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Indefinite theta functions arising in Gromov-Witten theory of elliptic orbifolds. (English) Zbl 1408.14176

Summary: In this paper, we consider natural geometric objects coming from Lagrangian Floer theory and mirror symmetry. S.-C. Lau and J. Zhou [Commun. Number Theory Phys. 9, No. 2, 345–385 (2015; Zbl 1347.81068)] showed that some of the explicit Gromov-Witten potentials computed by C.-H. Cho et al. [Adv. Math. 306, 344–426 (2017; Zbl 1355.53073)] are essentially classical modular forms. Recent work by Zwegers and two of the authors determined modularity properties of several simpler pieces of the last, and most mysterious, function by developing several identities between functions with properties generalizing those of the mock modular forms in S. Zwegers’ thesis [Mock theta functions. Utrecht: Universiteit Utrecht (PhD Thesis) (2002)]. Here, we complete the analysis of all pieces of Cho, Hong, Kim, and Lau’s functions, inspired by recent work of S. Alexandrov et al. [Sel. Math., New Ser. 24, No. 5, 3927–3972 (2018; Zbl 1420.11078)] on similar functions. Combined with the work of Lau and Zhou [loc. cit.], as well as the aforementioned work of Zwegers and two of the authors, this affords a complete understanding of the modularity transformation properties of the open Gromov-Witten potentials of elliptic orbifolds of the form \(\mathbb{P}^1_{a,b,c}\) computed by Cho et al. [loc. cit.]. It is hoped that this will provide a fuller picture of the mirror-symmetric properties of these orbifolds in subsequent works.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
33E30 Other functions coming from differential, difference and integral equations
81T60 Supersymmetric field theories in quantum mechanics
11F99 Discontinuous groups and automorphic forms
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