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

Daniel Berwick-Evans and Dmitri Pavlov

Algebraic & Geometric Topology 23 (2023) 3707–3743
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.

Keywords
functorial field theory
Mathematical Subject Classification
Primary: 53C05, 53C29, 55R10, 57R56
References
Publication
Received: 16 August 2021
Revised: 23 December 2021
Accepted: 6 June 2022
Published: 5 November 2023
Authors
Daniel Berwick-Evans
Department of Mathematics
University of Illinois at Urbana-Champaign
Urbana, IL
United States
https://danbe.web.illinois.edu/
Dmitri Pavlov
Department of Mathematics and Statistics
Texas Tech University
Lubbock, TX
United States
https://dmitripavlov.org/

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