Higher Equations of Motion in Liouville Field Theory

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.