A Geometric Depiction of Solomon-Tukachinsky's Construction of Open GW-Invariants

TL;DR

This paper provides a geometric framework for Solomon-Tukachinsky's open Gromov-Witten invariants, extending their construction under weaker assumptions and over arbitrary rings, and relates these invariants to Welschinger's invariants.

Contribution

It offers a direct geometric analogue of Solomon-Tukachinsky's construction under weaker assumptions and extends it over arbitrary rings, enhancing the understanding of open Gromov-Witten invariants.

Findings

A geometric analogue of Solomon-Tukachinsky's construction is developed.

The construction is extended over arbitrary rings and without assumptions over rings containing rationals.

A relation between standard Gromov-Witten invariants and Solomon-Tukachinsky's invariants is identified.

Abstract

The 2016 papers of J. Solomon and S. Tukachinsky use bounding chains in Fukaya's $A_{\infty}$-algebras to define numerical disk counts relative to a Lagrangian under certain regularity assumptions on the moduli spaces of disks. We present a (self-contained) direct geometric analogue of their construction under weaker topological assumptions, extend it over arbitrary rings in the process, and sketch an extension without any assumptions over rings containing the rationals. This implements the intuitive suggestion represented by their drawing and P. Georgieva's perspective. We also note a curious relation for the standard Gromov-Witten invariants readily deducible from their work. In a sequel, we use the geometric perspective of this paper to relate Solomon-Tukachinsky's invariants to Welschinger's open invariants of symplectic sixfolds, confirming their belief and G. Tian's related expectation concerning K. Fukaya's earlier construction.