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 qleftrightarrow t^{-1} which correctly reproduces 3d Seiberg duality when q is a specific root of unity.
Highlights
Their new theory can be seen as a deformation of CS gauge theory known as refined Chern–Simons which depends on two parameters q and t that appear in the definition of the modular S and T matrices
This shows that refined Chern–Simons (rCS) matrix model enjoys a property know as superintegrability to what was shown to happen for other q, t-deformed matrix models [8]
In this paper we considered matrix models for a refinement of CS theory and ABJ theory and we showed that there is an action of the q-Virasoro algebra on their generating functions of observables
Summary
In the seminal paper [1], Witten gave a gauge field theory construction of the celebrated Jones polynomials for knots and links in three dimensions using the path integral of Chern–Simons (CS) theory. In the refined theory one substitutes Schur polynomials with their q, t deformation known as Macdonald polynomials which are known to play an important role in the theory of symmetric functions and in representation theory as well In their paper they gave a matrix model definition of the partition function of rCS on S3 arguing that (unknot) Wilson-loop observables are given by averages of Macdonald functions inside of the matrix integral. Computing the solution up to some finite order we are able to conjecture a closed formula for the average of Macdonald characters, and we find that it can be nicely written in terms of the same characters evaluated at some specific locus This shows that rCS matrix model enjoys a property know as superintegrability to what was shown to happen for other q, t-deformed matrix models [8]. The physical interpretation of such a symmetry can be traced back to a 3d version of Seiberg duality which for pure CS reduces to the well-known level-rank duality of the representation ring of u(N )k
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