector supersymmetries, can be related to the one-loop string- corrected counterterms. The paper is organized as follows: in Section 2, we briefly describe the formulation of the Spin(7)-invariant NT = 1, D = 8 SYM. By analyzing the structure of the observables it is argued that this theory permits the construction of a cohomological extension which relies on the existence of the eight-dimensional analogue of the Pontryagin invariant. In Section 3, by performing a dimensional reduction to D = 4, it is shown that the matter- independent part of the resulting half-twisted theory [12] can be obtained from the operator ˆ = 1 2 W0 2 tr φ by means of a relationship associated with the vector supersymmetry. In Section 4, by generalizing this relationship to D = 8, we give the cohomological extension of the Spin(7)-invariant theory in the Landau type gauge. Appendix A contains the Euclidean spinor conventions which are used. Appendix B lists the off-shell transformation rules for the matter-independent part of the half-twisted theory. Appendix C gives, in some detail, the derivation of the cohomological extension in the Feynman type gauge.

2. Spin(7)-invariant, D = 8 Euclidean super-Yang–Mills theory

An eight-dimensional analogue of Donaldson–Witten theory on Spin(7) holonomy Joyce manifold, which localizes onto the moduli space of octonionic instantons, has been constructed in [5,6]. This theory, just as Donaldson–Witten theory on SU(2) holonomy hyper-Kähler two-fold, is topological without twisting. In fact, it is invariant under metric D. Mülsch, B. Geyer / Nuclear Physics B 684 [PM] (2004) 351–368 353 variations preserving the Spin(7) structure. In flat space the theory arises from the N = 2, D = 8 Euclidean SYM by reducing the SO(8) rotation invariance to Spin(7).This reduction has been explicitly carried out in [13] (preserving Hermiticity and imposing some extra constraints) and in [14] (relaxing the reality conditions on fermions without introducing extra constraints). Thereby, extensive use is made of the octonionic algebra together with their compatibility with Spin(7) invariance, generalized self-duality and chirality. The action of the Spin(7)-invariant NT = 1, D = 8 Euclidean SYM is built up from ¯ the 10 bosonic scalar and vector fields φ, φ and Aa (a = 1,...,8), respectively, and from = 1 cd the 16 fermionic scalar, vector and self-dual tensor fields η, ψa and χab 6 Φabcdχ , respectively. Φabcd is the Spin(7)-invariant, completely antisymmetric self-dual Cayley = 1 ef g h tensor, Φabcd 24 abcdefghΦ , with abcdefgh being the Levi-Civita tensor in eight dimensions. All the fields take their values in the Lie algebra Lie(G) of some compact gauge group G. Adopting the notation of [14] the Spin(7)-invariant action reads 8 1 abcd a ¯ ab a S = d x tr Θ FabFcd − 2D φDaφ − 2χ Daψb + 2ηD ψa 8 E ¯ a 1 ab ¯ 2 + 2φ ψ ,ψa + φ χ ,χab + 2φ{η,η}−2[φ,φ] , (1) 4 where Fab = ∂[aAb] +[Aa,Ab] and Da = ∂a +[Aa, ·]. Here, the (unnormalized) projector [6]

1 ef Θabcd = δacδbd − δadδbc + Φabcd, Θabef Θcd = Θabcd, 8 projects any antisymmetric second rank tensor onto its self-dual part 7 according to the decomposition 28 = 7 ⊕ 21 of the adjoint representation of SO(8) ∼ SO(8)/Spin(7) ⊗ Spin(7). It satisﬁes the relations [14] 1 g g ΘabegΘcdf − ΘabfgΘcde =−Θef a c δbd + Θef a d δbc + Θef bc δad − Θef bd δac 2 + Θabceδdf − Θabdeδcf − Θabcf δde + Θabdf δce

− Θcdaeδbf + Θcdbeδaf + Θcdaf δbe − Θcdbf δae, 1 g g ΘabegΘcdf + ΘabfgΘcde = Θabcdδef , (2) 2 enclosing all the properties of the structure constants entering the octonionic algebra [15]. On-shell, upon using the equation of motion for χab, the action (1) can be recast into the Q-exact form, S = QΨ , where the gauge fermion 8 ab 1 cd a ¯ ¯ Ψ = d x tr χ Fab − ΘabcdF − 2ψ Daφ − 2[η,φ]φ (3) 16 E is uniquely ﬁxed by requiring its invariance under vector supersymmetry, QaΨ = 0. 354 D. Mülsch, B. Geyer / Nuclear Physics B 684 [PM] (2004) 351–368

The full set of supersymmetry transformations, generated by the supercharges Q, Qa = 1 cd and Qab 6 ΦabcdQ , which leave the action (1) invariant, are given by [14],

QAa = ψa,Qψa = Daφ, Qφ = 0,Qφ¯ = η, ¯ 1 cd Qη =[φ,φ],Qχab = ΘabcdF , (4) 4

1 Q A = δ η + χ ,Qψ = F − Θ F cd + δ [φ,φ¯], a b ab ab a b ab 4 abcd ab ¯ Qaφ = ψa,Qaφ = 0, ¯ b ¯ Qaη = Daφ, Qaχcd = ΘabcdD φ (5) and

d d QabAc =−Θabcdψ ,Qabψc = ΘabcdD φ, ¯ Qabφ = 0,Qabφ = χab, 1 cd 1 g ef ¯ Qabη =− ΘabcdF ,Qabχcd = ΘabegΘcdf F + Θabcd[φ,φ]. (6) 4 4 The on-shell algebra of the supercharges Q, Qa and Qab reads . . {Q, Q} =−2δG(φ), {Q, Qa} = ∂a + δG(Aa), . . {Q ,Q } =−2δ δ (φ),¯ {Q ,Q} = 0, a b ab G ab . d d . {Qab,Qc} = Θabcd ∂ + δG A , {Qab,Qcd } =−2ΘabcdδG(φ), (7) where δG(ϕ) denotes a gauge transformation with field-dependent parameter ϕ = (A ,φ,φ)¯ , being defined by δ (ϕ)A =−D ϕ and δ (ϕ)X =[ϕ,X] for all the other a . G a a G fields. (The symbol = means that the corresponding relation is fulfilled only on-shell.) The crucial hint, supporting our claim that (1) allows for a cohomological extension, comes from the structure of the topological observables. Namely, in terms of differential ˜ forms, the method of descent equation implies, starting from the primary operator W0 = 1 2 ˜ − 2 tr φ , the following ladder of k-forms Wk (4 k 8) with ghost number 8 k [5,6], 1 W˜ = Φ ∧ tr φ2 , 4 2 ˜ = ∧ W5 Φ tr(ψφ), 1 W˜ = Φ ∧ tr Fφ+ ψ ∧ ψ , 6 2 ˜ = ∧ ∧ + ∧ W7 Φ tr(F ψ ψ ψ), 1 W˜ = Φ ∧ tr F ∧ F , (8) 8 2 = a = 1 a ∧ b = 1 a ∧ b ∧ c ∧ d a = a µ with ψ ψae , F 2 Fabe e and Φ 24 Φabcde e e e (e eµ dx ,where a eµ is the 8-bein on the Spin(7)-holonomy Joyce manifold M endowed with metric gµν ). D. Mülsch, B. Geyer / Nuclear Physics B 684 [PM] (2004) 351–368 355

The trace is taken in Lie(G).Thesek-forms obey the following descent equations,

˜ ˜ ˜ ˜ 0 = QW4,dWk = QWk+1, 4 k 7,dW8 = 0, which are typical for any cohomological gauge theory. Hence, if γ is a k-dimensional homology cycle on M then the integrated descendants I˜ (γ ) = W˜ (4 k 7) and I˜ = W˜ are Q-invariant, QI˜ (γ ) = QW˜ = k γ k 8 M 8 k γ k ˜ = γ dWk−1 0. They depend, up to a Q-exact term, only upon the homology class of γ , i.e., ˜ when adding to γ a boundary term ∂α,then Ik(γ ) remains unaltered, modulo a Q-exact ˜ + = ˜ = ˜ + ˜ = ˜ + ˜ = ˜ term, Ik(γ ∂α) γ +∂α Wk Ik(γ ) α dWk Ik(γ ) α QWk+1 Ik(γ ). Here, the ˜ integrated descendant I8 is the Cayley invariant, being invariant under a certain class of metric variations which do not change the reduced Spin(7) structure. = 1 4 By means of the same method, starting from the primary operator W0 4 tr φ , one can 1 derive the following ladder of k-forms Wk (0 k 8) with ghost number 8 − k, 1 W = tr φ4 , 0 (s) 4 3 W1 = tr(s) ψφ , 1 W = tr Fφ3 + ψ ∧ ψφ2 , 2 (s) 2 2 W3 = tr(s) F ∧ ψφ + ψ ∧ ψ ∧ ψ , 1 1 W = tr F ∧ Fφ2 + F ∧ ψ ∧ ψφ + ψ ∧ ψ ∧ ψ ∧ ψ , 4 (s) 2 4 1 W = tr F ∧ F ∧ ψφ + F ∧ ψ ∧ ψ ∧ ψ , 5 (s) 2 1 W = tr F ∧ F ∧ Fφ+ F ∧ F ∧ ψ ∧ ψ , 6 (s) 2 W = tr (F ∧ F ∧ F ∧ ψ), 7 (s) 1 W = tr F ∧ F ∧ F ∧ F , (9) 8 (s) 4 which obey similar descendant equations as before. Again, we observe that the integrated descendant I8 = M W8 yields a topological invariant which is now unchanged under arbitrary metric variations and which may be regarded as the eight-dimensional analogue of the Pontryagin invariant. This suggests that the Spin(7)-invariant theory permits, in fact, a cohomological extension, Sext = QΨext, which possibly should be constructed by the help = 1 4 of the primary operator W0 4 tr φ and with Ψext being uniquely ﬁxed by the requirement under vector supersymmetry, QaΨext = 0.

