Note on large-$p$ limit of $(2,2p+1)$ minimal Liouville gravity and moduli space volumes

TL;DR

This paper explores the large-$p$ limit of correlation numbers in $(2,2p+1)$ minimal Liouville gravity, revealing connections to Weil-Petersson volumes and JT-gravity, with implications for geometric interpretations and fusion rules.

Contribution

It demonstrates that correlation numbers simplify to Weil-Petersson volumes in the large-$p$ limit and uncovers proportionality to conformal blocks, linking minimal models to geometric structures.

Findings

Correlation numbers reduce to Weil-Petersson volumes at large $p$.

Correlation number proportional to the number of conformal blocks for large $p$.

Connection established between minimal Liouville gravity and JT-gravity.

Abstract

In this note we report on some properties of correlation numbers for 2-dimensional Liouville gravity coupled with $(2,2p+1)$ minimal model at large $p$. In the limit $p \to \infty$, for some explicitly known examples in a particular region of parameter space correlation numbers are shown to reduce to Weil-Petersson volumes, analytically continued to imaginary geodesic lengths. This marks another connection of this limit with JT-gravity. We also comment on supposed geometric meaning of the obtained answers outside of this region, in particular, the meaning of the minimal model fusion rules. Another observation is the proportionality of correlation number to the number of conformal blocks when $p$ is big enough compared to parameters of the correlator. This proportionality is valid even without taking the limit.