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Gauge Fields, Scalars, Warped Geometry, and Strings

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Gauge Fields, Scalars, Warped Geometry, and Strings SLAC-PUB-8671 December 2000 hep-th/0010144 Gauge Fields, Scalars, Warped Geometry, and Strings Eva Silverstein Department of Physics Stanford University, Stanford, CA 94305 Stanford Linear Accelerator Center Stanford University, Stanford, CA 94309 Presented at the Strings 2000 Conference, 7/10/2000—7/15/2000, Ann Arbor, MI, USA Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309 Work supported by Department of Energy contract DE–AC03–76SF00515. hep-th/0010144 SLAC-PUB-8671 Gauge Fields, Scalars, Warp ed Geometry, and Strings EvaSilverstein DepartmentofPhysics and SLAC Stanford University Stanford, CA 94305/94309 We review results on several interesting phenomena in warp ed compacti cations of M theory,aspresented at Strings 2000. The b ehavior of gauge elds in dimensional reduction from d +1 to d dimensions in various backgrounds is explained from the p ointof view of the holographic duals (and a p oint raised in the question session at the conference is addressed). We summarize the role of additional elds (in particular scalar elds) in 5d warp ed geometries in making it p ossible for Poincare-invariant domain wall solutions to exist to a nontrivial order in a controlled approximation scheme without ne-tuning arXiv:hep-th/0010144 v2 19 Oct 2000 of parameters in the 5d action (and comment on the status of the singularities arising in the general relativistic description of these solutions). Finally,we discuss brie y the emergence of excitations of wrapp ed branes in warp ed geometries whose e ective thickness, as measured along the Poincare slices in the geometry, grows as the energy increases. Octob er 2000 1. Intro duction Generic general-relativistic spacetime backgrounds with d-dimensional Poincare in- variance have a metric of the form 2A(y ) I J e dx dx + H (y )dy dy (1:1) IJ where ; =0;:::;d 1. Canonical examples in M theory with a nontrivial warp factor 2A(y ) e include heterotic compacti cations with (0; 2) worldsheet sup ersymmetry [1], com- pacti ed Horava-Witten theory [2], AdS and its relevant deformations, linear dilaton d+1 theories such as for example the NS5-brane solution [3] and the conifold singularity [4][5], and no doubt many more solutions yet to b e discovered with less sup ersymmetry. Manywarp ed backgrounds have non-gravitational holographic duals [6], but most have no known equivalent \b oundary theory". Almost all of these backgrounds have curvature singularities and/or strong coupling at some nite prop er distance from a generic p oints I on the comp onent of the geometry parameterized by y , so that general relativity breaks down in this region of the background. It is imp ortantandinteresting to understand as precisely as p ossible the physics of this typ e of background, in particular to see if any new phenomena emerge from the warp ed shap e of the spacetime. In this talk I will review results on three asp ects of this physics: (1) The b ehavior of gauge elds (2) The role of for example scalar elds in making p ossible, to the leading order in a controlled approximation scheme, solutions with Poincare invariance even after some non- trivial quantum corrections to the vacuum energy have b een included, and (3) The b ehavior of massive states coming from wrapp ed branes in this sort of geometry: in particular one nds a new corner of the theory where excitations can b e seen to grow in size as they grow in energy as a consequence of the warping in the metric (1.1). 2. Gauge Fields If we fo cus on cases where the warping o ccurs along a single direction y , the low energy e ective action (to the extent that it is reliable) takes the form Z p d 2 2 S = d xdy g a()R + b()(r) + c()F () : (2:1) 1 If the integral over y of the Einstein term is nite one obtains a nite d-dimensional Planck scale and \trapp ed gravity" [7]. This holds also for a d-dimensional graviphoton that arises from a d + 1-dimensional two-form p otential, as demonstrated recently in [8]. On the other hand if there is a d + 1-dimensional vector p otential, the dimensional reduction of its kinetic term mightgive a divergence indep endent of what is going on with the graviton R d 2 kinetic term. In particular in the d xdy F term there are twopowers of the inverse d +1- dimensional metric as opp osed to the single p ower of g in the dimensional reduction of the Einstein term. R 1 d 2 The co ecientofthed-dimensional gauge kinetic term d xtr F is the inverse 2 e d e ective gauge coupling (charge) squared of the d-dimensional gauge theory. If the e ective gauge coupling in d-dimensions is zero, then one might naively infer that this symmetry b ehaves like a global symmetry rather than a gauge symmetry after dimensional reduction. This would b e very surprising since the black hole no-hair theorems, at least in contexts where they have b een studied, indicate that information ab out global charges is lost in pro cesses in which the black hole absorbs particles which carried this charge. On the other hand in this context, where the symmetry is a b ona de gauge symmetry in d +1 dimensions, the charges must b e conserved. 1 In [9] we found that in several known examples of warp ed geometries in which 2 e d diverges, this divergence is either indicative of a conventional screening e ect or the e ective theory (2.1) breaks down at some y<1 where new b ehavior takes over that also has a conventional b ehavior in the IR of the d-dimensional description of the physics. The simplest example is the cuto AdS background studied by Randall and Sundrum 5 [7]. In this background, A(y )=jy j=L where L is the curvature radius of AdS. The calculation giving the e ectivecharge in 4d is Z y 0 1 1 4y=L 2y=L 2 dy e (e ) y (2:2) 0 2 2 e e 0 4 5 where wehaveintro duced an IR momentum cuto p through the relation y Ll og (p L) 0 0 0 p g , and the last following from the metric. The rst factor in the integrand comes from 2 from the twopowers of the inverse metric involved in forming tr F . So we are nding 1 1 Ll og (p L) (2:3) 0 2 2 e e 4 5 This logarithmic IR divergence is the result one would exp ect from screening of the charge from the 4d description (as rst p ointed out to us by E. Witten, and as rst noted in 2 pap ers of Pomaral [10]). This interpretation is con rmed by an explicit calculation of the electrostatic p otential arising from a p ointsourceofcharge at y =0: QK (pL) Q 1 A (p; y =0)= ! (2:4) 0 p!0 2 2K (pL) p logp 0 Similarly one nds a generalization of the screening e ect to higher dimensions and higher-form elds. In dimension d, for a q -form eld strength, as a function of IR momen- tum cuto p ,we nd 0 R 1 1 (2q d) L / e 2q 6= d 2q d 2 e p q (2:5) 0 R log (p ) 2q = d 0 At the conference, M. Du asked ab out the consistency of this result with the p os- sibility of dualizing q -form eld strengths to 5 q -form eld strengths. For example a scalar eld with a 1-form eld strength would b e dual to a 3-form p otential eld C with a 4-form eld strength. The latter, from (2.5), gives a mo de with zero charge up on di- mensional reduction; whereas a scalar eld, likegravity, is left with nontrivial interactions after dimensional reduction. I b elieve the answer to this is as follows (this result was obtained in collab oration withM.Schulz). The equation for dualizing a form, for example d = dC , is a linear di erential equation which lo cally has a solution. There is no guarantee, however, that this solution is nonsingular everywhere. Consider the equation for a scalar eld in the background (1.1). A massless mo de in 4d satis es the equation 00 0 0 +4 A =0 (2:6) where primes denote di erentiation with resp ect to y . One obvious solution is the zero mo de, = (x) indep endentof y . This is the mo de which gets \trapp ed" up on dimensional reduction, with a nite kinetic term. As noted by Du in his question, this cannot b e the solution dual to the three-form p otential. There is another solution to (2.6), which b ecomes singular at the AdS horizon in the RS geometry.Integrating (2.6), this solution satis es 0 4A(y ) =^ (x)e (2:7) 3 For the RS geometry, this yields +4y=L ^(x)e (2:8) The dimensional reduction of the kinetic term for this scalar go es like Z y 0 4y=L 4y=L 2 4y =L 0 dy e (e ) e (2:9) 0 p +4y=L from one p ower of g and twopowers of e from the solution (2.8) app earing in the quadratic action for . This is the same divergence which arises for the dimensional reduction of the standard zero-mo de solution for the three-form p otential C , whichgoeslike Z y 0 4y=L 2y=L 4 dy e (e ) (2:10) 0 p g and four p owers of the inverse metric required to form the square from one p ower of of the four-form eld strength.
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