Quantum effective action from the AdS/CFT correspondence

We obtain an Einstein metric of constant negative curvature given an arbitrary boundary metric in three dimensions, and a conformally flat one given an arbitrary conformally flat boundary metric in other dimensions. In order to compute the on-shell value of the gravitational action for these solutions, we propose to integrate the radial coordinate from the boundary till a critical value where the bulk volume element vanishes. The result, which is a functional of the boundary metric, provides a sector of the quantum effective action common to all conformal field theories that have a gravitational description. We verify that the so-defined boundary effective action is conformally invariant in odd (boundary) dimensions and has the correct conformal anomaly in even (boundary) dimensions. In three dimensions and for arbitrary static boundary metric the bulk metric takes a rather simple form. We explicitly carry out the computation of the corresponding effective action and find that it equals the non-local Polyakov action. PACS number(s): 11.25.-w,04.62.+v,04.60.-m,11.25.Hf email:[email protected] email:[email protected] 1 There is accumulating evidence that certain conformal field theories (CFT) have a dual description in terms of string (or M) theory on anti-de Sitter (adS) spaces [1–3]. This is a strong/weak coupling duality, so perturbative calculations on one side of the correspondence yield strong coupling results on the other side. In particular, the strong coupling limit of these CFTs has a description in terms of supergravity theory. Certain aspects of the correspondence are rather universal and apply to all cases that such a duality exists. For example, the leading in the large N contribution to the Weyl anomaly is universal [4]. More generally, (certain) correlation functions of the energy momentum tensor at leading order can also be computed by using only the purely gravitational part of the corresponding supergravity and are therefore universal. To obtain this universal behavior one would need to solve Einstein’s equations with arbitrary Dirichlet boundary conditions (more precisely, one needs to determine an Einstein manifold of constant negative curvature given a conformal structure at infinity). This is a rather difficult problem whose solution in all generality has yet to be worked out. An existence theorem for such Einstein’s metrics has been proved by Graham and Lee [5] for manifolds Xd+1 that are topologically a (d+1)-ball, so the boundary at infinity is a d-sphere S, and conformal structures sufficiently close to the standard one. Moreover, one can explicitly obtain an asymptotic expansion of the bulk metric near infinity starting from any boundary metric [6, 4]. This information is sufficient in order to obtain the counterterms that render the effective action finite and the conformal anomaly [4]. However, it is not enough in order to obtain the finite on-shell value of the gravitational action. For this one would need the full solution, not just its asymptotic expansion near infinity. In this letter we obtain a three dimensional Einstein metric of constant negative curvature given any boundary metric and a (d+1)-dimensional (d > 2) conformally flat Einstein metric with conformal structure at infinity represented by an arbitrary conformally flat metric. In the latter case, had we picked the flat metric as a representative of the boundary conformal structure, our solution would reduce to the standard anti-de Sitter metric in Poincaré coordinates, so it is locally isometric to adS spacetime. The