Supersymmetry and trace formulas I. Compact Lie groups

TL;DR

This paper introduces a novel supersymmetric localization principle applicable to trace formulas in quantum mechanics, enabling the computation of supertraces of non-supersymmetric observables through fermionic zero modes.

Contribution

It develops a new localization method based on fermionic zero modes, extending standard techniques to include non-supersymmetric observables and higher derivative deformations.

Findings

Path integral localizes to periodic orbits, not just constant ones.

Derived bosonic trace formulas on compact Lie groups.

Included classical Jacobi inversion formula as an example.

Abstract

In the context of supersymmetric quantum mechanics we formulate new supersymmetric localization principle, with application to trace formulas for a full thermal partition function. Unlike the standard localization principle, this new principle allows to compute the supertrace of non-supersymmetric observables, and is based on the existence of fermionic zero modes. We describe corresponding new invariant supersymmetric deformations of the path integral; they differ from the standard deformations arising from the circle action and require higher derivatives terms. Consequently, we prove that the path integral localizes to periodic orbits and not not only on constant ones. We illustrate the principle by deriving bosonic trace formulas on compact Lie groups, including classical Jacobi inversion formula.