Joint IAS/Princeton University Symplectic Geometry Seminar

I will describe joint work-in-progress with Sheel Ganatra and Vivek Shende. We introduce a class of Liouville manifolds with boundary, called Liouville sectors, in which Floer theory is well behaved (a condition on the characteristic foliation of the boundary controls holomorphic curves from escaping). A corollary of this setup is a local-to-global argument for verifying Abouzaid's generation criterion for the wrapped Fukaya category. We verify this criterion for Liouville sectors associated to Nadler's arboreal singularities, and so any Liouville manifold covered by such sectors has a finite generating collection of Lagrangians (conjecturally, all Weinstein manifolds admit such a cover). We expect the language of Liouville sectors also suffices to formulate and prove statements such as "the wrapped Fukaya category is a homotopy cosheaf" with respect to some reasonable class of open covers.