Boundary Liouville Theory and 2D Quantum Gravity

TL;DR

This paper establishes a correspondence between boundary correlation functions in Liouville theory and 2D quantum gravity models, deriving functional identities and difference equations that demonstrate their agreement.

Contribution

It derives and compares boundary correlation functions and difference equations in both Liouville theory and discrete 2D quantum gravity models, confirming their consistency.

Findings

Functional identities for boundary structure constants in Liouville theory.

Difference equations for boundary correlators match in both approaches.

Complete agreement between Liouville theory predictions and discrete quantum gravity models.

Abstract

We study the boundary correlation functions in Liouville theory and in solvable statistical models of 2D quantum gravity. In Liouville theory we derive functional identities for all fundamental boundary structure constants, similar to the one obtained for the boundary two-point function by Fateev, Zamolodchikov and Zamolodchikov. All these functional identities can be written as difference equations with respect to one of the boundary parameters. Then we switch to the microscopic realization of 2D quantum gravity as a height model on a dynamically triangulated disc and consider the boundary correlation functions of electric, magnetic and twist operators. By cutting open the sum over surfaces along a domain wall, we derive difference equations identical to those obtained in Liouville theory. We conclude that there is a complete agreement between the predictions of Liouville theory and the discrete approach.