Refined Chern-Simons theory and (q,t)-deformed Yang-Mills theory: Semi-classical expansion and planar limit

TL;DR

This paper explores the deep connections between refined Chern-Simons theory and (q,t)-deformed Yang-Mills theory, providing explicit formulas, analyzing phase structures, and linking to topological string theory and black hole microstates.

Contribution

It introduces a generalized Weyl character formula using Macdonald polynomials and formulates q-deformed matrix models for refined Chern-Simons theory.

Findings

Derived the instanton partition function for (q,t)-deformed Yang-Mills.

Established a link between refined gauge theories and topological string theory.

Analyzed large N phase transitions and matched them with string theory predictions.

Abstract

We study the relationship between refined Chern-Simons theory on lens spaces S^3/Z_p and (q,t)-deformed Yang-Mills theory on the sphere S^2. We derive the instanton partition function of (q,t)-deformed U(N) Yang-Mills theory and describe it explicitly as an analytical continuation of the semi-classical expansion of refined Chern-Simons theory. The derivations are based on a generalization of the Weyl character formula to Macdonald polynomials. The expansion is used to formulate q-generalizations of beta-deformed matrix models for refined Chern-Simons theory, as well as conjectural formulas for the chi_y-genus of the moduli space of U(N) instantons on the surface O(-p)--->P^1 for all p which enumerate black hole microstates in refined topological string theory. We study the large N phase structures of the refined gauge theories, and match them with refined topological string theory on the resolved conifold.