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Two-dimensional conformal field theory is a powerful tool to understand thegeometry of surfaces. Here, we study Liouville conformal field theory in theclassical (large central charge) limit, where it encodes the geometry of themoduli space of Riemann surfaces. Generalizing previous work, we employ this tostudy moduli spaces of higher genus surfaces, surfaces with boundaries, andsurfaces with cone points. In each case, the knowledge of classical conformalblocks provides an extremely efficient approximation to the Weil-Peterssonmetric on moduli space. We find detailed agreement with analytic results forvolumes and geodesic lengths on moduli space.

Liouville theory and the Weil-Petersson geometry of moduli space: bordered, conic, and higher genus surfaces