Black hole determinants and quasinormal modes

TL;DR

This work derives a general, analytically tractable representation of one-loop determinants in thermal spacetimes as products over quasinormal modes, with a local polynomial Pol(Δ) fixed by heat-kernel data. The authors demonstrate the approach in thermal AdS, BTZ, and de Sitter backgrounds, providing exact expressions and efficient routes for evaluating UV divergences via Pol and for extracting physically meaningful contributions from a few dominant QNMs. By linking determinants to the poles of retarded Green's functions in holographically dual theories, the paper offers a concrete framework for 1/N corrections and for interpreting thermal dynamics of strongly coupled systems. The results include new exact formulas in dS spaces and detailed BTZ/AdS$_3$ checks, along with practical numerical strategies and a broad generalization to other quantum numbers, with potential broad impact on holographic thermodynamics and quantum gravity calculations.

Abstract

We derive an expression for functional determinants in thermal spacetimes as a product over the corresponding quasinormal modes. As simple applications we give efficient computations of scalar determinants in thermal AdS, BTZ black hole and de Sitter spacetimes. We emphasize the conceptual utility of our formula for discussing `1/N' corrections to strongly coupled field theories via the holographic correspondence.