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

Berwick-Evans, Daniel; Pavlov, Dmitri

doi:10.2140/agt.2023.23.3707

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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

DOI: 10.2140/agt.2023.23.3707

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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Volume 23, issue 8 (2023)