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\({\widehat{Z}}\) at Large N: From Curve Counts to Quantum Modularity

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Abstract

Reducing a 6d fivebrane theory on a 3-manifold Y gives a q-series 3-manifold invariant \({\widehat{Z}}(Y)\). We analyse the large-N behaviour of \(F_K={\widehat{Z}}(M_K)\), where \(M_K\) is the complement of a knot K in the 3-sphere, and explore the relationship between an a-deformed (\(a=q^N\)) 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 and an order-by-order construction for other cases. They both 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.

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Notes

  1. A simple class of 3-manifolds for which the modular properties of \({\widehat{Z}} (M_3)\) have not yet been explicitly identified consists of surgeries on the figure-8 knot [GM19].

  2. Here we are using the reduced normalisation. For the unreduced normalisation, we should have, for instance,

    $$\begin{aligned} \lim _{q\rightarrow 1}F_K(x,q^N,q) = \left( \frac{x^{1/2}-x^{-1/2}}{\Delta _K(x)}\right) ^{N-1}, \end{aligned}$$

  3. In this “deformation” the m-th term is multiplied by \(x^m-x^{-m}\). Another version, that appears e.g. in [Par20b], is when the m-th term is multiplied just by \(x^m\); it gives a genuine deformation of the \(\eta \)-function, in which the latter is recovered by taking \(x \rightarrow 1\) and is related to the invariant \(F_K (x,q)\) by further anti-symmetrisation with respect to \(x \rightarrow x^{-1}\).

  4. [GM19] uses the unreduced normalisation. In the reduced normalisation, used in the major part of this paper, (22) reads \(F_K(x,q=e^{\hbar }) = \sum _{j\ge 0}\frac{p_j(x)}{\Delta _K(x)^{2j+1}}\frac{\hbar ^j}{j!}.\)

  5. To be more precise, it is a \(GL(N,\mathbb {C})\) connection, but the dynamics of the \(GL(1,\mathbb {C})\) sector is different from that of the \(SL(N,\mathbb {C})\) sector and can be decoupled.

  6. Strictly speaking, here \(A_{K}\) denotes the augmentation polynomial, which is however conjectured to coincide with the A-polynomial. For details and checks of the conjecture see [AV12, FGS13, FGS13, AEN14, GKS16, KS16, AMM14].

  7. Formula (73) differs from [FGS13] by the rescaling of \({\hat{x}}\) by q mentioned earlier, and of \({\hat{y}}\) by a/q due to the omitted prefactor.

  8. Note that there is a minor technicality when \(\Delta _{K}(x)\) is not monic as in that case we should get \(\frac{\Delta _{K}(x)}{a}\) where a is the leading coefficient. However this case doesn’t occur as when \(\Delta _{K}(x)\) is not monic the abelian branch should be degenerate.

  9. Again we rescale \({\hat{x}}\) by q, \({\hat{y}}\) by a/q and remove common factors of a, q. The A-polynomial we use corresponds to the reduced normalisation.

  10. The prefactor \(x^{\frac{\log a}{\hbar }-1}\) is omitted.

  11. In general, we cannot simply set \(f_0=1\); it should be determined by means other than recursion.

  12. The prefactor \(x^{\frac{\log a}{\hbar }-1}\) is omitted.

  13. In this refined case we rescale \({\hat{y}}\) by \(-at/q\), \({\hat{x}}\) by q and then we remove common factors of a, q, t.

  14. Actually it is interesting for non-fibered amphichiral knots too, as it will give us a q-series which is invariant under \(q\leftrightarrow q^{-1}\).

  15. One way to see this is to consider a simple example, e.g. a solid torus \(\cong S^3 {\setminus } \text {unknot}\), and realise the parity transformation as a sign change of one of the coordinates along the boundary.

  16. Since \({{\hat{y}}}\)-coefficients of \({{\hat{A}}}_K ({{\hat{x}}}, {{\hat{y}}},a,q)\) are rational functions of \({{\hat{x}}}\), a, and q, in the quantum A-polynomial one can also simply replace \(({{\hat{x}}}, a, q) \mapsto ({{\hat{x}}}^{-1}, a^{-1}, q^{-1})\).

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Acknowledgements

We would like to thank Sibasish Banerjee, Miranda Cheng, Luis Diogo, Boris Feigin, Francesca Ferrari, Sarah Harrison, Jakub Jankowski, Pietro Longhi, Ciprian Manolescu, Marko Sto\(\check{\text {s}}\)i\(\acute{\text {c}}\), 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).

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Authors and Affiliations

  1. Department of Mathematics, Uppsala University, Lägerhyddsvägen 1, 752 37, Uppsala, Sweden

    Tobias Ekholm

  2. Institut Mittag-Leffler, Auravägen 17, 182 60, Djursholm, Sweden

    Tobias Ekholm

  3. Division of Physics, Mathematics and Astronomy, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA, 91125, USA

    Angus Gruen, Sergei Gukov, Piotr Kucharski, Sunghyuk Park & Piotr Sułkowski

  4. Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093, Warsaw, Poland

    Piotr Kucharski & Piotr Sułkowski

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  1. Tobias Ekholm
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Ekholm, T., Gruen, A., Gukov, S. et al. \({\widehat{Z}}\) at Large N: From Curve Counts to Quantum Modularity. Commun. Math. Phys. 396, 143–186 (2022). https://doi.org/10.1007/s00220-022-04469-9

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