Three-point functions in c <= 1 Liouville theory and conformal loop ensembles

TL;DR

This paper demonstrates that the recently extended $c \,\leq\, 1$ Liouville Conformal Field Theory can be understood through microscopic loop models, establishing a correspondence between geometrical operators and vertex operators in the theory.

Contribution

It introduces a geometric interpretation of the $c \,\leq\, 1$ Liouville theory via lattice loop models and confirms the operator algebra matches that of vertex operators.

Findings

Operator algebra matches that of vertex operators in $c \,\leq\, 1$ Liouville.

Efficient lattice algorithms compute three-point functions.

Geometric interpretation of the limit $\,\hat{\alpha} \to 0$ is provided.

Abstract

The possibility of extending the Liouville Conformal Field Theory from values of the central charge $c \geq 25$ to $c \leq 1$ has been debated for many years in condensed matter physics as well as in string theory. It was only recently proven that such an extension -- involving a real spectrum of critical exponents as well as an analytic continuation of the DOZZ formula for three-point couplings -- does give rise to a consistent theory. We show in this Letter that this theory can be interpreted in terms of microscopic loop models. We introduce in particular a family of geometrical operators, and, using an efficient algorithm to compute three-point functions from the lattice, we show that their operator algebra corresponds exactly to that of vertex operators $V_{\hat{\alpha}}$ in $c \leq 1$ Liouville. We interpret geometrically the limit $\hat{\alpha} \to 0$ of $V_{\hat{\alpha}}$ and explain why it is not the identity operator (despite having conformal weight $\Delta=0$).