Small Mass Expansion of Functional Determinants on the Generalized Cone

TL;DR

This paper derives a small mass expansion for the functional determinant of a scalar Laplacian on a generalized cone, providing explicit formulas for specific dimensions and boundary conditions using zeta function regularization.

Contribution

It introduces a general method to compute the small mass expansion of functional determinants on generalized cones for various boundary conditions and dimensions.

Findings

Explicit formulas for the functional determinant in dimensions 2, 3, 4, 5.

General expression valid for any dimension and boundary condition.

Application to the case of a spherical base manifold.

Abstract

In this paper we compute the small mass expansion for the functional determinant of a scalar Laplacian defined on the bounded, generalized cone. In the framework of zeta function regularization, we obtain an expression for the functional determinant valid in any dimension for both Dirichlet and Robin boundary conditions in terms of the spectral zeta function of the base manifold. Moreover, as a particular case, we specify the base to be a $d$-dimensional sphere and present explicit results for $d=2,3,4,5$.