Refined Chern-Simons Theory and Knot Homology

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.

Based on talks presented by M.A. at several conferences and workshops, including String-Math 2011 Conference at U. of Pennsilvania

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Cited by in corpus (20)

• Tao Probing the End of the World
• Refined Chern-Simons Theory and Topological String
• Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants
• Gauge/Liouville Triality
• Racah matrices and hidden integrability in evolution of knots
• Differential expansion and rectangular HOMFLY for the figure eight knot
• Refined Black Hole Ensembles and Topological Strings
• On $W$-representations of $\beta$- and $q,t$-deformed matrix models
• Refined 3d-3d Correspondence
• Perspectives of differential expansion
• On Factorization of Generalized Macdonald Polynomials
• Extension of KNTZ trick to non-rectangular representations
• On exclusive Racah matrices $\bar{S}$ for rectangular representations
• Can tangle calculus be applicable to hyperpolynomials?
• On matrix-model approach to simplified Khovanov-Rozansky calculus
• KNTZ trick from arborescent calculus and the structure of differential expansion
• Colored knot amplitudes and Hall-Littlewood polynomials
• Exact Chern-Simons / Topological String duality
• Hopf superpolynomial from topological vertices
• Topological Entanglement Entropy and Braids in Chern-Simons Theory