Open Gromov-Witten Invariants in Genus Zero

Apart from the usual transversality problems in defining curve counting invariants, when defining the open Gromov-Witten invariants of a Lagrangian one has to deal with the fact that the moduli spaces have boundary. Thus a homological (virtual) fundamental class is usually not available. Assuming regularity, Solomon-Tukachinsky provide a way around this (in genus zero) by first constructing a weakly curved A_infity algebra and then extracting from this enumerative invariants. In this talk I will discuss how their framework can be generalised to deal with general (relatively spin, closed) Lagrangian, using the construction of a global Kuranishi chart for moduli spaces of open stable maps. Our construction can be extended to Lagrangian cobordisms and I will explain how these invariants behave with respect to the relation of being Lagrangian cobordant. Time permitting, I will also discuss some results in higher genus. This is joint work in progress with Kai Hugtenburg.