Cones, Spins and Heat Kernels

TL;DR

The paper investigates how conical singularities modify heat-kernel traces for fields of spins 0, 1/2, 1, 3/2, and 2, by performing a mode-by-mode analysis in 2D and extending to arbitrary dimensions. It identifies explicit surface (delta-function) corrections to the first Schwinger–DeWitt coefficient $A_1^{(j)}$ due to the singularity, with spins 0, 1/2, and 1 following scalar-like behavior, while spins 3/2 and 2 acquire novel contributions arising from broken local translational isometries near the cone, which persist in the limit $eta\to 2\pi$. The authors provide closed-form expressions for the conical corrections $A_{\\beta,1}^{(j)}$ and discuss their implications for one-loop ultraviolet divergences and the renormalization of black-hole entropy, highlighting that the graviton and gravitino sectors involve surface terms that cannot be absorbed by standard renormalization. The results offer a framework for understanding quantum corrections to gravity on singular spaces and motivate further checks in higher dimensions and for more general backgrounds.

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.