A Geometric Depiction of Solomon–Tukachinsky’s Construction of Open Gromov–Witten Invariants

Abstract

The 2016 papers of Solomon and 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 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 Tian’s related expectation concerning Fukaya’s earlier construction.

Funding

Directorate for Mathematical and Physical Sciences(1901979)