Trieste Meeting of the TMR Network on beyond the SM

PROCEEDINGS

On the perturbative corrections around D-

A.B. Hammou∗ SISSA/ISAS, Trieste, Italy, via Beirut 2-4, 34013 E-mail: [email protected]

Abstract: We study F 4-threshold corrections in an eight dimensional S-dual pair of string theories, as a prototype of dual string vacua with sixteen supercharges. We show that the CFT description of D-string instantons gives rise to a perturbative expansion similar to the one appearing on the fundamental string side. By an explicit calculation, using the Nambu-Goto action in the static gauge, we show that the first subleading term agrees precisely on the two sides. We then give a general argument to show that the agreement extends to all orders.

FL 1. Introduction tifold actions are defined by the orbifold (−) σV (“fundamental side”), where all the R-R states to During the last years, it have been an enormous which the D- usually couple are remouved progress in our understanding of the non pertur- and ΩσV (“type I side”), correspond- bative aspects of superstring theories. The key ing to a generalized Ω-projection of the closed to this development is the discovery of duality oriented type IIB string giving rise to vanishing symmetries, which relate the strong and weak massless tadpoles thus preventing the introduc- coupling limits of apparently different string the- tion of the open-string excitations, with σV a ories. This give us a way to compute certain shift of order two in the two torus. The complete strong coupling quantities in one matching of the BPS states on the two sides of by mapping it to a weak coupling result in a this duality map was computed [3], and it have dual string theory. The consistency checks of been shown that N winding modes on the funda- this non-perturbative dualities are strangest for mental side corresponds to bound states of ND- effective couplings called BPS-saturated effective strings on the type one side. Moreover, the mod- couplings, some times they are linked to anoma- uli dependence for O(2,2) F 4 gauge couplings, lies. Duality can be used to calculate their non- for the gauge fields coming from the KK reduc- perturbative corrections. One can then identify tion of the metric and antisymmetric tensor, were the non-perturbative effects responsible for such considered. These F 4 couplings have there ana- corrections. For theories with more than N=2 logues in the heterotic/type I duality for toroidal such non-perturbative effects are compactification to D ≥ 5 [4]. In that context due to instantons, which can be associated in these F 4 couplings were special since they are string theory to Euclidean wrapped around related by supersymmetry to CP-odd couplings, an appropriate compact manifold [2]. Here we for which it is well known that only one-loop cor- will be interested in a particular test of this strong- rections are possible [5]. The analysis in the het- weak duality in the simplest context of the eight erotic/type I duality was made possible since for dimensional dual pair with sixteen supercharges D ≥ 5, on the heterotic side, the BPS spectrum is obtained via an orbifold/orientifold of type IIB completely perturbative. Therefore, allowing for theory considered in [3]. The dual orbifold–orien- an exact treatment which yields an exact one- ∗This work has been done in collaboration with loop result even non-perturbatively. A similar E.Gava,J.F.Morales and K.S.Narain [1] Trieste Meeting of the TMR Network on Physics beyond the SM A.B. Hammou analysis for the type II models under consider- same structure as the one appearing on the fun- ation here, has not been done, but we expect damental string side. In particular, we will show similar result to be true. We will assume that a perfect agreement for the first subleading cor- this is the case, i.e. that the one-loop formula rection. obtained for the moduli dependence of some F 4 2 FL terms in the fundamental side is exact. As it will 2. Type IIB on T /(−) σV turn out that the non-perturbative result on the 4 dual type I side will support this assumption. The F couplings we are interested in are ob- The one-loop formula can be expressed [6, tained from the one-loop string amplitudes: 7, 3], in the type I variables, as a sum of a fi- L ` L 4−` A` = h(V8 ) (V9 ) i (2.1) nite number of perturbative corrections and an with infinite series of D-string contributions. Z Indeed, for D ≥ 5, the D-string instanton is the 2 1 V L = d z(G + iB )(∂Xµ − p SγµνS) only source of non-perturbative contributions on i µi µi 4 ν the type I side. The form of the N-instanton i 1 iρ ipX (∂X¯ − pρSγ˜ S˜)e (2.2) contribution in this sum suggests that it can be 4 read off from the O(N) describ- the vertex operators of the left O(2, 2) gauge ing N nearby D-strings, whose worldsheets are fields arising from the Fi ≡ Gµi + iBµi com- wrapped on a two-torus [7]. More precisely, the ponents of the metric and antisymmetric tensor. leading behavior in the volume of the compactifi- Here and in the following we will denote with 4 4 cation torus for the SO(32) F and R gauge and capital indices M =(µ, i) the ten dimensional gravitational couplings respectively were shown noncompact µ =0,1,...7andcompacti=8,9 in agreement [7] with the exact formula found directions. The combinations with the minus in a perturbative computation in the heterotic sign represent the graviphoton vertex operators, side [6]. Similar results were found in [3]. In all which carry always additional power of momenta the cases, the D-string instanton coupling to the and are therefore irrelevant for the computation gauge fields are obtained from the classical D- of F 4 couplings. It is convenient to define a gen- string instanton action, while the contributions erating function Z(νi,τ,τ¯), in term of which the from quantum fluctuations around the D-string above amplitudes read: Z instanton background are encoded in the elliptic 2 ` 4−` d τ 4 ∂ ∂ A ˜8 genus of the corresponding O(N) gauge theory. ` = t 2 τ2 ` 4−` Z(νi,τ,τ¯) (2.3) F τ2 ∂ν8 ∂ν The perturbative computations of [6, 3] show that, 9 ` 4−` on top of the leading D-string instanton correc- with t˜8 = t8F8 F9 , t8 is the tensor arising from tions, there are a finite number of contributions the trace over the right-moving fermions zero- corresponding to perturbative corrections in the modes and F the fundamental domain for the

