The trace of the heat kernel on a compact hyperbolic 3-orbifold

TL;DR

This paper analyzes the heat kernel trace on compact hyperbolic 3-orbifolds, showing how elliptic and hyperbolic elements influence heat coefficients and discussing physical implications.

Contribution

It provides explicit evaluation of heat coefficients on hyperbolic 3-orbifolds with elliptic and hyperbolic elements, highlighting their distinct effects on the heat kernel trace.

Findings

Hyperbolic elements cause exponentially small corrections.

Elliptic elements modify all heat coefficients except the Weyl term.

The Weyl term remains unaffected by the group elements.

Abstract

The heat coefficients related to the Laplace-Beltrami operator defined on the hyperbolic compact manifold $H^3/\Ga$ are evaluated in the case in which the discrete group $\Ga$ contains elliptic and hyperbolic elements. It is shown that while hyperbolic elements give only exponentially vanishing corrections to the trace of the heat kernel, elliptic elements modify all coefficients of the asymptotic expansion, but the Weyl term, which remains unchanged. Some physical consequences are briefly discussed in the examples.