Floer-Fukaya theory and topological elliptic objects

Abstract

Inspired by Segal-Stolz-Teichner project for geometric construction of elliptic (tmf) cohomology, and ideas of Floer theory and of Hopkins-Lurie on extended TFT's, we geometrically construct some $Ring$-valued representable cofunctors on the homotopy category of topological spaces. Using a classical computation in Gromov-Witten theory due to Seidel we show that for one version of these cofunctors $\pi_{2}$ of the representing space is non trivial, provided a certain categorical extension of Kontsevich conjecture holds for the symplectic manifold $ \mathbb{CP} ^{n}$, for some some $n \geq 1$. This gives further evidence for existence of generalized cohomology theories built from field theories living on a topological space.