Higher Weil-Petersson volumes of the moduli space of super Riemann surfaces

TL;DR

This work extends the Stanford–Witten recursion for super Weil-Petersson volumes by incorporating higher-degree kappa classes through Norbury’s Theta-class framework. By embedding the problem in the KdV hierarchy and Virasoro constraints, the authors derive two main recursion formulas (RecursionA and RecursionB) for mixed kappa-psi intersections and show how these yield broader recursions for super volumes without psi insertions. They further develop two volume recursions (superHWP1 and superHWP2) for $V_g(\kappa(\mathbf{b}))$ and $V_{g,n+1}(\kappa(\mathbf{b}))$, respectively, linking them to genus-zero/dilaton-type relations. A Virasoro-revisited framework expresses these relations as differential operators acting on a Tau-function, with $G^{\Theta}$ identified as a shifted BGW/KdV tau-function, reinforcing the connection between super moduli-space intersection theory and integrable systems. The results deepen computational tools for super moduli spaces and strengthen ties to JT supergravity via refined volume recurrences.

Abstract

Inspired by the theory of JT supergravity, Stanford-Witten derived a remarkable recursion formula of Weil-Petersson volumes of moduli space of super Riemann surfaces. It is the super version of the celebrated Mirzakhani's recursion formula. In this paper, we generalize Stanford-Witten's formula to include high degree kappa classes.

Theorems & Definitions (27)

• Theorem 1.1: Norbury norbury2020enumerative
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Proposition 2.1
• proof
• Proposition 2.2