Infinity Inner Products and Open Gromov-Witten Invariants

The open Gromov-Witten (OGW) potential is a function defined  on the Maurer-Cartan space of a closed Lagrangian submanifold in a  symplectic manifold with values in the Novikov ring. From the values  of the OGW potential, one can extract so-called open Gromov-Witten invariants, which count pseudoholomorphic disks with boundary on the Lagrangian. Standard definitions of the OGW potential only allow for  the construction of OGW invariants with values in the real or complex numbers. In this talk, we will present a construction of the OGW potential which gives invariants valued in any field of characteristic zero. The main algebraic input for our construction is a homotopy cyclic inner product on the (curved) Fukaya A-infinity algebra, which generalizes the notion of a cyclically symmetric inner product and is determined by a proper Calabi-Yau structure on the Fukaya category.