Communications in Mathematical Physics

, Volume 369, Issue 2, pp 675–719 | Cite as

Open Gromov–Witten Theory of \(K_{{\mathbb {P}}^2}, K_{{{\mathbb {P}}^1}\times {{\mathbb {P}}^1}}, K_{W{\mathbb {P}}\left[ 1,1,2\right] }, K_{{{\mathbb {F}}}_1}\) and Jacobi Forms

  • Bohan FangEmail author
  • Yongbin Ruan
  • Yingchun Zhang
  • Jie Zhou


It was known through the efforts of many works that the generating functions in the closed Gromov–Witten theory of \(K_{{\mathbb {P}}^2}\) are meromorphic quasi-modular forms (Coates and Iritani in Kyoto J Math 58(4):695–864, 2018; Lho and Pandharipande in Adv Math 332:349–402, 2018; Coates and Iritani in Gromov–Witten invariants of local \({\mathbb {P}}^{2}\) and modular forms, arXiv:1804.03292 [math.AG], 2018) basing on the B-model predictions (Bershadsky et al. in Commun Math Phys 165:311–428, 1994; Aganagic et al. in Commun Math Phys 277:771–819, 2008; Alim et al. in Adv Theor Math Phys 18(2):401–467, 2014). In this article, we extend the modularity phenomenon to \(K_{{{\mathbb {P}}^1}\times {{\mathbb {P}}^1}}, K_{W{\mathbb {P}}[1,1,2]}, K_{{\mathbb {F}}_1}\). More importantly, we generalize it to the generating functions in the open Gromov–Witten theory using the theory of Jacobi forms where the open Gromov–Witten parameters are transformed into elliptic variables.



B. F. would like to thank Chiu-Chu Melissa Liu and Zhengyu Zong for enlightening discussion. J. Z. would like to thank Murad Alim, Florian Beck, Kathrin Bringmann, Xiaoheng Jerry Wang and Baosen Wu for useful discussions. The authors are very grateful to the anonymous referee for the great improvement of this article. Y. R. is partially supported by NSF Grant DMS 1405245 and NSF FRG Grant DMS 1159265. Y. Z. is supported by China Scholarship Council Grant No. 201706010026. J. Z. ’s work was done while he was a postdoc at the University of Cologne and was partially supported by German Research Foundation Grant CRC/TRR 191.


