This article deals with the Floer cohomology (with Z 2 coefficients) between torus fibers and the real Lagrangian in Fano toric manifolds. We first investigate the conditions under which the Floer cohomology is defined, and then develop a combinatorial description of the Floer complex based on the polytope of the toric manifold. This description is used to show that if the Floer cohomology is defined, and the Floer cohomology of the torus fiber is non-zero, then the Floer cohomology of the pair is non-zero. Finally, we develop some applications to non-displaceability and the minimum number of intersection points under Hamiltonian isotopy.We do not have a general formula for the rank of the Floer cohomology. However, motivated by [4] and [8], we associate a potential function W c to each torus fiber L c and then are able to show Theorem 1.2. The Floer cohomology of the pair (R, (L c , L ρ )) is well-defined if and only if W c (ρ) = 0. Furthermore, HF (R, (L c , L ρ )) = 0 if and only if ∇W c (ρ) = 0.This fact explains why we use values in F 2 rather than Z 2 : Z 2 is usually not big enough to find solutions to the equations W c = 0, ∇W c = 0. As mentioned before, it is shown in [4] that the condition ∇W c (ρ) = 0 is equivalent to the non-vanishing of HF (L c , L ρ ). Therefore Theorem 1.2 implies that, when defined, the cohomology HF (R, (L c , L ρ )) = 0 if and only if HF (L c , L ρ ) = 0.When ∇W c (ρ) = 0 but W c (ρ) = 0, the Floer cohomology HF (R, (L c , L ρ )) is not defined. Nevertheless, we can still show that R and L c are non-displaceable. Following an idea of Miguel Abreu and Leonardo Macarini we consider the product of the toric manifold with itself. Then HF (R × R, (L c × L c , L ρ ⊕ L ρ )) is well defined and we obtain Theorem 1.3. Suppose there exists a locally constant sheaf L ρ such that ∇W c (ρ) = 0 and let φ be a Hamiltonian diffeomorphism such that φ(R) and L c intersect transversely. ThenThe fact that F 2 is algebraically closed allows us to obtain one more result.Corollary 1.4. Suppose X P is monotone and L 0 is the unique monotone torus fiber. Then there exists L ρ such that ∇W 0 (ρ) = 0. Therefore the Lagrangians R and L 0 are non-displaceable, i.e. for any Hamiltonian diffeomorphism φ, φ(R) ∩ L c = ∅.

We define the tensor product of filtered A∞-algebras, establish some of its properties and give a partial description of the space of bounding cochains in the tensor product. Furthermore we show that in the case of classical A∞-algebras our definition recovers the one given by Markl and Shnider. We also give a criterion that implies that a given A∞-algebra is quasi-isomorphic to the tensor product of two subalgebras. This will be used in a sequel to prove a Künneth Theorem for the Fukaya algebra of a product of Lagrangian submanifolds. arXiv:1404.7184v2 [math.SG] 30 May 2016 (Â, m x ) ⊗ ∞ (B, m y ) ( A ⊗ ∞ B, m ⊗,x y ).

Given a compact Lagrangian submanifold L of a symplectic manifold (M, ω), Fukaya, Oh, Ohta and Ono construct a filtered A∞-algebra F(L), on the cohomology of L, which we call the Fukaya algebra of L. In this paper we describe the Fukaya algebra of a product of two Lagrangians submanifolds L1 ×L2. Namely, we show that F(L1 × L2) is quasi-isomorphic to F(L1) ⊗∞ F(L2), where ⊗∞ is the tensor product of filtered A∞-algebras defined in [2]. As a corollary of this quasi-isomorphism we obtain a description of the bounding cochains on F(L1 × L2) and of the Floer cohomology of L1 × L2.

In this article, we construct a 2-category of Lagrangians in a fixed shifted symplectic derived stack S. The objects and morphisms are all given by Lagrangians living on various fiber products. A special case of this gives a 2-category of n-shifted symplectic derived stacks Symp n . This is a 2-category version of Weinstein's symplectic category in the setting of derived symplectic geometry. We introduce another 2-category Symp or of 0-shifted symplectic derived stacks where the objects and morphisms in Symp 0 are enhanced with orientation data. Using this, we define a partially linearized 2-category LSymp. Joyce and his collaborators defined a certain perverse sheaf on any oriented (−1)-shifted symplectic derived stack. In LSymp, the 2-morphisms in Symp or are replaced by the hypercohomology of the perverse sheaf assigned to the (−1)-shifted symplectic derived Lagrangian intersections. To define the compositions in LSymp we use a conjecture by Joyce, that Lagrangians in (−1)-shifted symplectic stacks define canonical elements in the hypercohomology of the perverse sheaf over the Lagrangian. We refine and expand his conjecture and use it to construct LSymp and a 2-functor from Symp or to LSymp. We prove Joyce's conjecture in the most general local model. Finally, we define a 2-category of d-oriented derived stacks and fillings. Taking mapping stacks into a n-shifted symplectic stack defines a 2-functor from this category to Symp n−d .

Given two cyclic A ∞ -algebras A and B, in this paper we prove that there exists a cyclic A ∞ -algebra structure on their tensor product A ⊗ B which is unique up to a cyclic A ∞ -quasi-isomorphism. Furthermore, the Kontsevich class of A ⊗ B is equal to the cup product of the Kontsevich classes of A and B on the moduli space of curves.