Abstracts for Bonn symplectic geometry seminar

BPS invariants and cohomology are central objects in the study of (Kontsevich-Soibelman) Hall algebras or in enumerative geometry of Calabi-Yau 3-folds. In joint work with Yukinobu Toda, we introduce and study a categorical version of BPS cohomology for local K3 surfaces, called quasi-BPS categories. For a generic stability condition, we construct semiorthogonal decompositions of (Porta-Sala) Hall algebra of a K3 surface in products of quasi-BPS categories.

The SYZ version of Kontsevich's Homological Mirror Symmetry conjecture predicts that if X is a symplectic manifold with the structure of a (possibly singular) Lagrangian torus fibration and with a Lagrangian section, then the split-closed derived Fukaya category should be equivalent to the bounded derived category of coherent sheaves on a "dual" fibration. The latter category has a monoidal structure coming from the tensor product, which raises a natural question: in this geometric situation, does the Fukaya category carry a monoidal structure, ideally on the chain level?

Gromov-Witten theory defines a cohomological field theory (CohFT). Teleman classified semi-simple CohFTs in terms of an ‘R-matrix’. In particular, this allows one to compute higher genus Gromov-Witten invariants of spaces with semi-simple quantum cohomology from their genus 0, 3 point invariants. In this talk I will show that this approach might extend to the non semi-simple setting. The inspiration for this is the Fukaya category and the cyclic open closed map.

For a Liouville manifold and a pair of Liouville subdomains intersecting nicely, we prove a Mayer-Vietoris exact sequence relating the symplectic cohomology groups of the Liouville subdomains, their intersection and their union. This generalizes Cieliebak-Oancea's previous work on filled Liouville cobordisms. The proof uses an invariant called relative symplectic cohomology over integers. Joint work with U. Varolgunes.