Heat-kernel coefficients for oblique boundary conditions

TL;DR

This paper computes heat-kernel coefficients for a U(1) bundle on a 4-Ball with oblique boundary conditions, revealing potential ellipticity breakdowns and constraining coefficient forms.

Contribution

It provides explicit calculations of heat-kernel coefficients under oblique boundary conditions, a novel analysis in this geometric setting.

Findings

Calculated coefficients up to a2 for the specified boundary conditions.

Identified conditions leading to breakdown of ellipticity.

Placed restrictions on the general form of heat-kernel coefficients.

Abstract

We calculate the heat-kernel coefficients, up to $a_2$, for a U(1) bundle on the 4-Ball for boundary conditions which are such that the normal derivative of the field at the boundary is related to a first-order operator in boundary derivatives acting on the field. The results are used to place restrictions on the general forms of the coefficients. In the specific case considered, there can be a breakdown of ellipticity.