P–Branes and the Field Theory Limit

Quantum aspects of gauge theories, supersymmetry and uniﬁcation

PROCEEDINGS

P–branes and the ﬁeld theory limit

Jan Pieter van der Schaar∗ Institute for Theoretical Physics, University of Groningen Nyenborgh 4, 9747 AG Groningen, The Netherlands E-mail: [email protected]

Abstract: We investigate Domain–Wall (DW)/Quantum Field Theory (QFT) correspondences in various dimensions. Our general analysis does not only cover the well–studied cases in ten and eleven dimensions but also enables us to discuss DW/QFT correspondences in lower dimensions. New examples include ‘d–branes’ in six dimensions preserving 8 supersymmetries.

Keywords: AdS/CFT correspondence, p–branes.

1. Introduction 2. P –brane near–horizon geometries

Our starting point is the D-dimensional action Z √ h Anti–de Sitter (AdS) supergravity has attracted −g S(D, a, p)= dDx −R− 4 (∂φ)2 much attention due to the conjectured correspon- 2κ2 D−2 D dence to a conformal field theory (CFT) on the 2k φ a i − gs e 2 boundary of the AdS spacetime [1, 2, 3] . Rec- Fd+1 , (2.1) 2(d +1)! gs ognizing that this correspondence has its roots in the special properties of D–branes, this cor- which contains three independent parameters: the respondence was soon extended to include (most target spacetime dimension D, the dilaton cou- of) the non–conformal ten–dimensional Dp–branes pling parameter a and a parameter p specifying − − [4, 5] (for a review, see [6]). This led to the more the rank D p 2 of the field strength F .The general conjecture that there exists a correspon- parameter k is given by dence between Domain–Wall (DW) supergravity a p+1 k(D, a, p)= +2 . (2.2) and (Yang–Mills) quantum field theory (QFT). 2 D−2 This can also be understood by noting that AdS We have furthermore introduced two useful de- spacetimes are special examples of the more gen- pendent parameters d and d˜ which are defined eral DW spacetimes [7]. In this talk we will ex- by tend the discussion of [4, 5] to general two–block p–branes in various dimensions. d = p + 1 wv dimension , (2.3) d˜ = D − p − 3 dual wv dimension . In Section 2 we will derive the near–horizon geometry of a generic two–block p–brane and in We next consider the following class of diago- Section 3 we discuss the low energy field theory nal “two-block” p–brane solutions (using the Ein- limit and restrict ourselves to (intersections of) stein frame)1: D–branes. We will end that Section by focusing − 4d˜ 4d 2 (D−2)∆ 2 (D−2)∆ 2 on a six–dimensional example and in Section 4 ds = H dxd + H dxd˜+2 , conclude with a summary and discussion. (D−2)a φ 4∆ e = gs H , (2.4) 1For later convenience, we give the solution in terms of the magnetic potential of rank D − p − 3. The p-brane ∗Based on work done in collaboration with Klaus solution is electrically charged with respect to the p +1– Behrndt, Eric Bergshoeff and Rein Halbersma. form potential. Quantum aspects of gauge theories, supersymmetry and unification Jan Pieter van der Schaar

r (2−k) 4 electric (p + 1)–form potential Cp+1 is propor- ˜ ··· ··· gs Fm1 md˜+1 = m1 md˜+1m∂mH, −k ∆ tional to gs , which follows from the action (2.1), where we ﬁnd that φ a 1 (2−k) e k ∗ τp ∼ . (2.12) ˜ p+1 k gs F = gs F. (2.5) `s g gs s ∼ D−2 2 We use a constant, i.e. metric independent, Levi- Comparing with (2.9) and using κD ls gs we φ(H=1) deduce that, for a single brane, Civita tensor. Furthermore, gs = e and ∆ is deﬁned by [8, 9] d˜ r0 ∼ 2−k gs . (2.13) 1 2 2dd˜ `s ∆= 8(D−2)a + , (2.6) D − 2 The two-block solutions (2.4) include (super- which is invariant under reductions and oxida- symmetric) domain-wall spacetimes. They corre- tions (in the Einstein frame). The function H is spond to the case d˜= −1, ε = −1andr0 =1/m. harmonic over the d˜+ 2 transverse coordinates The solutions also include the known branes in and, assuming that ten and eleven dimensions (M2, M5, Dp,F1,NS5 etc.) as well as branes in lower dimensions. If d˜=6 −2, 0 , (2.7) the branes under consideration preserve any su- (i.e. no constant or logarithmic harmonic) this persymmetries we can set [8, 9] harmonic function is given by 4 ∆= , (2.14) ˜ n r0 d H =1+ . (2.8) n r where generically 32/2 is the number of unbro- ken supersymmetries. Here r0 is an integration constant with the di- We now consider the limit for which the con- mensions of length. It is related to the mass and stant part in the harmonic function is negligible. charge of the p–brane a follows. The mass τp per We make a co-ordinate transformation and go to unit p–volume is given by the ADM–formula: Z the dual frame 1 − −2 D p m n − b r0 − ( a ) τp = d Σ ∂ hmn ∂mh b λ/r0 d˜ φ 2 = e g∗ = e gE . (2.15) 2κD ∂MD−p−1 r − − 2(D p 3) D−p−3 After these manipulations we can write the near– = 2 r0 ΩD−p−2 . (2.9) ∆κD horizon metric as

