On the construction of the correlation numbers in Minimal Liouville Gravity

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.

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Cited by in corpus (10)

• Conformal bootstrap to Rényi entropy in 2D Liouville and super-Liouville CFTs
• The complex Liouville string: the worldsheet
• Liouville field theory and log-correlated Random Energy Models
• Five-point correlation numbers in minimal Liouville gravity and matrix models
• Operator Product Expansion in Liouville Field Theory and Seiberg type transitions in log-correlated Random Energy Models
• Open minimal strings and open Gelfand-Dickey hierarchies
• Minimal gravity and Frobenius manifolds: bulk correlation on sphere and disk
• Dual Frobenius manifolds of minimal gravity on disk
• $x-y$ swap for $(2,2p+1)$ minimal string
• Disordered statistical physics in low dimensions: extremes, glass transition, and localization