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

TL;DR

This paper characterizes the quantum group symmetries of q-deformed 2D Yang-Mills theory, linking it to matrix models, topological strings, and higher-dimensional gauge theories, with new classification and refinement results.

Contribution

It develops a classification framework for quantum deformations and refinements of 2D gauge theories, connecting them to matrix models and topological string amplitudes.

Findings

Reconstruction of q-Yang-Mills amplitudes via quantum group categories

Refined amplitudes expressed through generalized characters

Connections established between gauge theories and matrix models

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.