TY - JOUR
T1 - Smooth one-dimensional topological field theories are vector bundles with connection
AU - Berwick-Evans, Daniel
AU - Pavlov, Dmitri
N1 - We thank Peter Teichner for many enlightening discussions about field theories, Ralph Cohen for his encouragement, Konrad Waldorf for helpful correspondence about representations of path groupoids, and Gabriel Drummond-Cole for feedback on an earlier draft. Pavlov was partially supported by the SFB 878 grant.
PY - 2023
Y1 - 2023
N2 - 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.
AB - 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.
KW - functorial field theory
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U2 - 10.2140/agt.2023.23.3707
DO - 10.2140/agt.2023.23.3707
M3 - Article
AN - SCOPUS:85176469767
SN - 1472-2747
VL - 23
SP - 3707
EP - 3743
JO - Algebraic and Geometric Topology
JF - Algebraic and Geometric Topology
IS - 8
ER -