Refined and Generalized $\hat{Z}$ Invariants for Plumbed $3$-Manifolds

Song Jin Ri

Symmetry, integrability and geometry: methods and applications, Tome 19 (2023)

Résumé

We introduce a two-variable refinement $\hat{Z}_a(q,t)$ of plumbed $3$-manifold invariants $\hat{Z}_a(q)$, which were previously defined for weakly negative definite plumbed $3$-manifolds. We also provide a number of explicit examples in which we argue the recovering process to obtain $\hat{Z}_a(q)$ from $\hat{Z}_a(q,t)$ by taking a limit $ t\rightarrow 1 $. For plumbed $3$-manifolds with two high-valency vertices, we analytically compute the limit by using the explicit integer solutions of quadratic Diophantine equations in two variables. Based on numerical computations of the recovered $\hat{Z}_a(q)$ for plumbings with two high-valency vertices, we propose a conjecture that the recovered $\hat{Z}_a(q)$, if exists, is an invariant for all tree plumbed $3$-manifolds. Finally, we provide a formula of the $\hat{Z}_a(q,t)$ for the connected sum of plumbed $3$-manifolds in terms of those for the components.

Détail

Keywords: $q$-series, $\hat{Z}$ invariants, plumbed $3$-manifolds.

Bibliographie

[1] Akhmechet R., Johnson P. K., Krushkal V., “Lattice cohomology and $q$-series invariants of 3-manifolds”, J. Reine Angew. Math., 796 (2023), 269–299, arXiv: 2109.14139 | DOI | MR

[2] Berndt B. C., Hafner J. L., “Two remarkable doubly exponential series transformations of Ramanujan”, Proc. Indian Acad. Sci. Math. Sci., 104 (1994), 245–252 | DOI | MR | Zbl

[3] Cheng M. C.N., Chun S., Ferrari F., Gukov S., Harrison S. M., “3d modularity”, J. High Energy Phys., 2019:10 (2019), 010, 93 pp., arXiv: 1809.10148 | DOI | MR

[4] Costantino F., Gukov S., Putrov P., “Non-semisimple TQFT's and BPS $q$-series”, SIGMA, 19 (2023), 010, 71 pp., arXiv: 2107.14238 | DOI | MR

[5] Ekholm T., Gruen A., Gukov S., Kucharski P., Park S., Sułkowski P., “$\widehat{Z}$ at large $N$: from curve counts to quantum modularity”, Comm. Math. Phys., 396 (2022), 143–186, arXiv: 2005.13349 | DOI | MR

[6] Gukov S., Manolescu C., “A two-variable series for knot complements”, Quantum Topol., 12 (2021), 1–109, arXiv: 1904.06057 | DOI | MR | Zbl

[7] Gukov S., Pei D., Putrov P., Vafa C., “BPS spectra and 3-manifold invariants”, J. Knot Theory Ramifications, 29 (2020), 2040003, 85 pp., arXiv: 1701.06567 | DOI | MR | Zbl

[8] Gukov S., Putrov P., Vafa C., “Fivebranes and 3-manifold homology”, J. High Energy Phys., 2017:7 (2017), 071, 81 pp., arXiv: 1602.05302 | DOI | MR

[9] Khovanov M., “A categorification of the Jones polynomial”, Duke Math. J., 101 (2000), 359–426, arXiv: math.QA/9908171 | DOI | MR | Zbl

[10] Lickorish W. B.R., “A representation of orientable combinatorial $3$-manifolds”, Ann. of Math., 76 (1962), 531–540 | DOI | MR | Zbl

[11] Neumann W. D., “A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves”, Trans. Amer. Math. Soc., 268 (1981), 299–344 | DOI | MR

[12] Ozsváth P., Szabó Z., “On the Floer homology of plumbed three-manifolds”, Geom. Topol., 7 (2003), 185–224, arXiv: math.SG/0203265 | DOI | MR | Zbl

[13] Park S., “Higher rank $\hat{Z}$ and $F_K$”, SIGMA, 16 (2020), 044, 17 pp., arXiv: 1909.13002 | DOI | MR | Zbl

[14] Reshetikhin N., Turaev V. G., “Invariants of $3$-manifolds via link polynomials and quantum groups”, Invent. Math., 103 (1991), 547–597 | DOI | MR | Zbl

[15] Sawilla R. E., Silvester A. K., Williams H. C., “A new look at an old equation”, Algorithmic Number Theory, Lecture Notes in Comput. Sci., 5011, Springer, Berlin, 2008, 37–59 | DOI | MR | Zbl

[16] Wallace A. H., “Modifications and cobounding manifolds”, Canadian J. Math., 12 (1960), 503–528 | DOI | MR

[17] Witten E., “Quantum field theory and the Jones polynomial”, Comm. Math. Phys., 121 (1989), 351–399 | DOI | MR | Zbl