Boundary Compactified Imaginary Liouville Theory

TL;DR

This paper extends Compactified Imaginary Liouville Theory to surfaces with boundaries, establishing its mathematical foundation and verifying it satisfies conformal field theory axioms, thus enabling further boundary studies.

Contribution

It generalizes boundary CILT construction from closed surfaces, incorporating curvature and exponential potentials, and proves it meets CFT axioms.

Findings

Proved boundary CILT satisfies Segal's axioms.

Defined curvature and potential terms using imaginary Gaussian multiplicative chaos.

Established a mathematical foundation for boundary CILT.

Abstract

We generalize the construction of Compactified Imaginary Liouville Theory (CILT), a non-unitary logarithmic Conformal Field Theory (CFT) defined on closed surfaces, to surfaces with boundary. Starting from a compactified Gaussian Free Field (GFF) with Neumann boundary condition, we perturb it by adding in curvature terms and exponential potentials on both the bulk and the boundary. In physics, this theory is conjectured to describe the scaling limit of loop models such as the Potts and $O(n)$ models. To define it mathematically, the curvature terms require a detailed analysis of the topology, and the potential terms are defined using the imaginary Guassian Multiplicative Chaos (GMC). We prove that the resulting probabilistic path integral satisfies the axioms of CFT, including Segal's gluing axioms. This work provides the foundation for future studies of boundary CILT and will also help with the understanding of CILT.