100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

Space Time in String Theory 329domain wall, the solution approaches the CFT dual to the AdS stationarypoint through one of its irrelevant directions. The domain wall is thusinterpreted as holographic renormalization group (HRG) flow. The spatialdirection z along which the domain wall varies, is mapped into the scale ofthe boundary field theory.The holographic renormalization group interpretation of these solutions,removes a paradox, which would have disturbed us if we had tried to makea more conventional interpretation of the wall as interpolating between twovacua of the same theory. Two vacua of the same theory always have thesame high energy behavior. But the CFT on the right hand side of thedomain wall has a more negative cosmological constant, and therefore asmaller density of states at high energy, than that on the left hand side. Inthe HRG interpretation, the loss of degrees of freedom is the usual decreaseof entropy we expect when we go into the infrared along a RG trajectory.Indeed, it has been shown that the holographic version of Zamolodchikov’sC theorem 32 holds for these flows, whenever the bulk SUGRA theory satisfiesthe dominant energy condition 31 .In this context then, stationary points at negative values of the effectivepotential do correspond to theories of quantum gravity in AdS space(under certain conditions), but not to different states of the same theory.Rather they are different quantum field theories, connected by RG flow.This interpretation makes sense when the less negative stationary point isa maximum, and both stationary points have curvature satisfying the BFbound.Note that in general, the functions which appear in RG equations arerenormalization scheme dependent. They do not have any invariant meaningapart from the number of fixed points and the spectrum of dimensions ateach fixed point. In the holographic RG, fixed points are stationary pointsof the bulk supergravity potential. Furthermore, the only domain wall solutionsthat have been given an RG interpretation in AdS/CFT, correspondto flows from a BF allowed saddle point to a stable AdS minimum or otherBF allowed saddle. Other minima of the potential, particularly those withpositive energy density, have not been given a reasonable interpretation inthe boundary field theory.5.2. SUSY and large radius AdS spaceThe relation between black hole and CFT entropy tells us that any CFTwith large entropy would, if it were dual to quantum gravity in an AdS