Open Gromov-Witten theory of \(K_{\mathbb {P}^2}\), \(K_{\mathbb {P}^1\times \mathbb {P}^1}\), \(K_{W\mathbb {P}[ 1,1,2]}\), \(K_{\mathbb{F}_1}\) and Jacobi forms. (English) Zbl 1440.14250
Summary: 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
[T. Coates and H. Iritani, Kyoto J. Math. 58, No. 4, 695–864 (2018; Zbl 1423.14313); H. Lho and R. Pandharipande, Adv. Math. 332, 349–402 (2018; Zbl 1423.14317);
T. Coates and H. Iritani, “Gromov-Witten invariants of local \(\mathbb {P}^2\) and modular forms”, Preprint, arXiv:1804.03292]
basing on the B-model predictions
[M. Bershadsky et al., Commun. Math. Phys. 165, No. 2, 311–427 (1994; Zbl 0815.53082); M. Aganagic et al., Commun. Math. Phys. 277, No. 3, 771–819 (2008; Zbl 1165.81037);
M. Alim et al., Adv. Theor. Math. Phys. 18, No. 2, 401–467 (2014; Zbl 1314.14081)]. 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.
MSC:
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
| 11F50 | Jacobi forms |
| 11F03 | Modular and automorphic functions |
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