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

Understand

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.