2d˜ On the other hand, the charge µp per unit p- 2 −2(1− ∆ )λ/r0 2 2 2 2 ds∗ = e dxd +dλ +r0dΩ ˜+1 , volume is given, in terms of the same integration d (D − 2)ad˜ constant r0, by the Gauss-law formula φ = −λ , (2.16) Z 4∆r0 1 D−p−2 m1···m ˜ (2−k) µ = (d Σ) d+1 g F˜ ··· 2 ˜ p 2 s m1 mD−p−2 which generically (when 1 − d =6 0) describes an 2κD ∆ r d˜+1 AdSd+1 ⊗S geometry and a linear (in λ) dila- ∆ = τp . (2.10) ton background. Another useful and standard 4 ˜ parametrisation, when 1 − 2d =6 0, can be ob- Hence, the p–brane solution satisﬁes the BPS ∆ tained by introducing a radial parameter u with bound r dimensions of mass 4 | | τp = µp . (2.11) β ˜ ∆ r 2d− u = β+1 ,β= 1 (2.17) To derive an expression for r0 in terms of the r0 ∆ string parameters gs and `s, which ﬁxes the scal- −2 and the metric after a rescaling of xd → β xd, ing of H, one must add a source term to the su- can be written as " ( ) # pergravity bulk action. Using the no–force con- 2 dition2 and the fact that in the string frame the 2 2 −2 2 2 du 2 ds∗ = r0 β u dxd + +dΩd˜+1 . 2Alternatively, one can use a scaling argument, see u Appendix B of [10]. (2.18)

2 Quantum aspects of gauge theories, supersymmetry and uniﬁcation Jan Pieter van der Schaar

Reducing over the d˜+ 1 angular variables of Here N denotes the number of stacked branes the sphere we end up with a gauged supergrav- and the scaling of the coupling constant with gs −1 ity in d+1 dimensions supporting a domain–wall is as the inverse tension τp . The undetermined solution. The precise relation between the pa- numerical factor cp and the parameter x depend rameters of the domain–wall supergravity action on the specific field theory under consideration. and its solution in terms of those of the p–brane For a rank q tensor field theory in p + 1 dimen- action (2.1) and its solution (2.16) can be found sions, x is in fact equal to p − 2q − 1. Depending in [10]. on whether x is positive or negative, the factor Summarizing, in this Section we showed that in (3.1) involving N and gs should either become in the dual frame, defined by (2.15), all p–branes large or small in the low energy limit in order to 2 solutions (2.4) have a near–horizon background keep gf fixed. In order for the theory to decou- d˜+1 DWd+1 ⊗ S . The domain–wall metric has all ple from the bulk supergravity we will assume the isometries of an AdS space. These isometries that gs 1(soκD →0 in the low energy limit) are broken in the full background because of the and therefore we take N →∞when gsN has to presence of a non–trivial dilaton. become large to fix gf . As explained in [11] there are two natural energy scales we can keep fixed in the low energy 3. The field theory limit limit, depending on how we probe the collection of D–branes. They concluded that probing a col- In this section we will set up the framework for lection of D–branes with supergravity fields is the the DW/QFT duality similar to the analysis of natural thing to do in the context of holography, [5] but for arbitrary dimensions. As we showed instead of probing the system with another D– in the previous section, the near–horizon geom- brane. In [5] it was observed that in the dual etry of a general p–brane in the dual frame is frame, where the metric describes an AdS space- equivalent to that of a non–dilatonic p–brane. It time, the radial parameter u (2.17) in the metric is therefore natural to assume that the duality (2.18) naturally corresponds to the energy scale might be extended. The presence of the dila- of a supergravity field probe. Clearly this ob- ton turns the AdS background into a DW back- servation can now be extended to all two–block ground and the conformal field theory into a non– p–branes and we will use the energy scale u as conformal QFT, hence the name DW/QFT dual- the fixed holographic energy scale. ity. In the following we will restrict ourselves to We demand the two different energy scales (intersecting) D–branes reduced over all relative to be related by some fixed quantity (in the low 3 transverse directions , giving a particular dila- energy field theory limit). After all, using one tonic p–brane solution in lower dimensions. This or the other probe should not make a (singular) restriction to D–branes (or equivalently k =1), difference. As is explained in [10] this gives the as one might expect, is necessary to obtain non– following constraint on x singular dilaton expressions after taking the limit − ˜ [10]. x =∆ d. (3.2) For the field theory to be non–trivial we need This tells us which coupling constant in the field that at least one coupling constant is fixed in the theory we should keep fixed. When ∆ = 4 and → low energy limit ls 0. Based on dimensional d˜=7−pwe find that x =3−p, which is the scal- analysis and the scaling of the effective tension ing of a Yang–Mills coupling constant appropri- with gs (2.12) we can deduce that the p–brane ate for D–branes in D =10[4,5].When∆=2 worldvolume field theory has a (’t Hooft) cou- and D = 6 we find x =1−p, which is the appro- 2 pling constant gf which can be written as follows: priate scaling of a scalar field coupling constant (and the conformal p = 1 case related to the D1– 2 k x gf = cpNgs ls . (3.1) D5 intersection was already discussed in [1]). 3This also means that these intersecting D–brane con- Rewriting the supergravity background so- figurations are delocalized. lutionintermsofthefixedfieldtheoryquanti-

