The Heat Kernel on AdS_3 and its Applications

TL;DR

This work develops a group-theoretic harmonic analysis framework to compute heat kernels for tensor fields of arbitrary spin on S^3 and Euclidean AdS_3, including their thermal quotients. By exploiting homogeneous space structure and SU(2)×SU(2) (and its SL(2, C) continuation) representations, the authors obtain explicit eigenfunctions, heat kernels, and coincident kernels, enabling clean one-loop determinant calculations. The resulting one-loop partition function for N=1 supergravity on thermal AdS_3 factorizes into left- and right-moving super Virasoro characters, corroborating Maloney–Witten-type arguments and matching known gravity results in the bosonic sector. The approach provides compact, analytically tractable expressions and suggests broader applications to string theory on AdS_3 and potential extensions to other AdS spacetimes via similar symmetry-based methods.

Abstract

We derive the heat kernel for arbitrary tensor fields on S^3 and (Euclidean) AdS_3 using a group theoretic approach. We use these results to also obtain the heat kernel on certain quotients of these spaces. In particular, we give a simple, explicit expression for the one loop determinant for a field of arbitrary spin s in thermal AdS_3. We apply this to the calculation of the one loop partition function of N=1 supergravity on AdS_3. We find that the answer factorizes into left- and right-moving super Virasoro characters built on the SL(2, C) invariant vacuum, as argued by Maloney and Witten on general grounds.