Unitary Chern-Simons matrix model and the Villain lattice action

TL;DR

This paper explores the connections between the Villain approximation, the Gross-Witten model, and Chern-Simons theory, revealing how different regimes relate to q-deformed 2D Yang-Mills and matrix models, with implications for understanding gauge theories.

Contribution

It demonstrates the relationship between the Gross-Witten model and Chern-Simons matrix models via the Villain approximation, and elucidates the emergence of q-deformed 2D Yang-Mills theory from lattice actions.

Findings

Weak-coupling limit corresponds to q->1 in Chern-Simons theory.

Strong-coupling limit relates to q->0 with a logarithmic coupling relationship.

Character expansion yields q-deformation of the heat kernel in 2D Yang-Mills.

Abstract

We use the Villain approximation to show that the Gross-Witten model, in the weak- and strong-coupling limits, is related to the unitary matrix model that describes U(N) Chern-Simons theory on S^3. The weak-coupling limit corresponds to the q->1 limit of the Chern-Simons theory while the strong-coupling regime is related to the q->0 limit. In the latter case, there is a logarithmic relationship between the respective coupling constants. We also show how the Chern-Simons matrix model arises by considering two-dimensional Yang-Mills theory with the Villain action. This leads to a U(1)^N theory which is the Abelianization of 2d Yang-Mills theory with the heat-kernel lattice action. In addition, we show that the character expansion of the Villain lattice action gives the q deformation of the heat kernel as it appears in q-deformed 2d Yang-Mills theory. We also study the relationship between the unitary and Hermitian Chern-Simons matrix models and the rotation of the integration contour in the corresponding integrals.