One-loop Partition Functions of 3D Gravity

TL;DR

This paper computes the one-loop partition function for free scalar, gauge, and graviton fields in locally AdS3 backgrounds using heat-kernel techniques. It demonstrates that the one-loop structure reproduces the Brown-Henneaux Virasoro description, yielding explicit BTZ and higher-genus corrections to the gravity partition function, such as $Z_{gravity}(\tau,\bar{\tau}) = |q|^{-2k} \prod_{m=2}^{\infty} 1/|1-q^{m}|^{2}$. By applying the heat-kernel method to quotient spaces $\mathbb{H}_3/\Gamma$, the work connects bulk quantum gravity in AdS3 to boundary CFT data and provides a framework for all-genus (and potentially higher-loop) analyses via Selberg-type traces and geodesic data. The results solidify the emergence of conformal field theory structure from AdS3 quantum gravity and lay groundwork for exploring operator product expansions from gravitational dynamics on hyperbolic manifolds.

Abstract

We consider the one-loop partition function of free quantum field theory in locally Anti-de Sitter space-times. In three dimensions, the one loop determinants for scalar, gauge and graviton excitations are computed explicitly using heat kernel techniques. We obtain precisely the result anticipated by Brown and Henneaux: the partition function includes a sum over "boundary excitations" of AdS3, which are the Virasoro descendants of empty Anti-de Sitter space. This result also allows us to compute the one-loop corrections to the Euclidean action of the BTZ black hole as well its higher genus generalizations.