On the quantum entropy function in 4d gauged supergravity

TL;DR

This work applies Sen's quantum entropy function to BPS black holes in 4d ${\cal N}=2$ gauged supergravity using bulk localization within the Euclidean conformal supergravity framework. By freezing the Weyl multiplet and localizing the matter sector around the half-BPS AdS$_2\times$S$^2$ attractor, the full path integral reduces to a finite-dimensional integral over near-horizon moduli $\phi_+^I$ with a universal delta constraint, plus an undetermined one-loop factor $Z^{\rm reg}_{\rm ind}(\phi_+)$. The renormalized action on the localization locus yields a simple saddle structure that reproduces the attractor entropy, while holographic comparisons with topologically twisted indices in dual field theories show agreement at leading order in large $N$ across several AdS/CFT examples, including CK, BB, GR, and massive IIA setups. The approach provides a clear route to finite-$N$ corrections through $Z^{\rm reg}_{\rm ind}$ and motivates deeper exploration of Weyl-multiplet fluctuations and exact holographic matches. Overall, the paper establishes a gravitational counterpart to exact supersymmetric partition function evaluations in holographically dual theories and outlines the key steps required for a full finite-$N quantum entropy program.

Abstract

We analyze BPS black hole attractors in the conformal 4d gauged supergravity formalism and apply the technique known as supergravity localization in order to evaluate Sen's quantum entropy function in the $\mathrm{AdS}_2\!\times\!\mathrm{S}^2$ near-horizon geometry. Under certain assumptions, we reduce the exact expression of the functional integral to a finite-dimensional integral for a number of supersymmetric black holes in gauged supergravity with AdS asymptotics subject to a holographic description via a dual field theory. Examples include the asymptotically $\mathrm{AdS}_4\!\times\!\mathrm{S}^7$ Cacciatori-Klemm black holes in M-theory and the asymptotically $\mathrm{AdS}_5\!\times\!\mathrm{S}^5$ generalizations of Gutowski-Reall black holes and Benini-Bobev black strings in type IIB, as well as the recently constructed asymptotically $\mathrm{AdS}_4\!\times\!\mathrm{S}^6$ solutions in massive type IIA. Our results provide an important first step towards a gravitational counterpart to the exact evaluation of supersymmetric partition functions at finite $N$ for the holographically dual field theories in these examples.