Dirichlet Scalar Determinants On Two-Dimensional Constant Curvature Disks

TL;DR

This paper calculates scalar determinants on two-dimensional disks with constant curvature using zeta-function regularization, providing explicit formulas for specific mass values, with applications to quantum gravity.

Contribution

It introduces explicit formulas for scalar determinants on curved disks with Dirichlet boundary conditions, including special mass cases, advancing understanding in quantum gravity contexts.

Findings

Explicit determinant formulas for specific mass values.

Elementary function expressions involving Euler Gamma functions.

Applications to Liouville and Jackiw-Teitelboim gravity.

Abstract

We compute the scalar determinants $\det(\Delta+M^{2})$ 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 $\zeta$-function prescription. When $M^{2}=\pm q(q+1)$, $q\in\mathbb N$, a simple expression involving only elementary functions and the Euler $\Gamma$ function is found. Applications to two-dimensional Liouville and Jackiw-Teitelboim quantum gravity are presented in a separate paper.