Generating functions of planar polygons from homological mirror symmetry of elliptic curves

K. Bringmann et al. Res. Number Theory (2020)6:28 https://doi.org/10.1007/s40993-020-00199-w RESEARCH Generating functions of planar polygons from homological mirror symmetry of elliptic curves Kathrin Bringmann1 , Jonas Kaszian2* * Correspondence: 2 Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany Full list of author information is available at the end of the article and Jie Zhou3 Abstract We study generating functions of certain shapes of planar polygons arising from homological mirror symmetry of elliptic curves. We express these generating functions in terms of rational functions of the Jacobi theta function and Zwegers’ mock theta function and determine their (mock) Jacobi properties. We also analyze their special values and singularities, which are of geometric interest as well. Keywords: Elliptic curves, Generating functions, Homological mirror symmetry, Jacobi forms, Mock theta functions Mathematics Subject Classification: 11F12, 11F37, 11F50, 14N35, 53D37 1 Introduction and statement of results Elliptic curves provide a fertile ground for the study of the homological mirror symmetry conjecture [10], which relates interesting algebraic structures occurring in the symplectic geometry and complex geometry of different manifolds. They are very simple manifolds that nevertheless exhibit surprisingly rich connections to many fields including Hodge theory, modular forms, and mathematical physics. Of central importance in this subject are the generating functions arising from the open Gromov-Witten theory of elliptic curves. They give the structure constants for the A∞ -structure (i.e., the homotopy version of associative algebra structure) in the Fukaya category (whose objects are Lagrangian submanifolds carrying vector bundles over them, and whose morphisms concern relations among the vector bundles). On the one hand, having a clear understanding of these functions is very useful to verify ideas and conjectures in homological mirror symmetry for elliptic curves and even for more general manifolds. On the other hand, these functions frequently exhibit transformation properties of mock modular forms and Jacobi forms that are interesting to study on their own. Specifically, they provide natural examples of mock modular forms of higher depth. Mock modular forms are holomorphic parts of so-called harmonic Maass forms, which are nonholomorphic generalizations of modular forms. Higher depths forms require additional differential operators. The generating functions arising in this context are very concrete 123 © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 0123456789().,–: volV 28 Page 2 of 19 K. Bringmann et al. Res. Number Theory (2020)6:28 objects and can be expressed using elementary geometric objects. By definition they enumerate holomorphic disks on elliptic curves bounded by a given set of Lagrangians, with appropriate weights specified for example by the area of the holomorphic disks. Due to the simplicity of the universal cover of the elliptic curve, the Lagrangians are represented by straight lines on the universal cover, holomorphic disks are then represented by polygons whose edges lie on these straight lines. This allows the reduction of the enumeration of these geometrical objects to a combinatorial problem. The resulting generating functions may then be written down and turn out to be indefinite theta functions [12,13,15,16], see also [4,9]. In particular, it was found in [16] that the enumeration of triangles yields Jacobi theta functions. The enumeration of parallelograms [13,15,16] gives the GöttscheZagier series [8], while that of more general shapes of 4-gons give the Appell-Lerch sums studied by Kronecker that describe sections of rank two vector bundles on the elliptic curve as shown by [14]. Interestingly, while the former only involves the usual Jacobi theta functions, the latter are related to the mock theta functions. Recently there also have been some works considering the genus zero open GromovWitten invariants of the quotient of elliptic curves called elliptic orbifolds [2,3,5,6,11]. A detailed study of the mock modularity of some generating functions arising from this context was performed in [2,3,11]. We remark that the objects studied in the present work differ from those in the above mentioned papers in that the occurring generating functions are different: the former mainly works with fixed Lagrangians, while in the present work deformations of the Lagrangians are considered as set up originally in [16]. In this paper, we follow the lines in [13,15,16] and study the generating functions arising from the enumeration of particular shapes of 4-gons and 5-gons. The main result of this paper is the following (see (6.1) and (7.1) for the generating functions and Theorem 6.3 and Theorem 7.4 for the mock Jacobi properties). Theorem 1.1 The functions f3 and f4 satisfy mock Jacobi properties. A careful analysis of the modular behavior of the generating functions reveals the global properties of the Gromov-Witten theory on the geometric side. Moreover, the study of special values and singularities can be used to detect what happens in the geometric context, which are otherwise very hard to approach (for example, when the Lagrangians do not intersect transversally). While the study of these very special shapes are already interesting, we hope to extend our investigation to include more general shapes of 5-gons and 6-gons in future work. The paper is organized as follows. In Sect. 2 we provide some preliminary results and conventions on Jacobi theta functions and mock theta functions of Zwegers. In Sect. 3 we review the geometric construction of the generating functions. We then study the generating functions case by case in Sect. 4 to 7. We conclude with some discussions and a conjecture in the final section. 2 Preliminaries In this section we recall some modular forms and generalizations thereof, which we require for this paper. Note that we frequently suppress τ in the notation of functions f : CN × H → C, (z, τ )  → f (z) = f (z; τ ) if it is viewed as fixed. Here and throughout we write components of vectors w ∈ CN as w1 (...truncated)