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

TL;DR

This paper establishes deep connections between two-dimensional Yang-Mills theory, the six-vertex model, and Painleve equations, revealing new integrable structures and phase behavior insights through matrix models and tau-functions.

Contribution

It demonstrates the mapping of Yang-Mills partition functions to six-vertex models and Painleve equations, introducing a unified integrable framework and matrix model representations.

Findings

Partition function mapped to six-vertex model with domain wall boundary conditions.

Partition function expressed as a tau-function of Painleve V equation.

Thermodynamic phases described by q-deformations of Yang-Mills theory.

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 the sphere and the identification of the partition function as a tau-function of the Painleve V equation. We describe the role played by generalized non-chiral Yang-Mills theory on the sphere 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.