Lax matrix solution of c=1 Conformal Field Theory

TL;DR

This paper introduces a Lax matrix approach to c=1 conformal field theory, linking correlation functions to isomonodromic differential equations and Painlevé VI solutions, enabling new computations of conformal blocks and three-point functions.

Contribution

It presents a novel Lax matrix framework for c=1 CFT, connecting conformal blocks to isomonodromic equations and providing a method to compute three-point functions directly.

Findings

Established a matrix differential equation for c=1 correlators.

Linked conformal blocks to Painlevé VI solutions.

Enabled direct computation of Runkel-Watts three-point functions.

Abstract

To a correlation function in a two-dimensional conformal field theory with the central charge $c=1$, we associate a matrix differential equation $\Psi' = L \Psi$, where the Lax matrix $L$ is a matrix square root of the energy-momentum tensor. Then local conformal symmetry implies that the differential equation is isomonodromic. This provides a justification for the recently observed relation between four-point conformal blocks and solutions of the Painlev\'e VI equation. This also provides a direct way to compute the three-point function of Runkel-Watts theory -- the common $c\rightarrow 1$ limit of Minimal Models and Liouville theory.