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Supersymmetry and trace formulas I. Compact Lie groups

Changha Choi, Leon A. Takhtajan

TL;DR

This work develops a novel localization principle in supersymmetric quantum mechanics that leverages fermionic zero modes to localize path integrals for bosonic partition functions onto periodic geodesics, even when the index vanishes. The method is applied to two canonical settings: a free particle on the circle, yielding an exact Jacobi inversion formula for the heat trace, and a supersymmetric particle on a compact Lie group, producing an exact derivation of Eskin's trace formula from localization of the Kostant cubic Dirac operator. By introducing higher-derivative invariant deformations that saturate zero modes, the authors extend localization beyond constant loops to closed geodesics and carefully treat singular cases. The results bridge spectral data (heat kernels and representation theory) with geometric sums over geodesics and lattices, offering a rigorous, supersymmetric-path-integral route to classical trace formulas with potential broad impact in geometric analysis and quantum mechanics.

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.

Supersymmetry and trace formulas I. Compact Lie groups

TL;DR

This work develops a novel localization principle in supersymmetric quantum mechanics that leverages fermionic zero modes to localize path integrals for bosonic partition functions onto periodic geodesics, even when the index vanishes. The method is applied to two canonical settings: a free particle on the circle, yielding an exact Jacobi inversion formula for the heat trace, and a supersymmetric particle on a compact Lie group, producing an exact derivation of Eskin's trace formula from localization of the Kostant cubic Dirac operator. By introducing higher-derivative invariant deformations that saturate zero modes, the authors extend localization beyond constant loops to closed geodesics and carefully treat singular cases. The results bridge spectral data (heat kernels and representation theory) with geometric sums over geodesics and lattices, offering a rigorous, supersymmetric-path-integral route to classical trace formulas with potential broad impact in geometric analysis and quantum mechanics.

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.
Paper Structure (20 sections, 7 theorems, 214 equations)

This paper contains 20 sections, 7 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. General Remarks on New Localization Principle
  4. Standard Localization Principle
  5. New Supersymmetric Localization Principle
  6. Localization Proof of the Trace formula on $S^1$
  7. Localization Proof of the Eskin Trace formula on $G$
  8. Supersymmetric Particle on $G$
  9. Eskin's Trace Formula on $G$
  10. Singular Trace Formula on $G$
  11. Critical points
  12. Quadratic form
  13. Semi-simple Lie groups and algebras
  14. Basic facts on $\frak{g}$
  15. Basic facts on $G$
  16. ...and 5 more sections

Key Result

Lemma 1

Let $V\in\Omega^{1}_{S^{1}}(N)$ be such that $i_{u}V=0$ and $i_{u}DV=0$. Then for all $s$

Theorems & Definitions (25)

  • Lemma 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition
  • Proposition 1
  • proof
  • Remark 4
  • Remark 5
  • ...and 15 more