On refined Chern-Simons and refined ABJ matrix models
arXiv:2107.07525 · doi:10.1007/s11005-022-01518-1
Abstract
We consider the matrix model of refined Chern-Simons theory on for the unknot. We derive a -difference operator whose insertion in the matrix integral reproduces an infinite set of Ward identities which we interpret as -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 -difference operator for a similar refinement of ABJ theory and show that the corresponding -Virasoro constraints are equal to those of refined Chern-Simons for a gauge super-group . Our equations and solutions are manifestly symmetric under Langlands duality which correctly reproduces 3d Seiberg duality when is a specific root of unity.
v2: 30 pages, minor revisions, added comments on relations to quantum mirror curves in Section 2.5, added comments on ABJ integrals in Appendix C, Lett.Math.Phys. version
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