In this talk we will consider the question of defining descendant invariants in open Gromov-Witten theory. In the closed Gromov-Witten theory, descendant invariants are constructed from Chern classes of certain tautological lines bundles which live on the moduli space of stable curves. The intersection numbers obtained from those classes (and other classes) can be incorporated in a generating function that satisfies various partial differential equations reflecting recurrence relations and which can sometimes be used to calculate the numbers explicitly.
We will start by briefly reviewing the definition of descendant invariants and their properties in the closed Gromov-Witten theory. We will then describe an approach of defining open Gromov-Witten invariants based on the notions of $A_{\infty}$ algebras and bounding chains due to Jake Solomon and Sara Tukachinsky. This involves constructing an $A_{\infty}$-algebra associated to the open moduli space and using a notion of bounding chain to define the Gromov-Witten invariants via a generating function.
Finally, we will discuss a joint work in progress with Jake Solomon on incorporating descendant invariants in the $A_{\infty}$-framework. This involves constructing connections on the tautological line bundles that satisfy various recursive compatibility conditions and using their curvatures to deform the $A_{\infty}$-structure.
I will (honestly!) try to make the talk as accessible as possible for general audience.