Title: DAHA Approach in the Refined Chern-Simons Theory and the BPS Theory of Invariants of Torus Knots
Speaker: Ivan Cherednik, University of North Carolina
Date: Tuesday, December 18, 2012
Time: 1:00pm – 2:00pm
Place: Lecture Hall 102, Simons Center
Abstract: The Chern-Simons gauge theory (Witten and others) for the Wilson loops associated with the knots on a torus results in the formulas for their Jones and Quantum Group invariants where the key ingredient is the Verlinde S-operator. M.Aganagic and S.Shakirov suggested to replace here S by its “refined” version due to Kirillov and Cherednik in terms of the Macdonald polynomials at roots of unity. They demonstrated that for the torus knots they considered (mainly for the trefoil), such expressions are related to the Poincare polynomials of stable Khovanov-Rozansky triple-graded homology, which presumably coincide with those in the BPS theory. Managing arbitrary torus knots, replacing roots of unity by generic q, the stabilization in terms of the matrix dimension and adding colors were not clarified in this work.
In the speaker’s IMRN paper, to be mainly discussed in this lecture, these problems were resolved for arbitrary root systems (the rank stabilization holds only for the classical ones) using DAHA, double affine Hecke Algebras. The machinery created there allowed to reproduce the formulas obtained by E.Gorsky in terms of the combinatorics of the rational DAHA and in quite a few (mainly physics) papers devoted to explicit formulas of superpolynomials. The construction is quite interesting even in the simplest case of Quantum Group invariants (only “q”). We will provide exact definitions and discuss the main facts and the conjectures on the DAHA superpolynomials (type A) and hyperpolynomials (any classical root systems). However the mathematics and physics origins and links will be touched upon only a little in this lecture.
The hyperpolynomials of torus knots may depend on up to 6 parameters, “a,q”, and four “t”. The DAHA superpolynomials (type A, one “t”) are expected to give the stable Khovanov-Rozansky polynomials and to be related to the categorification of Quantum Groups. The connection with the theory of Oblomkov, Rasmussen, Shende and Gorsky (type A, no colors) will be discussed a little, but there are no solid links so far in spite of impressive match at the level of formulas (and using DAHA in both approaches). The positivity fails in type A, e.g. for the “3-hook”, and even for the minuscule weights in the D-case, generalizing the Kauffman polynomials; this will be discussed. Other known and expected properties of the KhR/BPS homology presumably hold for the DAHA-hyperpolynomials, including the duality in types A-C (A was checked), similar to that conjectured by Gukov-Stosic for A.