Liouville theory with a central charge less than one

TL;DR

The paper addresses defining Liouville CFT for all complex central charges, proving a consistent theory for $c\le 1$ and showing timelike Liouville is inconsistent. It uses conformal bootstrap with a diagonal continuous spectrum, degenerate crossing equations, and the special function $\Upsilon_b$ to construct the three-point constants; $C^{DOZZ}$ for $c>1$ and $C^{c\le 1}$ for $c\le 1$. Non-analytic Liouville theories exist at rational $c$ as limits producing $C^{non-analytic}=C^{c\le 1}\,\sigma$; crossing symmetry is numerically validated, and all code is released. The results enable precise predictions for correlation functions across all $c$, with potential applications to Potts models and higher-dimensional theories; the work also clarifies the nonexistence of consistent timelike Liouville.

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.