Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 018, 52 pages      arXiv:2304.03934      https://doi.org/10.3842/SIGMA.2024.018

Quantum Modular $\widehat Z{}^G$-Invariants

Miranda C.N. Cheng abc, Ioana Coman bd, Davide Passaro b and Gabriele Sgroi b
a) Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The Netherlands
b) Institute of Physics, University of Amsterdam, Amsterdam, The Netherlands
c) Institute for Mathematics, Academica Sinica, Taipei, Taiwan
d) Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Japan

Received May 25, 2023, in final form February 07, 2024; Published online March 09, 2024

Abstract
We study the quantum modular properties of $\widehat Z{}^G$-invariants of closed three-manifolds. Higher depth quantum modular forms are expected to play a central role for general three-manifolds and gauge groups $G$. In particular, we conjecture that for plumbed three-manifolds whose plumbing graphs have $n$ junction nodes with definite signature and for rank $r$ gauge group $G$, that $\widehat Z{}^G$ is related to a quantum modular form of depth $nr$. We prove this for $G={\rm SU}(3)$ and for an infinite class of three-manifolds (weakly negative Seifert with three exceptional fibers). We also investigate the relation between the quantum modularity of $\widehat Z{}^G$-invariants of the same three-manifold with different gauge group $G$. We conjecture a recursive relation among the iterated Eichler integrals relevant for $\widehat Z{}^G$ with $G={\rm SU}(2)$ and ${\rm SU}(3)$, for negative Seifert manifolds with three exceptional fibers. This is reminiscent of the recursive structure among mock modular forms playing the role of Vafa-Witten invariants for ${\rm SU}(N)$. We prove the conjecture when the three-manifold is moreover an integral homological sphere.

Key words: 3-manifolds; quantum invariants; higher depth quantum modular forms; low-dimensional topology.

