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BPS counting for knots and combinatorics on words

Kucharski, Piotr and Sułkowski, Piotr (2016) BPS counting for knots and combinatorics on words. Journal of High Energy Physics, 2016 (11). Art. No. 120. ISSN 1126-6708. https://resolver.caltech.edu/CaltechAUTHORS:20160824-093108910

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Abstract

We discuss relations between quantum BPS invariants defined in terms of a product decomposition of certain series, and difference equations (quantum A-polynomials) that annihilate such series. We construct combinatorial models whose structure is encoded in the form of such difference equations, and whose generating functions (Hilbert-Poincaré series) are solutions to those equations and reproduce generating series that encode BPS invariants. Furthermore, BPS invariants in question are expressed in terms of Lyndon words in an appropriate language, thereby relating counting of BPS states to the branch of mathematics referred to as combinatorics on words. We illustrate these results in the framework of colored extremal knot polynomials: among others we determine dual quantum extremal A-polynomials for various knots, present associated combinatorial models, find corresponding BPS invariants (extremal Labastida-Mariño-Ooguri-Vafa invariants) and discuss their integrality.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1007/JHEP11(2016)120DOIArticle
http://link.springer.com/article/10.1007%2FJHEP11%282016%29120PublisherArticle
http://arxiv.org/abs/1608.06600arXivDiscussion Paper
https://static-content.springer.com/esm/art%3A10.1007%2FJHEP11%282016%29120/MediaObjects/13130_2016_5060_MOESM1_ESM.nbPublisherSupplementary Material
ORCID:
AuthorORCID
Kucharski, Piotr0000-0002-9599-5658
Sułkowski, Piotr0000-0002-6176-6240
Additional Information:© The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Article funded by SCOAP3. Received: August 24, 2016. Accepted: November 13, 2016. Published: November 21, 2016. We thank Markus Reineke and Marko Stošić for discussions on these and related topics. 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.
Group:Walter Burke Institute for Theoretical Physics
Funders:
Funding AgencyGrant Number
European Research Council (ERC)335739
Foundation for Polish ScienceUNSPECIFIED
Subject Keywords:Chern-Simons Theories, Non-Commutative Geometry, Topological Field Theories, Topological Strings
Other Numbering System:
Other Numbering System NameOther Numbering System ID
CALT-TH2016-022
Issue or Number:11
Record Number:CaltechAUTHORS:20160824-093108910
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20160824-093108910
Official Citation:Kucharski, P. & Sułkowski, P. J. High Energ. Phys. (2016) 2016: 120. doi:10.1007/JHEP11(2016)120
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:69889
Collection:CaltechAUTHORS
Deposited By: Joy Painter
Deposited On:24 Aug 2016 22:18
Last Modified:11 Feb 2020 19:16

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