Approaches to Quantum Gravity

174 G. Horowitz and J. Polchinskiwith the existence of another important set of gauge theory observables, the onedimensionalWilson loops. The Wilson loop can be thought of as creating a stringat the AdS 5 boundary, whose world-sheet then extends into the interior [31; 35].We now drop the pretense of not knowing string theory, and outline the originalargument for the duality in [30]. Maldacena considered a stack of N c parallelD3-branes on top of each other. Each D3-brane couples to gravity with a strengthproportional to the dimensionless string coupling g s , so the distortion of the metricby the branes is proportional to g s N c .Wheng s N c ≪ 1 the spacetime is nearly flatand there are two types of string excitations. There are open strings on the branewhose low energy modes are described by a U(N c ) gauge theory. There are alsoclosed strings away from the brane. When g s N c ≫ 1, the backreaction is importantand the metric describes an extremal black 3-brane. This is a generalization of ablack hole appropriate for a three dimensional extended object. It is extremal withrespect to the charge carried by the 3-branes, which sources the five form F 5 . Nearthe horizon, the spacetime becomes a product of S 5 and AdS 5 . (This is directlyanalogous to the fact that near the horizon of an extremal Reissner–Nordstromblack hole, the spacetime is AdS 2 × S 2 .) String states near the horizon are stronglyredshifted and have very low energy as seen asymptotically. In a certain low energylimit, one can decouple these strings from the strings in the asymptotically flatregion. At weak coupling, g s N c ≪ 1, this same limit decouples the excitationsof the 3-branes from the closed strings. Thus the low energy decoupled physics isdescribed by the gauge theory at small g s and by the AdS 5 × S 5 closed string theoryat large g s , and the simplest conjecture is that these are the same theory as seen atdifferent values of the coupling. 3 This conjecture resolved a puzzle, the fact thatvery different gauge theory and gravity calculations were found to give the sameanswers for a variety of string–brane interactions.In the context of string theory we can relate the parameters on the two sides of theduality. In the gauge theory we have gYM 2 and N c. The known D3-brane Lagrangiandetermines the relation of couplings, gYM2 = 4πg s. Further, each D3-brane is a∫source for the five-form field strength, so on the string side N c is determined bySF 5 5 ; this integrated flux is quantized by a generalization of Dirac’s argumentfor quantization of the flux ∫ SF 2 2 of a magnetic monopole. The supergravity fieldequations give a relation between this flux and the radii of curvature of the AdS 5and S 5 spaces, both being given byl = (4πg s N c ) 1/4 l s . (10.4)3 The U(1) factor in U(N c ) = SU(N c ) × U(1) also decouples: it is Abelian and does not feel the strong gaugeinteractions.