# One-loop determinants for black holes in 4d gauged supergravity

## TL;DR

This paper advances the calculation of quantum entropy for supersymmetric black holes in 4d gauged supergravity by evaluating one-loop determinants, crucial for understanding quantum corrections to black hole entropy.

## Contribution

It provides a detailed analysis of one-loop determinants for vector and hypermultiplets on non-compact spaces, filling gaps in previous localization approaches.

## Key findings

- Consistent evaluation of one-loop determinants using multiple techniques.
- Insights into logarithmic corrections to black hole entropy.
- Enhanced understanding of quantum effects in supergravity black holes.

## Abstract

We continue the effort of defining and evaluating the quantum entropy function for supersymmetric black holes in 4d ${\cal N} = 2$ gauged supergravity, initiated in [1803.05920]. The emphasis here is on the missing steps in the previous localization analysis, mainly dealing with one-loop determinants for abelian vector multiplets and hypermultiplets on the non-compact space $\mathbb{H}_2 \times \Sigma_{\rm g}$ with particular boundary conditions. We use several different techniques to arrive at consistent results, which have a most direct bearing on the logarithmic correction terms to the Bekenstein-Hawking entropy of said black holes.