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Differential forms and 0-dimensional supersymmetric field theories. (English) Zbl 1236.19008
The authors review the notion of field theories as functors, pioneered by Atiya, Kontsevich in topological case and Segal in the conformal case. They give precise meaning to the notion of smoothness for such functors by introducing family versions of the relevant bordism categories. The authors generalize these notions to supersymmetric field theories, also valid in higher dimensions, which they then continue to study in the simplest case that of dimension zero. The main contribution of the paper is to make all new mathematical notions regarding supersymmetric field theories precise. The authors observe that concordance gives an equivalence relation which can be defined for geometric objects over manifolds for which “pullbacks” and “isomorphisms” make sense. By Stokes’ theorem two closed \(n\)-forms on \(X\) are concordant if and only if they represent the same de Rham cohomology class; two vector bundles with connections are concordant if and only if they are isomorphic as vector bundles. Passing from a Euclidean field theory (EFT) over \(X\) to its concordance class forgets the geometric information while retaining the topological information: differential forms of a specific degree \(n\) arise by forgetting the Euclidean geometry (on superpoints) and working with topological field theories (TFTs) instead. As a consequence, concordance classes of such field theories are shown to represent de Rham cohomology.
The authors’ results are consistent with the formal group point of view towards (complex oriented) cohomology theories, where the additive formal group gives an ordinary rational cohomology, the multiplicative group gives \(K\)-theory and the formal groups associated to elliptic curves lead to elliptic cohomology.

MSC:
19L99 Topological \(K\)-theory
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
47A53 (Semi-) Fredholm operators; index theories
57R56 Topological quantum field theories (aspects of differential topology)
81T99 Quantum field theory; related classical field theories
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