We use @math algebras to define open Gromov-Witten invariants with both boundary and interior constraints, associated to a Lagrangian submanifold @math of arbitrary odd dimension. The boundary constraints are bounding chains, which are shown to behave like points. The interior constraints are arbitrary even degree classes in the cohomology of @math relative to @math We show the invariants satisfy analogs of the axioms of closed Gromov-Witten theory. Our definition of invariants depends on the vanishing of a series of obstruction classes. One way to show vanishing is to impose certain cohomological conditions on @math Alternatively, if there exists an anti-symplectic involution fixing @math , then part of the obstructions vanish a priori, and weaker cohomological conditions suffice to guarantee vanishing of the remaining obstructions. In particular, our definition generalizes both Welschinger's and Georgieva's real enumerative invariants.
Welschinger @cite_14 @cite_19 corrects disk bubbling by taking into account linking numbers, thus obtaining invariants in dimensions @math and @math . We expect this approach is closely related to the present paper. Another related approach is being developed by Tessler @cite_10 . Zernik @cite_20 @cite_34 follows an approach closely related to the present work to define equivariant open Gromov-Witten invariants and give an equivariant localization formula for them. Biran-Cornea @cite_27 define an invariant counting disks of a given degree through three points, arising as the discriminant of a quadratic form associated to the Lagrangian quantum product. Cho @cite_43 gives an example of an open Gromov-Witten invariant for the Clifford torus in @math . It would be interesting to find a connection between either of these invariants and the present paper.
@cite_14 @cite_19 @cite_27 @cite_43 @cite_34 @cite_10 @cite_20
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Given a closed orientable Lagrangian surface L in a closed symplectic four-manifold X together with a relative homology class d in H_2 (X , L ; Z) with vanishing boundary in H_1 (L ; Z), we prove that the algebraic number of J-holomorphic discs with boundary on L, homologous to d and passing through the adequate number of points neither depends on the choice of the points nor on the generic choice of the almost-complex structure J. We furthermore get analogous open Gromov-Witten invariants by counting, for every non-negative integer k, unions of k discs instead of single discs. Let (L ) be a closed orientable Lagrangian submanifold of a closed symplectic six-manifold ((X , ) ). We assume that the first homology group (H_1 (L ; A) ) with coefficients in a commutative ring (A ) injects into the group (H_1 (X ; A) ) and that (X ) contains no Maslov zero pseudo-holomorphic disc with boundary on (L ). Then, we prove that for every generic choice of a tame almost-complex structure (J ) on (X ), every relative homology class (d H_2 (X , L ; Z ) ) and adequate number of incidence conditions in (L ) or (X ), the weighted number of (J )-holomorphic discs with boundary on (L ), homologous to (d ), and either irreducible or reducible disconnected, which satisfy the conditions, does not depend on the generic choice of (J ), provided that at least one incidence condition lies in (L ). These numbers thus define open Gromov–Witten invariants in dimension six, taking values in the ring (A ). We use the "pearl" machinery in our previous work to study certain enumerative invariants associated to monotone Lagrangian submanifolds. We first compute three-point open Gromov-Witten numbers of Lagrangian torus fibers in toric Fano manifolds, and show that they depend on the choice of three points, hence they are not invariants. We show that for a cyclic A-infinity algebras, such counting may be defined up to Hochschild or cyclic boundary elements. In particular we obtain a well-defined function on Hochschild or cyclic homology of a cyclic A-infinity algebra, which has invariance property under cyclic A-infinity homomorphism. We set up an algebraic framework for the study of pseudoholomorphic discs bounding nonorientable Lagrangians, as well as equivariant extensions of such structures arising from a torus action. First, we define unital cyclic twisted @math algebras and prove some basic results about them, including a homological perturbation lemma which allows one to construct minimal models of such algebras. We then construct an equivariant extension of @math algebras which are invariant under a torus action on the underlying complex. Finally, we construct a homotopy retraction of the Cartan-Weil complex to equivariant cohomology, which allows us to construct minimal models for equivariant cyclic twisted @math algebras. In a forthcoming paper we will use these results to define and obtain fixed-point expressions for the open Gromov-Witten theory of @math , as well as its equivariant extension.