Liouville theory and the Weil-Petersson geometry of moduli space

Abstract

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 $$ \mathcal{M} $$ 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.