3 Quantum aspects of gauge theories, supersymmetry and unification Jan Pieter van der Schaar ties and taking the low energy limit gives us the Small curvature can be translated into the fol- near–horizon metric (2.18) (all ls dependence dis- lowing constraint appears) and the non–trivial dilaton becomes 2 3 φ / τD = dpNe 1. (3.5) " !# −(D−2)a β+1 1/d˜ 8 ( β ) φ 1 2 1/x dp This is the effective tension of closed strings in e = (gf ) uβ 1/x . N cp the dual frame and a particle (supergravity) ap- (3.3) proximation is valid when this tension is large. So the complete background solution is well be- We denote the crossover from a supergravity ap- haved everywhere except for the special points proximation to a string theory by the point uN u =0,∞where the dilaton either vanishes or and the crossover from a perturbative quantum blows up. The conjecture is that there is a cor- mechanics model to a strongly coupled quantum respondence between string theory on this DW mechanics model by ug. background and the p–brane theory in the low In the case of the d0–brane the supergravity energy field theory limit. We note that (as in approximation can be used in the following IR D = 10) string quantum effects are 1/N effects energy regime and a supergravity approximation can be used 2 −4 2 3 when the string coupling and the (string) space- uN = gf N u ug = gf , (3.6) time curvature are both small. However, the con- which can only be satisfied for large N.Inthe formal invariance which in the AdS/CFT dual- UV we can use perturbative field theory ity facilitates computations in the strongly coupled field theory is now broken so that any di- u ug , (3.7) rect check of a DW/QFT duality is ruled out. Let us end this section by discussing the six– which in this case reduces to a quantum me- dimensional d0–brane example. For details about chanics model. We therefore find the typical other new examples (like the D8–brane in D =10 DW/QF T behavior that the supergravity regime and the d4–brane in D =64) we refer to [10]. and the perturbative field theory regime do not In D = 6 we keep the scalar coupling con- overlap, avoiding inconsistencies. In the far IR, stant fixed. This also means that the Yang–Mills when coupling constant diverges. Assuming we are on u uN , (3.8) the Higgs branch, giving the gauge fields masses the string coupling becomes large and we could proportional to the diverging Yang–Mills cou- try to use an S–dual description. The different pling constant, the massive gauge fields will de- regimes are plotted in Figure 1. couple and we are left with Higgs branch physics only (also assuming that the Higgs branch de- S-dualSUGRA PQFT couples from the Coulomb branch as claimed in IR UV uuN g [12, 13]). g s >>1 R >>1 Let us investigate where we can trust the perturbative field theory and the supergravity ap- Figure 1: Different regimes in the energy plot for proximation. Clearly the perturbative field the- d0–branes with N 1. ory has become a quantum mechanics model and The d0–brane is related to the D4–brane by the dimensionless effective coupling constant gov- considering a IIA compactification on a K3man- erning the perturbative expansion in this case is ifold. It is conjectured (and by now well estab- defined as lished) that Type IIA superstring theory on a g2 = g2 u. (3.4) eff f K3 manifold is S–dual to Heterotic superstring 4 These cases are special because they require N → theory on a T 4 [14]. On the Heterotic side the ∞ to decouple gravity and the correspondence seems to S–dual soliton solution would be a fundamental indicate that perturbative QFT and DW supergravity are valid in the same energy regime, which at first sight seems state (k = 0) and has a curvature singularity contradictory. at u = 0, so a supergravity approximation will

