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 equivalent to 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 version 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.

23 pages. Comments and questions are welcome. v2: Completely rewritten, replacing fibrations with presheaves. v3: Identical to the journal version except for formatting and style

References in corpus (2)

• A note on the $(\infty ,n)$-category of cobordisms
• A Framework for Geometric Field Theories and their Classification in Dimension One

Cited by in corpus (4)

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• Two-dimensional algebra in lattice gauge theory
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