Supersymmetry and trace formulas III. Frenkel trace formula
Changha Choi, Leon A. Takhtajan
TL;DR
The paper presents two exact, supersymmetric localization-based path integral derivations of the non-chiral Frenkel trace formula for a general semisimple compact Lie group $G$. One approach uses a supersymmetric nonlinear sigma model on $G$ with left-right twists, while the other employs a gauged sigma model on $G\times G$ and the isomorphism $(G\times G)/G\simeq G$, paralleling methods used for Eskin and Selberg trace formulas. Both routes culminate in a unified Frenkel-type trace formula expressed in terms of the Weyl group $W$, the coroot/coweight structure, the Weyl vector $\rho$, and the characteristic lattice $\Gamma$, with consistency shown to the classic Frenkel result for simply connected $G$. The Harish-Chandra orbital integral formula plays a central role in translating group-theoretic data into the trace, and the two pictures provide complementary semiclassical interpretations that connect prior work by Choi et al. to a general, model-independent framework. This work thereby bridges different trace-formula derivations and broadens the applicability of localization techniques in representation-theoretic contexts.
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}.