D-string instanton background. One can ask the modulus of world-sheet torus. Z(νi,τ,τ¯)isthe question whether the CFT describing the infrared partition function arising from a perturbed Polyakov limit of the D-string gauge theory captures some action, whose bosonic part is: Z information about these subleading terms. The 2π √ S(ν )= d2σ( gG gαβ∂ Xµ∂ Xν aim of this letter is to show that this is indeed i α0 µν α β 0 the case. NS √ αβ µ ν α ¯ i Let us first start by briefly reviewing the +iBµν ε ∂αX ∂βX + g νi∂X ),(2.4) 2πτ2 computation of the moduli dependence of F 4 cou- and ∂¯ = 1 (∂ − τ∂ ). The geometry of the plings in the the fundamental side and how the τ2 σ2 σ1 result agrees, at the leading order in a large vol- target torus are described as usual by the two ume expansion, with what one gets from the CFT complex moduli √ describing the D-string system. We will subse- 1 N T = T1 + iT2 = 0 (B89 + i G) quently show that the CFT description also gives α √ rise to a perturbative expansion which has the U = U1 + iU2 =(G89 + i G)/G88 , (2.5)

2 Trieste Meeting of the TMR Network on Physics beyond the SM A.B. Hammou

N where Gij and Bij are the σ-model metric and The integration over the fundamental domain NS-NS antisymmetric tensor. Since the theory can be done using the standard trick [9], which we are considering involves half shift along X8 di- reduces the sum over all wrapping mode configu- rection, it is convenient to replace the radius R8 rations in a given SL(2, Z) orbit to a single inte- by 2R8. Indeed this new radius is what appears gration over an unfolded domain. The unfolded on the dual IIB/ΩσV theory (upto the scaling domains are either the strip or the upper half by the string coupling constant). As a result T , plane depending on whether the orbit is degener- U and ν8 are replaced by 2T , U/2and2ν8 respec- ate (det W =0)or non degenerate (det W =0).6 tively and furthermore all the windings along σ1 In [6, 3] these contributions were identified with and σ2 are valued. the perturbative and D-string instanton correc-

