Spectral Geometry and One-loop Divergences on Manifolds with Conical Singularities

TL;DR

The work addresses one-loop divergences produced by conical singularities on manifolds of the form M_alpha = C_alpha × Sigma. It employs the DeWitt-Schwinger proper-time formalism and a Sommerfeld-like integral to obtain a heat-kernel expansion with explicit surface corrections: Tr K_{M_alpha}(s) = (4π s)^{-d/2} ∑ (a_n + a_{alpha,n}) s^n, where a_{alpha,0}=0 and the first two surface terms a_{alpha,1}, a_{alpha,2} depend on the deficit angle α and the near-Sigma geometry. The coefficients are given explicitly in terms of α through c1(α) and c2(α), with c1(α) = 1/6((2π/α)^2 - 1) and c2(α) = 1/15 c1(α)((2π/α)^2 + 11), and involve contractions of the Riemann tensor with normals to Σ. These results are connected to known special cases and imply the necessity of surface counterterms in the effective action, highlighting the impact of conical singularities on renormalization and black hole entropy calculations, and they extend conceptually to higher spin fields.

Abstract

Geometrical form of the one-loop divergences induced by conical singularities of background manifolds is studied. To this aim the heat kernel asymptotic expansion on spaces having the structure $C_α\times Σ$ near singular surface $Σ$ is analysed. Surface corrections to standard second and third heat coefficients are obtained explicitly in terms of angle $α$ of a cone $C_α$ and components of the Riemann tensor. These results are compared to ones to be already known for some particular cases. Physical aspects of the surface divergences are shortly discussed.