On refined Chern-Simons and refined ABJ matrix models

TL;DR

This paper derives and solves $q$-difference equations for refined Chern-Simons and ABJ matrix models, revealing $q$-Virasoro constraints, superintegrability, and dualities, advancing understanding of refined gauge theories and their symmetries.

Contribution

It introduces $q$-difference operators and $q$-Virasoro constraints for refined Chern-Simons and ABJ models, providing new recursive solutions and symmetry insights.

Findings

Derived $q$-difference operators for refined models

Rewritten constraints as difference equations for Wilson loops

Revealed symmetry under Langlands duality and connections to 3d Seiberg duality

Abstract

We consider the matrix model of $U(N)$ refined Chern-Simons theory on $S^3$ for the unknot. We derive a $q$-difference operator whose insertion in the matrix integral reproduces an infinite set of Ward identities which we interpret as $q$-Virasoro constraints. The constraints are rewritten as difference equations for the generating function of Wilson loop expectation values which we solve as a recursion for the correlators of the model. The solution is repackaged in the form of superintegrability formulas for Macdonald polynomials. Additionally, we derive an equivalent $q$-difference operator for a similar refinement of ABJ theory and show that the corresponding $q$-Virasoro constraints are equal to those of refined Chern-Simons for a gauge super-group $U(N|M)$. Our equations and solutions are manifestly symmetric under Langlands duality $q\leftrightarrow t^{-1}$ which correctly reproduces 3d Seiberg duality when $q$ is a specific root of unity.