Knot Homology and Refined Chern–Simons Index

Abstract

We formulate a refinement of SU(N) Chern–Simons theory on a three-manifold M via an index in the (2, 0) theory on N M5 branes. The refined Chern–Simons theory is defined on any M 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. Refined Chern–Simons theory gives rise to new topological invariants of Seifert three-manifolds and torus knots inside them. We conjecture that the invariants are certain indices on knot homology groups. For torus knots in S3 colored by fundamental representation, the index equals the Poincare polynomials of the knot homology theory categorifying the HOMFLY polynomial. As a byproduct, we show that our theory on S3 has a large-N dual which is the refined topological string on \({X=\mathcal{O}(-1) \oplus \mathcal{O}(-1) \rightarrow {\rm I\!P}^1}\) ; this supports the conjecture by Gukov, Schwarz and Vafa relating the spectrum of BPS states on X to SLN knot homology. We also provide a matrix model description of some amplitudes of the refined Chern–Simons theory on S3.