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References

  1. Alexandrov S., Rank $N$ Vafa-Witten invariants, modularity and blow-up, Adv. Theor. Math. Phys. 25 (2021), 275-308, arXiv:2006.10074.
  2. Alexandrov S., Vafa-Witten invariants from modular anomaly, Commun. Number Theory Phys. 15 (2021), 149-219, arXiv:2005.03680.
  3. Alexandrov S., Banerjee S., Manschot J., Pioline B., Multiple D3-instantons and mock modular forms I, Comm. Math. Phys. 353 (2017), 379-411, arXiv:1605.05945.
  4. Alexandrov S., Banerjee S., Manschot J., Pioline B., Indefinite theta series and generalized error functions, Selecta Math. (N. S.) 24 (2018), 3927-3972, arXiv:1606.05495.
  5. Alexandrov S., Banerjee S., Manschot J., Pioline B., Multiple D3-instantons and mock modular forms II, Comm. Math. Phys. 359 (2018), 297-346, arXiv:1702.05497.
  6. Alexandrov S., Manschot J., Pioline B., S-duality and refined BPS indices, Comm. Math. Phys. 380 (2020), 755-810, arXiv:1910.03098.
  7. Alexandrov S., Pioline B., Black holes and higher depth mock modular forms, Comm. Math. Phys. 374 (2020), 549-625, arXiv:1808.08479.
  8. Alim M., Haghighat B., Hecht M., Klemm A., Rauch M., Wotschke T., Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes, Comm. Math. Phys. 339 (2015), 773-814, arXiv:1012.1608.
  9. Benetti Genolini P., Cabo-Bizet A., Murthy S., Supersymmetric phases of ${\rm AdS}_4/{\rm CFT}_3$, J. High Energy Phys. 2023 (2023), no. 6, 125, 71 pages, arXiv:2301.00763.
  10. Bringmann K., Folsom A., Ono K., Rolen L., Harmonic Maass forms and mock modular forms: theory and applications, Amer. Math. Soc. Colloq. Publ., Vol. 64, American Mathematical Society, Providence, RI, 2017.
  11. Bringmann K., Kaszian J., Milas A., Higher depth quantum modular forms, multiple Eichler integrals, and $\mathfrak{sl}_3$ false theta functions, Res. Math. Sci. 6 (2019), 20, 41 pages, arXiv:1704.06891.
  12. Bringmann K., Kaszian J., Milas A., Vector-valued higher depth quantum modular forms and higher Mordell integrals, J. Math. Anal. Appl. 480 (2019), 123397, 22 pages, arXiv:1803.06261.
  13. Bringmann K., Kaszian J., Milas A., Nazaroglu C., Integral representations of rank two false theta functions and their modularity properties, Res. Math. Sci. 8 (2021), 54, 31 pages, arXiv:2101.02902.
  14. Bringmann K., Kaszian J., Milas A., Nazaroglu C., Higher depth false modular forms, Commun. Contemp. Math. 25 (2023), 2250043, 53 pages, arXiv:2109.00394.
  15. Bringmann K., Kaszián J., Rolen L., Indefinite theta functions arising in Gromov-Witten theory of elliptic orbifolds, Camb. J. Math. 6 (2018), 25-57, arXiv:1608.08588.
  16. Bringmann K., Mahlburg K., Milas A., Higher depth quantum modular forms and plumbed 3-manifolds, Lett. Math. Phys. 110 (2020), 2675-2702, arXiv:1906.10722.
  17. Bringmann K., Mahlburg K., Milas A., Quantum modular forms and plumbing graphs of 3-manifolds, J. Combin. Theory Ser. A 170 (2020), 105145, 32 pages, arXiv:1810.05612.
  18. Bringmann K., Nazaroglu C., An exact formula for $\rm U(3)$ Vafa-Witten invariants on $\mathbb P^2$, Trans. Amer. Math. Soc. 372 (2019), 6135-6159, arXiv:1803.09270.
  19. Bringmann K., Nazaroglu C., A framework for modular properties of false theta functions, Res. Math. Sci. 6 (2019), 30, 23 pages, arXiv:1904.05377.
  20. Bringmann K., Ono K., The $f(q)$ mock theta function conjecture and partition ranks, Invent. Math. 165 (2006), 243-266.
  21. Bruinier J.H., Funke J., On two geometric theta lifts, Duke Math. J. 125 (2004), 45-90, arXiv:math.NT/0212286.
  22. Cappelli A., Itzykson C., Zuber J.B., The ${\rm A}$-${\rm D}$-${\rm E}$ classification of minimal and $A^{(1)}_1$ conformal invariant theories, Comm. Math. Phys. 113 (1987), 1-26.
