Compactified imaginary Liouville theory

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 and we construct the correlation functions for this CFT, involving electro-magnetic operators. This CFT has several conjectural exotic features: most importantly, it should be non-unitary with the structure of a logarithmic CFT. Our motivation is thus to provide a concrete setup for the mathematical study of logarithmic CFTs.