Zeta function for the Laplace operator acting on forms in a ball with gauge boundary conditions

TL;DR

This paper computes the zeta function, functional determinants, and Casimir energies for the Laplace operator on antisymmetric tensor fields in a Euclidean ball with gauge boundary conditions, providing explicit results for various dimensions.

Contribution

It explicitly derives the eigenfunctions and zeta function for the Laplace operator on forms with gauge boundary conditions in a ball, including exact evaluations of $ ext{zeta}(0)$ and $ ext{zeta}'(0)$ for multiple dimensions.

Findings

Explicit eigenfunctions for all D

Exact expressions for $ ext{zeta}(0)$ and $ ext{zeta}'(0)$

Computed Casimir energies for D=3 to 6

Abstract

The Laplace operator acting on antisymmetric tensor fields in a $D$--dimensional Euclidean ball is studied. Gauge-invariant local boundary conditions (absolute and relative ones, in the language of Gilkey) are considered. The eigenfuctions of the operator are found explicitly for all values of $D$. Using in a row a number of basic techniques, as Mellin transforms, deformation and shifting of the complex integration contour, and pole compensation, the zeta function of the operator is obtained. From its expression, in particular, $\zeta (0)$ and $\zeta'(0)$ are evaluated exactly. A table is given in the paper for $D=3, 4, ...,8$. The functional determinants and Casimir energies are obtained for $D=3, 4, ...,6$.