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Ẑ at Large N: From Curve Counts to Quantum Modularity

Ekholm, Tobias and Gruen, Angus and Gukov, Sergei and Kucharski, Piotr and Park, Sunghyuk and Sułkowski, Piotr (2022) Ẑ at Large N: From Curve Counts to Quantum Modularity. Communications in Mathematical Physics . ISSN 0010-3616. doi:10.1007/s00220-022-04469-9. (In Press)

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Reducing a 6d fivebrane theory on a 3-manifold Y gives a q-series 3-manifold invariant ẑ(Y). We analyse the large-N behaviour of F_K = Ẑ(M_K), where M_K is the complement of a knot K in the 3-sphere, and explore the relationship between an a-deformed (a = qᴺ) version of F_K and HOMFLY-PT polynomials. On the one hand, in combination with counts of holomorphic annuli on knot complements, this gives an enumerative interpretation of F_K in terms of counts of open holomorphic curves. On the other, it leads to closed form expressions for a-deformed F_K for (2,2p+1)-torus knots. They suggest a further t-deformation based on superpolynomials, which can be used to obtain a t-deformation of ADO polynomials, expected to be related to categorification. Moreover, studying how F_K transforms under natural geometric operations on K indicates relations to quantum modularity in a new setting.

Item Type:Article
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URLURL TypeDescription ReadCube access Paper
Ekholm, Tobias0000-0003-2060-8462
Gruen, Angus0000-0003-0284-009X
Gukov, Sergei0000-0002-9486-1762
Kucharski, Piotr0000-0002-9599-5658
Park, Sunghyuk0000-0002-6132-0871
Sułkowski, Piotr0000-0002-6176-6240
Additional Information:© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022. Received: 7 June 2020 / Accepted: 29 June 2022. We would like to thank Sibasish Banerjee, Miranda Cheng, Luis Diogo, Boris Feigin, Francesca Ferrari, Sarah Harrison, Jakub Jankowski, Pietro Longhi, Ciprian Manolescu, Marko Stošić, Cumrun Vafa, and Don Zagier for insightful discussions and comments on the draft. The work of T.E. is supported by the Knut and Alice Wallenberg Foundation KAW2020.0307 and by the Swedish Research Council VR2020-04535. The work of S.G. is supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award No. DE-SC0011632, and by the National Science Foundation under Grant No. NSF DMS 1664240. The work of P.K. is supported by the Polish Ministry of Science and Higher Education through its programme Mobility Plus (decision no. 1667/MOB/V/2017/0). The research of S.P. is supported by Kwanjeong Educational Foundation. The work of P.S. is supported by the TEAM programme of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund (POIR.04.04.00-00-5C55/17-00).
Funding AgencyGrant Number
Knut and Alice Wallenberg FoundationKAW2020.0307
Swedish Research CouncilVR2020-04535
Department of Energy (DOE)DE-SC0011632
Ministry of Science and Higher Education (Poland)1667/MOB/V/2017/0
Kwanjeong Educational FoundationUNSPECIFIED
Foundation for Polish ScienceUNSPECIFIED
European Regional Development FundPOIR.04.04.00-00-5C55/17-00
Record Number:CaltechAUTHORS:20220812-829381000
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Official Citation:Ekholm, T., Gruen, A., Gukov, S. et al. Ẑ at Large N: From Curve Counts to Quantum Modularity. Commun. Math. Phys. (2022).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:116274
Deposited By: George Porter
Deposited On:16 Aug 2022 21:40
Last Modified:16 Aug 2022 21:40

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