Infinity inner products and open Gromov--Witten invariants

TL;DR

This paper introduces a new, field-independent definition of open Gromov--Witten invariants using Lagrangian Floer theory, expanding their applicability beyond real or complex fields and connecting to Calabi--Yau structures.

Contribution

It provides an alternative, invariant-based definition of OGW potential over any characteristic zero field, generalizing previous real or complex field restrictions.

Findings

Defines OGW potential over arbitrary characteristic zero fields.

Constructs a cyclic open-closed map in Lagrangian Floer theory.

Recovers Solomon and Tukachinsky's OGW potential via de Rham cohomology.

Abstract

The open Gromov--Witten (OGW) potential is a function from the set of weak bounding cochains on a closed Lagrangian in a closed symplectic manifold to the Novikov ring. Existing definitions of the OGW potential assume that the ground field of the Novikov ring is either $\mathbb{R}$ or $\mathbb{C}$. In this paper, we give an alternate definition of the OGW potential in the pearly model for Lagrangian Floer theory which yields an invariant valued in the Novikov ring over any field of characteristic zero. We work under simplifying regularity hypotheses which are satisfied, for instance, by any monotone Lagrangian. Our OGW potential is defined in terms of an appropriate weakening of a strictly cyclic pairing on a curved $A_{\infty}$-algebra, which can be thought of as a version of a proper Calabi--Yau structure. Such a structure is obtained by constructing a version of the cyclic open-closed map on the pearly Lagrangian Floer cochain complex. We also explain an analogue of our construction in de Rham cohomology, and show that it recovers the OGW potential constructed by Solomon and Tukachinsky.