  23. Chauhan S., Ramadevi P., $\hat Z$-invariant for ${\rm SO}(3)$ and ${\rm OSp}(1|2)$ groups, Ann. Henri Poincaré 24 (2023), 3347-3371, arXiv:2209.00095.
  24. Cheng M.C.N., Chun S., Feigin B., Ferrari F., Gukov S., Harrison S.M., Passaro D., 3-manifolds and VOA characters, arXiv:2201.04640.
  25. Cheng M.C.N., Chun S., Ferrari F., Gukov S., Harrison S.M., 3d modularity, J. High Energy Phys. 2019 (2019), no. 10, 010, 93 pages, arXiv:1809.10148.
  26. Cheng M.C.N., Duncan J.F.R., Rademacher sums and Rademacher series, in Conformal Field Theory, Automorphic Forms and Related Topics, Contrib. Math. Comput. Sci., Vol. 8, Springer, Heidelberg, 2014, 143-182, arXiv:1210.3066.
  27. Cheng M.C.N., Ferrari F., Sgroi G., Three-manifold quantum invariants and mock theta functions, Philos. Trans. Roy. Soc. A 378 (2020), 20180439, 15 pages, arXiv:1912.07997.
  28. Choi D., Lim S., Rhoades R.C., Mock modular forms and quantum modular forms, Proc. Amer. Math. Soc. 144 (2016), 2337-2349.
  29. Chung H.-J., BPS invariants for Seifert manifolds, J. High Energy Phys. 2020 (2020), no. 3, 113, 66 pages, arXiv:1811.08863.
  30. Chung H.-J., BPS invariants for 3-manifolds at rational level $K$, J. High Energy Phys. 2021 (2021), no. 2, 083, 22 pages, arXiv:1906.12344.
  31. Chung H.-J., Resurgent analysis for some 3-manifold invariants, J. High Energy Phys. 2021 (2021), no. 5, 106, 39 pages, arXiv:2008.02786.
  32. Chung H.-J., BPS invariants for a knot in Seifert manifolds, J. High Energy Phys. 2022 (2022), no. 12, 122, 23 pages, arXiv:2201.08351.
  33. Dabholkar A., Gomes J., Murthy S., Nonperturbative black hole entropy and Kloosterman sums, J. High Energy Phys. 2015 (2015), no. 3, 074, 35 pages, arXiv:1404.0033.
  34. Dabholkar A., Putrov P., Three avatars of mock modularity, Internat. J. Modern Phys. A 36 (2021), 2130020, 37 pages, arXiv:2110.09790.
  35. Dabholkar A., Putrov P., Witten E., Duality and mock modularity, SciPost Phys. 9 (2020), 072, 45 pages, arXiv:2004.14387.
  36. de Boer J., Cheng M.C.N., Dijkgraaf R., Manschot J., Verlinde E., A Farey tail for attractor black holes, J. High Energy Phys. 2006 (2006), no. 11, 024, 28 pages, arXiv:hep-th/0608059.
  37. Denef F., Moore G.W., Split states, entropy enigmas, holes and halos, J. High Energy Phys. 2011 (2011), no. 11, 129, 152 pages, arXiv:hep-th/0702146.
  38. Dijkgraaf R., Maldacena J., Moore G., Verlinde E., A black hole farey tail, arXiv:hep-th/0005003.
  39. Eichler M., Eine Verallgemeinerung der Abelschen Integrale, Math. Z. 67 (1957), 267-298.
  40. Ekholm T., Gruen A., Gukov S., Kucharski P., Park S., Stošić M., Sułkowski P., Branches, quivers, and ideals for knot complements, J. Geom. Phys. 177 (2022), 104520, 75 pages, arXiv:2110.13768.
  41. Ekholm T., Gruen A., Gukov S., Kucharski P., Park S., Sułkowski P., $\hat{Z}$ at large $N$: from curve counts to quantum modularity, Comm. Math. Phys. 396 (2022), 143-186, arXiv:2005.13349.
  42. Ellegaard Andersen J., Elbaek Mistegrard W., Resurgence analysis of quantum invariants of Seifert fibered homology sphere, J. Lond. Math. Soc. 105 (2022), 709-764, arXiv:1811.05376.
  43. Feigin B.L., Tipunin I.Yu., Logarithmic CFTs connected with simple Lie algebras, arXiv:1002.5047.
  44. Ferrari F., Putrov P., Supergroups, $q$-series and 3-manifolds, Ann. Henri Poincaré, to appear, arXiv:2009.14196.
  45. Ferrari F., Reys V., Mixed Rademacher and BPS black holes, J. High Energy Phys. 2017 (2017), no. 7, 094, 36 pages, arXiv:1702.02755.
  46. Gukov S., Manolescu C., A two-variable series for knot complements, Quantum Topol. 12 (2021), 1-109, arXiv:1904.06057.
