On the construction of the correlation numbers in Minimal Liouville Gravity

TL;DR

This paper addresses discrepancies in calculating correlation numbers in Minimal Liouville Gravity by incorporating discrete terms in the operator product expansion, achieving full consistency between two different computational approaches.

Contribution

It proposes a modified expression for four-point correlation numbers that reconciles direct CFT-based and discrete string equation methods in Minimal Liouville Gravity.

Findings

Full agreement between approaches achieved after including discrete terms.

Modified correlation number expression removes restrictions on conformal blocks.

Enhanced understanding of operator product expansion in Liouville theory.

Abstract

The computation of the correlation numbers in Minimal Liouville Gravity involves an integration over moduli spaces of complex curves. There are two independent approaches to the calculation: the direct one, based on the CFT methods and Liouville higher equations of motion, and the alternative one, motivated by discrete description of 2D gravity and based on the Douglas string equation. However these two approaches give rise to the results that are not always consistent among themselves. In this paper we explore this problem. We show that in order to reconcile two methods the so-called discrete terms in the operator product expansion in the underlying Liouville theory must be properly taken into account. In this way we propose modified version of the expression for four-point correlation number and find full agreement between direct and alternative approaches. Our result allows to consider correlators without any restrictions on the number of conformal blocks contributing to the matter sector correlation function.