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Large genus asymptotics of super Weil-Petersson volumes

Xuanyu Huang

TL;DR

This work advances the understanding of super geometric volumes arising in JT supergravity by establishing a complete large-genus expansion for super Weil-Petersson volumes with Theta insertions. It develops an explicit algorithm to compute expansion coefficients, proves their polynomial dependence on π^{-1} and boundary data, and confirms the Griguolo–Papalini–Russo–Seminara conjectures up to a universal constant. The leading term is shown to be V^{\\Theta}_{g,n} ~ C (2g-3+n)! π^{2g+n} / (2^n √g) with C = π^{-9/2}, and the paper provides concrete expressions for the first few correction terms, including a closed-form for e^1, e^2, a^1_n, a^2_n, b^1_n, b^2_n. Additionally, the authors analyze asymptotics as the number of boundaries n grows with genus, establishing robust bounds when n = o(√g) and laying groundwork for a full polynomial-structure picture of super intersection numbers in large-genus limits.

Abstract

In this paper, we obtain the asymptotic expansions of super intersection numbers and prove that the associated coefficients are polynomials. Moreover, we give an algorithm which can explicitly compute these coefficients. As an application, we prove the existence of a complete asymptotic expansion of super Weil-Petersson volumes in the large genus. This generalizes the celebrated work of Mirzakhani-Zograf. We also confirm two conjectural formulae proposed by Griguolo-Papalini-Russo-Seminara.

Large genus asymptotics of super Weil-Petersson volumes

TL;DR

This work advances the understanding of super geometric volumes arising in JT supergravity by establishing a complete large-genus expansion for super Weil-Petersson volumes with Theta insertions. It develops an explicit algorithm to compute expansion coefficients, proves their polynomial dependence on π^{-1} and boundary data, and confirms the Griguolo–Papalini–Russo–Seminara conjectures up to a universal constant. The leading term is shown to be V^{\\Theta}_{g,n} ~ C (2g-3+n)! π^{2g+n} / (2^n √g) with C = π^{-9/2}, and the paper provides concrete expressions for the first few correction terms, including a closed-form for e^1, e^2, a^1_n, a^2_n, b^1_n, b^2_n. Additionally, the authors analyze asymptotics as the number of boundaries n grows with genus, establishing robust bounds when n = o(√g) and laying groundwork for a full polynomial-structure picture of super intersection numbers in large-genus limits.

Abstract

In this paper, we obtain the asymptotic expansions of super intersection numbers and prove that the associated coefficients are polynomials. Moreover, we give an algorithm which can explicitly compute these coefficients. As an application, we prove the existence of a complete asymptotic expansion of super Weil-Petersson volumes in the large genus. This generalizes the celebrated work of Mirzakhani-Zograf. We also confirm two conjectural formulae proposed by Griguolo-Papalini-Russo-Seminara.
Paper Structure (19 sections, 23 theorems, 198 equations)

This paper contains 19 sections, 23 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Super Weil-Petersson volumes
  3. Large genus asymptotics for intersection numbers
  4. Main results
  5. Basic estimates
  6. Recursion formulae
  7. Numerical properties of $\mathbf{a_n}$
  8. Basic properties for super intersection numbers
  9. Estimates for the join part
  10. Algorithm for computing coefficients in the expansions
  11. Calculation algorithm.
  12. First three terms
  13. Constant term
  14. Coefficients of $\frac{1}{g}$.
  15. Coefficients of $\frac{1}{g^2}$.
  16. ...and 4 more sections

Key Result

Theorem 1.2

We have the following asymptotic expansions as $g\to\infty$:

Theorems & Definitions (45)

  • Conjecture 1.1: GPRS griguolo2024asymptotics
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Conjecture 1.8
  • Theorem 1.9
  • Remark 2.1
  • ...and 35 more