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.
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Acknowledgments
We thank J. Chandra, S. Collier, K. Colville, T. Hartman, K. Namjou, S. Shenker, and H. Verlinde for helpful conversations. We also thank the referee for helpful comments on the manuscript. The work of S.M.H. is supported by the National Science and Engineering Council of Canada and the Canada Research Chairs program. Research of AM is supported in part by the Simons Foundation Grant No. 385602 and the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number SAPIN/00047-2020. TN is supported by MEXT KAKENHI Grant-in-Aid for Transformative Research Areas A “Extreme Universe” Grant Number 22H05248.
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Harrison, S.M., Maloney, A. & Numasawa, T. Liouville theory and the Weil-Petersson geometry of moduli space. J. High Energ. Phys. 2023, 227 (2023). https://doi.org/10.1007/JHEP11(2023)227
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DOI: https://doi.org/10.1007/JHEP11(2023)227