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Knot Homology and Refined Chern–Simons Index

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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 S 3 colored by fundamental representation, the index equals the Poincaré polynomials of the knot homology theory categorifying the HOMFLY polynomial. As a byproduct, we show that our theory on S 3 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 SL N knot homology. We also provide a matrix model description of some amplitudes of the refined Chern–Simons theory on S 3.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, USA

    Mina Aganagic & Shamil Shakirov

  2. Center for Theoretical Physics, University of California, Berkeley, USA

    Mina Aganagic

  3. Institute for Theoretical and Experimental Physics, Moscow, Russia

    Shamil Shakirov

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  1. Mina Aganagic
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  2. Shamil Shakirov
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Correspondence to Mina Aganagic.

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Communicated by N. A. Nekrasov

The paper is a version of arXiv:1105.5117 prepared for CMP.

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Aganagic, M., Shakirov, S. Knot Homology and Refined Chern–Simons Index. Commun. Math. Phys. 333, 187–228 (2015). https://doi.org/10.1007/s00220-014-2197-4

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