  1. [ABK08]
    Aganagic, M., Bouchard, V., Klemm, A.: Topological strings and (almost) modular forms. Commun. Math. Phys. 277, 771–819 (2008)ADSMathSciNetzbMATHGoogle Scholar
  2. [AKMV05]
    Aganagic, M., Klemm, A., Marino, M., Vafa, C.: The topological vertex. Commun. Math. Phys. 254, 425–478 (2005)ADSMathSciNetzbMATHGoogle Scholar
  3. [AKV02]
    Aganagic, M., Klemm, A., Vafa, C.: Disk instantons, mirror symmetry and the duality web. Z. Naturforsch. A 57(1–2), 1–28 (2002)ADSMathSciNetzbMATHGoogle Scholar
  4. [ALM10]
    Alim, M., Länge, J.D., Mayr, P.: Global properties of topological string amplitudes and orbifold invariants. JHEP 1003, 113 (2010)ADSMathSciNetzbMATHGoogle Scholar
  5. [ASYZ14]
    Alim, M., Scheidegger, E., Yau, S.-T., Zhou, J.: Special polynomial rings, quasi modular forms and duality of topological strings. Adv. Theor. Math. Phys. 18(2), 401–467 (2014)MathSciNetzbMATHGoogle Scholar
  6. [AV00]
    Aganagic, M., Vafa, C.: Mirror symmetry, D-branes and counting holomorphic discs. arXiv:hep-th/0012041 (2000)
  7. [Bat93]
    Batyrev, V.V.: Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori. Duke Math. J. 69(2), 349–409 (1993)MathSciNetzbMATHGoogle Scholar
  8. [BB91]
    Borwein, J.M., Borwein, P.B.: A cubic counterpart of Jacobi’s identity and the AGM. Trans. Am. Math. Soc. 323(2), 691–701 (1991)MathSciNetzbMATHGoogle Scholar
  9. [BBG94]
    Borwein, J.M., Borwein, P.B., Garvan, F.G.: Some cubic modular identities of Ramanujan. Trans. Am. Math. Soc. 343(1), 35–47 (1994)MathSciNetzbMATHGoogle Scholar
  10. [BBG95]
    Berndt, B.C., Bhargava, S., Garvan, F.G.: Ramanujan’s theories of elliptic functions to alternative bases. Trans. Am. Math. Soc. 347(11), 4163–4244 (1995)MathSciNetzbMATHGoogle Scholar
  11. [BCOV93]
    Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Holomorphic anomalies in topological field theories. Nucl. Phys. B 405, 279–304 (1993)ADSMathSciNetzbMATHGoogle Scholar
  12. [BCOV94]
    Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Kodaira–Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311–428 (1994)ADSMathSciNetzbMATHGoogle Scholar
  13. [BKMnP09]
    Bouchard, V., Klemm, A., Mariño, M., Pasquetti, S.: Remodeling the B-model. Commun. Math. Phys. 287(1), 117–178 (2009)ADSMathSciNetzbMATHGoogle Scholar
  14. [BKMnP10]
    Bouchard, V., Klemm, A., Mariño, M., Pasquetti, S.: Topological open strings on orbifolds. Commun. Math. Phys. 296(3), 589–623 (2010)ADSMathSciNetzbMATHGoogle Scholar
  15. [CI18a]
    Coates, T., Iritani, H.: A Fock sheaf for Givental quantization. Kyoto J. Math. 58(4), 695–864 (2018a)Google Scholar
  16. [CI18b]
    Coates, T., Iritani, H.: Gromov–Witten invariants of local \({\mathbb{P}}^{2}\) and modular forms. arXiv:1804.03292 [math.AG] (2018b)
  17. [CKYZ99]
    Chiang, T.M., Klemm, A., Yau, Shing-Tung, Zaslow, E.: Local mirror symmetry: calculations and interpretations. Adv. Theor. Math. Phys. 3, 495–565 (1999)MathSciNetzbMATHGoogle Scholar
  18. [CLS11]
    Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties, Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence, RI (2011)Google Scholar
  19. [Con96]
    Connell, I.: Elliptic Curve Handbook. McGill University, Montreal. (1996)
  20. [CP12]
    Cho, C.-H., Poddar, M.: Holomorphic orbidiscs and Lagrangian Floer cohomology of symplectic toric orbifolds. J. Differ. Geom. 98(1), 21–116 (2014)zbMATHGoogle Scholar
  21. [DMZ12]
    Dabholkar, A., Murthy, S., Zagier, D.: Quantum black holes, wall crossing, and mock modular forms. arXiv:1208.4074 [hep-th] (2012)
  22. [Dol97]
    Dolgachev, I.V.: Lectures on modular forms. Fall (1997/1998).
  23. [Dub94]
    Dubrovin, B.: Geometry of 2D topological field theories. In: Integrable Systems and Quantum Groups, pp. 120–348. Springer, Berlin (1996)Google Scholar
  24. [EO07]
    Eynard, B., Orantin, N.: Invariants of algebraic curves and topological expansion. Commun. Number Theory Phys. 1(2), 347–452 (2007)MathSciNetzbMATHGoogle Scholar
  25. [EMO07]
    Eynard, B., Orantin, N., Marino, M.: Holomorphic anomaly and matrix models. J. High Energy Phys. 2007(06), 058 (2007)MathSciNetGoogle Scholar
  26. [EZ85]
    Eichler, M., Zagier, D.: The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA (1985)Google Scholar
  27. [Fay77]
    Fay, J.D.: Fourier coefficients of the resolvent for a Fuchsian group. J. reine angew. Math. 293, 143–203 (1977)MathSciNetzbMATHGoogle Scholar
  28. [FLT13]
    Fang, B., Liu, C.-C., Tseng, H.-H.: Open–closed Gromov–Witten invariants of 3-dimensional Calabi–Yau smooth toric DM stacks. arXiv:1212.6073 [math.AG] (2012)
  29. [FLZ16]
    Fang, B., Liu, C.-C.M., Zong, Z.: On the remodeling conjecture for toric Calabi–Yau 3-orbifolds. arXiv:1604.07123 [math.AG] (2016)
  30. [GKMW07]
    Grimm, T.W., Klemm, A., Marino, M., Weiss, M.: Direct integration of the topological string. JHEP 0708, 058 (2007)ADSMathSciNetzbMATHGoogle Scholar
  31. [HKR08]
    Haghighat, B., Klemm, A., Rauch, M.: Integrability of the holomorphic anomaly equations. JHEP 0810, 097 (2008)ADSMathSciNetzbMATHGoogle Scholar
