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Gauge Symmetry and Localized Gravity in M Theory

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Gauge Symmetry and Localized Gravity in M Theory SLAC-PUB-8482 December 2000 hep-th/0006192 SU-ITP-00/17 Gauge Symmetry and Localized Gravity in M Theory Nemanja Kaloper1, Eva Silverstein 1,2, and Leonard Susskind1 1Department of Physics Stanford University, Stanford, CA 94305 2Stanford Linear Accelerator Center Stanford University, Stanford, CA 94309 Submitted to Journal of High Energy Physics (JHEP) Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309 Work supported by Department of Energy contract DE–AC03–76SF00515. hep-th/0006192 SLAC-PUB-8482 SU-ITP-00/17 Gauge Symmetry and Lo calized Gravity in M Theory 1 1;2 1 Nemanja Kalop er ,Eva Silverstein , and Leonard Susskind 1 DepartmentofPhysics Stanford University Stanford, CA 94305 2 Stanford Linear Accelerator Center Stanford University Stanford, CA 94309 We discuss the p ossibilityofhaving gravity \lo calized" in dimension d in a system where gauge b osons propagate in dimension d+1. In such a circumstance|dep ending arXiv:hep-th/0006192 v2 26 Jun 2000 on the rate of fallo of the eld strengths in d dimensions|one might exp ect the gauge symmetry in d+1 dimensions to b ehavelike a global symmetry in d dimensions, despite the presence of gravity. Naive extrap olation of warp ed long-wavelength solutions of general relativity coupled to scalars and gauge elds suggests that such an e ect might b e p ossible. However, in some basic realizations of such solutions in M theory,we nd that this e ect do es not p ersist microscopically. It turns over either to screening or the Higgs mechanism at long distances in the d-dimensional description of the system. We brie y discuss the physics of charged ob jects in this typ e of system. June 2000 1. Intro duction and Summary In the study of string dualities and the relation of string theory to eld theory, the lo calization of gauge dynamics (or more general quantum eld theory dynamics) to a sub- manifold of spacetime has b een analyzed in many contexts. More recently the p ossibility of lo calizing gravity has emerged in the study of cut-o AdS spaces [1]. It is natural to wonder therefore whether it is p ossible to lo calize gravityalonga d-dimensional slice of spacetime in a system where the gauge elds of some symmetry group G propagate in d +1 dimensions. In the most extreme imaginable versions of such a situation, in a d-dimensional descrip- tion the symmetry G would app ear more like a global symmetry than a gauge symmetry. Field lines would fall o faster than appropriate for a gauge symmetry in d-dimensions. On the other hand the conservation of the asso ciated charge would b e guaranteed by the fact that the symmetry was gauged in d + 1 dimensions. Because gravity is lo calized to d dimensions, one would then have a global symmetry in the presence of gravity. Since the no-hair theorems for black holes in ordinary d-dimensional e ective eld theory suggest strongly that such an e ect is imp ossible [2], it is likely that this situation could only o ccur, if at all, in systems with an in nite numb er of degrees of freedom in a d-dimensional de- scription. In systems with a holographic description in terms of a low-energy worldvolume quantum eld theory on (d-1)-branes, the physics must have a conventional interpretation in d dimensions. In this pap er we will obtain results consistent with this exp ectation by analyzing various systems in M theory. At the level of low-energy e ective eld theory in d +1 dimensions, the e ect app ears p ossible, even generic. Consider a d + 1-dimensional system involving a gauge theory with eld strength F coupled to a scalar and gravity. The action takes the form Z p d 2 2 S = d xdr g a()R + b()(r) + c()F () : (1:1) Here a; b; c and are general functions of . Manysuchsystemshave\warp ed" solutions which give a lo calized graviton in d dimensions up on integrating over the d + 1st co ordinate 2 r in (1.1). The dimensional reduction of the F term can in general b ehavedi erently from that of the Einstein term, since its kinetic term involves an extra p ower of the inverse metric g relativetothatofgravity, and since its coupling to the scalar will in general di er. At the level of (1.1) (without worrying ab out its emb edding into quantum gravity) there 1 2 will b e a large class of systems for which the co ecientoftheF term will diverge. This indicates that the long-distance b ehavior of the F eldisweaker than that of an ordinary unscreened un-Higgsed gauge eld in d dimensions. 