4 Quantum aspects of gauge theories, supersymmetry and unification Jan Pieter van der Schaar not make sense. The situation resembles the D1– theory should then also be the one living on the brane case in D = 10 Type IIB theory. There the lower–dimensional D–brane. Although the field curvature singularity in the S–dual F1–brane so- theory limit in that case fixes the Yang–Mills cou- lution was resolved by the strong coupling confor- pling constant, there could be a connection with mal fixed point of the (1 + 1)–dimensional gauge the results presented here in the sense that both theory. It is suggestive to propose the occurrence investigations start off with the same intersecting of a similar phenomenon in this case. It would D–brane system. therefore be interesting to determine the strongly coupled IR limit of the corresponding quantum Acknowledgements mechanics model. Let us remark however that in the context of The work reported here is based on the DW/QFT correspondence the quantum me- hep-th/9907006. I thank my collaborators Klaus chanical cases are not well understood [6, 15, 16]. Behrnd, Eric Bergshoeff and Rein Halbersma for This has to do with the fact that quantum me- many discussions. This work is part of the re- chanics does not have ”internal” worldline dy- search program of the “Stichting voor Funda- namics and only when we take the 0–branes apart menteel Onderzoek der Materie” (FOM) and also do we expect to obtain a dynamical model. How- supported by the European Commission TMR ever, when we take the 0–branes apart it is not programme ERBFMRX-CT96-0045, in which J.P. clear what the corresponding dual supergravity v.d. S. is associated to the University of Utrecht. background should be. References 4. Conclusions and discussion [1] J. Maldacena, The large n limit of At the end of this talk let us make the following superconformal field theories and supergravity, remarks. We did not present a detailed investi- Adv. Theor. Math. Phys. 2 (1998) 231, gation of the p–brane worldvolume field theories. [hep-th/9711200]. Our discussion was focussed on the p–brane ge- [2] E. Witten, Anti-de sitter space and holography, ometries in the field theory limit and the search Adv. Theor. Math. Phys. 2 (1998) 253–291, for well–defined supergravity regions. We showed [hep-th/9802150]. that the near–horizon geometry of the D =6 [3]S.S.Gubser,I.R.Klebanov,andA.M. d0–brane indeed has regions where a supergrav- Polyakov, Gauge theory correlators from ity approximation seems valid and the analysis non-critical string theory, Phys. Lett. B428 is strikingly similar to that of the D =10Dp– (1998) 105, [hep-th/9802109]. branes with p<3. [4] N. Itzhaki, J. M. Maldacena, J. Sonnenschein, We did make some general remarks on the and S. Yankielowicz, Supergravity and the large nature of the field theory, which is governed by n limit of theories with sixteen supercharges, scalar dynamics, presumably in the Higgs branch Phys. Rev. D58 (1998) 046004, of the p+1–dimensional gauge theory, consisting [hep-th/9802042]. of supersymmetry vector multiplets and hyper- [5] H. J. Boonstra, K. Skenderis, and P. K. multiplets. This suggests a possible relation with Townsend, The domain wall/qft Matrix theory descriptions of the M5–brane [13] correspondence, JHEP 01 (1999) 003, where a similar limit is described. It would be [hep-th/9807137]. interesting to pursue this connection further. [6]O.Aharony,S.S.Gubser,J.Maldacena, We want to point out that other work was H. Ooguri, and Y. Oz, Large n field theories, done on localized Dp–brane intersections and the string theory and gravity, hep-th/9905111. field theory limit [17, 18, 19]. In these investi- [7] H. Lu, C. N. Pope, and P. K. Townsend, gations a limit is considered taking one into the Domain walls from anti-de sitter spacetime, near–horizon geometry of the lower–dimensional Phys. Lett. B391 (1997) 39–46, D–brane in the intersection and the dual field [hep-th/9607164].

5 Quantum aspects of gauge theories, supersymmetry and uniﬁcation Jan Pieter van der Schaar

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