Abstract

I will outline a construction of a triangulated Fukaya category directly from geometric data in the case of relative wrapped Fukaya categories of surfaces. These pop up in two ways. One is in homological mirror symmetry, where they allow to interpret braid group actions on derived categories as mapping class group actions. Another, conjectured by Lekili-Segal, is that they are equivalent to wrapped Fukaya categories of higher-dimensional symplectic manifolds admitting certain algebraic torus fibrations.

The construction is explicit, meaning that all objects of the category are just curves with additional data, and does not involve any choices such as a coherent perturbation scheme. The key ingredients to achieve that are to work with a version of a Chekanov-Eliashberg algebra of a Legendrian lift of a curve in the surface as obstruction algebra and with A-infinity pre-categories.

I will explain, without assuming familiarity with Fukaya categories, what the objects, morphisms, and A-infinity operations in a relative wrapped Fukaya category of surface are, why it is triangulated, what the correct notion of isotopy invariance in this context is, and how to find finitely many curves that generate the category.