Supersymmetry and trace formulas II. Selberg trace formula

TL;DR

This paper extends a supersymmetric localization approach to derive the Selberg trace formula on compact Riemann surfaces and locally symmetric spaces, broadening its applicability to vector-valued automorphic forms and Maass Laplacians.

Contribution

It introduces a path integral derivation of the Selberg trace formula using supersymmetric localization, generalizing to arbitrary vector-valued automorphic forms and locally symmetric spaces.

Findings

Path integral derivation of the Selberg trace formula

Extension to vector-valued automorphic forms and Maass Laplacians

Generalization to compact locally symmetric spaces

Abstract

By extending the new supersymmetric localization principle introduced in \cite{Choi:2021yuz}, we present a path integral derivation of the Selberg trace formula on arbitrary compact Riemann surfaces, including the case of arbitrary vector-valued automorphic form and weight corresponding to Maass Laplacian. We also generalize the method to formulate the Selberg trace formula on generic compact locally symmetric space.