q-deformations of two-dimensional Yang-Mills theory: Classification, categorification and refinement

Abstract

We characterise the quantum group gauge symmetries underlying q-deformations of two-dimensional Yang-Mills theory by studying their relationships with the matrix models that appear in Chern-Simons theory and six-dimensional N=2 gauge theories, together with their refinements and supersymmetric extensions. We develop uniqueness results for quantum deformations and refinements of gauge theories in two dimensions, and describe several potential analytic and geometric realisations of them. We reconstruct standard q-deformed Yang-Mills amplitudes via gluing rules in the representation category of the quantum group associated to the gauge group, whose numerical invariants are the usual characters in the Grothendieck group of the category. We apply this formalism to compute refinements of q-deformed amplitudes in terms of generalised characters, and relate them to refined Chern-Simons matrix models and generalized unitary matrix integrals in the quantum beta-ensemble which compute refined topological string amplitudes. We also describe applications of our results to gauge theories in five and seven dimensions, and to the dual superconformal field theories in four dimensions which descend from the N=(2,0) six-dimensional superconformal theory.

71 pages; v2: references added; final version to be published in Nuclear Physics B

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• Supersymmetric $U(N)$ Chern-Simons-matter theory and phase transitions
• Large $N$ phase transition in $T\overline{T}$-deformed $2d$ Yang-Mills theory on the sphere
• $T\overline{T}$-deformation of $q$-Yang-Mills theory
• Quantum exponentials for the modular double and applications in gravity models
• Torus knot polynomials and susy Wilson loops
• Refined Chern-Simons theory and (q,t)-deformed Yang-Mills theory: Semi-classical expansion and planar limit
• Chern-Simons theory encoded on a spin chain
• Wilson punctured network defects in 2D q-deformed Yang-Mills theory
• Supersymmetric gauge theories, Coulomb gases and Chern-Simons matrix models
• Five-dimensional cohomological localization and squashed $q$-deformations of two-dimensional Yang-Mills theory
• Chiral expansion and Macdonald deformation of two-dimensional Yang-Mills theory
• Semi-phenomenological classification models of the genetic code(s) using q-deformed numbers