Knot Homology from Refined Chern-Simons Theory

TL;DR

This paper develops a refined SU(N) Chern-Simons theory linked to topological strings and M5 branes, providing new invariants for three-manifolds and knots, and conjecturing a connection to knot homology theories like Khovanov-Rozansky.

Contribution

It introduces a one-parameter refinement of Chern-Simons theory, relates it to Macdonald polynomials, and proposes a conjecture connecting knot invariants to sl(n) homology.

Findings

Derived explicit refined S and T matrices for Chern-Simons theory.

Obtained new invariants for Seifert three-manifolds and torus knots.

Confirmed that for many torus knots, invariants match Khovanov-Rozansky Poincare polynomials.

Abstract

We formulate a refinement of SU(N) Chern-Simons theory on a three-manifold via the refined topological string and the (2,0) theory on N M5 branes. The refined Chern-Simons theory is defined on any three-manifold with a semi-free circle action. We give an explicit solution of the theory, in terms of a one-parameter refinement of the S and T matrices of Chern-Simons theory, related to the theory of Macdonald polynomials. The ordinary and refined Chern-Simons theory are similar in many ways; for example, the Verlinde formula holds in both. We obtain new topological invariants of Seifert three-manifolds and torus knots inside them. We conjecture that the knot invariants we compute are the Poincare polynomials of the sl(n) knot homology theory. The latter includes the Khovanov-Rozansky knot homology, as a special case. The conjecture passes a number of nontrivial checks. We show that, for a large number of torus knots colored with the fundamental representation of SU(N), our knot invariants agree with the Poincare polynomials of Khovanov-Rozansky homology. As a byproduct, we show that our theory on S^3 has a large-N dual which is the refined topological string on X=O(-1)+O(-1)->P^1; this supports the conjecture by Gukov, Schwarz and Vafa relating the spectrum of BPS states on X to sl(n) knot homology. We also provide a matrix model description of some amplitudes of the refined Chern-Simons theory on S^3.