Exact Expressions For Infinitely Many Weil-Petersson Volumes

TL;DR

This paper derives exact formulas for generalized Weil-Petersson volumes, connecting moduli space geometry with matrix models and string theory, including superstring cases with flux.

Contribution

It introduces a method to compute generalized Weil-Petersson volumes for low genus and arbitrary boundaries using matrix model solutions, extending previous geometric results.

Findings

Explicit formulas for genus 0 and 1 volumes with any number of boundaries.

Connection between volumes and matrix model perturbative expansions.

Applicability to superstring models with Ramond-Ramond flux.

Abstract

Weil-Petersson volumes are the volumes of the moduli spaces of bordered Riemann surfaces and have played an important role in the relationship between two-dimensional quantum gravity and algebraic geometry. In the last couple years progress has been made to understand their role in the context of matrix models, where it is possible to define a generalization of the volumes in terms of an infinite set of coupling constants $t_k$. Using a recent open string matrix model construction we calculate the generalized Weil-Petersson volumes for fixed genus $g = 0,1$ and an arbitrary number of boundaries $n$. Both results are expressed in terms of the perturbative expansion of the solution to the string equation of the matrix model in the closed string sector. The formalism has the added benefit of applying to type 0A superstring matrix models with nonzero Ramond-Ramond flux.