1 In order to condense the notation, we introduced a symmetrized trace, tr(s), which is deﬁned in Appendix C. 356 D. Mülsch, B. Geyer / Nuclear Physics B 684 [PM] (2004) 351–368

3. Dimensional reduction to four dimensions

In order to get an idea how such an extension Sext could be constructed with the = 1 4 help of the primary operator W0 4 tr φ , we look at the matter-independent part of the topologically half-twisted theory which, similarly to Ref. [12], is obtained by reducing to four dimensions the Spin(7)-invariant theory. To this end, for 1 a,b 4, we group the components of Aa, ψa and χab into a vector isosinglet AA, a vector isosinglet ψA and = 1 CD = a self-dual tensor isosinglet χAB 2 ABCDχ (A,B 1, 2, 3, 4), respectively, and for i i 5 a,b 8, into a left-handed two-spinor isodoublet Gα , a left-handed isodoublet λα i and a right-handed isodoublet ζα˙ (i = 1, 2), respectively (for the two-spinor conventions, ij i i see, Appendix A). The index i is raised and lowered as follows: ϕj = ϕ and ϕ ij = ϕj , 12 where ij is the invariant tensor of the group SU(2), = 12 = 1. Then, for that action of the half-twisted theory one gets, according to the prediction [5], 1 1 S = S + d4x tr DAGα D G i + Gα ,Gβ G i ,G j red 0 2 i A α 4 i j α β E α ¯ i ¯ α i αi˙ − 2 G i, φ Gα ,φ + 2φ λ i ,λα + 2φ ζ ,ζαi˙ − 1 AB αβ i − αi˙ A β χ (σAB ) Gα ,λβi 2iζ σ αβ˙ DAλ i 2 + αi˙ A β + α i 2iζ σ αβ˙ G i ,ψA 2η G i,λα , (10) where the matter-independent part S0 is just the Donaldson–Witten action [4], 4 1 ABCD A ¯ AB A S = d x tr Θ FAB FCD − 2D φDAφ − 2χ DAψB + 2ηD ψA 0 8 E ¯ A 1 AB ¯ 2 + 2φ ψ ,ψA + φ χ ,χAB + 2φ{η,η}−2[φ,φ] . (11) 2 Hence, the result of compactifying the Spin(7)-invariant theory (1) to four dimensions gives the Donaldson–Witten theory with matter in the adjoint representation. On the other hand, Donaldson–Witten theory with matter, after some rearrangements of the spinor ﬁelds, so that Sred becomes real, gets uniﬁed in eight dimensions (the resulting theory is very similar to the non-Abelian version of the Seiberg–Witten monopole theory). The (unnormalized) projector

1 EF ΘABCD = δACδBD − δADδBC + ABCD, ΘABEF ΘCD = ΘABCD, (12) 4 projects any antisymmetric second rank tensor onto its self-dual part 3 according to the decomposition 6 = 3 ⊕ 3 of the adjoint representation of SO(4) ∼ Spin(3) ⊗ Spin(3).It obeys the relations D. Mülsch, B. Geyer / Nuclear Physics B 684 [PM] (2004) 351–368 357

 1 G G ΘABEGΘCDF − ΘABF GΘCDE 2 = ΘEFACδBD − ΘEFADδBC − ΘEFBCδAD + ΘEFBDδAC

= ΘABCEδDF − ΘABDE δCF − ΘABCF δDE + ΘABDF δCE, 1 G G ΘABEGΘCDF + ΘABF GΘCDE = ΘABCDδEF , (13) 2 with ABCD being the Levi-Civita tensor in four dimensions. Furthermore, for the dimensionally reduced transformation rules, generated by Q, QA = 1 CD and QAB 2 ABCDQ , from (4)–(6) one gets

QAA = ψA,

QψA = DAφ, Qφ = 0, Qφ¯ = η, Qη =[φ,φ¯ ],

1 CD 1 αβ i QχAB = ΘABCDF + (σAB ) Gα ,Gβi , 4 4 QG i = λ i, α α i i Qλα = Gα ,φ , i 1 A βi Qζα˙ = i σ DAG , (14) 2 βα˙

QAAB = δABη + χAB ,

1 CD ¯ 1 αβ i QAψB = FAB − ΘABCDF + δAB [φ,φ]− (σAB ) Gα ,Gβi , 4 4 QAφ = ψA, ¯ QAφ = 0, ¯ QAη = DAφ, B ¯ QAχCD = ΘABCDD φ, i = βi˙ QAGα i(σA)αβ˙ζ ,

i 1 i 1 B βi QAλα = DAGα − (σAB )αβ D G , 2 2 i βi ¯ QAζα˙ = i(σA)βα˙ G , φ , (15) and

D QAB AC =−ΘABCDψ , D QAB ψC = ΘABCDD φ,

QAB φ = 0, ¯ QAB φ = χAB, 358 D. Mülsch, B. Geyer / Nuclear Physics B 684 [PM] (2004) 351–368

1 CD 1 αβ i QAB η =− ΘABCDF − (σAB ) Gα ,Gβi , 4 4 1 EF ¯ QAB χCD = ΘABEGΘCDFGF + ΘABCD[φ,φ] 4

1 α βγ i + (σAB ) γ (σCD) Gα ,Gβi , 4 Q G i = (σ ) λβi, AB α AB αβ i βi QAB λα =−(σAB )αβ G ,φ , i 1 C D βi QAB ζα˙ = i σ ΘABCDD G . (16) 4 βα˙ = 1 CD Notice that by introducing an auxiliary ﬁeld BAB 2 ABCDB the matter-independent part of (14)–(16) can be closed off-shell (see, Appendix B). The crucial point is that, on-shell, the matter-independent part S0 can be obtained from ˆ = 1 2 the operator W0 2 tr φ as follows, . 4D 1 ABCD 4 1 2 . S = S − QAQB QC QD d x tr φ = QΨ , (17) 0 top 4! 2 0 E where 4D 4 1 ABCD S = d x tr FAB FCD (18) top 8 E is the four-dimensional Pontryagin invariant. Moreover, on-shell, the gauge fermion Ψ0 = 1 ¯2 can be obtained from the prepotential V0 2 tr φ according to . 1 B C A 4 1 ¯2 Ψ = QA QB QC d x tr φ . (19) 0 4! 2 E On the other hand, the topological observables of the Donaldson–Witten theory can also ˆ ˆ be obtained from W0 by means of the following ladder of k-forms Wk (0 k 4) with ghost number 4 − k [4], 1 Wˆ = tr φ2 , 0 2 ˆ = W1 tr(ψφ), 1 Wˆ = tr Fφ+ ψ ∧ ψ , 2 2 ˆ = ∧ + ∧ W3 tr(F ψ ψ ψ), 1 Wˆ = tr F ∧ F . (20) 4 2 Hence, because (9) is just the eight-dimensional analogue of (20) the cohomological extension Sext of the action (1) should be precisely the eight-dimensional extension of (17)–(19). D. Mülsch, B. Geyer / Nuclear Physics B 684 [PM] (2004) 351–368 359

4. Cohomological extension Sext in the Landau type gauge

As anticipated, for the cohomological extension Sext we are looking for we make the following ansatz . 8D 1 abcdefgh 8 1 4 . S = S − QaQbQcQd QeQf QgQh d x tr φ = QΨ , (21) ext top 8! 4 ext E with . 1 b c d e f g a 8 1 ¯4 Ψ = Qa Qb Qc Qd Qe Qf Qg d x tr φ , (22) ext 8! 4 where 1 S8D = d8x tr abcdefghF F F F (23) top 64 ab cd ef gh E is the eight-dimensional analogue of the Pontryagin invariant. Let us briefly comment on the possibilities to determine Ψext either from (21) or from (22). The most favorable way seems to be to construct Sext by means of Q and Qab (and not by means of Qa) because then Ψext can be obtained directly from the prepotential = 1 ¯4 V0 4 tr φ . However, despite the fact that the transformation rules (4)–(6) look very similar to those of the matter-independent part of (4)–(6), it is impossible to close the latter off-shell with a finite number of auxiliary fields. Apart from the general arguments given in [16], this may be traced back to the fact that in the former case there are fewer algebraic identities among the Spin(7) projection operator Θabcd than for the Spin(3) projection operator ΘABCD in the latter case (cf. Eq. (2) with Eq. (13)). More precisely, we were able to find an off-shell realization only for the subset Q, Qab (see, Appendix C), but not for the subset Q, Qa . For that reason, we do not further pursue the possibility to determine Ψext by means of (22). Hence, for our purpose only the relationship (21) is really eligible—after recasting it into an Q-exact form. However, proceeding in that less ideal way one is confronted with the tricky problem to verify the Q-exactness of (21) and, therefore, the exposition of Ψext becomes very complex (see, Appendix C). Obviously, when evaluating (21) in the Feynman type gauge one gets a huge number of terms belonging to the non-minimal sector which, however, are of no particular interest. Therefore, focusing only on terms belonging to minimal sector, we shall restrict ourselves to the Landau type gauge. In that gauge the action (1) simplifies as follows: 8 ab a ¯ S = QΨ ,Ψ= d x tr χ Fab − 2ψ Daφ , (24) E where the first term of Ψ enforces the localization into the moduli space whereas the second term ensures that pure gauge degrees of freedom are projected out. One easily verifies that Ψ is invariant under the vector supersymmetry Qa (in the Landau type 360 D. Mülsch, B. Geyer / Nuclear Physics B 684 [PM] (2004) 351–368 gauge) = ¯ = QaAb 0,Qaφ 0, = = ¯ Qaφ ψa,Qaη Daφ, = = b ¯ Qaψb Fab,Qaχcd ΘabcdD φ. (25) For the cohomological extension Sext of (24), whose computation is postponed to Appen- dix C, one obtains

S = QΨ , ext ext = 8 [abcd ef ] − e ¯ ab cd Ψext α d x tr(s) Φ χ FabFcd Fef 12ψ Φ[abcdDe]φF F , (26) E α being an arbitrary constant. Ψext is invariant under the vector supersymmetry (25) as well.2 In order to check this crucial property one needs the following identities:

1 abcdefgh abcd e f bcd[e f ] a 1 cd[ef a] b Φmngh = Φ δ[ δ − Φ δ δ ] + Φ δ δ ] 2 m n] [m n 2 [m n 1 d[ef a b] c 1 [ef a b c] d 1 [abcd e f ] − Φ δ δ ] + Φ δ δ ] = Φ δ[ δ , 6 [m n 24 [m n 48 m n] 1 abcdefgh 1 [abcd e] Φmf g h = Φ δ , 6 24 m 1 abcdefgh abcd Φef g h = Φ , (27) 24 where the last one expresses the self-duality of Φabcd. It is amusing to see that, in the Landau type gauge, the extension of Ψ into Ψext is rather simple. Namely, it obtains either by performing in (24) the replacements

ab ab abcd Φabef F → Φabef F + αΦ F[abFcd Fef ], e e [abcd e] D ψe → D ψe + 6αΦ D (ψeFabFcd + FabψeFcd + FabFcd ψe), or, equivalently, by changing (formally) in the same manner the self-duality gauge ab e condition Φabef F = 0 and the ghost gauge condition D ψe = 0. Summarizing, we have shown that the Spin(7)-invariant super-Yang–Mills theory, which relies on the existence of the Cayley invariant, permits the construction of a = 1 4 cohomological extension by the help of the operator W0 4 tr φ , which relies on the existence of the eight-dimensional analogue of the Pontryagin invariant.

