2016/08/31 by Jake P. Solomon, Sara B. Tukachinsky
Mathematics · Physics and Astronomy · #Bounding overwatch #Computer science #Equivalence (formal languages) #Gauge symmetry #Gauge theory #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical physics #Mathematics #Pure mathematics #Submanifold #Symplectic geometry #Symplectic manifold #hep-th #math.AG #math.SG #msc:14N10 #msc:14N35 #msc:53D12 #msc:53D37 #msc:53D45
paper · pdf · doi:10.1007/s00039-021-00583-3
published as Geom. Funct. Anal. 31 (2021), 1245-1320 · 66 pages, 2 figures; includes summary of results needed from arXiv:1608.01304; added explanations, details, and references; minor corrections
arxiv created 2021/09/12 · openalex publication_date 2021/10/01 · arxiv updated 2021/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We present a solution to the problem of defining genus zero open Gromov-Witten invariants with boundary constraints for a Lagrangian submanifold of arbitrary dimension. Previously, such invariants were known only in dimensions 2 and 3 from the work of Welschinger. Our approach does not require the Lagrangian to be fixed by an anti-symplectic involution, but can use such an involution, if present, to obtain stronger results. Also, non-trivial invariants are defined for broader classes of interior constraints and Lagrangian submanifolds than previously possible even in the presence of an anti-symplectic involution. The invariants of the present work specialize to invariants of Welschinger, Fukaya, and Georgieva in many instances. The main obstacle to defining open Gromov-Witten invariants with boundary constraints in arbitrary dimension is the bubbling of J-holomorphic disks. Unlike in low dimensions or for interior constraints, disk bubbles do not cancel in pairs by anti-symplectic involution symmetry. Rather, we use the technique of bounding chains introduced in Fukaya-Oh-Ohta-Ono's work on Lagrangian Floer theory to cancel disk bubbling. At the same time and independently, gauge equivalence classes of bounding chains play the role of boundary constraints, in place of the cohomology classes that usually serve as constraints in Gromov-Witten theory. A crucial step in our construction is to identify a canonical up to gauge equivalence family of "point-like" bounding chains, which specialize in dimensions 2 and 3 to the point constraints considered by Welschinger.