Floer-Fukaya theory and topological elliptic objects

TL;DR

This paper constructs ring-valued cofunctors inspired by elliptic cohomology and topological field theories, providing evidence for generalized cohomology theories from field theories on topological spaces.

Contribution

It geometrically constructs new cofunctors related to elliptic cohomology using Floer and TFT ideas, and links their properties to conjectures in symplectic geometry.

Findings

pi_2 of the representing space is non-trivial under certain conjectural conditions

Provides evidence for the existence of generalized cohomology theories from field theories

Connects Floer theory, TFTs, and elliptic cohomology in a geometric framework

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.