2 The square bracket antisymmetrization is iteratively deﬁned as

[ab]=ab − ba, [abc]=a[bc]+b[ca]+c[ab],

[abcd]=a[bcd]−b[cda]+c[dab]−d[abc], etc. D. Mülsch, B. Geyer / Nuclear Physics B 684 [PM] (2004) 351–368 361

With regard to this, a couple of interesting questions is left still open deserving a further study. So far, the Spin(7)-invariant theory was considered in flat space only. But, according to Berger’s classification [17] a metric with Spin(7) holonomy on the simply connected eight-dimensional Riemannian manifold M with Euclidean signature admits a covariantly- constant spinor ζ .Ifsuchζ exists, the metric is automatically Ricci-flat. In addition, such metric has the Spin(7)-invariant closed Cayley four-form Φ (for a given choice of the orientation of M). Conversely, if Φ with respect to a metric on M is closed, then the metric has Spin(7) holonomy. Hence, the action (24) and its extension (26) can be considered on a curved manifold M with Spin(7) holonomy. As is widely believed, super-Yang–Mills theory in eight dimensions may arise as low- energy effective world volume theory on a Euclidean 7-brane in Type IIB string theory [9]. Thus, as it was pointed out in [6], the Spin(7)-invariant theory in curved space can be considered as a theory being obtained by wrapping an Euclidean 7-brane of Type IIB string theory around a manifold with Spin(7) holonomy. However, such a theory is not renormalizable and, therefore, extra degrees of freedom are needed at high energies. It seems to be natural to assume that such extra degrees of freedom are given by the counterterms arising from string theory after compactification to eight dimensions. In another context, in [11] some arguments were given that all the string-corrected eight- dimensional counterterms are still Q-exact. So, one may ask whether in the Landau type gauge the cohomological extension (26) in curved space differs from the string-corrected one-loop counterterm (identifying α with the string tension) only by replacing the Levi- 2 Civita tensor abcdefgh through a certain SO(8)-invariant tensor tabcdefgh(φ ) involving = 1 2 the operator W0 2 tr φ (see, e.g., [18]). Another question is whether, in a similar way, = 1 6 a cohomological extension can be constructed from the operator W0 6 tr φ , too, which should be related to the string-corrected two-loop counterterms, and so on.

Appendix A

The Euclidean two-spinor conventions adopted in this paper are similar to those of αβ˙ Ref. [19, Appendix E]. The numerically invariant tensors (σA) and (σA)αβ˙ are the 1 1 C Clebsch–Cordon coefﬁcients relating the ( 2 , 2 ) representation of SL(2, ) to the vector representation of SO(4), αβ˙ = − − − := γδ˙ = ∗ αβ˙ (σA) ( iσ1, iσ2, iσ3,I2), (σA)αβ˙ (σA) γ˙α˙ δβ σA , αβ˙ αγ β˙δ˙ ∗ ˙ = := ˙ = (σA)αβ (iσ1,iσ2,iσ3,I2), (σA) (σA)γ δ σA αβ˙,

αβ˙ (σA)αβ˙ and (σA) being the corresponding complex conjugate coefﬁcients. Thereby, σ1, σ2, σ3 are the Pauli matrices, which satisfy the Clifford algebra

αγ˙ αγ˙ α (σA) (σB )γβ˙ + (σB ) (σA)γβ˙ = 2δABδ β , γ β˙ + γ β˙ = α˙ (σA)αγ˙ (σB ) (σB )αγ˙ (σA) 2δABδ β˙, 362 D. Mülsch, B. Geyer / Nuclear Physics B 684 [PM] (2004) 351–368 and, in addition, the completeness relations, ˙ B βα˙ B A γ δ δ˙ γ (σA)αβ˙ σ = 2δA ,(σA)αβ˙ σ = 2δα˙ δβ , ˙ A αβ˙ A γ δ αγ β˙δ˙ ˙ = ˙ ˙ = (σA)αβ σ γδ˙ 2αγ βδ,(σA) σ 2 . αγ β αβ γ The spinor index α (analogously α˙ )israisedandloweredby ϕγ = ϕ and ϕα γβ = = 12 = ϕαβ ,whereαβ (analogously α˙β˙) is the invariant tensor of the group SU(2), 12 = 1˙2˙ = 1˙2˙ 1. The selfdual and antiselfdual SO(4) generators, (σAB )αβ and (σAB )α˙ β˙ , and the various Clebsch–Gordon coefﬁcients are related by the properties

αγ˙ β αβ αβ (σA) (σB )γ˙ = (σAB ) − δAB , ˙ αγ˙ β˙ D αβ (σC ) (σAB )γ˙ = (δACδBD − δBCδAD − ABCD) σ , α˙γ˙ β˙ α˙ β˙ α˙ β˙ (σCD) (σAB )γ˙ = (δACδBD − δBCδAD − ABCD) − δ[C[A(σB]D]) , γ (σA)αγ˙ (σB ) ˙ = (σAB ) ˙ ˙ + δAB ˙ ˙, β αβ αβ γ D ˙ = − + (σC )αγ (σAB ) β (δACδBD δBCδAD ABCD) σ αβ˙ , γ (σCD)αγ (σAB ) β = (δACδBD − δBCδAD + ABCD)αβ − δ[C[A(σB]D])αβ . Finally, some often used identities are B =− − (σAB )α˙ β˙ σ ˙ 2(σA)αδ˙ β˙γ˙ α˙ β˙(σA)γδ˙ , γδ AB = − σ ˙ ˙(σAB )γ˙δ˙ 8α˙ γ˙ β˙δ˙ 4α˙β˙γ˙δ˙, αβ B = + (σAB )αβ σ ˙ 2(σA)αδ˙βγ αβ (σA)γ δ˙, γ δ AB = − σ αβ (σAB )γδ 8αγ βδ 4αβ γδ.

Appendix B

In this appendix we give the off-shell transformation rules of the matter-independent part of (14)–(16),

QAA = ψA,

QψA = DAφ, Qφ = 0, Qφ¯ = η, Qη =[φ,φ¯ ],

QχAB = BAB ,

QBAB =[χAB,φ], (B.1) D. Mülsch, B. Geyer / Nuclear Physics B 684 [PM] (2004) 351–368 363

QAAB = δABη + χAB , ¯ QAψB = FAB − BAB + δAB[φ,φ],

QAφ = ψA, ¯ QAφ = 0, ¯ QAη = DAφ, Q χ = Θ DB φ,¯ A CD ABCD B ¯ B QABCD = DAχCD − ΘABCD ψ , φ + D η , (B.2)

D QAB AC =−ΘABCDψ , D QAB ψC = ΘABCDD φ,

QAB φ = 0, ¯ QAB φ = χAB, =− QAB η BAB , 1 1 Q χ =−Θ E F − B + Θ E F − B AB CD ABC DE 2 DE ABD CE 2 CE + Θ [φ,φ¯ ], ABCD E 1 E 1 QAB BCD = ΘABC D[DψE] − [χDE,φ] − ΘABD D[C ψE] − [χCE,φ] 2 2

− ΘABCD[η,φ], (B.3) where the operator ΘABCD was introduced in Eq. (12).

Appendix C

In this appendix we prove the off-shell Q-exactness of the cohomological extension (21) and we show that, by choosing the Landau type gauge, one exactly reproduces the expression (26) for the gauge fermion Ψext. As already emphasized in Section 4, we are forced to start with the complicated expression 8D 1 abcdefgh 8 1 4 S = S − QaQbQcQd QeQf QgQh d x tr φ , (C.1) ext top 8! 4 E 8D where Stop is the eight-dimensional topological invariant (23). Namely, we found only the off-shell extension of the scalar and vector supersymmetry transformations (3) and (4),

QAa = ψa,Qψa = Daφ, Qφ = 0,Qφ¯ = η, ¯ Qη =[φ,φ],Qχab = Bab,

QBab =[χab,φ], (C.2) 364 D. Mülsch, B. Geyer / Nuclear Physics B 684 [PM] (2004) 351–368 and ¯ QaAb = δabη + χab,Qaψb = Fab − Bab + δab[φ,φ], ¯ Qaφ = ψa,Qaφ = 0, Q η = D φ,¯ Q χ = Θ Dbφ,¯ a a a cd abcd b ¯ b QaBcd = Daχcd − Θabcd ψ , φ + D η , (C.3) = 1 cd where Bab 6 ΦabcdB is the antiﬁeld of χab. One simply veriﬁes that Q and Qa satisfy the following superalgebra off-shell,

{Q, Q}=−2δG(φ), {Q, Qa}=∂a + δG(Aa), ¯ {Qa,Qb}=−2δabδG(φ). (C.4)

In order to recast (C.1) into the Q-exact form Sext = QΨext it is convenient to exploit only the algebraic relations (C.4) among Q and Qa , but not their explicit realizations (C.2) and (C.3). Otherwise, owing to the increasing number of terms which arise by evaluating (C.1), it is nearly impossible to determine Ψext explicitly. = 1 4 To begin with, by repeated application of Qa on the primary operator W0 4 tr φ we decompose (C.1) into a form where the different operators Qa act only on the scalar ﬁeld φ. After straightforward calculations one obtains 8D 1 abcdefgh 8 3 2 S = S − d x tr ψabcdefghφ + 8ψabcdefgψhφ ext top 8! (s) E 2 2 + 28ψabcdef ψghφ + 56ψabcdef ψgψhφ + 56ψabcdeψfghφ 2 + 168ψabcdeψfgψhφ + 336ψabcdeψf ψgψh + 35ψabcdψef g h φ

+ 280ψabcdψef g ψhφ + 420ψabcdψef ψghφ + 840ψabcdψef ψgψh + 280ψ ψ ψ φ + 560ψ ψ ψ ψ + 1680ψ ψ ψ ψ abc def gh abc def g h abc de fg h + 630ψabψcd ψef ψgh , (C.5) where

ψa = Qaφ, ψab = Qaψb,ψabc = Qaψbc, etc., (C.6) and where, for the sake simplicity, we have introduced a symmetrized trace, tr(s).Itis deﬁned as follows: ﬁrst, for every monomial X1X2X3X4 in the integrand of (C.5) which, due to its origin from φ4, always consists of exactly 4 factors, we consider all those graded permutations (with repetitions) of the various factors Xi = (φ, ψa,ψab,...) which do not correspond to an antisymmetrization of their space indices; owing to the presence of the Levi-Civita tensor in (C.5) we can ignore such permutations. After that, we take the trace of the sum of all these permuted polynomials and drop their largest common factor. As an illustration let us give some examples: tr ψ ψ φ2 = tr ψ ψ φ2 + φψ φ + φ2ψ , (s) abcdefg h abcdefg h h h 2 2 2 tr(s) ψabcdψef g h φ = tr ψabcd ψef g h φ + φψef g h φ + φ ψef g h ,

tr(s){ψabcdeψf ψgψh}=tr{ψabcdeψf ψgψh}. D. Mülsch, B. Geyer / Nuclear Physics B 684 [PM] (2004) 351–368 365

