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

TL;DR

The authors address UV divergences and conformal anomalies for vector fields on spherical domains with conical singularities by deriving the exact spectrum of the vector Laplacian on $S^d_\beta$. They decompose into transversal and longitudinal parts, relate the spectrum to the scalar case, and compute the heat-kernel coefficients via ζ-function residues, obtaining explicit expressions for $\bar{A}_1^{(1)}$ and a detailed form for $\bar{A}_2^{(1)}$ with conical corrections. They also propose a generalization of the second coefficient to arbitrary ${\cal M}_\beta$ and show that gauge-field one-loop divergences can be renormalized by standard gravitational couplings to first order in the deficit angle, mirroring scalar results. The findings underpin off-shell black-hole entropy analyses and offer a framework for extending spectral methods to higher-rank tensors on singular spaces. Overall, the work provides exact spectral data and concrete heat-kernel results that connect geometric singularities to quantum-field theoretic renormalization and conformal anomalies.

Abstract

The spherical domains $S^d_β$ 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 $S^d_β$ 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 $ζ$-function. In particular, the second coefficient, defining the conformal anomaly, is explicitly calculated on $S^d_β$ 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.