Exact Expressions For Infinitely Many Weil-Petersson Volumes

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.