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Published December 15, 2017 | Submitted + Published
Journal Article Open

BPS states, knots and quivers

Abstract

We argue how to identify the supersymmetric quiver quantum mechanics description of BPS states, which arise in string theory in brane systems representing knots. This leads to a surprising relation between knots and quivers: to a given knot, we associate a quiver, so that various types of knot invariants are expressed in terms of characteristics of a moduli space of representations of the corresponding quiver. This statement can be regarded as a novel type of categorification of knot invariants, and among its various consequences we find that Labastida-Mariño-Ooguri-Vafa (LMOV) invariants of a knot can be expressed in terms of motivic Donaldson-Thomas invariants of the corresponding quiver; this proves integrality of LMOV invariants (once the corresponding quiver is identified), conjectured originally based on string theory and M-theory arguments.

Additional Information

© 2017 American Physical Society. Received 1 March 2017; published 27 December 2017. We thank Sergei Gukov, Satoshi Nawata, Miłosz Panfil, Yan Soibelman, Richard Thomas, Cumrun Vafa, and Paul Wedrich for discussions and comments on the manuscript. 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, and by Fundação para a Ciência e a Tecnologia (FCT), through the FCT Investigador Grant.

Attached Files

Published - PhysRevD.96.121902.pdf

Submitted - 1707.02991.pdf

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August 19, 2023
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