Liouville theory with a central charge less than one

TL;DR

This paper fully characterizes Liouville theory for all complex central charges less than or equal to one, clarifying its spectrum, correlation functions, and the absence of a consistent timelike version, supported by numerical checks.

Contribution

It completes the definition of Liouville theory across all complex central charges and explores non-analytic cases at rational values, providing computational tools.

Findings

Spectrum is always spacelike

No consistent timelike Liouville theory exists

Numerical validation of crossing symmetry

Abstract

We determine the spectrum and correlation functions of Liouville theory with a central charge less than (or equal) one. This completes the definition of Liouville theory for all complex values of the central charge. The spectrum is always spacelike, and there is no consistent timelike Liouville theory. We also study the non-analytic conformal field theories that exist at rational values of the central charge. Our claims are supported by numerical checks of crossing symmetry. We provide Python code for computing Virasoro conformal blocks, and correlation functions in Liouville theory and (generalized) minimal models.