Heat-kernel Coefficients and Spectra of the Vector Laplacians on Spherical Domains with Conical Singularities

TL;DR

This paper computes the spectrum and heat-kernel coefficients of the vector Laplacian on spherical domains with conical singularities, providing explicit formulas for the conformal anomaly and implications for gauge field renormalization.

Contribution

It offers an exact spectral analysis of the vector Laplacian on conical spherical domains and derives explicit heat-kernel coefficients, including the conformal anomaly, for arbitrary dimensions.

Findings

Exact spectrum of vector Laplacian on $S^d_eta$ calculated.

Explicit second heat-kernel coefficient related to conformal anomaly derived.

Renormalization of gauge fields effectively removes divergences up to first order in conical deficit.

Abstract

The spherical domains $S^d_\beta$ with conical singularities are a convenient arena for studying the properties of tensor Laplacians on arbitrary manifolds with such a kind of singular points. In this paper the vector Laplacian on $S^d_\beta$ is considered and its spectrum is calculated exactly for any dimension $d$. This enables one to find the Schwinger-DeWitt coefficients of this operator by using the residues of the $\zeta$-function. In particular, the second coefficient, defining the conformal anomaly, is explicitly calculated on $S^d_\beta$ and its generalization to arbitrary manifolds is found. As an application of this result, the standard renormalization of the one-loop effective action of gauge fields is demonstrated to be sufficient to remove the ultraviolet divergences up to the first order in the conical deficit angle.