2 In the ﬁrst example, the 24 possible graded permutations of the monomial ψabcdefgψhφ , after taking the trace over their sum lead to only 3 different monomials with the common factor 8 which has to be dropped. In the same way one performs the symmetrized trace of the other examples. Notice, that in the second example one has to ignore the 2 permutation of ψabcd and ψef g h in ψabcdψef g h φ because it can be reversed through an antisymmetrization of their space indices. In the last example one even has to ignore all possible graded permutations. With that deﬁnition of the symmetrized trace the various factors in front of the monomials in (C.5) agree precisely with the number of permutations which are necessary in order to recast the symmetrized trace of these polynomials into a fully antisymmetized 2 form. For example, in order to recast ψabcdψef g h φ into a totally antisymmetrized form one has to perform 8!/4!4!=70 permutations, thereby taking into account that, owing to the presence of the Levi-Civita tensor, only the completely antisymmetrized part of ψabcd and ψef g h appears in (C.5). By taking the symmetrized trace of that polynomial this number is further reduced to 35 since, under that trace, one can permute ψabcd and ψef g h thereby dividing the total number of permutations by two. Hence, one has to perform only 35 permutations in accordance with the prefactor of that polynomial in (C.5). As a next step, owing to the Q-exactness of ψa we can split off from each of the higher rank objects ψab, ψabc,...in (C.6) an Q-exact term by making use of the second relation (C.4), as a result of which one gets the following decompositions,

ψab =−Qλab + Fab,

ψabc = Qλabc − 3Daλbc,

ψabcd =−Qλabcd + 4Daλbcd − 3{λab,λcd },

ψabcde = Qλabcde − 5Daλbcde + 10[λab,λcde],

ψabcdef =−Qλabcdef + 6Daλbcdef + 10[λabc,λdef ]−15{λab,λcdef },

ψabcdefg = Qλabcdefg − 7Daλbcdefg − 35[λabc,λdefg]+21[λab,λcdefg],

ψabcdefgh =−Qλabcdefgh + 8Daλbcdefgh − 28{λab,λcdefgh}+56[λabc,λdefgh]

− 35{λabcd,λef g h }, (C.7) where

λab = QaAb,λabc = Qaλbc,λabcd = Qaλbcd, etc. (C.8) Thereby, for the sake simplicity, we still have performed some replacements on the right- hand side of (C.7). For example, let us derive the decomposition of ψabc,

ψa = Qaφ = QAa,

ψab = Qaψb = Qa(QAb) =−Q(QaAb) + Fab =−Qλab + Fab, ψabc = Qaψbc =−Qa(Qλbc − Fbc) = Q(Qa λbc) − Daλbc − D[c(QaAb])

⇒ Qλabc − 3Daλbc, where in the last relation we have replaced D[c(QaAb]) = Dcλab −Dbλac through 2Daλbc since only the completely antisymmetric part of ψabc enters into (C.5). In exactly the 366 D. Mülsch, B. Geyer / Nuclear Physics B 684 [PM] (2004) 351–368 same way one can derive iteratively all the other decompositions in (C.7). Notice, that the various factors in front of the different terms in (C.7) agree precisely with the number of permutations which must be performed in order to fully antisymmetrize these terms, thereby taking into account that after inserting the decompositions (C.7) into (C.5) only the fully antisymmetric part of λab, λabc,...contribute. We are now faced with the difﬁcult problem to rewrite (C.5) in an Q-exact form. This is owing to the fact that after inserting the decompositions (C.7) into (C.5) one gets a huge number of symmetrized terms and, therefore, the computational effort in order to expose the gauge fermion is considerably. By making use of the ﬁrst relation (C.4), after a straightforward but lengthy algebraic computation for Ψext one obtains 1 Ψ = abcdefgh d8x tr λ φ3 − 8λ ψ φ2 ext 8! (s) abcdefgh abcdefg h E 2 − 28λabcdef (Qλgh − Fgh)φ + 56λabcdef ψgψhφ − 56λ (Qλ − 3D λ )φ2 − 336λ ψ ψ ψ abcde fgh f gh abcde f g h + 168λ ψ (Qλ − F )φ − 35λ Qλ − 8D λ abcde f gh gh abcd ef g h e fgh 2 + 6{λef ,λgh} φ + 280λabcdψe(Qλfgh − 3Df λgh)φ

+ 420λabcd(Qλef − Fef )(Qλgh − Fgh)φ − 840λabcdψeψf (Qλgh − Fgh)

+ 280λabc(Qλdef − 6Dd λef )(Qλgh − Fgh)φ

− 560λabcψd ψe(Qλfgh − 6Df λgh)

− 1680λabcψd (Qλef − Fef )(Qλgh − Fgh) 2 + 280λabc[λdef ,λgh]φ + 840λabcψd {λef ,λgh}φ − 420λab{λcd ,λef }Fghφ

+ 315λab{λcd ,λef }Qλghφ − 630λabQλcd Qλef Qλgh

− 560λabcψd Deλfghφ + 1680λabψcDd λef Qλgh

− 840λabψcψd {λef ,λgh}+840λabQλcd Qλef Fgh + 1680λ D λ D λ φ − 1260λ Qλ F F − 2520λ ψ D λ F ab g cd h ef ab cd ef gh ab c d ef gh + 2520λabFcd Fef Fgh , (C.9) which is the eight-dimensional extension of the gauge fermion Ψ0 of the Donaldson–Witten theory. Here, a remark is in order. In (C.5) the symmetrized trace was introduced for monomials consisting of exactly 4 factors Xi . After substituting for ψab, ψabc,...the decompositions (C.7) we introduce, besides the ﬁeld strength Fab, also the covariant derivative Da and some graded commutators of the higher rank objects λab, λabc,... as well as their Q-transforms. Thus, in order not to spoil the deﬁnition of the symmetrized trace, one has to view these new objects, in particular all the graded commutators of λab, λabc,...,as single factors, i.e., new objects like Xi ! Let us still give some example of the correct use of the symmetrized trace in (C.9), 2 2 2 tr(s) [λab,λcde]λfghφ = tr [λab,λcde] λfghφ + φλfghφ + φ λfgh , D. Mülsch, B. Geyer / Nuclear Physics B 684 [PM] (2004) 351–368 367 tr λ {λ ,λ }φ2 = tr λ {λ ,λ }φ2 + φ{λ ,λ }φ + φ2{λ ,λ } , (s) abcd ef gh abcd ef gh ef gh ef gh tr λ ψ (D λ )Qλ (s) ab c d ef gh = tr λ ψ (D λ )Qλ + (Qλ )ψ D λ − (D λ )(Qλ )ψ ab c d ef gh gh c d ef d ef gh c − (Dd λef )ψcQλgh + ψc(Qλgh)Dd λef − (Qλgh)(Dd λef )ψc . In order to pick out from (C.9) all the terms belonging to the minimal sector, we have to evaluate the higher rank objects λab, λabc,... in (C.8) explicitly. By making use of (C.3) after a simple calculation one obtains

λab = χab, m ¯ λabc = Φabc Dmφ, ¯ m ¯ λabcd =−Φabcd[η,φ]−Φabc [χdm, φ], λ =−5Φ [D φ,¯ φ¯], abcde abcd e λ =−5Φ [χ , φ¯], φ¯ , abcdef abcd ef λ =−5Φ Φ m [D φ,¯ φ¯], φ¯ , abcdefg abcd ef g m ¯ ¯ ¯ m ¯ ¯ ¯ λabcdefgh = 5ΦabcdΦef g h [η,φ], φ , φ + 5ΦabcdΦef g [χhm, φ], φ , φ , (C.10) with

Qλ = B , ab ab Qλ = Φ m [ψ , φ¯]+D η , abc abc m m ¯ ¯ m ¯ Qλabcd =−Φabcd [φ,φ], φ −{η,η} − Φabc [Bdm, φ]−{χdm,η} , (C.11) where we have carried out similar replacements as in (C.7) and omitted all the terms which do not contribute to (C.9) (remind that only the fully antisymmetric part of λab, λabc,... enters). Then,byinsertinginto(C.9)forλab, λabc,... the expressions (C.10) and (C.11) and taking into account (27), together with the following basic identity [15],

abcm [a b c] 1 [ab c] Φ Φdefm = δ δ δ + Φ [deδ , d e f 4 f ] ¯ after a tedious calculation we could express Ψext in terms of Aa, ψa , χab, η, φ, φ and Bab. ¯ In order to select from Ψext the terms belonging to the minimal sector, we rescale χab, η, φ and Bab as well as Ψext with the gauge parameter ξ and 1/ξ, respectively. Then, by putting ξ equal to zero, i.e., by choosing the Landau type gauge, Ψext considerably simpliﬁes into 1 abcdefgh 8 Ψ = d x tr {2520λabFcd Fef Fgh − 1680λabcψd Fef Fgh}. (C.12) ext 8! (s) E Moreover, it is easily seen that the further evaluation of the right-hand side of (C.12) reveals precisely the expression (14) we are looking for. 368 D. Mülsch, B. Geyer / Nuclear Physics B 684 [PM] (2004) 351–368

