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2D dilaton gravity and the Weil-Petersson volumes with conical defects

Lorenz Eberhardt, Gustavo J. Turiaci

TL;DR

This work derives a complete Weil-Petersson measure for moduli spaces of hyperbolic surfaces with conical defects of arbitrary angles, including a uniquely fixed cohomology class for the WP form and associated string and dilaton equations. It then applies these mathematical results to two-dimensional dilaton gravity, showing how a gas of defects deforms JT gravity and can be encoded in a corresponding matrix integral, with checks against simple geometries and links to the minimal string. The paper provides detailed calculations for blunt and sharp defects, introduces a defect-generating function, and demonstrates agreement between gravity path integrals (via localization in regimes without separating geodesics) and matrix-model predictions in several cases. It also clarifies the role of odd versus even sectors in the large-$p$ limit of the minimal string and its relation to hyperbolic geometry. Overall, the results offer a solvable framework for 2d pure dilaton gravity with a broad class of potentials and defects, connecting moduli-space geometry, JT gravity, and random-matrix theory.

Abstract

We derive the Weil-Petersson measure on the moduli space of hyperbolic surfaces with defects of arbitrary opening angles and use this to compute its volume. We conjecture a matrix integral computing the corresponding volumes and confirm agreement in simple cases. We combine this mathematical result with the equivariant localization approach to Jackiw-Teitelboim gravity to justify a proposed exact solution of pure 2d dilaton gravity for a large class of dilaton potentials.

2D dilaton gravity and the Weil-Petersson volumes with conical defects

TL;DR

This work derives a complete Weil-Petersson measure for moduli spaces of hyperbolic surfaces with conical defects of arbitrary angles, including a uniquely fixed cohomology class for the WP form and associated string and dilaton equations. It then applies these mathematical results to two-dimensional dilaton gravity, showing how a gas of defects deforms JT gravity and can be encoded in a corresponding matrix integral, with checks against simple geometries and links to the minimal string. The paper provides detailed calculations for blunt and sharp defects, introduces a defect-generating function, and demonstrates agreement between gravity path integrals (via localization in regimes without separating geodesics) and matrix-model predictions in several cases. It also clarifies the role of odd versus even sectors in the large- limit of the minimal string and its relation to hyperbolic geometry. Overall, the results offer a solvable framework for 2d pure dilaton gravity with a broad class of potentials and defects, connecting moduli-space geometry, JT gravity, and random-matrix theory.

Abstract

We derive the Weil-Petersson measure on the moduli space of hyperbolic surfaces with defects of arbitrary opening angles and use this to compute its volume. We conjecture a matrix integral computing the corresponding volumes and confirm agreement in simple cases. We combine this mathematical result with the equivariant localization approach to Jackiw-Teitelboim gravity to justify a proposed exact solution of pure 2d dilaton gravity for a large class of dilaton potentials.
Paper Structure (35 sections, 108 equations, 4 figures)

This paper contains 35 sections, 108 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Weil-Petersson form on the moduli space of cone surfaces
  3. Sharp and blunt defects
  4. The cohomology class of the Weil-Petersson form
  5. Uniqueness.
  6. Check of properties.
  7. The dilaton operator
  8. String equation.
  9. Dilaton equation.
  10. Some simple examples
  11. $g=0$, $n=4$.
  12. $g=0$, $n=5$.
  13. $g=1$, $n=2$.
  14. Two dimensional gravity and random matrices
  15. JT gravity and matrix integrals
  16. ...and 20 more sections

Figures (4)

  • Figure 1: The two behaviors of two merging conical singularities.
  • Figure 2: The boundary divisor for the class $\delta_{1,\{1,3,6\}}=\delta_{1,\{2,4,5\}} \in \mathop{\mathrm{H}}\nolimits^2(\overline{\mathcal{M}}_{2,6})$.
  • Figure 3: Pair of pants decomposition of a genus 2 hyperbolic surface with a cusp.
  • Figure 4: The disk moduli space with $n$ marked points (here $n=4$). The surface is unchanged under $\text{U}(1)$ rotations when all points coincide at the center of the disk and the asymptotic cutoff for the Schwarzian mode is round.