The partition function Z(νi,τ,τ¯)isgivenby tions respectively in the type I side. The result Z Z 2 2 X for the non-degenerate contributions were writ- d τ ' d τ  A 2 Z(νi,ττ¯) 2 Γ2,2(νi) (2.6) ten as [3] F τ F τ 2 2  V h ` 4−`i 10 ` 4−` 1 with (F8) (F9) nondeg = 2 0 4 t8F8 F9 (4π α ) T2  2 X πT τ X X 4− 4T2 2 det − 2 (1 ) ` `  πiT W τ2U2 U W −1 n ∂ ∂ Γ2,2(νi)= e e A In(ν8,ν9) (2.11) 2  ` 4−` U ∂ν8 ∂ν9 W,  n, m1,n1,n2 π τ − (ν8 ν9)W e τ2 −1 (2.7) An where  are the coefficient in theq ¯ expansion of the modular forms (2.8) and I are the result and A the anti-holomorphic BPS partition n  of the modular integrations functions 1 4 4 n +U n 1 ϑ2(¯q) 1 ϑ4(¯q) (U2T2) 2 −2 −2πin( 1 1 2 ) A ≡ A ≡ πiT1m1n2 m1 +− 12 , −+ 12 , In = e e 2 η(¯q) 2 η(¯q) m1 r √ 4 nU2 8( − 1) π −2 1 ϑ3(¯q) πiν m1T2 m βγ A−− ≡ (2.8) e e (2.12) 2 η(¯q)12 β

The Γ2,2 has been written in (2.7) as a with sum over all possible world-sheet instantons 2        2 U2ν8 8 1 1 β = π(n2U2T2 + ν9n2 − ν8U1n2 − ) X σ m1 n1 σ 4T2 9 = W 2 ≡ 2 (2.9) X σ m2 n2 σ πT2 nU2 2 γ = (m1 + ) . (2.13) U2 T2m1 with worldsheet and target space coordinates σ1,σ2 8 9 and X ,X respectively, both taking values in Here the sum over m1 and n1 are over even and the interval (0,1]. The entries m1,n1 are even or odd , depending on the -sectors, and for odd integers depending on the specific orbifold afixedm1 the range of n1 goes from 0 to m1 − 1. sector, while m2,n2 run over all integers. We denote the three relevant sectors: n1 odd, m1 2 odd and both odd by  =+−,−+,−− respec- 3. Type IIB on T /ΩσV tively. The untwisted sector,  = ++, will not 2 contribute, since it has too many zero modes to In terms of the type IIB on T /ΩσV variables be soaked by the four vertex insertions at this this result should come from D-instanton contri- F I order in the momenta. Note that, due to the butions since the duality relations (T2 = T2 /λI ) NTI − 2 existence of these different sectors, the duality 1 λ implies that it goes like e I . Indeed it has group Γ on U is not the full SL(2,Z) group but been shown in [3] that its leading order in the T2 the defined by the matrices   1where the subscripts “F ”and“I” are used to distin- 2 F 2 ab guish orbifold (T /(−) L σV ) and orientifold (T /ΩσV ) ; b∈2Z (2.10) cd compactifications of the type IIB string.

3 Trieste Meeting of the TMR Network on Physics beyond the SM A.B. Hammou expansion In terms of these new variables, after a long but

` 4−` straightforward algebra, one can express, to the h(F8) (F9) i = nondeg order of interest, theν ¯-expansion of the gener- X X 4 ` P V10 4− 1 m1 U n ` ` ating function I(ν, ν¯)= A In of (2.11) for- 2 0 4 t8F8 F9 4 n  (4π α ) T2 m1|n2| U2  m1,n1,n2 mally as: n1 + Un¯ 2   −2πiTm1|n2| X4 r e A( ) (3.1) ν¯ 1 m1 I r I (ν, ν¯)= DU¯ ν +1 0(ν) r! (T2 − )r is exactly reproduced by a direct computation of r=0 2π 5 D-instanton contributions in the type I side. In +O(¯ν ), (4.2) this identification the D-instanton number N = where LM is mapped to the determinant m1n2 of the ¯ V X X −2πiTm1n2 representative matrices M, while different wrap- I 10 e 0(ν)= 2 4 5 ping mode configurations labelled by m1,n1,n2 (4π α0) m1n2  m1,n1,n2 are mapped to L, s, M, describing the different ¯ n1 +Un2 2m1n2ν sectors of the orbifold conformal field theory to A( )e (4.3) m1 which the effective O(N) gauge theory flows in ¯ the infrared. The coupling of the N D-instanton and DU¯ is the U-covariant derivative, which act- background to the F8,F9 gauge fields is read from ing on a modular form of weight −2r gives a the the classical D-instanton action weight −2r + 2 form

