Two-dimensional Yang-Mills theory, Painleve 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 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.

27 pages, 1 figure; v2: Presentation of Section 2 improved; Final version to be published in Journal of Physics A

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• Emptiness formation probability of the six-vertex model and the sixth Painlevé equation
• New differential equations in the six-vertex model
• Continuous representations of scalar products of Bethe vectors
• A symmetry reduction technique for higher order Painlevé systems
• Quantum Mechanics of Plancherel Growth