Topological Field Theories and Harrison Homology

TL;DR

This paper adapts Costello's tools to graph spaces to prove a version of Deligne's conjecture, showing Harrison homology of homotopy commutative algebras forms a module over a 3-manifold cobordism category.

Contribution

It introduces a novel approach connecting graph spaces with topological field theories to establish a new module structure for Harrison homology.

Findings

Harrison homology is a module over a 3D cobordism category.

The adaptation of Costello's methods to Outer Spaces.

Proof of a version of Deligne's conjecture.

Abstract

The tools and arguments developed by Kevin Costello are adapted to families of "Outer Spaces" or spaces of graphs. This allows us to prove a version of Deligne's conjecture: the Harrison homology associated to a homotopy commutative algebra is naturally a module over a particular cobordism category of 3-manifolds.