F πN S − =2πNT − p i r D inst 0 ν ¯ ¯ − 4α U2λI DU Φr =( ∂U )Φr (4.4) 2 cm µν cm π πU (Gµ9 + UGµ8)S0 γ S0 + .... (3.2) √ P In (4.2) the first 4 terms in the series have been F 1 i cm 1 N i where T = 0 (B89+ G)andS0 = S α λI N i=1 explicitly indicated, being the only relevant ones are the fermionic zero modes corresponding to to the present discussion, The function appear- the center of mass of the N copies of D-string ing there has no definite modular transforma- worldsheets. Indeed, bringing down four pow- r tion properties, and the covariant derivative DU¯ ers of these terms, we soak the eight fermionic should be understood as acting on the coefficient zero modes and combining with theP quantum D- 4−r 1 −4 of ν in the expansion of the function inside instanton partition function M r 4−r NT2 M,L,s the brackets in (4.2). This gives the F¯ F A s+UM¯ ( L ), the perturbative result (3.1) is repro- gauge couplings we are interested in, with F = 1 1 duced. Note that the center of mass fermions (F9 + UF8)andF¯= (F9+UF¯ 8): 2U2 2U2 here are defined with unconventional normaliza- tion in order to make the comparison with the V X −2πiNT¯ X4−r fundamental string side more transparent. h ¯r 4−ri 10 e 1 F F = 2 4 +1 (4π α0) NTr (NT2)k N 2 k=0

4. Perturbative corrections around (r +k)! r H (D ¯ A−+) (4.5) D-string instantons k!(r!)2(4 − r − k)! N U

The exact formula (2.11) shows the presence of where HN is the Hecke operator for the subgroup T2-subleading terms which should correspond to Γ: a finite number of perturbative corrections around XXLX−1 r 4−2r the N D-instanton background. In order to make HN(DU¯A−+)= L more transparent the translation in terms of the  N|Ls=0

r ¯ ¯ type I variables it is convenient to use the com- DU¯A(U)| ¯ MU+s (4.6) U= L plex source ν, and its complex conjugateν ¯,de- fined as: where (L, s) are (even,odd), (odd,even) and (odd,odd) − − −− 1 in the three  sectors + , +and respec- ≡ r ν (ν9 + Uν8), (4.1) −+ ¯ 2U2 tively. Note that DU¯ A (U) is a modular form

4 Trieste Meeting of the TMR Network on Physics beyond the SM A.B. Hammou

of weight 2r − 4 with respect to the subgroup Γ. stant values of Fµνz¯ will solve the equations of The above definition of the Hecke operator coin- motion since the resulting stress energy tensor is cides with the standard one appearing for exam- zero. ii) the first order inν ¯ already comes with 2 ple in [10] upto an overall N dependent factor. 1/T2 and therefore does not require an expan- − ν 2 We would like now to show how these correc- sion of the term 1/(T2 2π ) . This means that tions can be reproduced by the effective D-string in the action we need to include only the first or- instanton action. To this aim, we start with der term in both the F ’s which certainly solves the D-string instanton action in the presence of the linearized equations of motion. All the ex- constant field strength background for the gauge pressions in the following should be understood

