Two-dimensional Yang–Mills theory, Painlevé equations and the six-vertex model

Abstract

We show that the chiral partition function of two-dimensional Yang–Mills theory on the sphere can be mapped to the partition function of the homogeneous six-vertex model with domain wall boundary conditions in the ferroelectric phase. A discrete matrix model description in both cases is given by the Meixner ensemble, leading to a representation in terms of a stochastic growth model. We show that the partition function is a particular case of the z-measure on the set of Young diagrams, yielding a unitary matrix model for chiral Yang–Mills theory on S2 and the identification of the partition function as a tau-function of the Painlevé V equation. We describe the role played by generalized non-chiral Yang–Mills theory on S2 in relating the Meixner matrix model to the Toda chain hierarchy encompassing the integrability of the six-vertex model. We also argue that the thermodynamic behaviour of the six-vertex model in the disordered and antiferroelectric phases are captured by particular q-deformations of two-dimensional Yang–Mills theory on the sphere.