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Higher depth quantum modular forms and plumbed 3-manifolds

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Abstract

In this paper, we study new invariants \(\widehat{Z}_{{{\varvec{a}}}}(q)\) attached to plumbed 3-manifolds that were introduced by Gukov, Pei, Putrov, and Vafa. These remarkable q-series at radial limits conjecturally compute WRT invariants of the corresponding plumbed 3-manifold. Here, we investigate the series \(\widehat{Z}_{0}(q)\) for unimodular plumbing H-graphs with six vertices. We prove that for every positive definite unimodular plumbing matrix, \(\widehat{Z}_{0}(q)\) is a depth two quantum modular form on \({\mathbb {Q}}\).

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Notes

  1. In [15], M is negative definite, which we account for by replacing it with \(-M\) when referring to their work.

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Acknowledgements

The authors thank S. Chun, S. Gukov, and C. Manolescu for helpful discussion on some aspects of [15]. Moreover, we thank the anonymous referees for their helpful comments.

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Author notes

  1. Antun Milas

    Present address: Department of Mathematics and Statistics, SUNY-Albany, Albany, NY, 12222, USA

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Cologne, Weyertal 86-90, 50931, Cologne, Germany

    Kathrin Bringmann

  2. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Karl Mahlburg

  3. Max Planck Institute for Mathematics, Vivatsgasse 7, 53111, Bonn, Germany

    Antun Milas

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  1. Kathrin Bringmann
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  2. Karl Mahlburg
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  3. Antun Milas
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Correspondence to Antun Milas.

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The research of the first author was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp foundation, and the research leading to these results receives funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement no. 335220 - AQSER. The third author was supported by NSF-DMS Grant 1601070 and a stipend from the Max Planck Institute for Mathematics, Bonn.

Appendix: Data for positive unimodular matrices

Appendix: Data for positive unimodular matrices

Here, we list all positive unimodular matrices of the form (5.1) and the corresponding parameters that appear in Z(q) (see (4.1) and Proposition 5.3). In each case, one can directly check that the assumptions in Sect. 4 are satisfied.

The value of c and the quadratic form Q are given below, and the data for \(\mathcal {S}_j\) are presented in condensed form.

  • 1. M(2, 3, 7, 1, 2, 3)

    $$\begin{aligned} Q({{\varvec{n}}})= n_1^2 + 12 n_1 n_2 + 37 n_2^2, \ c=\frac{5}{6}, \ N_1=N_2=12, \ r_1=r_2=1, \ s_1=s_2=5. \end{aligned}$$

  • 2. M(2, 7, 4, 1, 5, 2)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 21 n_1^2 + 140 n_1 n_2 + 235 n_2^2,\ c=\frac{47}{70}, \ N_1=28, N_2=20, \\ r_1= & {} 5, s_1=9, r_2=3, s_2=7. \end{aligned}$$

  • 3. M(6, 31, 3, 1, 2, 7)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 465 n_1^2 + 2604 n_1 n_2 + 3647 n_2^2, c=\frac{274}{651}, N_1=372, N_2=28, \\ r_1= & {} 149, s_1=161, r_2=5, s_2=23. \end{aligned}$$

  • 4. M(7, 18, 3, 1, 2, 7)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 45 n_1^2 + 252 n_1 n_2 + 353 n_2^2, c=\tfrac{53}{126}, N_1=252, N_2=28, \\ r_1= & {} 101, s_1=115, r_2=5, s_2=9. \end{aligned}$$

  • 5. M(3, 11, 3, 1, 2, 9)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 77 n_1^2 + 396 n_1 n_2 + 510 n_2^2, c=\tfrac{205}{396}, N_1=66, N_2=36, \\ r_1= & {} 19, s_1=25, r_2=7, s_2=11. \end{aligned}$$

  • 6. M(2, 19, 3, 1, 2, 11)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 171 n_1^2 + 836 n_1 n_2 + 1023 n_2^2, c=\frac{239}{418}, N_1=76, N_2=44, \\ r_1= & {} 17, s_1=21, r_2=9, s_2=13. \end{aligned}$$

  • 7. M(2, 3, 3, 1, 2, 27)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 25 n_1^2 + 108 n_1 n_2 + 117 n_2^2, c=\frac{37}{54}, N_1=12, N_2=108, \\ r_1= & {} 1, s_1=5, r_2=25, s_2=29. \end{aligned}$$

  • 8. M(2, 3, 3, 1, 3, 5)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 14 n_1^2 + 60 n_1 n_2 + 65 n_2^2, c=\frac{41}{60}, N_1=12, N_2=30, \\ r_1= & {} 1, s_1=5, r_2=7, s_2=13. \end{aligned}$$

  • 9. M(2, 11, 3, 1, 3, 4)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 55 n_1^2 + 264 n_1 n_2 + 318 n_2^2, c=\frac{155}{264}, N_1=44, N_2=24, \\ r_1= & {} 9, s_1=13, r_2=5, s_2=11. \end{aligned}$$

