From double-scaled SYK correlators to Weil-Petersson volumes

TL;DR

This paper rigorously connects double-scaled SYK correlators, represented by certain polynomials, to Weil-Petersson volumes, revealing a quantum geometric structure and bridging combinatorics with geometry.

Contribution

It provides a rigorous proof that Okuyama's polynomials relate to $q$-deformed Weil-Petersson volumes, establishing a mathematical link between SYK correlators and moduli space geometry.

Findings

Okuyama's polynomials are the top degree part of the $q$-deformed Weil-Petersson volumes.

The work confirms the $q o 1$ limit recovers classical Weil-Petersson volumes.

Establishes a connection suggesting a quantum Weil-Petersson geometry.

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.