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From double-scaled SYK correlators to Weil-Petersson volumes

Norman Do, Paul Norbury

TL;DR

This work rigorously proves two key phenomena connecting the double-scaled SYK model to Weil-Petersson geometry: (i) Okuyama's $q$-dependent discrete volumes $N^q_{g,n}(b_1,\dots,b_n)$ have coefficients that are polynomials in even $q$-zeta values $\zeta_q(2k)$ and recover WP volumes in the $q\to1$ limit, and (ii) the top-degree part of these polynomials matches newly defined $q$-deformations $V^q_{g,n}(L_1,\dots,L_n)$ of WP volumes. The authors establish these results via a detailed analysis of Okuyama's spectral curve using topological recursion, and they introduce a top-degree spectral curve $\mathcal{S}^{\mathrm{top}}$ whose TR-led correlators encode the leading terms of $N^q_{g,n}$. This creates a bridge between quantum (or $q$-deformed) Weil-Petersson geometry and the combinatorics of DS-SYK correlators, suggesting a quantum WP framework and a combinatorial-geometric approach to DS-SYK. The paper also provides explicit $q$-deformed WP volumes, such as $V^q_{2}(L)$ and $V^q_{3}(L)$, and demonstrates that they converge to the classical WP volumes as $q\to1$.

Abstract

Okuyama introduced a family of polynomials, whose coefficients depend on a parameter $q$, in his study of correlators in the double-scaled SYK model. He verified in small cases that their coefficients can be expressed in terms of certain $q$-zeta values and that the polynomials recover the Weil-Petersson volumes of moduli spaces studied by Mirzakhani under a certain $q \to 1$ limit. In this paper, we provide mathematically rigorous proofs of these two phenomena. The authors previously defined natural $q$-deformations of the Weil-Petersson volumes of moduli spaces of curves. We prove that these polynomials appear as the top degree part of Okuyama's polynomials. Our work provides a link between the two topics of the title, which hints at a ``quantum'' Weil-Petersson geometry and a combinatorial-geometric approach to double-scaled SYK correlators.

From double-scaled SYK correlators to Weil-Petersson volumes

TL;DR

This work rigorously proves two key phenomena connecting the double-scaled SYK model to Weil-Petersson geometry: (i) Okuyama's -dependent discrete volumes have coefficients that are polynomials in even -zeta values and recover WP volumes in the limit, and (ii) the top-degree part of these polynomials matches newly defined -deformations of WP volumes. The authors establish these results via a detailed analysis of Okuyama's spectral curve using topological recursion, and they introduce a top-degree spectral curve whose TR-led correlators encode the leading terms of . This creates a bridge between quantum (or -deformed) Weil-Petersson geometry and the combinatorics of DS-SYK correlators, suggesting a quantum WP framework and a combinatorial-geometric approach to DS-SYK. The paper also provides explicit -deformed WP volumes, such as and , and demonstrates that they converge to the classical WP volumes as .

Abstract

Okuyama introduced a family of polynomials, whose coefficients depend on a parameter , in his study of correlators in the double-scaled SYK model. He verified in small cases that their coefficients can be expressed in terms of certain -zeta values and that the polynomials recover the Weil-Petersson volumes of moduli spaces studied by Mirzakhani under a certain limit. In this paper, we provide mathematically rigorous proofs of these two phenomena. The authors previously defined natural -deformations of the Weil-Petersson volumes of moduli spaces of curves. We prove that these polynomials appear as the top degree part of Okuyama's polynomials. Our work provides a link between the two topics of the title, which hints at a ``quantum'' Weil-Petersson geometry and a combinatorial-geometric approach to double-scaled SYK correlators.

Paper Structure

This paper contains 13 sections, 18 theorems, 107 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Topological recursion
  4. The double-scaled SYK model
  5. Okuyama's spectral curve
  6. Weil--Petersson volumes
  7. Discrete volumes
  8. The appearance of q-zeta values
  9. Weil--Petersson volumes in the limit
  10. Top degree terms
  11. The $q$-deformation of Weil--Petersson volumes
  12. The $q$-deformation of classical Weil--Petersson volumes
  13. Table of $q$-Weil--Petersson volumes

Key Result

Theorem 1.1

For $(g,n) \neq (0,1)$ or $(0,2)$, Okuyama's discrete volume $N^q_{g,n}(b_1, \ldots, b_n)$ is a quasi-polynomial in $b_1^2, \ldots, b_n^2$ with coefficients in $\mathbb{Q}[\zeta_q(2), \zeta_q(4), \zeta_q(6), \ldots]$ of degree $6g-6+2n$, using the degree defined above.

Figures (1)

  • Figure 1: The fifteen 3-chord diagrams, which contribute to the calculation of $\langle \, \mathop{\mathrm{Tr}}\nolimits H^6 \, \rangle_J = q^3 + 3q^2 + 6q + 5$.

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Proposition 2.2: Berkooz, Isachenkov, Narovlansky and Torrents BINT19
  • Theorem 2.3: Norbury and Scott nor-sco13
  • Theorem 2.4: Eynard and Orantin eyn-ora07b
  • Theorem 3.1
  • Proposition 3.2: Do, Leigh and Norbury do-lei-nor16
  • Lemma 3.3
  • ...and 19 more