Dirichlet Scalar Determinants On Two-Dimensional Constant Curvature Disks

Abstract

We compute the scalar determinants $\det (\hat{I}”+M^2)$ on the two-dimensional round disks of constant curvature $R=0$, $\mp 2$, for any finite boundary length $\ell$ and mass $M$, with Dirichlet boundary conditions, using the $\hat{I}\P$-function prescription. When $M^2=\pm q(q+1)$, $q\in \mathbb{N}$, a simple expression involving only elementary functions and the Euler $\hat{I}“$ function is found. Applications to two-dimensional Liouville and Jackiw-Teitelboim quantum gravity are presented in a separate paper.

40 pages; minor changes have been made in the abstract and the introduction, a subsection has been added with the exact computation of the determinant on a hemisphere, version matches with the one to be published