  47. Gukov S., Marino M., Putrov P., Resurgence in complex Chern-Simons theory, arXiv:1605.07615.
  48. Gukov S., Park S., Putrov P., Cobordism invariants from BPS $q$-series, Ann. Henri Poincaré 22 (2021), 4173-4203, arXiv:2009.11874.
  49. Gukov S., Pei D., Putrov P., Vafa C., BPS spectra and 3-manifold invariants, J. Knot Theory Ramifications 29 (2020), 2040003, 85 pages, arXiv:1701.06567.
  50. Gukov S., Putrov P., Vafa C., Fivebranes and 3-manifold homology, J. High Energy Phys. 2017 (2017), no. 7, 071, 80 pages, arXiv:1602.05302.
  51. Gupta R.K., Murthy S., Nazaroglu C., Squashed toric manifolds and higher depth mock modular forms, J. High Energy Phys. 2019 (2019), no. 2, 064, 44 pages, arXiv:1808.00012.
  52. Hikami K., Mock (false) theta functions as quantum invariants, Regul. Chaotic Dyn. 10 (2005), 509-530, arXiv:math-ph/0506073.
  53. Hikami K., On the quantum invariant for the Brieskorn homology spheres, Internat. J. Math. 16 (2005), 661-685, arXiv:math-ph/0405028.
  54. Hikami K., Quantum invariant, modular form, and lattice points, Int. Math. Res. Not. 2005 (2005), 121-154, arXiv:math-ph/0409016.
  55. Lawrence R., Zagier D., Modular forms and quantum invariants of $3$-manifolds, Asian J. Math. 3 (1999), 93-107.
  56. Maldacena J., Strominger A., ${\rm AdS}_3$ black holes and a stringy exclusion principle, J. High Energy Phys. 1998 (1998), no. 12, 005, 24 pages, arXiv:hep-th/9804085.
  57. Males J., Mono A., Rolen L., Higher depth mock theta functions and $q$-hypergeometric series, Forum Math. 33 (2021), 857-866, arXiv:2101.04991.
  58. Manschot J., Vafa-Witten theory and iterated integrals of modular forms, Comm. Math. Phys. 371 (2019), 787-831, arXiv:1709.10098.
  59. Matsusaka T., Terashima Y., Modular transformations of homological blocks for Seifert fibered homology $3$-spheres, arXiv:2112.06210.
  60. Minahan J.A., Nemeschansky D., Vafa C., Warner N.P., ${\rm E}$-strings and $N=4$ topological Yang-Mills theories, Nuclear Phys. B 527 (1998), 581-623, arXiv:hep-th/9802168.
  61. Murakami Y., A proof of a conjecture of Gukov-Pei-Putrov-Vafa, arXiv:2302.13526.
  62. Murthy S., Pioline B., A Farey tale for ${\mathcal N}=4$ dyons, J. High Energy Phys. 2009 (2009), no. 9, 022, 27 pages, arXiv:0904.4253.
  63. Niebur D., Construction of automorphic forms and integrals, Trans. Amer. Math. Soc. 191 (1974), 373-385.
  64. Park S., Higher rank $\hat{Z}$ and $F_K$, SIGMA 16 (2020), 044, 17 pages, arXiv:1909.13002.
  65. Park S., Inverted state sums, inverted Habiro series, and indefinite theta functions, arXiv:2106.03942.
  66. Shimura G., Sur les intégrales attachées aux formes automorphes, J. Math. Soc. Japan 11 (1959), 291-311.
  67. Skoruppa N.-P., Über den Zusammenhang zwischen Jacobiformen und Modulformen halbganzen Gewichts, Bonn. Math. Schr., Vol. 159, Universität Bonn, Bonn, 1985.
  68. Skoruppa N.-P., Jacobi forms of critical weight and Weil representations, in Modular Forms on Schiermonnikoog, Cambridge University Press, Cambridge, 2008, 239-266, arXiv:0707.0718.
  69. Zagier D., Valeurs des fonctions zêta des corps quadratiques réels aux entiers négatifs, Astérisque 41-42 (1977), 135-151.
  70. Zagier D., Ramanujan's mock theta functions and their applications (after Zwegers and Ono-Bringmann), Astérisque (2009), Exp. No. 986, 143-164.
  71. Zagier D., Quantum modular forms, in Quanta of Maths, Clay Math. Proc., Vol. 11, American Mathematical Society, Providence, RI, 2010, 659-675.
  72. Zagier D., Holomorphic quantum modular forms,Talk at Hausdorff Center for Mathematics, 2020, available at https://www.youtube.com/watch?v=2Rj_xh3UKrU.
  73. Zwegers S., Mock theta functions, arXiv:0807.4834.

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