References

[1] D. Birmingham, M. Blau, M. Rakowski, G. Thompson, Phys. Rep. 209 (1991) 129; S. Cordes, G. Moore, S. Ramgoolam, Lectures on 2D Yang–Mills Theory, Equiariant Cohomology and Topological Field Theories, in: Les Houches Session LXII, hep-th/9411210. [2] R. Dijkgraaf, E. Verlinde, H. Verlinde, Nucl. Phys. B 352 (1991) 246. [3] W. Lerche, C. Vafa, N.P. Warner, Nucl. Phys. B 324 (1989) 427; B. Green, C. Vafa, N.P. Warner, Nucl. Phys. B 324 (1989) 371; P. Candelas, G. Horowitz, A. Strominger, E. Witten, Nucl. Phys. B 258 (1985) 46. [4] E. Witten, Commun. Math. Phys. 117 (1988) 353. [5] L. Baulieu, H. Kanno, I.M. Singer, Commun. Math. Phys. 194 (1998) 149. [6] B.S. Acharya, M. O’Loughlin, B. Spence, Nucl. Phys. B 503 (1997) 657. [7] M. Blau, G. Thompson, Phys. Lett. B 415 (1997) 242; L. Baulieu, A. Losev, N. Nekrasov, Nucl. Phys. B 522 (1998) 82; L. Baulieu, P. West, Phys. Lett. B 436 (1998) 97; B.S. Acharya, J.F. Figueroa-O’Farrill, M. O’Loughlin, B. Spence, Nucl. Phys. B 514 (1998) 583; J.M. Figueroa-O’Farrill, A. Imaanpur, J. McCarty, Phys. Lett. B 419 (1998) 167. [8] D. Mülsch, B. Geyer, Euclidean super-Yang–Mills theory on a hyper-Kähler eightfold, hep-th/0304098. [9] E. Witten, Nucl. Phys. B 460 (1996) 335. [10] W. Lerche, S. Stieberger, Adv. Theor. Math. Phys. 2 (1998) 1105; C. Bachas, C. Fabre, E. Kiritsis, N.A. Obers, P. Vanhove, Nucl. Phys. B 509 (1998) 33; W. Lerche, N. Warner, S. Stieberger, Adv. Theor. Math. Phys. 3 (1999) 1575–1613. [11] L. Baulieu, S.L. Shatashvili, JHEP 9903 (1999) 011. [12] J. Yamron, Phys. Lett. B 213 (1988) 325. [13] H. Nishino, S. Rajpoot, Octonions, G2 symmetry, generalized self-duality and supersymmetries in dimensions D 8, hep-th/0210132. [14] D. Mülsch, B. Geyer, G2-invariant 7D super-Yang–Mills theory as a higher-dimensional analogue of 3D super-BF theory, hep-th/0310237. [15] R. Dündarer, F. Gürsey, C.H. Tze, J. Math. Phys. 25 (1984) 1496. [16] W. Siegel, M. Rocek, Phys. Lett. B 105 (1981) 275; V. Rivelles, J.G. Taylor, J. Phys. A: Math. Gen. A 15 (1982) 163. [17] M. Berger, Bull. Soc. Math. France 83 (1955) 279. [18] W. Lerche, Nucl. Phys. B 308 (1988) 101. [19] P. van Nieuwenhuizen, Phys. Rep. 68 (1981) 189. Nuclear Physics B 684 [PM] (2004) 369–383 www.elsevier.com/locate/npe

Homotopy and duality in non-Abelian lattice gauge theory

Romain Attal

Laboratoire de Physique Théorique et Hautes Énergies, Université Pierre et Marie Curie 4, Place Jussieu, F-75252 Paris cedex 05, France Received 21 November 2003; received in revised form 13 January 2004; accepted 21 January 2004

Abstract We propose an approach of lattice gauge theory based on a homotopic interpretation of its degrees of freedom. The basic idea is to dress the plaquettes of the lattice to view them as elementary homotopies between nearby paths. Instead of using a unique G-valued field to discretize the connection 1-form, A,weuseanAut(G)-valued field U on the edges, which discretizes the 1-form adA,andaG-valued field V on the plaquettes, which corresponds to the Faraday tensor, F .The 1-connection, U, and the 2-connection, V , are then supposed to have a 2-curvature which vanishes. This constraint determines V as a function of U up to a phase in Z(G), the center of G. The 3- curvature around a cube is then Abelian and is interpreted as the magnetic charge contained inside this cube. Promoting the plaquettes to elementary homotopies induces a chiral splitting of their usual Boltzmann weight, w = vv¯, defined with the Wilson action. We compute the Fourier transform, vˆ,of this chiral Boltzmann weight on G = SU3 and we obtain a finite sum of generalized hypergeometric ˆ functions. The dual model describes the dynamics of three spin fields: λP ∈ G and mP ∈ Z(G) Z3, on each oriented plaquette P ,andεab ∈ Out(G) Z2, on each oriented edge (ab). Finally, we sketch a geometric interpretation of this spin system in a fibered category modeled on the category of representations of G.  2004 Elsevier B.V. All rights reserved.

PACS: 11.15.Ha; 12.38.Aw; 12.90.+b

Keywords: Lattice gauge theory; Kramers–Wannier duality; Homotopy; Fibered categories

E-mail address: [email protected] (R. Attal).

1. Introduction

A discretization of Yang–Mills gauge theory on a lattice has been defined by Wilson in order to find an analytic criterion for the confinement of quarks [13]. This approach suggests that confinement is a consequence of the non-Abelian nature of the structure group (usually G = SUN ) [14]. The gauge invariant quantities, i.e., the physical observables of the theory, are obtained by averaging products of traces of holonomies of a connection on a vector bundle, along spin networks. The latter are directed graphs whose edges carry a representation of G and whose vertices carry a homomorphism from the tensor product of the incoming representations to the tensor product of the outgoing ones. These observables can be expanded as power series whose coefficients are sums over surfaces made of plaquettes [6,9,13]. In this framework, the correspondence between weak and strong coupling is expressed by a Fourier transformation on the space of gauge fields and is a non-Abelian generalization of the Kramers–Wannier duality [8]. The main algebraic difficulty is the resolution of the dual Bianchi constraint [4]. In [11], the dual theory has been formulated as a spin-foam model. In the present work, we propose to make a further step in the study of non-Abelian duality by using homotopic methods. Our paper is organized as follows. In Section 2, we define the dressed cells of our lattice in order to view them as elementary homotopies. In Section 3, we define the gauge fields of our system and their partition function, in which the 2-curvature is forced to vanish. Since Wilson’s action is proportional to the real part of the trace of the holonomy along a little square loop, the Boltzmann weight, w, of an unoriented plaquette is the product of two complex conjugate weights, v and v¯, each one being associated to an orientation of this plaquette. This chiral splitting allows to associate independent representations to the 2-cells of the dual lattice which correspond to oppositely oriented plaquettes. In Section 4, we express the partition function of the dual model in terms of three fields of representations (or spin fields) taking their values in Gˆ , Z(G) and Out(G). Finally, in Section 5, we draw some conclusions and future prospects. In the appendices, we collect some mathematical results. In Appendix A, we recall some useful results about the structure of SU3 and its characters. In Appendix B, we give a dual interpretation of Wilson’s action by proving that wˆ satisfies a diffusion equation on the weight lattice of SU3. In a Minkowskian space– time, this corresponds to the motion of a free particle following a discretized Schrödinger’s equation. The application of the spectral theorem to the corresponding discrete Laplacian provides us with a direct proof of Fourier’s inversion formula. In Appendix C, we compute explicitly the Fourier coefficients of the chiral Boltzmann weight v:thisisalinear combination of generalized hypergeometric series.

2. Homotopy on a lattice

In this section, we formulate some ideas of lattice gauge theories [7,12,13] in terms of 4 4 parametrized loops. We consider the lattice Z ⊂ R , with its canonical basis (ei)1i4. For p = 0,...,4, we define the set of (oriented) p-cells by d Cp := σ =[u, i1 ···ip]: u ∈ Z ,ir ∈{1, 2, 3, 4},ir = is if r = s /Ap, R. Attal / Nuclear Physics B 684 [PM] (2004) 369–383 371 where Ap denotes the alternate group acting on {1,...,p}, i.e., we identify the p-tuples of indices related by an even permutation. The geometric realization of σ =[u, i1 ···ip] | |:={ + +···+ ∈[ ]} is the set σ u t1ei1 tpeip : t1,...,tp 0, 1 and the p-skeleton, skp(C), is the set of cells of dimension p. C∗ will denote the cubic complex of the dual lattice, Z + 1 4 ( 2 ) . The set of paths is defined by 4 P1 := (a0,...,ap): ai ∈ Z , ai+1 − ai=1 . = = = = If γ (a0,...,ap) and γ (a0,...,aq ) are two paths and if t0(γ) s0(γ ), i.e., ap a0, · = then we can compose them: γ γ (a0,...,ap,a1,...,aq ). The composition of γ with −1 the reversed family, γ = (ap,...,a0), provides a path which is homotopic to (a0,a0) in ˜ sk1(C). The elementary 1-paths are simply the oriented edges and form the set C1. We define recursively the p-paths as the finite families of (p − 1)-paths which have the same (p − 2)-source and the same (p − 2)-target and such that the composite of any component with the inverse of the precedent one is homotopic, in skp−1(C),tothe boundary of a p-cell. The (p − 1)-source of a p-path is its first component, its (p − 1)- target is the last one, and, for q

We could use also (0 → 4) or (2 → 2) homotopies, i.e., diagonal motions, but the motions parallel to the lattice axes are sufﬁcient to connect all the paths and they respect the symmetries of Z4. We will also write abcd for (ad, abcd),andabcd for (abcd, ad). The set of 3-paths is P := = −1 · ∈ P ∈ 3 τ (σ1,...,σn): σi σi+1 2 ∂Qi,σi 2,Qi C3 and the elementary 3-paths, i.e., the couples of shortest 2-paths which bound a single cube, ˜ form the set C3. For example τ = (ad,abcd),(ad,ahed,ahgfed,ahgbcfed,abcfed,abcd) splits into two parts the boundary of a cube [a,i1i2i3]: 372 R. Attal / Nuclear Physics B 684 [PM] (2004) 369–383

Starting from the path (ad) = s1(τ), if we want to reach the path (abcd) = t1(τ), we can sweep either the 2-path (ad, abcd) = s2(τ) or the ﬁve other plaquettes, via (ad,ahed,ahgfed,ahgbcfed,abcfed,abcd)= t2(τ).

3. Gauge ﬁelds and curvatures

In the standard formulation of lattice gauge theories, the connection is represented = −1 (locally) by a map, g, from the set of edges to the structure group, G, such that gyx gxy . Then, to each oriented plaquette with a base point, is associated the holonomy of g along its boundary. However, in the homotopic context, we need to separate the degrees of freedom associated to the 1-cells from those associated to the 2-cells. This leads us to use two ﬁelds ˜ U : C1 −→ Aut(G), ˜ V : C2 −→ G called the 1-connection and the 2-connection, and satisfying the relations

= −1 Uba Uab , = −1 Vabcd Vabcd, Vdabc = Uda(Vabcd),

Vdcba = Uda( Vabcd ) which ensure that we have a natural representation of the space of paths. Indeed, the first relation means that the motion along ba is represented by the inverse of the automorphism corresponding to ab. The second relation means that (3 → 1) moves and (1 → 3) moves are represented by inverse elements. The third relation means that a rotation of angle π/2 applied to abcd replaces the flux Vabcd ∈ G by an equivalent flux, Vdabc, obtained by applying the automorphism Uda, which corresponds to the motion of the 0-target along da. The fourth relation means that the flux swept by the flipped 2-path, dcba,isthecomplex conjugate of the flux swept by abcd, pulled back to s0(dcba) = t0(abcd) = d. Thus, we have only one independent variable, V , per plaquette, the others being obtained by the action of U or by inversion. We can now define two kinds of curvature: the 2-curvature on the elementary 2-paths and the 3-curvature on the elementary 3-paths. R. Attal / Nuclear Physics B 684 [PM] (2004) 369–383 373

Deﬁnition 1. Ig denoting the conjugation by g, the 2-curvature is the map ˜ Φ : C2 −→ Aut(G), −→ = abcd Φabcd IVabcd UabUbcUcd Uda, −→ = −1 abcd Φabcd Φabcd.