1 One such case is the cuto AdS system with d =4,whichwestudyinx2. In this 5 2 system, however, the divergence in the F term is logarithmic in the energy scale of the dual conformal eld theory coupled to gravity, so that the e ect is identical to that of 2 screening of the charge in 4 dimensions. We calculate the electrostatic p otential for a test charge at the \Planck brane" and nd a result consistent with this interpretation. In realizations of this system in typ e I IB sup ergravity, the charged matter that leads to the screening is evident, although the e ect arises from geometry indep endent of assuming the presence of charged matter in the bulk. We also study the pattern of infrared divergences in kinetic terms for arbitrary q - form gauge p otentials in arbitrary dimension. This suggests a \screening" phenomenon for higher-form elds in a range of dimensions. A more interesting case is a linear dilaton solution of string theory, whichwe consider in x3. We study in particular the typ e I IA and typ e I IB Neveu-Schwarz vebrane solutions 2 (for which d = 6). In this case, the string theory solution again has a diverging F term while the Einstein term survives with a nite co ecient up on dimensional reduction from 7d to 6d. The gauge eld propagates as if in 7d at space in the linear dilaton solution. However, the I IA solution gets corrected to one lo calized along the eleventh dimension of M-theory [5], so that the symmetry is e ectively Higgsed. In the I IB case one also nds a lifting of the RR two-form p otential from e ects o ccurring in a region where the linear dilaton solution has broken down. In x 4we discuss some asp ects of the physics of charged black holes that make the d + 1 and d dimensional descriptions of these systems consistent. Finally in x5we discuss other long-wavelength solutions which naively exhibit this e ect and discuss prosp ects for realizing them in M theory. One p ossible application is to the problem of compactifying matrix theory down to four dimensions. This requires considering D0-branes in a I IA compacti cation down to three dimensions. This is problematic in part b ecause of the in nite classical electrostatic self-energy of the D0-brane in 3d. If the electric eld lives in 4d, while gravity is lo calized to 3d, this problem maybeavoided. 1 Gauge elds in the bulk of the Randall-Sundrum approach to the hierarchy problem [3] were studied by [4]. 2 We thank E. Witten for p ointing this out. 2 2. Cuto AdS Space In AdS spaces cut o bya Poincare-invariant \Planck brane", the metric can b e written 2 2jr j=L 2 2 ds = e dx + dr (2:1) jj where x refers to the dimensions along the brane. The dimensional reduction of the jj Einstein term in the 5d action gives a nite 4d PlanckscaleM [1]: 4 Z 1 2 3 2r=L M / M dr e (2:2) 4 5 0 where M is the 5d Planck scale. On the other hand, if weintro duce a Maxwell eld, its 5 kinetic term in 4d is divergent. In terms of an infrared cuto R, the corresp onding integral is Z R 1 dr / R; (2:3) 2 e 0 which diverges as R !1.From the metric (2.1), this cuto scale R on the co ordinate r 1 corresp onds to -Llog (k L) where k L is an infrared momentum cuto along the four 0 0 dimensions parameterized by x . So this e ect is quite conventional in four dimensions: jj the charge is screened by the charged matter in the 4d conformal eld theory dual to this 3 background [6]. This result agrees with that obtained by a direct calculation of the electrostatic p o- tential of a charge lo calized on the Planck brane. (See [4] for similar calculations for gauge elds in [3], and for example [7][8][9] for analogous calculations of corrections to the grav- itational propagator.) The electrostatic p otential is found byintegrating the Maxwell's equation MN N r F = QJ (2:4) M in the background geometry (2.1) in the electrostatic approximation, when the vector p o- 1 M (4) p tential is A =(; 0; 0; 0; 0) and the density currentisJ = (x x )(1; 0; 0; 0; 0) M 0 g (here x stands for all spatial co ordinates in (2.1)). To compute the p otential, it is con- venienttocho ose the co ordinates such that the metric (2.1) is conformally at. De ning jz j + L = L exp (jr j=L), the metric b ecomes 2 L 2 2 2 + dz ); (2:5) ds = (dx jj 2 (jz j + L) 5 2 3 In the realizati on of AdS in the I IB compacti cation on S , there is a factor of N in the 5 2 5 8 2 2 3 expression for the renormalized 1=e .
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