Localization of the action in AdS/CFT

Abstract

We derive a simple formula for the action of any supersymmetric solution to minimal gauged supergravity in the AdS 4 /CFT 3 correspondence. Such solutions are equipped with a supersymmetric Killing vector, and we show that the holographically renormalized action may be expressed entirely in terms of the weights of this vector field at its fixed points, together with certain topological data. In this sense, the classical gravitational partition function localizes in the bulk. We illustrate our general formula with a number of explicit examples, in which exact dual field theory computations are also available, which include supersymmetric Taub-NUT and Taub-bolt type spacetimes, as well as black hole solutions. Our simple topological formula also allows us to write down the action of any solution, provided it exists.

Euclidean black saddles and AdS4 black holes

“…We illustrate this in figure 1. The regularized on-shell action I for the Euclidean black saddle solutions in (2.13) was computed recently in [30]. 7 The result is that for any value of Q one finds the simple answer 16) and so it is independent of Q.…”

Topologically twisted index in the ’t Hooft limit and the dual AdS4 black hole entropy

“…Alternatively, and perhaps more conveniently, we might simply take the four-dimensional point of view. Most of the analysis of the solution was already performed in [11] and rigorous approaches to holographic renormalization of this background were recently presented in [38][39][40]. The bottom line is that given the prepotential the 4d…”

Unreasonable Effectiveness of Higher Derivative Supergravity in AdS4 Holography

“…As shown in Ref. [22], the two-derivative on-shell action F localizes on the fixed points of the preserved equivariant supercharges, and this principle allows for its explicit evaluation for generic NUT or Bolt solutions. Our result (11) demonstrates that the higher-derivative corrections to the on-shell action can also be written purely in terms of topological fixed point data.…”

“…It should therefore be possible to repeat the analysis of Ref. [22] for the most general supersymmetric solutions of the four-derivative action in Eq. (5) and establish a higher-derivative generalization of their localization results.…”

“…11 Table II. There are two types of contributions to the on-shell action of supersymmetric solutions, which can be classified by the dimension of the fixed loci under the isometry required by supersymmetry [21,22]. As shown in Ref.…”