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RESEARCH

Generating functions of planar polygons from homological mirror symmetry of elliptic curves Kathrin Bringmann1 , Jonas Kaszian2* * Correspondence:

[email protected] 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

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