Surface Operators and Knot Homologies

Creators: Gukov, Sergei

Other:: Sidoravičius, V.

Abstract

Topological gauge theories in four dimensions which admit surface operators provide a natural framework for realizing homological knot invariants. Every such theory leads to an action of the braid group on branes on the corresponding moduli space. This action plays a key role in the construction of homological knot invariants. We illustrate the general construction with examples based on surface operators in N=2 and N=4 twisted gauge theories which lead to a categorification of the Alexander polynomial, the equivariant knot signature, and certain analogs of the Casson invariant.

Additional Information

© 2009 Springer Science + Business Media B.V. I would like to thank N. Dunfield, T. Hausel, A. Kapustin, M. Khovanov, M. Mariño, T. Mrowka, J. Roberts, and C. Vafa for clarifying discussions and comments. I am specially indebted to E. Witten for many useful discussions and for his observations on a preliminary version of this paper. This work is supported in part by DOE grant DE-FG03-92-ER40701, in part by RFBR grant 07-02-00645, and in part by the grant for support of scientific schools NSh-8004.2006.2. This paper is an extended version of the talk delivered at the International Congress on Mathematical Physics 2006 (Rio de Janeiro) and at the RTN workshop "Constituents, Fundamental Forces and Symmetries of the Universe" (Napoli). I am very grateful to the organizers for the opportunity to participate in these meetings and to all the participants for providing a stimulating environment.

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