Knot Homology and Refined Chern–Simons Index

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• 作者：Mina Aganagic (1) (2)
  Shamil Shakirov (1) (3)
  
• 刊名：Communications in Mathematical Physics
• 出版年：2015
• 出版时间：January 2015
• 年：2015
• 卷：333
• 期：1
• 页码：187-228
• 全文大小：493 KB
• 参考文献：1. Witten E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351 (1989) CrossRef
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• 作者单位：Mina Aganagic (1) (2)
  Shamil Shakirov (1) (3)
  
  1. Department of Mathematics, University of California, Berkeley, USA
  2. Center for Theoretical Physics, University of California, Berkeley, USA
  3. Institute for Theoretical and Experimental Physics, Moscow, Russia
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mathematical and Computational Physics
  Quantum Physics
  Quantum Computing, Information and Physics
  Complexity
  Statistical Physics
  Relativity and Cosmology
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0916

文摘

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.