Proves uniqueness of solutions to constraints on (q,t)-deformed hypergeometric functions and derives superintegrability relations for a general (q,t)-deformed matrix model with allowed parameters.
Refined Chern-Simons Theory and Knot Homology
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abstract
The refined Chern-Simons theory is a one-parameter deformation of the ordinary Chern-Simons theory on Seifert manifolds. It is defined via an index of the theory on N M5 branes, where the corresponding one-parameter deformation is a natural deformation of the geometric background. Analogously with the unrefined case, the solution of refined Chern-Simons theory is given in terms of S and T matrices, which are the proper Macdonald deformations of the usual ones. This provides a direct way to compute refined Chern-Simons invariants of a wide class of three-manifolds and knots. The knot invariants of refined Chern-Simons theory are conjectured to coincide with the knot superpolynomials -- Poincare polynomials of the triply graded knot homology theory. This conjecture is checked for a large number of torus knots in S^3, colored by the fundamental representation. This is a short, expository version of arXiv:1105.5117, with some new results included.
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Refined Chern-Simons theory universality is restricted to simply laced Lie groups, unlike the original which applies to all simple Lie groups.
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Superintegrability for some $(q,t)$-deformed matrix models
Proves uniqueness of solutions to constraints on (q,t)-deformed hypergeometric functions and derives superintegrability relations for a general (q,t)-deformed matrix model with allowed parameters.
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A note on universality in refined Chern-Simons theory
Refined Chern-Simons theory universality is restricted to simply laced Lie groups, unlike the original which applies to all simple Lie groups.