When the lattice spacing goes to zero, Φ corresponds to the “fake curvature” 2-form ν as deﬁned in [5]. Since we want to represent the homotopy along abcd as a ﬂux which depends only on the direction of sweeping, we impose the relation

Vabcd = UabUbcUcdUda(Vabcd) which implies the vanishing of the 2-curvature: := = ∀ ∈ ˜ Φabcd IVabcd UabUbcUcd Uda 1Aut(G) abcd C2. (1) In order to describe the Yang–Mills–Wilson theory, we must impose another relation ˜ between U and V , which is a discrete version of Bianchi’s identity. For each τ ∈ C3, we want to represent in G the homotopy leading from s2(τ) to t2(τ). For this purpose, we make the product of the six ﬂuxes (pulled back to s0(τ) = a) which correspond to the six plaquettes swept in the corresponding order and we let

Ωτ := Uad(Vdefc)VahedUah(Vhgf e)Uahg(Vgbcf )VabghVadcb.

Deﬁnition 2. The 3-curvature is the map ˜ Ω : C3 −→ G,

τ −→ Ωτ , ¯ −→ = −1 τ Ωτ¯ Ωτ .

Remark 3. Bianchi’s identity can be stated as the vanishing condition for the 3-curvature (Ωτ = 1G). However, we do not impose this condition in our present model, because Ωτ is the magnetic charge inside τ and these topological defects could play a crucial role in the mechanism of quark conﬁnement. In fact, Bianchi’s identity is almost satisﬁed thanks to the following elementary lemma.

Lemma 4. If the 2-curvature vanishes then the 3-curvature is Abelian: ˜ ˜ Φabcd = 1Aut(G) ∀abcd ∈ C2 ⇒ Ωτ ∈ Z(G) ∀τ ∈ C3 .

˜ Proof. Let τ ∈ C3. After the deﬁnition of Ωτ ,wehave = IΩτ IUad(Vdefc)IVahed IUah(Vhgf e)IUahg(Vgbcf )IVabghIVadcb = (UadIVdefcUda)IVahed (UahIVhgf e Uha)(UahgIVgbcf Ugha)IVabghIVadcb . 374 R. Attal / Nuclear Physics B 684 [PM] (2004) 369–383

Using the relation (1) we obtain = IΩτ (UadUdcUcf UfeUedUda)(UadUdeUehUha)(UahUheUef UfgUghUha)

(UahgUgf UfcUcbUbgUgha)(UahUhg UgbUba)(UabUbcUcd Uda)

= 1Aut(G).

Therefore, Ωτ in the kernel of I, i.e., in the center of G. ✷

4. Partition function and duality

We are now ready to write down a partition function on U and V . Let tr denote the trace in the fundamental representation of G.Letα = (3g2T)−1,whereg is the Yang– Mills coupling and T denotes the temperature of the four-dimensional system. The usual Boltzmann weight w(M) = exp 2α Re tr(M − 1G) is associated to an unoriented plaquette. However, since the elementary 2-paths carry an orientation, we will use as Boltzmann weight v(M) := exp α tr(M − 1G) . 2 4 We thus have w =|v| .LetN ∈ N and let ΛN be the box deﬁned by ΛN := {x ∈ R : |xi| N, ∀i ∈ A}. The degrees of freedom contained in this box are described by the maps U and V which satisfy the appropriate relations and which are ﬁxed to unity outside of ΛN . These maps form compact groups on which we have the product Haar measures, DU and DV . The partition function of this system is Z := DU DV δ(ΦP )v(VP ),

P ∈C2 where δ(ΦP ) denotes the Dirac distribution at 1Aut(G).

4.1. Duality transformation

Let us apply Fourier’s inversion formula (Appendix B) v(M) = dλvˆλχ¯λ(M) λ∈Gˆ to express Z as the partition function of a spin system Z = DU DV δ(ΦP ) dλvˆλχ¯λ(VP ) P ∈C ˆ 2 λ∈G = ˆ ¯ dλP vλP DU DV δ(ΦP )χλP (VP ) , λ P ∈C2 P ∈C2 R. Attal / Nuclear Physics B 684 [PM] (2004) 369–383 375

ˆ vˆ is computed in Appendix C. λ now denotes a map from C2 to G so that λabcd and λdcba are independent degrees of freedom. The constraint of vanishing 2-curvature can be solved as follows. Aut(G) is a semi-direct product

Aut(G) (G/Z) Out(G). ε Thus, for each edge ab, Uab deﬁnes a unique outer automorphism c ab ,wherec ∈ Out(G) denotes the complex conjugation and εab ∈{0, 1}. Moreover, we can choose Tab ∈ G such that εab Uab = ZTab, c , where Zg is the class of g ∈ G in the coset G/Z.IntermsofT and ε, the 2-curvature becomes dεabcd Φabcd = ZVabcdTabTbcTcd Tda, c .

Since Φabcd = 1Aut(G), dε = 0 and, for each choice of Tab, there exists an antisymmetric ˆ map (m : C2 → Z) such that

2iπ Vabcd = exp mabcd TadTdcTcbTba. 3

Henceforth, we will suppose that ε = 0, i.e., that Uab lies in the connected component of 1Aut(G), because we want U to approximate a continuous ﬁeld. The phase of Tab is arbitrary and can be shifted freely by a gauge transformation on m which, however, leaves Ω unchanged: ˜ n : C1 −→ Z3,

(ab) −→ nab =−nba, 2iπ mab T ab = e 3 Tab, m abcd = mabcd + (dn)abcd, 2iπ d2n Ω = e 3 Ω = Ω. We can now express the partition function as a sum over (λ, m) constrained by an integral over T : Z = ˆ ¯ DT dλP vλP χλP VP (T , m) . λm P ∈C2 ˆ Then we use the 3-ality τλ ∈ Z Z3,deﬁnedbytherestrictionofRλ to Z = Z(G),orby ≡ − ¯ τλ p q mod 3, to separate the variables m and T in the decomposition of χλP (VP ):

d d d d λP λP λP λP 2iπ τλ mP χ¯λ (VP ) = e 3 P Rλ (Tab) Rλ (Tbc) P P qaqb P qbqc = = = = qa 1 qb 1 qc 1 qd 1 × Rλ (Tcd ) Rλ (Tda) . P qcqd P qd qa 376 R. Attal / Nuclear Physics B 684 [PM] (2004) 369–383 Let : =[a,i]∈C1 and let d = a + ei . Following [11], we evaluate dT: by collecting all the factors which contain T: in the integrand of Z.LetJλ: be the set of indices of an orthonormal basis adapted to the decomposition into irreducible components = RλP Sλi P ⊃: of the tensor product of the representations RλP associated to the twelve 2-cells which contain :. Then, we insert into Z this decomposition: 2iπ Z = ˆ 3 τλP mP dλP vλP e Kλ: , λm P ∈C2 :∈C1 = Kλ: dT: RλP (T:) . qr∈Jλ: P ⊃: qr

Let A = Calg(G) be the algebra generated by the matrix entries of all ﬁnite-dimensional representations of G. We can view a representation : G → GL(V)) as a vector in the A- module M = End(V) ⊗C A. The integral of R over G provides the orthogonal projector P onto the spin-0 subspace, V0: R = P ∈ End(V) ⊂ M. G [⊗ ] If we apply this result to dT: RλP , we obtain V = λ: VλP , P ⊃:

Pλ: ∈ End(Vλ:), Kλ: = [Pλ:]qr. qr∈Jλ:

Therefore, the partition function of the ﬁelds U and V is also that of a dual spin system, constrained by the projectors Pλ:: 2iπ Z = ˆ 3 τλP mP [P ] dλP vλP e λ: qr . (2) λm P ∈C2 :∈C1 qr∈Jλ:

We can deﬁne this ﬁeld theory on the dual lattice, C∗, by associating each representation ∗ ∈ ∗ RλP to the dual 2-cell P C2 . The dual of the 2-cells which contain : are the 2-cells of ∗ ∗ ∈ ∗ C which form the boundary of : C3 : R. Attal / Nuclear Physics B 684 [PM] (2004) 369–383 377

∗ → In order to extend the map (P RλP ∗ ) to the set of all 2-cells of the dual lattice, we need a dual 1-connection, i.e., maps associated to the dual edges such that the two maps associated to the source and target of a dual 2-cell, P ∗, are related by the conjugation by

RλP ∗ . This relation is a vanishing condition for the dual 2-curvature.

4.2. The dual connection

We sketch here very briefly a geometric interpretation of the dual spin system. In the initial gauge theory, we have replaced the usual G-valued 1-cochain (gab) by the (G → Aut(G))-valued 2-cocycle, (U, V ). After a duality transformation, the theory is described by a field of representations λP ) defined on the set of oriented plaquettes of the dual lattice. The integration over the group variables then constrains the dual 3-curvature to vanish. Let A denote the category of representations of G over finite-dimensional complex vector spaces. The A-valued 2-form R is closed, i.e., for each cube Q of the dual lattice, there exist non-zero intertwiners between P ⊂Q RλP and the trivial representation, R0. If Poincaré’s lemma extends to A-valued p-forms, then there exists an A-valued 1-form ρ and a field of non-zero intertwiners between dρ and R: ⊗ ⊗ ⊗ ρwx ρxy ρyz ρzw Rλ[wxyz] . If we integrate over such spin fields, ρ, then the dual Bianchi constraint is automatically satisfied. The field ρ can be viewed as a functorial connection in a fibered category with typical fiber φx = A, at each site x of the dual lattice. (ρxy : φy → φx) maps r to ρxy ⊗ r. The field R then appears as a flat 2-connection on a 2-stack and ρ is the 1-connection of a stack whose differential is the precedent 2-stack. The construction of these spaces requires a fully categorical approach to lattice gauge theory [3,10] as well as the extension of the geometry of G-gerbes [5] to stacks modeled on tensor categories.