  • 10. M(3, 4, 3, 1, 3, 4)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 5 n_1^2 + 24 n_1 n_2 + 29 n_2^2, c=\tfrac{155}{264}, N_1=N_2=24, \\ r_1= & {} 5, s_1=11, r_2=5, s_2=11. \end{aligned}$$

  • 11. M(3, 7, 2, 1, 3, 97)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 1337 n_1^2+4074 n_1 n_2+3104 n_2^2, c=\frac{835}{2037}, N_1=42, N_2=582, \\ r_1= & {} 11, s_1=17, r_2=191, s_2=197. \end{aligned}$$

  • 12. M(3, 8, 2, 1, 3, 56)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 109 n_1^2+ 336 n_1 n_2+ 259 n_2^2, c=\tfrac{17}{42}, N_1=48, N_2=336, \\ r_1= & {} 13, s_1=19, r_2=109, s_2=115. \end{aligned}$$

  • 13. M(3, 47, 2, 1, 3, 17)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 1457 n_1^2 + 4794 n_1 n_2 + 3944 n_2^2, c=\frac{895}{2397}, N_1=282, N_2=102, \\ r_1= & {} 91, s_1=97, r_2=31, s_2=37. \end{aligned}$$

  • 14. M(3, 88, 2, 1, 3, 16)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 319 n_1^2 + 1056 n_1 n_2 + 874 n_2^2, c=\frac{391}{1056}, N_1=528, N_2=96, \\ r_1= & {} 173, s_1=179, r_2=29, s_2=35. \end{aligned}$$

  • 15. M(4, 5, 2, 1, 3, 47)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 1820 n_1^2 + 5640 n_1 n_2 + 4371 n_2^2, c=\frac{2263}{5640}, N_1=40, N_2=282, \\ r_1= & {} 11, s_1=19, r_2=91, s_2=97. \end{aligned}$$

  • 16. M(4, 77, 2, 1, 3, 11)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 532 n_1^2 + 1848 n_1 n_2 + 1605 n_2^2 ,c=\frac{635}{1848}, N_1=616, N_2=66, \\ r_1= & {} 227, s_1=235, r_2=19, s_2=25. \end{aligned}$$

  • 17. M(5, 16, 2, 1, 3, 11)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 1520 n_1^2 + 5280 n_1 n_2 + 4587 n_2^2,c=\frac{1813}{5280}, N_1=160, N_2=66, \\ r_1= & {} 59, s_1=69, r_2=19, s_2=25. \end{aligned}$$

  • 18. M(7, 92, 2, 1, 3, 8)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 2093 n_1^2 +7728 n_1 n_2 + 7134 n_2^2,c=\frac{2365}{7728}, N_1=1288, N_2=48, \\ r_1= & {} 545, s_1=559, r_2=13, s_2=19. \end{aligned}$$

  • 19. M(8, 35, 2, 1, 3, 8)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 455 n_1^2 + 1680 n_1 n_2 + 1551 n_2^2),c=\frac{257}{840}, N_1=560, N_2=48, \\ r_1= & {} 237, s_1=253, r_2=13, s_2=19. \end{aligned}$$

  • 20. M(11, 16, 2, 1, 3, 8)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 286 n_1^2 + 1056 n_1 n_2 + 975 n_2^2,c=\frac{323}{1056}, N_1=352, N_2=48, \\ r_1= & {} 149, s_1=171, r_2=13, s_2=19. \end{aligned}$$

  • 21. M(12, 133, 2, 1, 3, 7)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 836 n_1^2 + 3192 n_1 n_2 + 3047 n_2^2,c=\frac{905}{3192}, N_1=3192, N_2=42, \\ r_1= & {} 1451, s_1=1475, r_2=11, s_2=17. \end{aligned}$$

  • 22. M(13, 72, 2, 1, 3, 7)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 3432 n_1^2 + 13104 n_1 n_2 + 12509n_2^2,c=\frac{3715}{13104},N_1=1872,N_2=42,\\ r_1= & {} 851, s_1=877, r_2=11, s_2=17. \end{aligned}$$

  • 23. M(3, 4, 2, 1, 4, 23)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 195 n_1^2 + 552 n_1 n_2 + 391 n_2^2, c=\frac{121}{276}, N_1=24, N_2=184, \\ r_1= & {} 5, s_1=11, r_2=65, s_2=73. \end{aligned}$$

  • 24. M(3, 10, 2, 1, 4, 9)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 115 n_1^2 + 360 n_1 n_2 + 282 n_2^2,c=\frac{143}{360}, N_1=60, N_2=72, \\ r_1= & {} 17, s_1=23, r_2=23, s_2=31. \end{aligned}$$

  • 25. M(3, 52, 2, 1, 4, 7)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 663 n_1^2 + 2184 n_1 n_2 + 1799 n_2^2 ,c=\frac{407}{1092}, N_1=312, N_2=56, \\ r_1= & {} 101, s_1=107, r_2=17, s_2=25. \end{aligned}$$