fields Gµi,Bµi under consideration. Each field to make sense only in the first subleading term strength appears in the action with two fermion in 1/T2. zero modes and therefore the amplitude is ob- The Green-Schwarz, covariant world-sheet ac- tained by extracting the 8 fermion zero modes in tion for a single string coupled to the N = 1, 10- the partition function. One immediately encoun- D multiplet, is given by [11] (α0 = 1 ters two problems: 2 ): s 1) One needs to keep in the action higher Z | ˆ | order terms in the fermion zero modes. Unfortu- 2 detGab 1 ab SD−inst =2π dσ[ +  Rab] nately in curved backgrounds the covariant Green- λI 2 Schwarz actions that exist in the literature in- + ... (4.7) clude only terms upto quartic in fermions, with The ... above indicate terms involving 6 and higher order terms in principle being calculable higher powers of fermion fields. Here we have di- using space-time supersymmetry and kappa sym- rectly rewritten the action of [11] in the Nambu- metry. On dimensional grounds, as will become Goto form, by using the equations of motion for clear in the following, each pair of fermion zero the world-sheet metric. The resulting induced modes appears with 1/T2 and therefore, such higher 2 metric Gˆab and the antisymmetric coupling Rab order terms can contribute to order 1/T2 correc- are given by: tion (compared to the leading order term). In ˆ M N M ¯ the absence of information on such higher or- Gab = ∂(aX ∂b)X GMN −2i∂(aX ΘγM Db)Θ der terms we will be restricted below to calcu- M −Θ¯γM DaΘΘ¯γ DbΘ late only the first subleading correction. How- M N 2i M ¯ ever later we will argue that the complete action Rab = i∂[aX ∂b]X BMN + ∂[aX ΘγMσ3Db]Θ λI should correctly reproduce the full perturbative 1 ¯ ¯ M + ΘγMD[aΘΘγ σ3Db]Θ. (4.8) expansion. λI 2) Even upto the terms quartic in fermion Here the covariant derivative Da is defined to be: fields, the Green-Schwarz action presupposes that − 1 Kˆ Lˆ Kˆ Lˆ N the background fields satisfy the supergravity equa- Da = ∂a γ ω˜N ∂aX , (4.9) tions of motion. Of course constant field strengths 4 ˆ ˆ ˆ ˆ ˆ ˆ KL KL λI KL satisfy the gauge field equations, however they whereω ˜N = ωN + 2 HN σ3,withωthe spin can contribute to the stress energy tensor and connection and H the 3-form field strength of therefore can modify the classical equations for the RR antisymmetric tensor B. The capital, and the 8-dimensional metric. However Latin indices are 10-dimensional curved indices, this problem will not appear if we restrict our- whereas hatted ones are SO(1, 9) flat indices. T selves to the first subleading correction in 1/T2. Θ =(θ, θ˜) denotes a doublet of SO(1, 9) Majorana- Indeed as seen from equation (4.5), the first sub- Weyl on which σ3 acts. θ and θ˜ have leading term appears in two ways: i) forν ¯ =0 the same SO(1, 9) chirality and are world-sheet T 0 T 0 (i.e. Fµνz = 0) there is a correction coming from scalars. Finally, Θ¯ ≡ (θ γ , θ˜ γ ). Note that − ν the first order expansion of 1/(T2 2π ). This can the fields appearing above are supposed to sat- get contribution from terms in the action that are isfy the 10-dimensional supergravity equations of quadratic in Fµνz¯.ForFµνz = 0 arbitrary con- motion.

5 Trieste Meeting of the TMR Network on Physics beyond the SM A.B. Hammou

We then proceed by choosing the static gauge, θ˜ is anti-periodic, it can be seen that the contri- which amounts to identifying the Euclidean world- butions of interest in (4.8) come from two kind of 1 1 2 z¯ Kˆ Lˆ Kˆ Lˆ µ Kˆ Lˆ sheet complex coordinates z ≡ √ (σ + τσ ), terms, θ¯0γ γ ω˜ θ0 for i = z,z¯ and θ¯0γ γ 2τ2 i z ≡ Kˆ Lˆ z¯, with the spacetime torus coordinates X ω˜ρ θ0. The first type of terms gives the follow- √1 (X8 + UX9), Xz¯ respectively and the com- ing fermionic bilinears 2τ2 plex structure of the world-sheet torus, τ,with i µνz¯ that of the spacetime torus, U. In addition, local η = ∂[ A ]¯θ¯0γ θ0 4 µ ν z kappa-symmetry allows to reduce the fermionic i µνz¯ 2 η¯ = ∂[ A ] θ¯0γ θ0, (4.12) degrees of freedom by imposing 4 µ ν z 1 γzθ ≡ √ (γ8 + Uγ9)θ =0 whereas the second type gives 2U2 i ¯ 1 8 9 ¯ µνz¯ z˜≡ √ ¯ ρµ = ∂[ν Aρ]¯zθ0γ θ0. (4.13) γ θ (γ +Uγ )θ = 0 (4.10) 4 2U2