5. Conclusions and perspectives

In the present work, we have laid the basis of a homotopic approach to non-Abelian lattice gauge theories where the field strength represents the homotopies along the 2-cells and the gauge field transports the field strength along the edge-paths. A difference with the usual approach [6,7,9,13] is in the usage of complex weights, v and v¯, associated to oppositely oriented 2-cells. We obtain a new closed formula for the Boltzmann weight of 378 R. Attal / Nuclear Physics B 684 [PM] (2004) 369–383 the dual spin system which couples three fields of representations: one for G, one for its center and one for its group of automorphisms. We have chosen to keep the latter vanishing in order to approximate continuous fields. In this dual spin model appears a projector PλQ on each 3-cell, Q, of the dual lattice. A careful definition shows that this projector depends on the way ∂Q is swept. When PλQ = 0, any tensor product of the six representations associated to the six oriented plaquettes of ∂Q admits a non-zero intertwiner onto the trivial representation of G. This is the condition of vanishing 2-curvature for the dual spin model, and it is satisfied when, for each plaquette P , there exists a non-vanishing morphism of representations between RλP and the curvature dρP of a Rep(G)-valued 1-form (ρxy). The problem is then to count, for each field of highest weights (λP ), the number of 1-forms (ρxy) for which such an intertwiner exists. A suitable version of Poincaré’s lemma would imply that this number is non-zero when the 2-form λP ) is closed and this number would give the weight of ρ in the expression of the unconstrained partition function. This is the analogue of the volume of the gauge group in the initial theory, formulated in terms of connection and curvature. The main lesson of the present work is that the spaces adapted to the formulation of non-Abelian duality are fibered categories modeled on categories of G-torsors and G-modules. We must therefore reformulate the present homotopic approach in a purely categorical language and we hope that these methods will provide a notion of duality adapted to the treatment of strongly coupled gauge theories.

Acknowledgements

I thank Olivier Babelon, Marc Bellon, Robert Oeckl and Hendryk Pfeiffer for useful discussions at various stages of this work.

Appendix A. SU3 and its characters

We recall here some standard results of group theory [1] to compute the Fourier transform of v.LetT be the maximal torus of SU3 whose elements are the diagonal iθ i(θ −θ ) −iθ matrices D = diag(e 1 ,e 2 1 ,e 2 ) with θ1,θ2 ∈[0, 2π[,andt be its Lie algebra. ∗ Let L1,L2,L3 ∈ t be the linear forms deﬁned by Li (d) = dii. The corresponding root ∗ system is the set Φ := {Li − Lj : i = j}⊂t . The adjoint representation of t on g provides a Cartan decomposition of g into eigenspaces of t: g = t ⊕ gα , α∈Φ t corresponds to the 0 eigenvalue and each gα is a one-dimensional representation of t:

[H,X]=α(H)X ∀H ∈ t, ∀X ∈ gα. We can choose in Φ a subset, Π, which is a basis of t and such that each root is a linear combination of elements of Π with coefﬁcients which are either all positive or all negative. Π is our system of fundamental roots. In the present case, we take Π ={α1,α2} with R. Attal / Nuclear Physics B 684 [PM] (2004) 369–383 379

α1 = L1 − L2 and α2 = L2 − L3. We associate to Π the fundamental weights 2α + α 2α + α ω = 1 2 and ω = 2 1 1 3 2 3 so that ωi ,αj =δij . The weight lattice of SU3 is generated by ω1 and ω2:

∗ Λw ={λ = pω1 + qω2: p,q ∈ Z}⊂t . Each ﬁnite-dimensional, C-linear, irreducible representation : G → GL(V )) is charac- terized by a highest weight, λ ∈ Λw, with positive coordinates on ω1 and ω2, and by a vector u ∈ V such that R(H )(u) = λ(H )u for all H ∈ t.ThesetGˆ of isomorphy classes of such representations is then parametrized by two positive integers: ˆ G {λ = pω1 + qω2: p,q ∈ N}. (A.1)

Appendix B. Free dynamics on Gˆ and Fourier’s inversion formula

We give here a short proof of the fact that the Fourier transform of w = vv¯ follows a free dynamics on the weight lattice of SU3. The deﬁnition of v and w implies

∂αv = 10 − 3)v,

∂αw = 10 + χ01 − 6)w. Moreover, a straightforward computation shows that

χ10 χpq = χp+1,q + χp,q−1 + χp−1,q+1,

χ01 χpq = χp,q+1 + χp−1,q + χp+1,q−1.

Consequently, vˆ, v¯ˆ and wˆ satisfy the following differential equations:

∂αvˆpq =ˆvp+1,q +ˆvp,q−1 +ˆvp−1,q+1 − 3vˆpq, ˆ ˆ ˆ ˆ ˆ ∂αv¯pq = v¯p,q+1 + v¯p−1,q + v¯p+1,q−1 − 3v¯pq,

∂αwˆ pq =ˆwp+1,q +ˆwp,q−1 +ˆwp−1,q+1 +ˆwp,q+1 +ˆwp−1,q +ˆwp+1,q−1 − 6wˆ pq. Let E be the Banach space :∞(N2, C) and let H be the Hilbert space :2(N2, C). N2 being endowed with the metric induced by the Killing form of t∗, we deﬁne the Laplace operator, H ∈ B(E),by

1 (Hφ)pq := (φp+ ,q + φp,q− + φp− ,q+ + φp,q+ + φp− ,q + φp+ ,q− ) − φpq, 6 1 1 1 1 1 1 1 1 H computes the mean value of a function at the six nearest neighbours of each site λ and substracts the value at λ. In a fundamental dual Weyl chamber, we represent H by the following diagram: 380 R. Attal / Nuclear Physics B 684 [PM] (2004) 369–383

Thus, wˆ satisﬁes a diffusion equation on N2 × R+:

∂αwˆ = 6Hwˆ (B.1) with the boundary conditions wˆ 0q =ˆwp0 = 0andwˆ pq(0) = δ0pδ0q . The eigenspaces p q of H in E are spanned by the eigenfunctions φxy(p, q) = x y and the corresponding = 1 + −1 + + −1 + + − eigenvalues are µxy 6 (x x y y x/y y/x) 1. The spectrum of H is = T2 =[−4 ] H σ(H) µ( ) 3 , 0 .SinceH is a bounded self-adjoint operator on , we can apply the spectral theorem [15] and expand vˆ on the eigenfunctions φ of H pq xy

vˆpq = dxdy φxy, vˆ φxy(p, q),

2 T φxy, vˆ = φxy(p, q)vˆpq, p,q1 φxy(p, q) = dpqχpq(M), where M = diag(x,y,(xy)−1) ∈ T , and we obtain Fourier’s inversion formula for the class functions: v = vˆλ dλ χ¯λ. (B.2) λ∈Gˆ

Appendix C. The Fourier transform of the chiral Boltzmann weight on SU3

The characters of the irreducible, ﬁnite dimensional representations of G can be computed by applying Weyl’s character formula as follows. For all p,q ∈ Z,letApq ∈ C(X, Y ) be the Laurent polynomial deﬁned by p+q+2 q+1 −q−1 p+1 −p−1 −p−q−2 Apq(X, Y ) = X Y + X Y + X Y − (X Y).

Apq is antisymmetric in X and Y and satisﬁes A−p,−q =−Aqp = Apq. The characters of G are the polynomials

Apq(L1,L3) χλ = χpq = . A00(L1,L3) R. Attal / Nuclear Physics B 684 [PM] (2004) 369–383 381

One easily checks the following identities

χ00 = 1,

χqp = χpq,

χp,−1 = χ−1,q = 0, −1 χ10(M) = x + y + (xy) = tr(M), χ + χ Re tr(M) = 01 10 (M). 2 Since G is compact, and v is continuous and invariant by conjugation, we can apply the Peter-Weyl theorem to approximate v uniformly on G by linear combinations of ˆ characters. The Fourier transform of v is the function (vˆ : G →⊕λ∈ˆvEnd(Vλ)) deﬁned ˆ = = by vλ G vπλ,whereRλ (Vλ,πλ) is a unitary representation of G for each class ˆ λ ∈ G and dλ = dim(Vλ).Sincev is a class function, Schur’s lemma implies that vˆpq is a dilatation on Vλ. Its coefﬁcient will still be written vˆpq. Moreover, Weyl’s integration formula expresses vˆpq as an integral over T : 1 2 vˆpq = χpqv = |A | χpq exp α(χ − 3) 6 00 10 G T e−3α = A Apq exp(αχ ) 6 00 10 T e−3α = Cλsrfrs(α). 6 r,s∈Z

The matrix Cλ has a ﬁnite number of non-zero entries and is obtained by expanding A00Apq: −r −s Cλsr = A00ApqX Y , T λ r s = −2 −1 + −1 + 2 − −2 −1 − −1 − 2 CsrX Y X Y XY XY Y X YX YX ∈Z r,s × Xp+q+2Y q+1 + X−q−1Y p+1 + X−p−1Y −p−q−2 + + + − − + − − − − − − Y p q 2Xq 1 + Y q 1Xp 1 + Y p 1X p q 2 .

The functions frs(α) are deﬁned, for all r, s ∈ Z,by = r s frs(α) L1L3 exp(αχ10) T 2π 2π dθ1 dθ2 − − − = ei(rθ1 sθ2) exp α eiθ1 + ei(θ2 θ1) + e iθ2 2π 2π 0 0 382 R. Attal / Nuclear Physics B 684 [PM] (2004) 369–383 ∞ n n k α n k + − + − − − = eiθ1(k : n r) eiθ2(2k : n s) n! k : n=0 k=0 :=0 ∞ n k αn = δ − − δ − − . (n − k)!:!(k − :)! r,n k : s,2k : n n=0 k=0 :=0 In the last sum, the non-zero terms are those which satisfy n − 2r − s 2n − r + s 0 : = k = n. 3 3 Our sums being indexed by the integers for which all the factorials are meaningful, we have αn frs(α) = ( n+r−s )!( n+r−s − r)!( n+r−s + s)! n∈3N−r+s 3 3 3 α3m−r+s = . m!(m − r)!(m + s)! mmax(0,r,−s) We recognize generalized hypergeometric series [2], deﬁned by µ ν µFν : C × C × C −→ C,

∞ n (a1)n(a2)n ···(aµ)n z (a,b,z)−→ µFν (a,b,z):= , (b1)n(b2)n ···(bν)n n! n=0 (ai)n = ai(ai + 1) ···(ai + n) for µ,ν ∈ N and 0 µ ν. We thus have

