Compactified Imaginary Liouville Theory

TL;DR

This paper constructs a rigorous mathematical model of a non-unitary, logarithmic conformal field theory on Riemann surfaces using a compactified Gaussian Free Field with imaginary parameters, relevant for critical loop models.

Contribution

It provides the first mathematical construction of a logarithmic conformal field theory satisfying Segal's axioms, incorporating imaginary parameters and complex operators.

Findings

Constructed a path integral based on imaginary Liouville action.

Proved the path integral satisfies conformal field theory axioms.

Developed correlation functions involving electromagnetic operators.

Abstract

On a given Riemann surface, we construct a path integral based on the Liouville action functional with imaginary parameters. The construction relies on the compactified Gaussian Free Field (GFF), which we perturb with a curvature term and an exponential potential. In physics this path integral is conjectured to describe the scaling limit of critical loop models such as Potts and O(n) models. The potential term is defined by means of imaginary Gaussian Multiplicative Chaos theory. The curvature term involves integrated 1-forms, which are multivalued on the manifold, and requires a delicate regularisation in order to preserve diffeomorphism invariance. We prove that the probabilistic path integral satisfies the axioms of Conformal Field Theory (CFT) including Segal's gluing axioms. We construct the correlation functions for this CFT, involving electro-magnetic operators. This CFT has several exotic features: most importantly, it is non unitary and has the structure of a logarithmic CFT. This is the first mathematical construction of a logarithmic CFT and therefore the present paper provides a concrete mathematical setup for this concept.