Overview
This work presents two complementary path integral derivations for a 'non-chiral' trace formula applicable to a semisimple compact Lie group $G$. This derived formula generalizes the existing Frenkel trace formula. The derivations utilize a new supersymmetric localization principle, drawing upon methodologies introduced in Choi:2021yuz and Choi:2023pjn. The physical systems involved in these derivations are a quantum mechanical sigma model on $G$ and a gauged sigma model on $G\times G$, respectively.
Research Context
The research builds upon a previously introduced supersymmetric localization principle detailed in Choi:2021yuz and Choi:2023pjn. Specifically, it aims to extend the Frenkel trace formula into a `non-chiral' form applicable to a semisimple compact Lie group $G$. The two approaches presented are conceptual extensions of techniques found in the Eskin trace formula Choi:2021yuz and the Selberg trace formula Choi:2023pjn.
Approach
The study employs two distinct, yet complementary, approaches for the path integral derivation. Both approaches rely on the supersymmetric localization principle. One approach corresponds to the quantum mechanical sigma model on $G$. The other approach utilizes a gauged sigma model on $G\times G$. These methodologies are structured to align with the principles underlying the Eskin trace formula as presented in Choi:2021yuz and the Selberg trace formula as presented in Choi:2023pjn, respectively.
Findings
The application of the supersymmetric localization principle yielded two distinct path integral derivations for a `non-chiral' trace formula for a semisimple compact Lie group $G$. This formula represents a generalization of the Frenkel trace formula. The derivations connect two previously distinct approaches:
- One derivation emerged from considerations of the quantum mechanical sigma model on $G$, following the spirit of Choi:2021yuz (Eskin trace formula).
- The second derivation originated from the gauged sigma model on $G\times G$, aligning with the spirit of Choi:2023pjn (Selberg trace formula).
These two methods provide a conceptual link between the independent derivations outlined in Choi:2021yuz and Choi:2023pjn.