αs−r f (α) = F Ø,(−r, s), α3 if r<0ands>0 rs L(−r)L(s) 0 2

|r+s|+Mrs α 3 = 0F2 Ø, |r + s|,Mrs ,α if r>0ors<0, L(|r + s|)L(Mrs) where Ø denotes the empty family and Mrs = max(r, −s). When the argument of one of the L functions vanishes, i.e., when r = 0ors = 0ors =−r<0, the L factor is replaced by 1 and the corresponding factorial coefﬁcient in the series starts at the value 1. By condensing our formulas, we obtain e−3α vˆλ = tr(Cλf). (C.1) 6

References

[1] J.F. Adams, Lectures on Lie Groups, University of Chicago Press, 1969. [2] G.E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge Univ. Press, 2001. [3] R. Attal, Combinatorics of non-Abelian gerbes with connection and curvature, math-ph/0203056, Ann. Fondation Louis de Broglie, in press. R. Attal / Nuclear Physics B 684 [PM] (2004) 369–383 383

[4] G.G. Batrouni, Plaquette formulation and the Bianchi identity for lattice gauge theories, Nucl. Phys. B 208 (1982) 467–483. [5] L. Breen, W. Messing, Differential geometry of Gerbes, math.AG/0106083. [6] J.-M. Drouffe, J.-B. Zuber, Strong coupling and mean-field methods in lattice gauge theories, Phys. Rep. 102 (1983) 1–119. [7] R. Gambini, J. Pullin, Loops, Knots, Gauge Theories and Quantum Gravity, Cambridge Univ. Press, 1996. [8] H.A. Kramers, G.H. Wannier, Statistics of the two-dimensional ferromagnet, Phys. Rev. 60 (1941) 252–262. [9] I. Montvay, G. Münster, Quantum Fields on a Lattice, Cambridge Univ. Press, 1994. [10] R. Oeckl, Generalized lattice gauge theory, spin foams and state sum invariants, J. Geom. Phys. 46 (2003) 308–354, hep-th/0110259. [11] R. Oeckl, H. Pfeiffer, The dual of pure non-Abelian lattice gauge theory as a spin-foam model, Nucl. Phys. B 598 (2001) 400–426, hep-th/0008095. [12] A.M. Polyakov, Gauge Fields and Strings, Harwood Academic, 1987. [13] K. Wilson, Confinement of quarks, Phys. Rev. D 10 (1974) 2445–2459. [14] L.G. Yaffe, Confinement in SU(N) lattice gauge theories, Phys. Rev. D 21 (1980) 1574–1590. [15] R.J. Zimmer, Essential Results of Functional Analysis, University of Chicago Press, 1990. Nuclear Physics B 684 (2004) 385–387 www.elsevier.com/locate/npe

CUMULATIVE AUTHOR INDEX B681–B684

Abel, S.A. B682 (2004) 183 Freyhult, L. B681 (2004) 65 Ahn, C. B683 (2004) 177 Fring, A. B682 (2004) 551 Alonso Izquierdo, A. B681 (2004) 163 Fukuma, M. B682 (2004) 377 Aoki, H. B684 (2004) 162 Arnowitt, R. B682 (2004) 347 García Fuertes, W. B681 (2004) 163 Attal, R. B684 (2004) 369 Gates Jr., S.J. B683 (2004) 67 Gegelia, J. B682 (2004) 367 Bajnok, Z. B682 (2004) 585 Gehrmann, T. B682 (2004) 265 Becker, M. B683 (2004) 67 Gehrmann-De Ridder, A. B682 (2004) 265 Beisert, N. B682 (2004) 487 Geng, C.Q. B684 (2004) 281 Bellucci, S. B684 (2004) 321 Geyer, B. B684 (2004) 351 Bjerrum-Bohr, N.E.J. B684 (2004) 209 Ghosh, P.K. B681 (2004) 359 Bœhm, C. B683 (2004) 219 Giannakis, I. B681 (2004) 120 Böhm, G. B682 (2004) 585 González León, M.A. B681 (2004) 163 Bonciani, R. B681 (2004) 261 Grinza, P. B682 (2004) 521 Grzelinska,´ A. B681 (2004) 230 Caffo, M. B681 (2004) 230 Guchait, M. B681 (2004) 31 Carmelo, J.M.P. B683 (2004) 387 Hardmeier, A. B682 (2004) 150 Castro-Alvaredo, O. B682 (2004) 551 Harmark, T. B684 (2004) 183 Chen, H. B683 (2004) 196 Hatsuda, M. B681 (2004) 152 Constantin, D. B683 (2004) 67 Havare, A. B682 (2004) 457 Cudell, J.R. B682 (2004) 391 Heinrich, G. B682 (2004) 265 Czy˙z, H. B681 (2004) 230 Ho, I.-L. B684 (2004) 281 Hou, H.-S. B683 (2004) 196 Dai, J. B684 (2004) 75 Hu, B. B682 (2004) 347 Dall’Agata, G. B682 (2004) 243 Datta, A. B681 (2004) 31 Intriligator, K. B682 (2004) 45 D’Auria, R. B682 (2004) 243 Iso, S. B684 (2004) 162 Delﬁno, G. B682 (2004) 521 Itzhaki, N. B684 (2004) 264 Derkachov, S. B681 (2004) 295 Ivanov, E. B684 (2004) 321 Djouadi, A. B681 (2004) 31 Dutta, B. B682 (2004) 347 Jaikumar, P. B683 (2004) 264 Janke, W. B682 (2004) 618 Eden, B. B681 (2004) 195 Johnston, D.A. B682 (2004) 618

Fayet, P. B683 (2004) 219 Karakhanyan, D. B681 (2004) 295 Feng, H. B683 (2004) 168 Kawai, H. B683 (2004) 27 Ferroglia, A. B681 (2004) 261 Kenna, R. B682 (2004) 618 Forde, D.A. B684 (2004) 125 K h o ze, V. V. B682 (2004) 217

0550-3213/2004 Published by Elsevier B.V. doi:10.1016/S0550-3213(04)00193-2 386 Nuclear Physics B 684 (2004) 385–387

Kirschner, R. B681 (2004) 295 Rastelli, L. B684 (2004) 264 Klebanov, I.R. B684 (2004) 264 Ravindran, V. B682 (2004) 421 Konishi, Y. B682 (2004) 465 Remiddi, E. B681 (2004) 230 Kono, Y. B682 (2004) 377 Remiddi, E. B681 (2004) 261 Körs, B. B681 (2004) 77 Ren, H.-C. B681 (2004) 120 Korunur, M. B682 (2004) 457 Román, J.M. B683 (2004) 387 Kostov, I.K. B683 (2004) 309 Ryzhov, A.V. B682 (2004) 45 Kovacs, S. B684 (2004) 3 Kraus, P. B682 (2004) 45 Sagnotti, A. B682 (2004) 83 Krivonos, S. B684 (2004) 321 Sakaguchi, M. B681 (2004) 137 Kuroki, T. B683 (2004) 27 Sakaguchi, M. B684 (2004) 100 Kuzenko, S.M. B683 (2004) 3 Sakai, K. B682 (2004) 465 Kyae, B. B683 (2004) 105 Sarma, D. B681 (2004) 351 Sato, T. B682 (2004) 117 Laliena, V. B683 (2004) 455 Scherer, S. B682 (2004) 367 Lechtenfeld, O. B684 (2004) 321 Schindler, M.R. B682 (2004) 367 Lee, C.-A. B683 (2004) 105 Schwarz, A. B681 (2004) 324 Linch III, W.D. B683 (2004) 67 Serban, D. B683 (2004) 309 Liu, J.T. B681 (2004) 120 Shaﬁ, Q. B683 (2004) 105 Lukyanov, S.L. B683 (2004) 423 Shigemori, M. B682 (2004) 45 Lunghi, E. B682 (2004) 150 Siegel, W. B681 (2004) 152 Siegel, W. B683 (2004) 168 Ma, W.-G. B683 (2004) 196 Signer, A. B684 (2004) 125 Martynov, E. B682 (2004) 391 Smith, J. B682 (2004) 421 Mastrolia, P. B681 (2004) 261 Sogut, K. B682 (2004) 457 Mateos Guilarte, J. B681 (2004) 163 Sommovigo, L. B682 (2004) 243 Mazumdar, A. B683 (2004) 264 Sonnenschein, J. B682 (2004) 3 McArthur, I.N. B683 (2004) 3 Soyez, G. B682 (2004) 391 Meessen, P. B684 (2004) 235 Splittorff, K. B683 (2004) 467 Merrell, W. B683 (2004) 67 Stanishkov, M. B683 (2004) 177 Miller, D.J. B681 (2004) 3 Steinhauser, M. B683 (2004) 277 Misiak, M. B683 (2004) 277 Sun, Y.-B. B683 (2004) 196 Miwa, A. B682 (2004) 377 Takács, G. B682 (2004) 585 Moortgat, F. B681 (2004) 31 Tsulaia, M. B682 (2004) 83 Morita, T. B683 (2004) 27 Movshev, M. B681 (2004) 324 Uzawa, K. B683 (2004) 122 Mülsch, D. B684 (2004) 351 Vafa, C. B682 (2004) 45 Nagao, K. B684 (2004) 162 Vaman, D. B682 (2004) 3 Nath, P. B681 (2004) 77 van der Bij, J.J. B681 (2004) 261 Necco, S. B683 (2004) 137 van Neerven, W.L. B682 (2004) 421 Nehme, A. B682 (2004) 289 Vaulà, S. B682 (2004) 243 Nevzorov, R. B681 (2004) 3 Verbaarschot, J.J.M. B683 (2004) 467 Vitchev, E.S. B683 (2004) 423 Obers, N.A. B684 (2004) 183 Ortín, T. B684 (2004) 235 Wang, W. B683 (2004) 48 Ouyang, P. B684 (2004) 264 Wang, X.-J. B683 (2004) 363 Owen, A.W. B682 (2004) 183 Wu, T.H. B684 (2004) 281 Wu, Y.-S. B683 (2004) 363 Pando Zayas, L.A. B682 (2004) 3 Wu, Y.S. B684 (2004) 75 Penc, K. B683 (2004) 387 Wyler, D. B682 (2004) 150 Phillips, J. B683 (2004) 67 Pirjol, D. B682 (2004) 150 Yaguna, C.E. B681 (2004) 247 Ponsot, B. B683 (2004) 309 Yamamoto, M. B683 (2004) 177 Profumo, S. B681 (2004) 247 Yetkin, T. B682 (2004) 457 Nuclear Physics B 684 (2004) 385–387 387

Yoshida, K. B681 (2004) 137 Zamolodchikov, A.B. B683 (2004) 423 Yoshida, K. B683 (2004) 122 Zerwas, P.M. B681 (2004) 3 Yoshida, K. B684 (2004) 100 Zhang, R.-Y. B683 (2004) 196 Yue, C. B683 (2004) 48 Zhou, P.-J. B683 (2004) 196