Homological algebra of knots and BPS states
- Creators
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Gukov, Sergei
- Stošić, Marko
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.
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.Attached Files
Published - Gukov_2012p125.pdf
Submitted - 1112.0030v2.pdf
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Additional details
- Eprint ID
- 36874
- Resolver ID
- CaltechAUTHORS:20130212-111358154
- Department of Energy (DOE)
- DE-FG03-92-ER40701
- NSF
- PHY-0757647
- Fundação para a Ciência e a Tecnologia (FCT)
- PTDC/MAT/101503/2008
- Ministry of Science (Serbia)
- 174012
- Created
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2013-02-13Created from EPrint's datestamp field
- Updated
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2021-11-09Created from EPrint's last_modified field
- Caltech groups
- Caltech Theory
- Series Name
- Proceedings of Symposia in Pure Mathematics
- Series Volume or Issue Number
- 85