Lax matrix solution of c = 1 conformal field theory

Abstract : To a correlation function in a two-dimensional conformal field theory with the central charge c = 1, we associate a matrix differential equation Ψ ′ = LΨ, 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 Painlevé VI equation. This also provides a direct way to compute the three-point function of Runkel-Watts theory – the common c → 1 limit of Minimal Models and Liouville theory.