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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We describe a correspondence between the Donaldson–Thomas invariants enumerating D0–D6 bound states on a Calabi–Yau 3-fold and certain Gromov–Witten invariants counting rational curves in a family of blowups of weighted projective planes. This is a variation on a correspondence found by Gross–Pandharipande, with D0–D6 bound states replacing representations of generalised Kronecker quivers. We build on a small part of the theories developed by Joyce–Song and Kontsevich–Soibelman for wall-crossing formulae and by Gross–Pandharipande–Siebert for factorisations in the tropical vertex group. Along the way we write down an explicit formula for the BPS state counts which arise up to rank 3 and prove their integrality. We also compare with previous “noncommutative DT invariants” computations in the physics literature. 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. Let @math be a smooth scheme over an algebraically closed field @math of characteristic zero and @math a regular function, and write @math Crit @math , as a closed subscheme of @math . The motivic vanishing cycle @math is an element of the @math -equivariant motivic Grothendieck ring @math defined by Denef and Loeser math.AG 0006050 and Looijenga math.AG 0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) @math depends only on the third-order thickenings @math of @math . (b) If @math is another smooth scheme, @math is regular, @math Crit @math , and @math is an embedding with @math and @math an isomorphism, then @math equals @math "twisted" by a motive associated to a principal @math -bundle defined using @math , where now we work in a quotient ring @math of @math . (c) If @math is an "oriented algebraic d-critical locus" in the sense of Joyce arXiv:1304.4508, there is a natural motive @math , such that if @math is locally modelled on Crit @math , then @math is locally modelled on @math . Using results from arXiv:1305.6302, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with "orientation data", as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory arXiv:0811.2435, and on intersections of oriented Lagrangians in an algebraic symplectic manifold. This paper is an analogue for motives of results on perverse sheaves of vanishing cycles proved in arXiv:1211.3259. We extend this paper to Artin stacks in arXiv:1312.0090. 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.