Gravity-Matter Couplings from Liouville Theory

TL;DR

This paper derives three-point functions for minimal models coupled to gravity using Liouville theory's operator approach, confirming results with matrix models and clarifying the structure of gravity dressing and symmetries.

Contribution

It introduces a new operator-based derivation of three-point functions in Liouville gravity, elucidating the role of quantum group symmetries and the structure of emission operators.

Findings

Agreement with matrix-model calculations on the sphere

Clarification of the gravity dressing structure

Identification of a symmetry explaining screening operator continuation

Abstract

The three-point functions for minimal models coupled to gravity are derived in the operator approach to Liouville theory which is based on its $U_q(sl(2))$ quantum group structure. The result is shown to agree with matrix-model calculations on the sphere. The precise definition of the corresponding cosmological constant is given in the operator solution of the quantum Liouville theory. It is shown that the symmetry between quantum-group spins $J$ and $-J-1$ previously put forward by the author is the explanation of the continuation in the number of screening operators discovered by Goulian and Li. Contrary to the previous discussions of this problem, the present approach clearly separates the emission operators for each leg. This clarifies the structure of the dressing by gravity. It is shown, in particular that the end points are not treated on the same footing as the mid point. Since the outcome is completely symmetric this suggests the existence of a picture-changing mechanism in two dimensional gravity.