Seminars: L. O. Chekhov, Fool's crowns, Schwarzians, and volumes of moduli spaces

SEMINARS


Seminar of the Department of Theoretical Physics, Steklov Mathematical Institute of RAS December 24, 2025 18:00, Moscow, online

Fool's crowns, Schwarzians, and volumes of moduli spaces L. O. Chekhov^ab ^a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow ^b Michigan State University
Abstract: In 2017, Stanford and Witten renewed the interest to Jackiw—Teitelboim (JT) gravity on "trumpets—parts of a one-sheet hyperboloid with geodesic boundary on one side and bounded on one side and "curved" boundary on the other side supplied with the Schwarzian action. Our goal is to obtain this action in the limit as $n\to\infty$ of "fool's crowns" — trumpets with $n$ horocycle-decorated boundary cusps. We start with finding volumes of the related moduli spaces since, unlike the celebrated Mirzakhani's construction for moduli spaces of hyperbolic Riemann surfaces, the moduli spaces of crowns have infinite volumes when integrating against the invariant measure, so it is necessary to introduce regularization, or action, to make these volumes finite. A variant of such action was proposed in arXiv:2411.03913 We evaluate the volumes of moduli spaces for arbitrary action coefficient $\kappa$ and show similarity with Mirzhakhani's volumes in the case $\kappa=1$. An interesting limit, however, is $n\to\infty$ with $\kappa/n\to\sigma>0$ in which we obtain the Schwarzian action. This is a joint work with Timothy Budd (Neijmegen Univ.)