5 research outputs found
Differential forms, Fukaya algebras, and Gromov-Witten axioms
Consider the differential forms on a Lagrangian submanifold . Following ideas of Fukaya-Oh-Ohta-Ono, we construct a family of
cyclic unital curved structures on parameterized by the
cohomology of relative to The family of structures
satisfies properties analogous to the axioms of Gromov-Witten theory. Our
construction is canonical up to pseudoisotopy. We work in the
situation that moduli spaces are regular and boundary evaluation maps are
submersions, and thus we do not use the theory of the virtual fundamental
class.Comment: 51 pages, 6 figures; improved exposition, added illustrations,
corrected minor errors, added reference
Open Gromov-Witten theory for cohomologically incompressible Lagrangians
This paper classifies separated bounding pairs for Lagrangian submanifolds
that are homologically trivial inside the ambient space, under the assumption
that restriction on cohomology from the ambient space to the Lagrangian is
surjective. As an application, open Gromov-Witten invariants are defined under
the above assumptions. When the Lagrangian is the fixed locus of an
anti-symplectic involution, the surjectivity assumption can be somewhat relaxed
while the classifying space needs to be modified.Comment: 34 page
Relative quantum cohomology
We establish a system of PDE, called open WDVV, that constrains the
bulk-deformed superpotential and associated open Gromov-Witten invariants of a
Lagrangian submanifold with a bounding chain. Simultaneously, we
define the quantum cohomology algebra of relative to and prove its
associativity. We also define the relative quantum connection and prove it is
flat. A wall-crossing formula is derived that allows the interchange of
point-like boundary constraints and certain interior constraints in open
Gromov-Witten invariants. Another result is a vanishing theorem for open
Gromov-Witten invariants of homologically non-trivial Lagrangians with more
than one point-like boundary constraint. In this case, the open Gromov-Witten
invariants with one point-like boundary constraint are shown to recover certain
closed invariants. From open WDVV and the wall-crossing formula, a system of
recursive relations is derived that entirely determines the open Gromov-Witten
invariants of with odd, defined in
previous work of the authors. Thus, we obtain explicit formulas for enumerative
invariants defined using the Fukaya-Oh-Ohta-Ono theory of bounding chains.Comment: 69 pages, 6 figures; corrected minor errors, improved expositio