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.
Bibliographical noteFunding Information:
The authors would like to thank M. Abouzaid, D. Auroux, P. Georgieva, D. Joyce, T. Kimura, E. Kosloff, M. Liu, L. Polterovich, E. Shustin, I. Smith, G. Tian, and A. Netser Zernik, for helpful conversations. The authors were partially supported by ERC starting Grant 337560 and ISF Grant 1747/13. The first author was partially supported by ISF Grant 569/18. The second author was partially supported by the Canada Research Chairs Program and NSF Grant No. DMS-163852.
© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
- A algebra
- Bounding chain
- Gromov–Witten axiom
- Lagrangian submanifold
- Open Gromov–Witten invariant
- Stable map