Unitary Chern-Simons matrix model and the Villain lattice action

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.

17 pages, Minor corrections to match the published version

References in corpus (7)

• Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function
• Chern-Simons Theory on S^1-Bundles: Abelianisation and q-deformed Yang-Mills Theory
• Unitary Integrals and Related Matrix Models
• Chern-Simons matrix models and Stieltjes-Wigert polynomials
• Notes on supersymmetric Wilson loops on a two-sphere
• Wall Crossing As Seen By Matrix Models
• Chern-Simons matrix models, two-dimensional Yang-Mills theory and the Sutherland model

Cited by in corpus (12)

• Supersymmetric $U(N)$ Chern-Simons-matter theory and phase transitions
• q-deformations of two-dimensional Yang-Mills theory: Classification, categorification and refinement
• The Heisenberg XX spin chain and low-energy QCD
• Matrix models for classical groups and Toeplitz$\pm$Hankel minors with applications to Chern-Simons theory and fermionic models
• Oscillatory matrix model in Chern-Simons theory and Jacobi-theta determinantal point process
• Torus knot polynomials and susy Wilson loops
• Multiple phases and meromorphic deformations of unitary matrix models
• Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels
• Villain model with long-range couplings
• Chern-Simons theory encoded on a spin chain
• Polynomial solution of quantum Grassmann matrices
• Complex (super)-matrix models with external sources and $q$-ensembles of Chern-Simons and ABJ(M) type