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Quadruply-graded colored homology of knots

Gorsky, Eugene and Gukov, Sergei and Stošić, Marko (2018) Quadruply-graded colored homology of knots. Fundamenta Mathematicae, 243 . pp. 209-299. ISSN 0016-2736.

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We conjecture the existence of four independent gradings in colored HOMFLYPT homology, and make qualitative predictions of various interesting structures and symmetries in the colored homology of arbitrary knots. We propose an explicit conjectural description for the rectangular colored homology of torus knots, and identify the new gradings in this context. While some of these structures have a natural interpretation in the physical realization of knot homologies based on counting supersymmetric configurations (BPS states, instantons, and vortices), others are completely new. They suggest new geometric and physical realizations of colored HOMFLYPT homology as the Hochschild homology of the category of branes in a Landau–Ginzburg B-model or, equivalently, in the mirror A-model. Supergroups and supermanifolds are surprisingly ubiquitous in all aspects of this work.

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Gukov, Sergei0000-0002-9486-1762
Additional Information:© 2018 Instytut Matematyczny PAN. Received 28 October 2014; revised 21 February 2017. Published online 3 September 2018. We are grateful to M. Abouzaid, M. Aganagic, J. M. Baptista, M. Bershtein, I. Cherednik, K. Costello, R. Elliot, P. Etingof, A. Gorsky, K. Hikami, M. Khovanov, B. Kim, A. N. Kirillov and A. A. Kirillov Jr., I. Losev, A. Morozov, H. Nakajima, A. Neguµ, N. Nekrasov, A. Oblomkov, A. Okounkov, J. Rasmussen, L. Rozansky, S. Shakirov, V. Shende, C. Vafa, O. Viro, E. Witten, and C. Woodward for useful discussions. E.G. would like to thank California Institute of Technology and Kyoto Research Institute for Mathematical Sciences for hospitality. The research of E.G. is partially supported by the NSF grant DMS-1559338, grants RFBR-10-01-678, NSh-8462.2010.1, Simons Foundation and Russian Academic Excellence Project 5-100. S.G. would like to thank Instituto Superior Técnico in Lisbon and the Simons Center for Geometry and Physics at Stony Brook for hospitality during the key stages of this work. The work of S.G. is supported in part by DOE grant DE-FG03-92-ER40701FG-02 and in part by NSF grant PHY-0757647. M.S. would like to thank the California Institute of Technology for hospitality while part of this work was done. The work of S.G. and M.S. was partially supported by ERC Starting Grant no. 335739 “Quantum fields and knot homologies” funded by the European Research Council under the European Union Seventh Framework Programme. M.S. was also partially supported by the Portuguese Fundação para a Ciência e a Tecnologia through the project PTDC/MAT/101503/2008, New Geometry and Topology, and by the Ministry of Science of Serbia, project no. 174012. Opinions and conclusions expressed here are those of the authors and do not necessarily reflect the views of the funding agencies.
Funding AgencyGrant Number
Russian Foundation for Basic ResearchRFBR-10-01-678
Russian Foundation for Basic ResearchNSh-8462.2010.1
Simons FoundationUNSPECIFIED
Russian Academic Excellence Project (RAEP)5-100
Department of Energy (DOE)DE-FG03-92-ER40701FG-02
European Research Council (ERC)335739
Fundação para a Ciência e a Tecnologia (FCT)PTDC/MAT/101503/2008
Ministry of Science (Serbia)174012
Subject Keywords:knot homology, colored HOMFLYPT invariants, BPS invariants, differentials, Lie superalgebras
Classification Code:2010 Mathematics Subject Classification: Primary 57M27; Secondary 81T30, 20C08
Record Number:CaltechAUTHORS:20160503-082624275
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:66605
Deposited By: Tony Diaz
Deposited On:03 May 2016 18:37
Last Modified:09 Mar 2020 13:18

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