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The Selberg trace formula for spin Dirac operators on degenerating hyperbolic surfaces

Rares Stan

TL;DR

This work analyzes the spin Dirac operator on finite-area hyperbolic surfaces undergoing pinching of simple geodesics, under the non-trivial spin condition along the pinched curves. It develops a Selberg trace formula tailored to the Dirac operator via a spin-structure class function {\varepsilon}, proves a non-standard $t\to0$ heat-trace expansion with a cusp term, and establishes a uniform Weyl law for degenerating families. It also proves a Dirac-analog of Huber’s isospectrality, provides explicit Dirac operator formulae, and demonstrates the meromorphic extension and pinching convergence of the Selberg zeta function $Z_{\varepsilon}(s,M)$, including a precise rescaling that yields a finite limit in the cusped case. Collectively, these results connect Dirac spectral data to geometric and topological invariants (length spectrum, cusp count, and spin data encoded by $\varepsilon$) and elucidate the analytic behavior of the Selberg zeta function under degeneration.

Abstract

We investigate the spectrum of the spin Dirac operator on families of hyperbolic surfaces where a set of disjoint simple geodesics shrink to $0$, under the hypothesis that the spin structure is non-trivial along each pinched geodesic. The main tool is a trace formula for the Dirac operator on finite area hyperbolic surfaces. We derive a version of Huber's theorem and a non-standard small-time heat trace asymptotic expansion for hyperbolic surfaces with cusps. As a corollary we find a simultaneous Weyl law for the eigenvalues of the Dirac operator which is uniform in the degenerating parameter. The main result is the convergence of the Selberg zeta function associated to the Dirac operator on such families of hyperbolic surfaces. A central role is played by a $\{ \pm 1 \}$-valued class function $\varepsilon$ determined by the spin structure.

The Selberg trace formula for spin Dirac operators on degenerating hyperbolic surfaces

TL;DR

This work analyzes the spin Dirac operator on finite-area hyperbolic surfaces undergoing pinching of simple geodesics, under the non-trivial spin condition along the pinched curves. It develops a Selberg trace formula tailored to the Dirac operator via a spin-structure class function {\varepsilon}, proves a non-standard heat-trace expansion with a cusp term, and establishes a uniform Weyl law for degenerating families. It also proves a Dirac-analog of Huber’s isospectrality, provides explicit Dirac operator formulae, and demonstrates the meromorphic extension and pinching convergence of the Selberg zeta function , including a precise rescaling that yields a finite limit in the cusped case. Collectively, these results connect Dirac spectral data to geometric and topological invariants (length spectrum, cusp count, and spin data encoded by ) and elucidate the analytic behavior of the Selberg zeta function under degeneration.

Abstract

We investigate the spectrum of the spin Dirac operator on families of hyperbolic surfaces where a set of disjoint simple geodesics shrink to , under the hypothesis that the spin structure is non-trivial along each pinched geodesic. The main tool is a trace formula for the Dirac operator on finite area hyperbolic surfaces. We derive a version of Huber's theorem and a non-standard small-time heat trace asymptotic expansion for hyperbolic surfaces with cusps. As a corollary we find a simultaneous Weyl law for the eigenvalues of the Dirac operator which is uniform in the degenerating parameter. The main result is the convergence of the Selberg zeta function associated to the Dirac operator on such families of hyperbolic surfaces. A central role is played by a -valued class function determined by the spin structure.
Paper Structure (26 sections, 30 theorems, 136 equations, 4 figures)

This paper contains 26 sections, 30 theorems, 136 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Degenerating surfaces
  3. Trace formula for the Dirac operator
  4. Huber's Theorem for the Dirac operator
  5. Heat trace asymptotics and uniform Weyl law
  6. The convergence of the Selberg Zeta function
  7. The Dirac operator
  8. The spinor bundle
  9. The Dirac operator
  10. Spin structures on hyperbolic surfaces
  11. Encoding the spin structure in a class function
  12. Non-trivial spin structures and discrete spectrum
  13. Explicit formulae for the Dirac operator
  14. Bounding the number of geodesics on a hyperbolic surface
  15. Trace formula on hyperbolic surfaces
  16. ...and 11 more sections

Key Result

Theorem 2

Two finite area hyperbolic surfaces, endowed with non-trivial spin structures are Dirac isospectral if and only if they have the same length spectrum and their respective class functions $\varepsilon$ coincide on the primitive length spectrum.

Figures (4)

  • Figure 1: Pair of pants
  • Figure 2: A fundamental domain of Figure \ref{['pairOfPants']}
  • Figure 3: The compact region $K(t)$ on the surface
  • Figure 4: The pair of pants $P$

Theorems & Definitions (60)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Proposition 6
  • proof
  • Definition 7
  • Lemma 8
  • proof
  • ...and 50 more