Higher algebraic structures in Hamiltonian Floer theory

Abstract

This is the second of two papers devoted to showing how the rich algebraic formalism of Eliashberg-Givental-Hofer's symplectic field theory (SFT) can be used to define higher algebraic structures on symplectic cohomology. Using the SFT of Hamiltonian mapping tori we show how to define the analogue of rational Gromov-Witten theory for open symplectic manifolds. More precisely, we show that their symplectic cohomology can be equipped with the structure of a so-called cohomology F-manifold. After discussing applications to the quantum cohomology ring of the quintic, we outline why the the classical (closedstring) mirror symmetry conjecture for open Calabi-Yau manifolds shall be formulated as an isomorphism of cohomology F-manifolds. O.ANote that here we view the small quantum product as a product on the tangent space to Q at q = 0,Algebraically the big quantum (cup) product can equivalently be viewed as product on the space of vector fields T (1,0) Q on Q,In other words, it is an element in the space T (1,2) Q of (1, 2)-tensor fields on Q. It is the key ingredient of the Frobenius manifold structure on Q ∼ = QH * (M ) defined by Dubrovin, see [22], [11].Generalizing from closed to open symplectic manifolds M , it is known that quantum cohomology needs to be replaced by the symplectic cohomology SH * (M ) of M , defined using Floer theory. While it is well-known that the small quantum product on QH * (M ) generalizes to the pair-of-pants product, it is the main goal of this paper to define the generalization of the big quantum product and the resulting Frobenius manifold structure. After defining a big pair-of-pants product on symplectic cohomology, we will prove Theorem 0.1. Generalizing Dubrovin's definition of Frobenius manifolds for closed symplectic manifolds, the symplectic cohomology of a Liouville manifold, Calabi-Yau or not, can be equipped with the structure of a cohomology F-manifold in such a way that, on the tangent space at zero, we recover the ring structure given by the pair-of-pants product.Cohomology F-manifolds are generalizations of Frobenius manifolds defined by Merkulov in [20], [21]. Among other things, the vector field product ⋆ now just lives on a differential graded manifold instead of a graded linear space and one drops the requirement for an underlying potential. In other words, a cohomology F-manifold is a differential graded manifold (Q, X) equipped with a graded commutative and associative product for vector fields ⋆ : T (1,0) Q X ⊗ T (1,0) Q X → T (1,0) Q X .Following [8] and [19] 1 , a differential graded manifold is given by a pair of a graded linear space Q (more generally, a formal pointed graded manifold

Versality in mirror symmetry

“…Analogously to §5.2, there exists an L ∞ algebra SC * (X \ D) which one expects should come with an L ∞ morphism (6.1) CO : SC * (X \ D) CC * (Fuk(X \ D)) (see [Fab13]), which can be hoped to be a quasi-isomorphism by analogy with work of Ganatra [Gan13]. The cochain complex (SC * (X \ D), 1 ) was introduced in [FH94] and its cohomology is called the symplectic cohomology SH * (X \ D) (references include [Vit99, Oan04, Sei11]).…”

“…. , γ s ) is the count of (pseudo-)holomorphic maps u : Σ → X with domain as in Figure 3, asymptotic to the orbits γ i as shown, weighted by q ω(u) · e 2πiB(u) (compare [Fab13]).…”

Quantum cohomology as a deformation of symplectic cohomology

“…Moreover, one expects that Floer-theoretic operations on quantum cohomology of M (such as the quantum cup product) are deformations of the corresponding operations on symplectic cohomology of X by β, c.f. [12].…”

Symplectic Criteria on Stratified Uniruledness of Affine Varieties and Applications

“…Remark 5.8. Symplectic cohomology has a fruitful structure, the L ∞ structure, from higher homotopy of Chas-Sullivan string topology on the loop space of a symplectic manifold [ [20], [40], [39], [127]]. Can we detect rationally connectedness of (X, D) by understanding obstruction of higher operations on symplectic cohomology which is from TQFT(Topological Quantum Field Theory) or from SFT(Symplectic Field Theory), which is symplectic invariant?…”