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 exam- ples 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, " = −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 en- ergy 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.