Smooth one-dimensional topological field theories are vector bundles with connection

Abstract

We prove that smooth 1-dimensional topological field theories over a manifold are the same as vector bundles with connection. The main novelty is our definition of the smooth 1-dimensional bordism category, which encodes cutting laws rather than gluing laws. We make this idea precise through a smooth generalization of Rezk's complete Segal spaces. With such a definition in hand, we analyze the category of field theories using a combination of descent, a smooth version of the 1-dimensional cobordism hypothesis, and standard differential geometric arguments.