We will start be explaining the difficulties in constructing enumerative open Gromov-Witten theories, and mention cases we can overcome these difficulties and obtain a rich enumerative structure.
We then restrict to one such case, and define the full genus 0 stationary open Gromov-Witten theory of maps to CP^1 with boundary conditions on RP^1, including descendents, together with its equivariant extension. We fully compute the theory.
In the last part of the talk, using ideas which involve moduli spaces of metric ribbon graphs, we extend the localization definition to all genera, and conjecture that this localization definition can really be obtained from a geometric definition. We prove a decomposition formula for the closed invariants in terms of the open. If time permits, we show how, based on the conjecture, an open Gromov-Witten/Hurwitz correspondence holds (in all genera).
Joint work with A. Buryak, R. Pandharipande and A. Zernik.