The Fukaya A-infinity algebra governs the interactions of pseudoholomorphic disks in a symplectic manifold with boundary in a Lagrangian submanifold. It appears in homological mirror symmetry, in the general definition of Floer cohomology and in the definition of open Gromov-Witten invariants. Previous work in the non-orientable case is limited. A joint work with Jake Solomon provides a definition of the Fukaya A-infinity algebra for non-orientable Lagrangian submanifolds by introducing a local system of non-commutative Novikov rings.
I will give an overview of past results and state the new contribution. I will sketch the A-infinity relations in a diagrammatic way and derive the standard definition from the sketch. Furthermore, I will explain what the obstacles are in the derivation for the non-orientable setting. I will present the definition of an orientor, a new algebraic gadget to handle orientation data. Orientors are the appropriate tool to deal with pushforward of differential forms with values in local systems, which is a crucial part of our definition.