The Selberg trace formula for Dirac operators

TL;DR

This paper derives a Selberg trace formula for Dirac operators on compact hyperbolic surfaces, linking their spectra to geometric and spectral properties, with implications for symmetry and multiplicity of eigenvalues.

Contribution

It introduces a novel Selberg trace formula specifically for Dirac operators on hyperbolic surfaces, expanding spectral analysis tools.

Findings

Derived a trace formula for Dirac operators

Identified symmetry effects on eigenvalue multiplicities

Connected Dirac spectra with Maass-Laplace spectra

Abstract

We examine spectra of Dirac operators on compact hyperbolic surfaces. Particular attention is devoted to symmetry considerations, leading to non-trivial multiplicities of eigenvalues. The relation to spectra of Maass-Laplace operators is also exploited. Our main result is a Selberg trace formula for Dirac operators on hyperbolic surfaces.