Supersymmetry and trace formulas III. Frenkel trace formula

TL;DR

This paper develops two supersymmetric localization-based methods to derive a generalized non-chiral trace formula for semisimple compact Lie groups, connecting different mathematical approaches and physical models.

Contribution

It introduces two complementary path integral derivations of the Frenkel trace formula using supersymmetric localization, bridging previous independent approaches.

Findings

Derivation via quantum sigma model on G

Derivation via gauged sigma model on G×G

Provides conceptual link between different trace formulas

Abstract

By applying the new supersymmetric localization principle introduced in \cite{Choi:2021yuz,Choi:2023pjn}, we present two complementary approaches for the path integral derivation of the `non-chiral' trace formula for a semisimple compact Lie group $G$, which generalizes the so-called Frenkel trace formula. Corresponding physical systems for each picture are the quantum mechanical sigma model on $G$ and the gauged sigma model on $G\times G$, and the approaches closely follow the spirit of the Eskin trace formula \cite{Choi:2021yuz} and the Selberg trace formula \cite{Choi:2023pjn} respectively. These methods provide a natural conceptual bridge between two seemingly independent derivations in \cite{Choi:2021yuz} and \cite{Choi:2023pjn}.