Liouville theory without an action

TL;DR

Liouville theory is a non-rational 2D CFT usually defined by its Liouville action, but this work shows that crossing symmetry of four-point functions already encodes enough information to determine the theory without invoking the Liouville interaction. By exploiting a degenerate field and the conformal bootstrap, the authors derive functional equations for the three-point structure constants and, together with the $b\leftrightarrow 1/b$ duality, recover the DOZZ formula and the relation between the cosmological constants $\mu$ and $\tilde{\mu}$. The key result is that the special constant $C_-(\alpha)$ can be obtained from bootstrap under BPZ assumptions, effectively solving Liouville theory in a Lagrangian-free framework. This action-free construction offers new avenues for constructive approaches to Liouville theory and non-rational CFT bootstrap methods.

Abstract

We show that the crossing symmetry of the four-point function in the Liouville conformal field theory on the sphere contains more information than what was hitherto considered. Under certain assumptions, it provides the special structure constants that were previously computed perturbatively and allows to solve the theory without using the Liouville interaction.