  Recently, Tsai-Tseng-Yau constructed new invariants of symplectic manifolds:
a sequence of Aoo-algebras built of differential forms on the symplectic
manifold. We show that these symplectic Aoo-algebras have a simple topological
interpretation. Namely, when the cohomology class of the symplectic form is
integral, these Aoo-algebras are equivalent to the standard de Rham
differential graded algebra on certain odd-dimensional sphere bundles over the
symplectic manifold. From this equivalence, we deduce for a closed symplectic
manifold that Tsai-Tseng-Yau's symplectic Aoo-algebras satisfy the Calabi-Yau
property, and importantly, that they can be used to define an intersection
theory for coisotropic/isotropic chains. We further demonstrate that these
symplectic Aoo-algebras satisfy several functorial properties and lay the
groundwork for addressing Weinstein functoriality and invariance in the smooth
category.
