Chiral expansion and Macdonald deformation of two-dimensional Yang-Mills theory

Abstract

We derive the analog of the large N Gross-Taylor holomorphic string expansion for the refinement of q-deformed U(N) Yang-Mills theory on a compact oriented Riemann surface. The derivation combines Schur-Weyl duality for quantum groups with the Etingof-Kirillov theory of generalized quantum characters which are related to Macdonald polynomials. In the unrefined limit we reproduce the chiral expansion of q-deformed Yang-Mills theory derived by de Haro, Ramgoolam and Torrielli. In the classical limit q = 1, the expansion defines a new β-deformation of Hurwitz theory wherein the refined partition function is a generating function for certain parameterized Euler characters, which reduce in the unrefined limit β = 1 to the orbifold Euler characteristics of Hurwitz spaces of holomorphic maps. We discuss the geometrical meaning of our expansions in relation to quantum spectral curves and β-ensembles of matrix models arising in refined topological string theory.

Symmetry defects and orbifolds of two-dimensional Yang-Mills theory

“…Yang-Mills theory on a Riemann surface has a long and rich history as an exactly solvable quantum gauge theory which is the first example of a non-abelian gauge theory that can be reformulated as a (topological) string theory (see [CMR95] for a review). Mathematically it has served as a tool for studying the topology of various moduli spaces of interest in geometry and dynamical systems, such as the moduli spaces of flat connections [AB82,Wit91,Wit92], the Hurwitz moduli spaces of branched coverings [CMR95, KSS16], and the principal moduli spaces of holomorphic differentials [GSS04,GSS05]. In the following we will study how some of these features are modified in the presence of domain walls, which in two dimensions can be thought of as symmetry twist branch cuts on the surface [BBCW14], corresponding to outer automorphisms of the gauge group.…”