Refined Chern-Simons Theory and Knot Homology

TL;DR

This paper introduces a refined version of Chern-Simons theory, connecting it to knot homology and superpolynomials, and provides computational methods and evidence for the conjectured equivalence.

Contribution

It formulates the refined Chern-Simons theory using Macdonald deformations and verifies its relation to knot superpolynomials for various torus knots.

Findings

Refined invariants computed via S and T matrices.

Verification of the conjecture for multiple torus knots.

New results supporting the refined theory's connection to 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.