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Supersymmetry and trace formulas II. Selberg trace formula

Changha Choi, Leon A. Takhtajan

TL;DR

The paper derives the Selberg trace formula on compact Riemann surfaces and general compact locally symmetric spaces from a path-integral perspective grounded in an extended supersymmetric localization principle. By gauging appropriate sigma-models, analyzing Wilson-loop defects, and applying a careful decomposition of loop spaces, it translates spectral data of Laplace-type operators into sums over conjugacy classes (identity, hyperbolic, elliptic) with precise weight factors and contour prescriptions. It generalizes to arbitrary representations and weights, including Maass Laplacians of weight $\pm\tfrac12$ and higher, and extends the setup to generic $\Gamma\backslash G/K$, with explicit treatment of compact hyperbolic 3-manifolds. This provides a unified, physically motivated route to the Eskin–Selberg-type trace formulas and clarifies the role of anomalies, holonomies, and fermionic zero modes in determining the spectral/geometric data. The approach has potential implications for connecting quantum chaos, spectral theory, and automorphic forms via localization techniques.

Abstract

By extending the new supersymmetric localization principle introduced in \cite{Choi:2021yuz}, we present a path integral derivation of the Selberg trace formula on arbitrary compact Riemann surfaces, including the case of arbitrary vector-valued automorphic form and weight corresponding to Maass Laplacian. We also generalize the method to formulate the Selberg trace formula on generic compact locally symmetric space.

Supersymmetry and trace formulas II. Selberg trace formula

TL;DR

The paper derives the Selberg trace formula on compact Riemann surfaces and general compact locally symmetric spaces from a path-integral perspective grounded in an extended supersymmetric localization principle. By gauging appropriate sigma-models, analyzing Wilson-loop defects, and applying a careful decomposition of loop spaces, it translates spectral data of Laplace-type operators into sums over conjugacy classes (identity, hyperbolic, elliptic) with precise weight factors and contour prescriptions. It generalizes to arbitrary representations and weights, including Maass Laplacians of weight and higher, and extends the setup to generic , with explicit treatment of compact hyperbolic 3-manifolds. This provides a unified, physically motivated route to the Eskin–Selberg-type trace formulas and clarifies the role of anomalies, holonomies, and fermionic zero modes in determining the spectral/geometric data. The approach has potential implications for connecting quantum chaos, spectral theory, and automorphic forms via localization techniques.

Abstract

By extending the new supersymmetric localization principle introduced in \cite{Choi:2021yuz}, we present a path integral derivation of the Selberg trace formula on arbitrary compact Riemann surfaces, including the case of arbitrary vector-valued automorphic form and weight corresponding to Maass Laplacian. We also generalize the method to formulate the Selberg trace formula on generic compact locally symmetric space.
Paper Structure (15 sections, 166 equations)

This paper contains 15 sections, 166 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Extended new supersymmetric localization principle
  4. Propagator on non-compact semisimple $G$
  5. Selberg trace formula on compact Riemann surfaces
  6. The gauged sigma model
  7. Wilson loops and defects
  8. Localization and the pre-trace formula
  9. The Selberg trace formula
  10. Generalization to arbitrary representations
  11. Generalization to arbitrary weights
  12. Selberg trace formula on general compact locally symmetric space
  13. General set-up for $\Gamma\backslash G/K$
  14. Selberg trace formula for compact hyperbolic 3-manifold
  15. Laplace operators on $\mathrm{SL}(2,\mathbb{R})$ and $\mathrm{SL}(2,\mathbb{R})/\mathrm{SO}(2)$

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8