Speaker:  Sergey Alexandrov (Université de Montpellier)

Abstract: The seminars aim to explain modular properties of rank 0 generalized Donaldson-Thomas (DT) invariants, how these properties can be used to compute the invariants, and their implications for other topological invariants such as Vafa-Witten and Gopakumar-Vafa.

1. In the first part, I'll explain basic facts about modular forms, mock modular forms, Jacobi forms and indefinite theta series. In the end, I'll switch the topic and explain a few facts about generalized DT invariants associated to Calabi-Yau threefolds.

2. In the second part, I'll explain how one can derive precise modular properties of the generating functions of rank 0 DT invariants which turn out to be (higher depth) mock modular forms. I'll present an equation encoding the modular anomaly, its extensions and how it can be solved for compact and non-compact CY threefolds.

3. In the compact case, the solution of the modular anomaly allows us to fix the generating functions up to a finite number of coefficients, the so-called polar terms. In the third part, I'll show how these terms can be computed using wall-crossing and direct integration of topological string, which for a set of compact one-parameter threefolds resulted in explicit modular and mock modular forms encoding rank 0 DT invariants. In turn, they have been used to generate new Gopakumar-Vafa invariants overcoming the limitations of the direct integration method.