Borel Summation and Analytic Continuation of the Heat Kernel on Hyperbolic Space

TL;DR

This paper explores the Borel summation and analytic continuation of the heat kernel on hyperbolic spaces, revealing dualities and connections to Schrödinger kernels and spherical heat kernels through resummation techniques.

Contribution

It introduces resummation methods using incomplete gamma functions to accurately extend the heat kernel beyond asymptotic regimes and uncovers dualities between short and long time behaviors.

Findings

Resummation provides accurate extrapolations of the heat kernel.

Analytic continuation relates hyperbolic heat kernels to spherical kernels.

Duality between short and long time heat kernels on hyperbolic space.

Abstract

The heat kernel expansion on even-dimensional hyperbolic spaces is asymptotic at both short and long times, with interestingly different Borel properties for these short and long time expansions. Resummations in terms of incomplete gamma functions provide accurate extrapolations and analytic continuations, relating the heat kernel to the Schrodinger kernel, and the heat kernel on hyperbolic space to the heat kernel on spheres. For the diagonal heat kernel there is also a duality between short and long times which mixes the scalar and spinor heat kernels.