Knot Homology from Refined Chern-Simons Theory

By Mina Aganagic, Sh. Shakirov. 2011

174 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8b5caeaf2e92d87c9220fa52752d1498b1830349

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. D. Gaiotto
2. E. Witten
3. I. Danilenko
4. I. Cherednik
5. Sergei Gukov
6. Tudor Dimofte
7. C. Vafa
8. Sungjay Lee

In Mathematical Physics

1. D. Gaiotto
2. E. Witten
3. I. Danilenko
4. I. Cherednik
5. Sergei Gukov
6. Tudor Dimofte
7. N. Nekrasov
8. Dan Xie

Citations in the corpus

• DAHA and iterated torus knots (2014, methodology): “A similar connection is also expected with the physics superpolynomials associated with the BPS states (see eg [15; 1; 16]) and those related to the Hilbert schemes of plane curve singularities and rational DAHA; see [33; 23; 22] and references therein.”
• Knot Invariants from Four-Dimensional Gauge Theory (2011, background): “See [5]-[13] for a sampling of additional developments.”
• Jones polynomials of torus knots via DAHA (2011, methodology): “7 of [1] upon the substitution from (2.”