Functional determinants, index theorems, and exact quantum black hole entropy

TL;DR

This paper computes the one-loop quantum corrections to BPS black hole entropy in four-dimensional  supergravity using localization and index theorems, providing an exact formula that extends previous large-charge results.

Contribution

It introduces a novel calculation of fluctuation determinants for supergravity multiplets, extending entropy results to finite charges and clarifying cancellations in higher supersymmetry cases.

Findings

Derived an exact quantum entropy formula for BPS black holes.

Extended large-charge entropy computations to finite charges.

Explained cancellations in -BPS black holes in  supergravity.

Abstract

The exact quantum entropy of BPS black holes can be evaluated using localization in supergravity. An important ingredient in this program, that has been lacking so far, is the one-loop effect arising from the quadratic fluctuations of the exact deformation (the $Q\mathcal{V}$ operator). We compute the fluctuation determinant for vector multiplets and hyper multiplets around $Q$-invariant off-shell configurations in four-dimensional $\mathcal{N}=2$ supergravity with $AdS_{2} \times S^{2}$ boundary conditions, using the Atiyah-Bott fixed-point index theorem and a subsequent zeta function regularization. Our results extend the large-charge on-shell entropy computations in the literature to a regime of finite charges. Based on our results, we present an exact formula for the quantum entropy of BPS black holes in $\mathcal{N}=2$ supergravity. We explain cancellations concerning $\frac18$-BPS black holes in $\mathcal{N}=8$ supergravity that were observed in arXiv:1111.1161. We also make comments about the interpretation of a logarithmic term in the topological string partition function in the low energy supergravity theory.