Higher Equations of Motion in Liouville Field Theory

TL;DR

This work proves an infinite family of higher equations of motion in Liouville field theory, labeled by degenerate pairs $(m,n)$, by exploiting exact Liouville structure constants and degenerate representations. It introduces logarithmic degenerate primaries $V'_{m,n}$ and demonstrates the operator relation $D_{m,n}\bar{D}_{m,n}V'_{m,n}=B_{m,n}\tilde{V}_{m,n}$ with an explicit coefficient $B_{m,n}$, derived from three-point functions and $\Upsilon$-function identities. The paper also derives explicit norms $r_{m,n}$ of logarithmic primaries, and shows that one-point functions on the Poincaré disk yield a geometry-independent ratio $r_{m,n}=2\,\prod (lb^{-1}+kb)$, reinforcing the universality of the HEMs. Finally, it outlines applications to minimal Liouville gravity, where these HEMs enable exact reductions of MG correlators to boundary contributions and connect to BRST/ground-ring structures, including a detailed $(1,2)$ example. Overall, the results extend the Liouville bootstrap by providing a robust algebraic framework for higher-order equations of motion with potential 2D gravity implications.

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.