The C_2 heat-kernel coefficient in the presence of boundary discontinuities

TL;DR

This paper derives the boundary discontinuity contributions to the C_2 heat-kernel coefficient on manifolds with piecewise smooth boundaries, using symmetry, specific manifold evaluations, and perturbation techniques.

Contribution

It provides the first detailed expression for boundary discontinuity contributions to the heat-kernel coefficient for scalar fields with various boundary conditions.

Findings

Derived geometrical quantities for boundary discontinuities.

Calculated contributions to smeared heat-kernel coefficient and cocycle function.

Developed a perturbation technique for Robin boundary conditions.

Abstract

We consider the heat-kernel on a manifold whose boundary is piecewise smooth. The set of independent geometrical quantities required to construct an expression for the contribution of the boundary discontinuities to the C_{2} heat-kernel coefficient is derived in the case of a scalar field with Dirichlet and Robin boundary conditions. The coefficient is then determined using conformal symmetry and evaluation on some specific manifolds. For the Robin case a perturbation technique is also developed and employed. The contributions to the smeared heat-kernel coefficient and cocycle function are calculated. Some incomplete results for spinor fields with mixed conditions are also presented.