Knots-quivers correspondence
Abstract
We introduce and explore the relation between knot invariants and quiver representation theory, which follows from the identification of quiver quantum mechanics in D‑brane systems representing knots. We identify various structural properties of quivers associated to knots, and identify such quivers explicitly in many examples, including some infinite families of knots, all knots up to 6 crossings, and some knots with thick homology. Moreover, based on these properties, we derive previously unknown expressions for colored HOMFLY‑PT polynomials and superpolynomials for various knots. For all knots, for which we identify the corresponding quivers, the LMOV conjecture for all symmetric representations (i.e. integrality of relevant BPS numbers) is automatically proved.
Additional Information
© 2020 by International Press. Published 15 May 2020. We thank Sergei Gukov, Satoshi Nawata,Mi losz Panfil, Yan Soibelman, Richard Thomas, Cumrun Vafa, and Paul Wedrich for comments and discussions. We are grateful to the American Institute of Mathematics (San Jose), the Isaac Newton Institute for Mathematical Sciences (Cambridge University), and Institut Henri Poincaré (Paris) for hospitality. The last author thanks the California Institute of Technology, the Matrix Institute (University of Melbourne, Creswick), University of California Davis, and the Isaac Newton Institute for Mathematical Sciences (Cambridge University) for hospitality and opportunity to present the results of this work during seminars and conferences within the last year. This work is supported by the ERC Starting Grant no. 335739 "Quantum fields and knot homologies" funded by the European Research Council under the European Union's Seventh Framework Programme, and the Foundation for Polish Science. M.S. is partially supported by the Ministry of Science of Serbia, project no. 174012.Attached Files
Published - ATMP-2019-0023-0007-a004.pdf
Submitted - 1707.04017.pdf
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Additional details
- Eprint ID
- 79160
- Resolver ID
- CaltechAUTHORS:20170718-135408851
- 335739
- European Research Council (ERC)
- 174012
- Ministry of Science (Serbia)
- Created
-
2017-07-18Created from EPrint's datestamp field
- Updated
-
2021-11-15Created from EPrint's last_modified field
- Caltech groups
- Walter Burke Institute for Theoretical Physics
- Other Numbering System Name
- CALT-TH
- Other Numbering System Identifier
- 2017-040