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

TL;DR

This paper develops a refined large N expansion for q-deformed 2D Yang-Mills theory using quantum group duality and Macdonald polynomials, connecting to Hurwitz theory and refined topological strings.

Contribution

It introduces a novel $eta$-deformation of Hurwitz theory linked to refined 2D Yang-Mills and derives a generalized string expansion incorporating Macdonald polynomials.

Findings

Reproduces unrefined chiral expansion of q-deformed Yang-Mills

Defines a new $eta$-deformation of Hurwitz theory

Connects expansions to quantum spectral curves and matrix models

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 $\beta$-deformation of Hurwitz theory wherein the refined partition function is a generating function for certain parameterized Euler characters, which reduce in the unrefined limit $\beta=1$ 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 $\beta$-ensembles of matrix models arising in refined topological string theory.