Higher equations of motion at level 2 in Liouville CFT
Guillaume Baverez, Baojun Wu
TL;DR
The article proves the higher equations of motion (HEMs) in Liouville CFT, extending the BPZ framework by exploiting poles of the Poisson operator on the Kac table. It develops a probabilistic, GMC-based representation of Liouville states and shows that the residues at the degenerate weights alpha_{1,2} and alpha_{2,1} yield explicit relations between descendant and primary states, both in the bulk and on the boundary. Central to the method are singular integral expressions, fusion estimates for gamma-insertions, and exact evaluations of Dotsenko–Fateev and Selberg-type integrals, together with meromorphic continuation of the Poisson operator. The results confirm Zamolodchikov’s predictions for the bulk (and most of the boundary) HEMs and provide a robust framework for relating conformal blocks, Virasoro representations, and GMC-based Liouville amplitudes. These findings have potential implications for minimal gravity, Liouville bootstrap, and boundary Liouville theory, and they open questions about critical regimes and general (r,s) HEMs across higher-rank theories.
Abstract
We prove conjectures of Zamolodchikov and Belavin-Belavin in Liouville conformal field theory (CFT), which are generalisations of the celebrated Belavin-Polyakov-Zamolodchikov equations known as the higher equations of motion. Algebraically, these equations give examples of non-zero singular states in Virasoro modules, which is a relatively rare phenomenon in the physical study of CFT. In probability theory, these equations and their variants have been instrumental in the rigorous derivation of the structure constants of Liouville CFT in the unit disc. The proof builds on a previous work of ours studying the analytic continuation of the Poisson operator of Liouville theory. The main novelty is that this operator admits poles on the Kac table, and the higher equations of motions are obtained via a residue computation.