  • 26. M(6, 67, 2, 1, 4, 5)

    $$\begin{aligned} Q_1({\varvec{n}})= & {} 2211n_1^2 + 8040n_1 n_2 + 7310n_2^2, c=\tfrac{2539}{8040}, N_1=804, N_2=40, \\ r_1= & {} 329, s_1=341, r_2=11, s_2=19. \end{aligned}$$

  • 27. M(2, 7, 2, 1, 4, 77)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 227 n_1^2 +616 n_1 n_2 + 418 n_2^2),c=\frac{279}{616}, N_1=28, N_2=616, \\ r_1= & {} 5, s_1=9, r_2=227, s_2=235. \end{aligned}$$

  • 28. M(7, 26, 2, 1, 4, 5)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 1001 n_1^2 + 3640 n_1 n_2 + 3310 n_2^2 ,c=\frac{1149}{3640}, N_1=364, N_2=40, \\ r_1= & {} 149, s_1=163, r_2=11, s_2=19. \end{aligned}$$

  • 29. M(2, 11, 2, 1, 4, 25)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 781 n_1^2 + 2200 n_1 n_2 + 1550 n_2^2 ,c=\frac{969}{2200}, N_1=44, N_2=200, \\ r_1= & {} 9, s_1=13, r_2=71, s_2=79. \end{aligned}$$

  • 30. M(2, 19, 2, 1, 4, 17)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 893 n_1^2 + 2584 n_1 n_2 + 1870 n_2^2 ,c=\frac{1113}{2584}, N_1=76, N_2=136, \\ r_1= & {} 17, s_1=21, r_2=47, s_2=55. \end{aligned}$$

  • 31. M(2, 71, 2, 1, 4, 13)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 2485 n_1^2 + 7384 n_1 n_2 + 5486 n_2^2 ,c=\frac{3105}{7384}, N_1=284, N_2=104, \\ r_1= & {} 69, s_1=73, r_2=35, s_2=43. \end{aligned}$$

  • 32. M(3, 7, 2, 1, 5, 7)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 69 n_1^2 + 210 n_1 n_2 + 160 n_2^2 ,c=\frac{43}{105}, N_1=42, N_2=70, \\ r_1= & {} 11, s_1=17, r_2=23, s_2=33. \end{aligned}$$

  • 33. M(2, 5, 2, 1, 5, 33)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 254 n_1^2 + 660 n_1 n_2 + 429 n_2^2,c=\frac{307}{660}, N_1=20, N_2=330, \\ r_1= & {} 3, s_1=7, r_2=127, s_2=137. \end{aligned}$$

  • 34. M(2, 7, 2, 1, 5, 16)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 413 n_1^2 + 1120 n_1 n_2 + 760 n_2^2 ,c=\frac{507}{1120}, N_1=28, N_2=160, \\ r_1= & {} 5, s_1=9, r_2=59, s_2=69. \end{aligned}$$

  • 35.M(2, 21, 2, 1, 5, 9)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 434 n_1^2 + 1260 n_1 n_2 + 915 n_2^2 ,c=\frac{541}{1260}, N_1=84, N_2=90, \\ r_1= & {} 19, s_1=23, r_2=31, s_2=41. \end{aligned}$$

  • 36. M(2, 55, 2, 1, 5, 8)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 297 n_1^2 + 880 n_1 n_2 + 652 n_2^2 ,c=\frac{371}{880}, N_1=220, N_2=80, \\ r_1= & {} 53, s_1=57, r_2=27, s_2=37. \end{aligned}$$

  • 37. M(2, 3, 2, 1, 8, 57)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 391 n_1^2 + 912 n_1 n_2 + 532 n_2^2 ,c=\frac{445}{912}, N_1=12, N_2=912, \\ r_1= & {} 1, s_1=5, r_2=391, s_2=407. \end{aligned}$$

  • 38. M(2, 3, 2, 1, 9, 32)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 247 n_1^2 + 576 n_1 n_2 + 336 n_2^2 ,c=\frac{281}{576}, N_1=12, N_2=576, \\ r_1= & {} 1, s_1=5, r_2=247, s_2=265. \end{aligned}$$

  • 39. M(2, 3, 2, 1, 12, 17)

    $$\begin{aligned} Q({{\varvec{n}}})= & {} 175 n_1^2 + 408 n_1 n_2 + 238 n_2^2 ,c=\frac{199}{408}, N_1=12, N_2=408, \\ r_1= & {} 1, s_1=5, r_2=175, s_2=199. \end{aligned}$$

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Bringmann, K., Mahlburg, K. & Milas, A. Higher depth quantum modular forms and plumbed 3-manifolds. Lett Math Phys 110, 2675–2702 (2020). https://doi.org/10.1007/s11005-020-01310-z

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