Quantum effective action from the AdS/CFT correspondence

TL;DR

This paper derives a boundary effective action from the AdS/CFT correspondence by computing the on-shell gravitational action for specific bulk solutions, revealing conformal invariance and anomalies, and explicitly obtaining the Polyakov action in three dimensions.

Contribution

It introduces a method to compute the quantum effective action from the gravitational dual, applicable to various boundary geometries, and verifies its conformal properties and anomalies.

Findings

The boundary effective action is conformally invariant in odd dimensions.

In even dimensions, it reproduces the correct conformal anomaly.

In three dimensions, the effective action matches the Polyakov action.

Abstract

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.