Cones, Spins and Heat Kernels

TL;DR

This paper investigates the heat kernels of Laplacians for various spin fields on manifolds with conical singularities, revealing how singularities affect quantum corrections to black hole entropy.

Contribution

It provides a detailed mode-by-mode analysis of heat kernels for multiple spins on conical manifolds and identifies new effects on quantum corrections due to singularities.

Findings

Corrections to Schwinger-DeWitt coefficients due to conical singularities are derived.

Heat kernel traces for spins 1/2 and 1 resemble the scalar case.

Unique features are found for spin 2 and 3/2 operators near conical singularities.

Abstract

The heat kernels of Laplacians for spin 1/2, 1, 3/2 and 2 fields, and the asymptotic expansion of their traces are studied on manifolds with conical singularities. The exact mode-by-mode analysis is carried out for 2-dimensional domains and then extended to arbitrary dimensions. The corrections to the first Schwinger-DeWitt coefficients in the trace expansion, due to conical singularities, are found for all the above spins. The results for spins 1/2 and 1 resemble the scalar case. However, the heat kernels of the Lichnerowicz spin 2 operator and the spin 3/2 Laplacian show a new feature. When the conical angle deficit vanishes the limiting values of these traces differ from the corresponding values computed on the smooth manifold. The reason for the discrepancy is breaking of the local translational isometries near a conical singularity. As an application, the results are used to find the ultraviolet divergences in the quantum corrections to the black hole entropy for all these spins.