In the above formulae and in the following γz The resulting gauge fixed fermions θ (θ˜) become and γz¯ denote SO(2) gamma matrices and the left-moving (right-moving) world-sheet fermions T2 dependence will be explicitly displayed. Con- in the 8s (8c)ofSO(8). 4 tributions to the F couplings will come from the We will be interested in a constant back- Gµi 3 D-instanton partition function once four powers ground for the field strength of Aµi ≡ +Bµi . λI of these combinations are brought down to soak To extract the relevant couplings it is convenient the 8 fermionic zero modes. This is why we re- to choose for the vielbeins the representation placed the fermions in (4.12,4.13) by their zero   j 4 ˆ mode part θ0. F couplings will be defined then L ei Gµi 2 eM = + O(Gµi) (4.11) 0 ηµν by the fourth order terms in the η,¯ηexpansion of the resulting effective action. The action (4.7) j 4 with ei the square root of thep torus metric (in canthenbewrittenas z z¯ I z z¯ the complex basis ez = ez¯ = T2 , ez¯ = ez =0). F S − =2πiT − 4πη Since we are interested in extracting four powers D instZ h of the field strengths Aµi from the expansion of 2 µ ¯ +2π d z ∂zX ∂z¯Xµ − 2iθγ˜ z¯∂zθ˜ − 2iθγ¯ z∂z¯θ (4.7) and then set Gµi to zero, we can set always  Gµi to zero unless a derivative hits on it. This is η¯ µ ¯ + (∂z¯X ∂z¯Xµ − 2iθγ˜ z¯∂z¯θ˜)+... (4.14) so because such terms would anyway disappear T2 − η in the final result due to gauge invariance. Fur- where ...represent higher quantum fluctuations, thermore since the connections involve at most and we have rescaled Xµ, θ˜ as one derivative of the metric, the terms contain- ing two or more derivatives of GµI are irrelevant µ → − 2 ν X (1 F )µν X µ =0,...7 for our present discussion. Here and in the fol- T2 η 1 lowing we will always set these components to ˜ → − 2 ˜ θα (1 F ) θα α =1,...8. (4.15) zero. T2 Using the ansatz (4.11), and recalling from We have considered so far the Nambu-Goto [3] that, after orientifolding, θ is periodic (and action for a single D-string. The results (4.14) therefore has 8 zero modes, denoted by θ0), whereas can however easily be generalized to the N D- 2 Strictly speaking, because of the Majorana-Weyl na- string case. The low energy effective action de- ture of fermions, the gauge fixing condition makes sense only in the Minkowski space-time (and world-sheet). We scribing the excitations of the N D-string sys- assume that X9 is time like in which case z andz ¯ are real tem is described by O(N) gauge theory studied light-like coordinates and τ2 and U2 are imaginary. At in [3]. As argued in that reference, after inte- the very end we will do the analytic continuation back to grating out the very massive degrees of freedom complex τ and U 3 4 Notice that precisely this combination appears in the We omit in this expression a trivial λI -rescaling of all connections in (4.9) bosonic Xµ and fermionic fields θ, θ˜

