Geometry & Physics Seminar

The Fukaya category on the elliptic curve equipped with Lagrangians furnishes an A-infinity structure. The structure constants, given by the enumeration of holomorphic disks, are simple to write down (following Polishchuk-Zaslow) by virtue of the nice combinatorial nature of the unformization of the elliptic curve and the Lagrangians. The resulting generating series from enumeration, called Fukaya series, are theta functions for indefinite lattices and are expected to belong to the larger class of the so-called mock theta functions. Establishing mock modularity of the A-infinity structure constants can then detect the global properties and symmetry of the latter, and can significantly help understand Fukaya category and moduli spaces of holomorphic maps themselves.

In this talk, I will talk about a joint work in progress with Kathrin Bringmann and Jonas Kaszián in which we compute and prove the mock modularity of some Fukaya series arising from this context. I will mainly try to explain what mock modularity is and why it shows up in geometric terms, as well as the connection to homological mirror symmetry.
Speaker(s): Jie Zhou (Tsinghua University)