Higher Equations of Motion in Liouville Field Theory

TL;DR

This paper establishes an infinite set of operator relations in Liouville field theory, generalizing the equation of motion, with potential applications in two-dimensional gravity.

Contribution

It introduces a new hierarchy of operator-valued relations in Liouville theory, extending the classical equation of motion to an infinite set of higher equations.

Findings

Proves an infinite set of operator relations in Liouville theory.

Uses exact structure constants in conformal field theory.

Discusses implications for 2D gravity.

Abstract

An infinite set of operator-valued relations in Liouville field theory is established. These relations are enumerated by a pair of positive integers $(m,n)$, the first $(1,1)$ representative being the usual Liouville equation of motion. The relations are proven in the framework of conformal field theory on the basis of exact structure constants in the Liouville operator product expansions. Possible applications in 2D gravity are discussed.