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Homological algebra of knots and BPS states

Gukov, Sergei and Stošić, Marko (2012) Homological algebra of knots and BPS states. In: String-Math 2011. Proceedings of Symposia in Pure Mathematics. No.85. American Mathematical Society , Providence, RI, pp. 125-171. ISBN 9780821872956. http://resolver.caltech.edu/CaltechAUTHORS:20130212-111358154

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Abstract

It is known that knot homologies admit a physical description as spaces of open BPS states. We study operators and algebras acting on these spaces. This leads to a very rich story, which involves wall crossing phenomena, algebras of closed BPS states acting on spaces of open BPS states, and deformations of Landau-Ginzburg models. One important application to knot homologies is the existence of "colored differentials" that relate homological invariants of knots colored by different representations. Based on this structure, we formulate a list of properties of the colored HOMFLY homology that categorifies the colored HOMFLY polynomial. By calculating the colored HOMFLY homology for symmetric and anti-symmetric representations, we find a remarkable "mirror symmetry" between these triply-graded theories.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1090/pspum/085DOIArticle
http://www.ams.org/books/pspum/085/PublisherArticle
http://arxiv.org/abs/1112.0030v2arXivDiscussion Paper
Additional Information:© 2012 American Mathematical Society. We would like to thank S. Cautis, M. Marino, K. Schaeffer, Y. Soibelman, C. Stroppel, C. Vafa, J. Walcher, and E. Witten for valuable discussions. The work of SG is supported in part by DOE Grant DE-FG03-92-ER40701 and in part by NSF Grant PHY-0757647. Opinions and conclusions expressed here are those of the authors and do not necessarily reflect the views of funding agencies. MS was partially supported by the Portuguese Fundação para a Ciência e a Tecnologia through ISR/IST plurianual funding and through the project number PTDC/MAT/101503/2008, New Geometry and Topology. MS was also partially supported by the Ministry of Science of Serbia, project no. 174012.
Group:Caltech Theory
Funders:
Funding AgencyGrant Number
Department of Energy (DOE)DE-FG03-92-ER40701
NSFPHY-0757647
Fundação para a Ciência e a Tecnologia (FCT)PTDC/MAT/101503/2008
Ministry of Science (Serbia)174012
Record Number:CaltechAUTHORS:20130212-111358154
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20130212-111358154
Official Citation:S. Gukov and M. Stosic. Homological algebra of knots and BPS states. Proc. Symp. Pure Math. (2012) 125-171.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:36874
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:13 Feb 2013 22:36
Last Modified:25 Feb 2016 00:13

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