Vector Fields on a Disk with Mixed Boundary Conditions

TL;DR

This paper investigates vector fields on a disk with mixed boundary conditions derived from BRST-invariance, providing harmonic expansions, eigenfunctions, heat kernel coefficients, and correcting previous analytical errors.

Contribution

It introduces a new approach to vector fields with mixed boundary conditions, linking BRST-invariance to de Rham complex and correcting prior analytical inaccuracies.

Findings

Eigenfunctions expressed in terms of Dirichlet and Robin boundary conditions

First coefficients of heat kernel expansion computed for 4D disk

Correction of previous analytical expression by Branson and Gilkey

Abstract

We study vector fields on a disk satisfying two types of mixed boundary conditions. These boundary conditions are selected by BRST-invariance in electrodynamics. They also appear in the de Rham complex. The manifest construction of the harmonic expansion is presented. The eigenfunctions of the vector Laplace operator are expressed in terms of fields satisfying pure Dirichlet or Robin boundary conditions. For the case of four-dimensional disk several first coefficients of the heat kernel expansion are computed. An error in the analitical expression by Branson and Gilkey is corrected.