6 Trieste Meeting of the TMR Network on Physics beyond the SM A.B. Hammou in the infrared limit, we are left with an effec- using: tive conformal field theory in terms of the di- µν i i i µ ν δ agonal multiplets X , θ and θ˜ , i =1,...,N, h∂ X (z)∂¯X (w)i = − . (4.17) µ z z 8π on which the orbifold permutation group SN is acting. Correspondingly, in (4.14)P the fermionic The claimed result follows after performing the N ¯i µνz¯ i z-integration and using δµν  = −η. We expect bilinears η andη ¯ will involve i=1 θ γ θ = µν cm µνz¯ cm cm Nθ¯0 γ θ0 + ..., θ0 being the fermionic zero that higher order terms reproduce the expansion 1 mode corresponding to the center of mass. Also, of F − . We can then write finally P T2 η N z z¯ I the volume factor i=1 Gzz¯∂zXi ∂z¯Xi = NT2 Dinst −SND−inst(η,0) pick up a factor of N. The effective action is I0 (η) ≡he i X X I V ¯ then given by simply replacing T2 ,η,η¯ and  by 10 1 −2πiTF LM I = 2 4 5 F e NT2 ,Nη,Nη¯ and N. The operators expressed (2π ) LM (T2 − η)  L,M,s in terms of the non-zero mode fields will be re- ¯ s + UM 4πLMη placed by the sum of the N copies of them, in- A( )e (4.18) i i ˜i L volving Xµ θ and θ . The moduli dependence of F 4 couplings in that reproduces the r = 0 term in (4.2) after the the N D-instanton background are then defined previous identifications. by the η4−r η¯r terms in the expansion of the expo- We can now go on and consider r η¯ insertions ?? nential of (4.14) written for ND-instantons. Iden- in (4.18). From theη ¯ coupling in ( ) one can see that each of these correspond to the insertionR of a tifying the fermionic bilinears 2πη, 2πη¯ with our 1 2 normalized stress-energy tensor F − d zT(¯z) previous sources ν, ν¯, one can show that the per-  T2 η  P i turbative results (4.2), (4.3) are reproduced. ≡ N µi i − ¯˜ ˜i with T (¯z) i=1 ∂z¯X ∂z¯Xµ(¯z) 2iθ γz¯∂z¯θ (¯z) . Let us start by considering theη ¯ =0case. But this is precisely the Virasoro generator of an Were it not for the rescaling of the Xµ, θ˜ fields infinitesimal shift in thez ¯ worldsheet coordinate, (4.15), we would have a free theory (upto Weyl L¯0, which couples to U¯. Therefore its insertions permutations) defining the orbifold partition func- P ¯ translate into covariant derivatives D ¯ acting on tion 1 M−4A(s+UM )e2Nη [3, 12]. Iden- U NT2 M,L,s L (4.18). Note that although the expressions above tifying as before L, M, s with m1,n2,n1 respec- were to make sense only to the first subleading tively we can see by a simple inspection of (4.2), term in 1/T2 thefactthatthe¯ηcoupling appears (4.3) that we reproduce theη ¯ = 0 term except 1 with the same normalization factor F as in 1 1 T2 −η for the fact that we get F instead of F − that T2 T2 η the perturbative result (4.2) is quite remarkable. appears in (4.2). The calculation above shows the precise match- We want now to show that, at least to first ing of the first subleading correction around D- 1 order in F , the rescaling of the fields (4.15) re- T2 string instantons with the exact result on the fun- sults in the desired renormalization of the inverse damental string side. How can we extend this 1 → 1 F area prefactor F F . Indeed, to the first to all orders in 1/T2 ? The steps involved are T2 T2 −η order, the Jacobian of the transformation (4.15) two fold. First of all, we will need terms upto is given by the following correlation function: 8th order in fermion zero modes in the covariant Green-Schwarz action. This can in principle be Z h i 2 µ ν ¯ z obtained by repeatedly using space-time super- d z 4 h∂ X ∂¯X i−2iηhθγ˜ ∂ θ˜i(4.16). µν z z z symmetry and kappa symmetry. Secondly, we need to solve the linearized equation forη ¯ in the This expression, which involves expectation val- presence of arbitrary η. This will involve turn- ues of the kinetic energies of Xµ and θ˜, needs ing on 8-dimensional gravitational and dilaton to be regularized. If we adopt a point-splitting fields to the first order inη ¯. Although straight- regularization, the term involving θ˜ is zero be- forward, the computation involved is rather cum- cause θ˜ has no zero modes. We are thus left with bersome. We will give here an alternative argu- the bosonic contribution, which can be evaluated ment to show that the complete action should

