Localization in Supergravity and Quantum $AdS_4/CFT_3$ Holography

TL;DR

This paper calculates the quantum gravity partition function in M-theory on $AdS_4 imes X_7$ using localization in supergravity, resulting in an exact expression matching boundary ABJM theory predictions.

Contribution

It introduces a localization technique in 4D gauged supergravity to compute the quantum gravity partition function in $AdS_4$, connecting bulk calculations with boundary CFT results.

Findings

Partition function reduces to a finite integral over instanton solutions.

Resulting partition function matches the Airy function form predicted by boundary theory.

The computation is nonperturbatively exact in Planck length, ignoring brane-instanton corrections.

Abstract

We compute the quantum gravity partition function of M-theory on $AdS_4 \times X_7 $ by using localization techniques in four-dimensional gauged supergravity obtained by a consistent truncation on the Sasaki-Einstein manifold $X_{7}$. The supergravity path integral reduces to a finite dimensional integral over two collective coordinates that parametrize the localizing instanton solutions. The renormalized action of the off-shell instanton solutions depends linearly and holomorphically on the "square root" prepotential evaluated at the center of $AdS_{4}$. The partition function resembles the Laplace transform of the wave function of a topological string and with an assumption about the measure for the localization integral yields an Airy function in precise agreement with the computation from the boundary ABJM theory on a 3-sphere. Our bulk quantum gravity computation is nonperturbatively exact in four-dimensional Planck length but ignores corrections due to brane-instantons.