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Link Homologies and the Refined Topological Vertex

Gukov, Sergei and Iqbal, Amer and Kozçaz, Can and Vafa, Cumrun (2010) Link Homologies and the Refined Topological Vertex. Communications in Mathematical Physics, 298 (3). pp. 757-785. ISSN 0010-3616.

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We establish a direct map between refined topological vertex and sl(N) homological invariants of the of Hopf link, which include Khovanov-Rozansky homology as a special case. This relation provides an exact answer for homological invariants of the Hopf link, whose components are colored by arbitrary representations of sl(N). At present, the mathematical formulation of such homological invariants is available only for the fundamental representation (the Khovanov-Rozansky theory) and the relation with the refined topological vertex should be useful for categorizing quantum group invariants associated with other representations (R _1, R _2). Our result is a first direct verification of a series of conjectures which identifies link homologies with the Hilbert space of BPS states in the presence of branes, where the physical interpretation of gradings is in terms of charges of the branes ending on Lagrangian branes.

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Gukov, Sergei0000-0002-9486-1762
Additional Information:© The Author(s) 2010. This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited. Received: 16 September 2009. Accepted: 14 December 2009. Published online: 20 April 2010. Communicated by N.A. Nekrasov. We would like to thank C. Doran, J. Rasmussen, and B.Webster for valuable discussions. It is our pleasure to thank the Stony Brook physics department and the 4th Simons Workshop in Mathematics and Physics for hospitality during the initial stages of this work. In addition, C.V. thanks the CTP at MIT for hospitality during his sabbatical leave. The work of S.G. is supported in part by DOE grant DE-FG03-92-ER40701, in part by RFBR grant 04-02-16880, and in part by the grant for support of scientific schools NSh-8004.2006.2. The work of C.V. is supported in part by NSF grants PHY-0244821 and DMS-0244464.
Funding AgencyGrant Number
Department of Energy (DOE)DE-FG03-92-ER40701
Russian Foundation for Basic Research (RFBR)04-02-16880
Issue or Number:3
Record Number:CaltechAUTHORS:20160505-114729379
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Official Citation:Link Homologies and the Refined Topological Vertex Sergei Gukov, Amer Iqbal, Can Kozçaz, Cumrun Vafa in Communications in Mathematical Physics (2010) September 2010, Volume 298, Issue 3, pp 757-785
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:66693
Deposited By: Ruth Sustaita
Deposited On:05 May 2016 21:53
Last Modified:09 Mar 2020 13:18

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