  32. [Hos04]
    Hosono, S.: Central charges, symplectic forms, and hypergeometric series in local mirror symmetry. In: Mirror Symmetry V, AMS/IP Studies in Advanced Mathematics, vol. 38, pp. 405–439. American Mathematical Society, Providence, RI (2006)Google Scholar
  33. [HV00]
    Hori, K., Vafa, C.: Mirror symmetry. arXiv:hep-th/0002222 [hep-th] (2000)
  34. [Iri09]
    Iritani, H.: An integral structure in quantum cohomology and mirror symmetry for toric orbifolds. Adv. Math. 222(3), 1016–1079 (2009)MathSciNetzbMATHGoogle Scholar
  35. [Kat76]
    Katz, N.M.: p-adic interpolation of real analytic Eisenstein series. Ann. Math. 104, 459–571 (1976)Google Scholar
  36. [KK03]
    Kokotov, A, Korotkin, D: Bergmann tau-function on Hurwitz spaces and its applications. ArXiv preprint arXiv:math-ph/0310008 (2003)
  37. [KK04a]
    Kokotov, A., Korotkin, D.: Tau-functions on Hurwitz spaces. Math. Phys. Anal. Geom. 7(1), 47–96 (2004)MathSciNetzbMATHGoogle Scholar
  38. [KK04b]
    Kokotov, A., Korotkin, D.: Tau-functions on spaces of Abelian differentials and higher genus generalizations of Ray–Singer formula. J. Differ. Geom. 82(1), 35–100 (2009)MathSciNetzbMATHGoogle Scholar
  39. [KL01]
    Katz, S., Liu, C.-C.M.: Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc. Adv. Theor. Math. Phys. 5(1), 1–49 (2001)MathSciNetzbMATHGoogle Scholar
  40. [KM10]
    Konishi, Y., Minabe, S.: Local B-model and mixed Hodge structure. Adv. Theor. Math. Phys. 14(4), 1089–1145 (2010)MathSciNetzbMATHGoogle Scholar
  41. [KZ95]
    Kaneko, M., Zagier, D.: A generalized Jacobi theta function and quasimodular forms. In: The Moduli Space of Curves (Texel Island, 1994), Progress in Mathematics, vol. 129, pp. 165–172. Birkhäuser, Boston (1995)Google Scholar
  42. [KZ99]
    Klemm, A., Zaslow, E.E.: Local mirror symmetry at higher genus. AMS IP Stud. Adv. Math. 23, 183–208 (2001)MathSciNetzbMATHGoogle Scholar
  43. [Lho18]
    Lho, H.: Gromov–Witten invariants of Calabi–Yau manifolds with two Kähler parameters. arXiv:1804.04399 [math.AG] (2018)
  44. [Liu02]
    Liu, C.-C.M.: Moduli of J-holomorphic curves with Lagrangian boundary conditions and open Gromov–Witten invariants for an \(S^1\)-equivariant pair. arXiv:math/0210257 [math.SG] (2002)
  45. [LP18]
    Lho, H., Pandharipande, R.: Stable quotients and the holomorphic anomaly equation. Adv. Math. 332, 349–402 (2018)Google Scholar
  46. [Mai09]
    Maier, R.S.: On rationally parametrized modular equations. J. Ramanujan Math. Soc. 24(1), 1–73 (2009)MathSciNetzbMATHGoogle Scholar
  47. [Mai11]
    Maier, R.S.: Nonlinear differential equations satisfied by certain classical modular forms. Manuscr. Math. 134(1–2), 1–42 (2011)MathSciNetzbMATHGoogle Scholar
  48. [Moh02]
    Mohri, K.: Exceptional string: instanton expansions and Seiberg–Witten curve. Rev. Math. Phys. 14, 913–975 (2002)MathSciNetzbMATHGoogle Scholar
  49. [Ran77]
    Rankin, R.A.: Modular Forms and Functions. Cambridge University Press, Cambridge (1977)zbMATHGoogle Scholar
  50. [Sch12]
    Schoeneberg, B.: Elliptic Modular Functions: An Introduction, vol. 203. Springer, Berlin (2012)zbMATHGoogle Scholar
  51. [Sil09]
    Silverman, J.H.: The Arithmetic of Elliptic Curves, vol. 106. Springer, Berlin (2009)zbMATHGoogle Scholar
  52. [Sti97]
    Stienstra, J.: Resonant hypergeometric systems and mirror symmetry. In: Proceedings of the Taniguchi Symposium 1997. Integrable Systems and Algebraic Geometry. World Scientific (1998)Google Scholar
  53. [Sti06]
    Stienstra, J.: Mahler measure variations, Eisenstein series and instanton expansions. In: Mirror Symmetry V, AMS/IP Studies in Advanced Mathematics, vol. 38, pp. 139–150. American Mathematical Society, Providence, RI (2006)Google Scholar
  54. [Tak01]
    Takhtajan, L.A.: Free bosons and tau-functions for compact Riemann surfaces and closed smooth Jordan curves. Current correlation functions. Lett. Math. Phys. 56(3), 181–228 (2001)MathSciNetzbMATHGoogle Scholar
  55. [Tyu78]
    Tyurin, A.N.: On periods of quadratic differentials. Russian Math. Surv. 33(6), 169–221 (1978)ADSMathSciNetzbMATHGoogle Scholar
  56. [Wit93]
    Witten, E.: Phases of \(N = 2\) theories in two dimensions. Nucl. Phys. B 403, 159–222 (1993)ADSMathSciNetzbMATHGoogle Scholar
  57. [YY04]
    Yamaguchi, S., Yau, S.-T.: Topological string partition functions as polynomials. JHEP 0407, 047 (2004)ADSMathSciNetGoogle Scholar
  58. [Zag08]
    Zagier, D.: Elliptic modular forms and their applications. In: The 1-2-3 of Modular Forms, pp. 1–103. Universitext, Springer, Berlin (2008)Google Scholar
  59. [Zho14]
    Zhou, J: Arithmetic Properties of Moduli Spaces and Topological String Partition Functions of Some Calabi-Yau Threefolds. Harvard Ph.D. Thesis (2014)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Beijing International Center for Mathematical ResearchPeking UniversityBeijingChina
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA
  3. 3.Yau Mathematical Sciences Center, Jinchunyuan West BuildingTsinghua UniversityBeijingChina

Personalised recommendations