Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

Mock modularity of Calabi-Yau threefolds

In the first part of the seminar, I'll explain basic facts about modular forms, mock modular forms and indefinite theta series. Then I'll switch the topic and consider generalized Donaldson-Thomas invariants associated to Calabi-Yau threefolds. In the second part, I'll explain how one can show that the generating functions of rank 0 DT invariants are (higher depth) mock modular forms. This allows to fix the generating functions up to a finite number of coefficients, the so-called polar terms, which in turn can be computed using wall-crossing and direct integration of topological string producing Gromov-Witten invariants. This program has been realized for compact one-parameter Calabi-Yau threefolds, resulting in explicit modular and mock modular forms encoding rank 0 DT invariants. In turn, they have been used to generate new boundary conditions for the holomorphic anomaly equation of topological string allowing to overcome the limitations of the direct integration method and find new GW invariants.