7 Trieste Meeting of the TMR Network on Physics beyond the SM A.B. Hammou

correctly reproduce the full perturbative expan- dence follows from theP definition of the fermion cm 1 i sion. For N = 1 the covariant D-string action is zero mode θ = N i θ . the same as the fundamental string action. The One can see from this general form of the ac- static gauge computation presented here should tion, that the leading term for each r comes with 1 r −+ be the same as the light cone gauge computation N HN (DU¯ A ) while in the subleading correc- F F F in the fundamental string side provided one re- tion in 1/T2 , the volume T2 is replaced by NT2 . stricts to detW = 1 in equation (4.5). This shows This exactly reproduces the exact formula (4.5) that at least for single instanton the two results of the fundamental string. should agree to all orders in 1/T2.ForN>1, the orbifold CFT description implies that the result References is obtained from the N = 1 result by applying the [1] E. Gava, A. B. Hammou, J. F.Morales, and K. Hecke operator HN , upto N dependent factors. 03 Indeed, by the zero mode argument of [12, 3], one S. Narain, JHEP (1999) 023 knows that only the twisted sectors of the form [2] E. kiritsis, Duality and Instantons in String The- ory, hep-th/9906018 (L)M contribute and the resulting expression in- [3] M. Bianchi, E. Gava, J. F. Morales and K. S. volves the sum over L, M, s. Modular covariance Narain, D-Strings in unconventional type I vac- ¯ in U under the subgroup Γ then fixes the pow- uum configurations, hep-th/9811013, to appear ers of L in the sum over the sectors, giving rise in Nucl. Phys. B. to Hecke operator HN . The only quantity which [4] C. Bachas and C. Fabre, Nucl. Phys. B476 4− is not fixed in the amplitude hF¯rF ri by this (1996) 418, hep-th/9605028; C. Bachas and E. argument is the N dependence of the leading as Kiritsis, Nucl. Phys. Proc. Suppl. B55 (1997) F well as the subleading terms in 1/T2 for each r. 194, hep-th/9611205. To determine this N dependence we note that [5]O.Yasuda,Phys.Lett.B215 (1988) 306; Phys. B218 the general form of the N D-string action in the Lett. (1989) 455; A. A. Tseytlin, Phys. B367 orbifold limit is schematically of the form Lett. (1996) 84, hep-th/9510173; Nucl. Phys. B467 (1996) 383, hep-th/9512081. X4 [6] C. Bachas, C. Fabre, E. Kiritsis, N. Obers, and F ak 2 k S − = NT (F¯θ ) B509 ND inst 2 F k z 0 P. Vanhove, Nucl. Phys. (1998) 33, hep- (T2 ) k=0 th/9707126; E. Kiritsis and N. Obers, J. High XN X4 XN Energy Phys. 10 (1997) 004, hep-th/9709058. 1 2 k i 1 2 + (F¯θ ) A + (F θ ) 68 F k z 0 k F z 0 [7] C. Bachas, Nucl. Phys. Proc. Suppl. (T2 ) T2 i=1 k=0 i=1 (1998) 348, hep-th/9710102; P. Vanhove, X3 BPS-saturated amplitudes and non-perturbative 1 2 k i (F¯θ ) B (4.19) F k z 0 k string theory, hep-th/9712079. (T2 ) k=0 [8] K. Foerger and S. Stieberger, Higher derivative couplings and heterotic-type I duality in eight di- where ak are numerical coefficients independent of N and Ai and Bi are dimension (1,1) and (0,2) mensions, hep-th/9901020. k k [9] L. J. Dixon, V. S. Kaplunovsky and J. Louis, operators (depending on X, θ˜ and the non-zero Nucl. Phys. B355 (1991) 649. modes of θ) for each of the N copies of the vari- [10] R. C. Gunning, Lectures on Modular Forms, ables. Here we have suppressed all 8-dimensional Princeton University Press, 1962. gamma matrices and Lorentz indices. Note that [11] E. S. Fradkin and A. A. Tseytlin, Phys. Lett. (4.14) obtained by including only upto quartic B160 (1985) 69. terms in fermion fields in the covariant Green- [12] E. Gava, J. F. Morales, K. S. Narain, and G. Schwarz action fixes a0,a1,a2,A0,A1 and B0 and Thompson, Nucl. Phys. B528 (1998) 95, hep- these were the objects that entered in the compu- th/9801183 tation of the first subleading corrections. In writ- ing (4.19) we have assumed that the higher di- mensional operators will decouple in the infrared F limit. The powers of T2 in the above are easily seen by scaling arguments while the N depen-

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