$p \to \infty$ limit of tachyon correlators in $(2,2p+1)$ minimal Liouville gravity from classical Liouville theory

Abstract

Previously it was suggested, motivated by correspondence with JT gravity, that tachyon correlators in $(2,2p+1)$ minimal Liouville gravity (MLG) in the $p\to \infty$ (semiclassical) limit should be interpreted as moduli space volumes for constant curvature surfaces with conical defects. In this work we propose that these volumes are associated with Kahler metrics on moduli spaces introduced by Zograf and Takhtajan, for which the classical Liouville action is a Kahler potential. We check this proposal by numerical calculation of these Kahler metrics and associated volumes for the simplest example of genus 0 surface with 4 conical defects, using conformal field theory. A peculiar property of MLG correlators is proportionality to number of conformal blocks in a certain region of parameter space; in a particular limiting case, we check this property for the volumes following from classical Liouville action and thus provide an analytic confirmation of our proposal.