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

Daniel Berwick-Evans, Dmitri Pavlov

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)3707-3743
Number of pages37
JournalAlgebraic and Geometric Topology
Issue number8
StatePublished - 2023


  • functorial field theory

ASJC Scopus subject areas

  • Geometry and Topology


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