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A convergent genus expansion for the plateau

Phil Saad, Douglas Stanford, Zhenbin Yang, Shunyu Yao

TL;DR

The paper proposes a convergent genus expansion for the spectral form factor in a tau-scaled limit of double-scaled matrix integrals, tying the expansion to low-energy JT gravity physics. By analyzing Airy and Kontsevich-graph descriptions, it connects the genus terms to semiclassical encounters, showing genus-1/2 and genus-1 contributions arise from large regions of moduli space where encounters are regulated at low energy. It demonstrates how cancellations evident at high energy in periodic-orbit theory are broken at low energy, yielding a finite plateau assembled from quantum-modified moduli-space integrals. The work points to a universal geometric structure in moduli space governing the plateau and suggests deep connections to sigma-model descriptions and action-correlations, with important implications for understanding discrete black-hole spectra through geometry.

Abstract

We conjecture a formula for the spectral form factor of a double-scaled matrix integral in the limit of large time, large density of states, and fixed temperature. The formula has a genus expansion with a nonzero radius of convergence. To understand the origin of this series, we compare to the semiclassical theory of "encounters" in periodic orbits. In Jackiw-Teitelboim (JT) gravity, encounters correspond to portions of the moduli space integral that mutually cancel (in the orientable case) but individually grow at low energies. At genus one we show how the full moduli space integral resolves the low energy region and gives a finite nonzero answer.

A convergent genus expansion for the plateau

TL;DR

The paper proposes a convergent genus expansion for the spectral form factor in a tau-scaled limit of double-scaled matrix integrals, tying the expansion to low-energy JT gravity physics. By analyzing Airy and Kontsevich-graph descriptions, it connects the genus terms to semiclassical encounters, showing genus-1/2 and genus-1 contributions arise from large regions of moduli space where encounters are regulated at low energy. It demonstrates how cancellations evident at high energy in periodic-orbit theory are broken at low energy, yielding a finite plateau assembled from quantum-modified moduli-space integrals. The work points to a universal geometric structure in moduli space governing the plateau and suggests deep connections to sigma-model descriptions and action-correlations, with important implications for understanding discrete black-hole spectra through geometry.

Abstract

We conjecture a formula for the spectral form factor of a double-scaled matrix integral in the limit of large time, large density of states, and fixed temperature. The formula has a genus expansion with a nonzero radius of convergence. To understand the origin of this series, we compare to the semiclassical theory of "encounters" in periodic orbits. In Jackiw-Teitelboim (JT) gravity, encounters correspond to portions of the moduli space integral that mutually cancel (in the orientable case) but individually grow at low energies. At genus one we show how the full moduli space integral resolves the low energy region and gives a finite nonzero answer.
Paper Structure (20 sections, 100 equations, 1 figure)

This paper contains 20 sections, 100 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Tau scaling of the spectral form factor
  3. Encounters in orbits and in JT
  4. Review of periodic-orbit theory
  5. Sieber-Richter pair
  6. Cancellation of encounters in GUE
  7. Sieber-Richter pair in JT
  8. Beyond encounters
  9. Kontsevich's decomposition of moduli space
  10. Genus one-half
  11. Genus one
  12. Encounters
  13. Beyond encounters
  14. Discussion
  15. Airy sigma model
  16. ...and 5 more sections

Figures (1)

  • Figure 1: The nine Kontsevich graphs with two boundaries and genus one consist of these five, together with another four given by interchanging the solid/red and dashed/black lines on the last four graphs. The dashed/black lines correspond to $1$ boundaries, and the solid/red lines correspond to $2$. The gray lines with arrows show what happens if we apply a cross operation to a given edge.