Liouville Theory and the Weil-Petersson Geometry of Moduli Space

TL;DR

This paper leverages Liouville theory to develop an efficient algorithm for computing the Weil-Petersson metric on moduli spaces of Riemann surfaces, providing high-accuracy numerical results and insights into spectral statistics.

Contribution

It introduces a novel computational approach using Liouville theory and Zamolodchikov's recursion to accurately determine Weil-Petersson metrics and spectral properties of moduli spaces.

Findings

High-accuracy numerical computation of Weil-Petersson metric on .

Eigenvalues of Weil-Petersson Laplacian follow GOE random matrix statistics.

Algorithm demonstrates efficiency in complex geometric calculations.

Abstract

Liouville theory describes the dynamics of surfaces with constant negative curvature and can be used to study the Weil-Petersson geometry of the moduli space of Riemann surfaces. This leads to an efficient algorithm to compute the Weil--Petersson metric to arbitrary accuracy using Zamolodchikov's recursion relation for conformal blocks. For example, we compute the metric on $\mathcal M_{0,4}$ numerically to high accuracy by considering Liouville theory on a sphere with four punctures. We numerically compute the eigenvalues of the Weil-Petersson Laplacian, and find evidence that the obey the statistics of a random matrix in the Gaussian Orthogonal Ensemble.