Factorization of log-corrections in AdS$_4$/CFT$_3$ from supergravity localization

TL;DR

This paper derives a universal formula for one-loop logarithmic corrections in AdS4/CFT3 using supergravity localization and the Atiyah-Singer index theorem, linking corrections to topological data and degrees of freedom.

Contribution

It introduces a fixed-point formula for logarithmic corrections in supersymmetric partition functions in AdS4 backgrounds, connecting them to topological invariants and dynamical coefficients.

Findings

Factorization of one-loop determinants on fixed points and two-manifolds.

A universal fixed-point formula for logarithmic corrections.

Corrections depend on topological data and degrees of freedom.

Abstract

We use the Atiyah-Singer index theorem to derive the general form of the one-loop corrections to observables in asymptotically anti-de Sitter (AdS$_4$) supersymmetric backgrounds of abelian gauged supergravity. Using the method of supergravity localization combined with the factorization of the supergravity action on fixed points (NUTs) and fixed two-manifolds (Bolts) we show that an analogous factorization takes place for the one-loop determinants of supergravity fields. This allows us to propose a general fixed-point formula for the logarithmic corrections to a large class of supersymmetric partition functions in the large $N$ expansion of a given 3d dual theory. The corrections are uniquely fixed by some simple topological data pertaining to a particular background in the form of its regularized Euler characteristic $\chi$, together with a single dynamical coefficient that counts the underlying degrees of freedom of the theory.