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Higher Rank Ẑ and F_K

Park, Sunghyuk (2020) Higher Rank Ẑ and F_K. Symmetry, Integrability and Geometry, Methods and Applications (SIGMA), 16 . Art. No. 044. ISSN 1815-0659. https://resolver.caltech.edu/CaltechAUTHORS:20200611-071349486

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Abstract

We study q-series-valued invariants of 3-manifolds that depend on the choice of a root system G. This is a natural generalization of the earlier works by Gukov-Pei-Putrov-Vafa [arXiv:1701.06567] and Gukov-Manolescu [arXiv:1904.06057] where they focused on G=SU(2) case. Although a full mathematical definition for these "invariants" is lacking yet, we define Ẑ^G for negative definite plumbed 3-manifolds and F^G_K for torus knot complements. As in the G=SU(2) case by Gukov and Manolescu, there is a surgery formula relating F^G_K to Ẑ^G of a Dehn surgery on the knot K. Furthermore, specializing to symmetric representations, F^G_K satisfies a recurrence relation given by the quantum A-polynomial for symmetric representations, which hints that there might be HOMFLY-PT analogues of these 3-manifold invariants.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.3842/sigma.2020.044DOIArticle
https://arxiv.org/abs/1909.13002arXivDiscussion Paper
Additional Information:© 2020 National Academy of Science of Ukraine. Received January 15, 2020, in final form May 11, 2020; Published online May 24, 2020. I would like to thank my advisor Sergei Gukov for his invaluable guidance, as well as Francesca Ferrari, Sarah Harrison, Ciprian Manolescu and Nikita Sopenko for helpful conversations. Special thanks go to Nikita Sopenko for his kind help with Mathematica coding. I would also like to thank the anonymous referees for useful comments that helped to improve the paper. The author was supported by Kwanjeong Educational Foundation.
Funders:
Funding AgencyGrant Number
Kwanjeong Educational FoundationUNSPECIFIED
Subject Keywords:3-manifold; knot; quantum invariant; complex Chern-Simons theory; TQFT; q-series; colored Jones polynomial; colored HOMFLY-PT polynomial.
Classification Code:2020 Mathematics Subject Classification: 57K16; 57K31; 81R50
Record Number:CaltechAUTHORS:20200611-071349486
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20200611-071349486
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:103835
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:11 Jun 2020 15:31
Last Modified:11 Jun 2020 15:31

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