Field theory configuration spaces for connective $\mathrm{ko}$–theory

Abstract

We describe a new $\Omega$–spectrum for connective $ko$–theory formed from spaces ${inf}_n$ of operators which have certain nice spectral properties, and which fulfill a connectivity condition. The spectral data of such operators can equivalently be described by certain Clifford-linear, symmetric configurations on the real axis; in this sense, our model for $ko$ stands between an older one by Segal, who uses nonsymmetric configurations without Clifford-structure on spheres, and the well-known Atiyah–Singer model for $KO$ using Clifford-linear Fredholm operators. Dropping the connectivity condition we obtain operator spaces ${Inf}_n$. These are homotopy equivalent to the spaces ${\mathcal{ℰ}\mathcal{ℱ}\mathcal{T}}_n$ of $1|1$–dimensional supersymmetric Euclidean field theories of degree $n$ which were defined by Stolz and Teichner (in terms of certain homomorphisms of super semigroups). They showed that the ${\mathcal{ℰ}\mathcal{ℱ}\mathcal{T}}_{-n}$ are homotopy equivalent to ${KO}_n$ and gave the idea for the connection between ${\mathcal{ℰ}\mathcal{ℱ}\mathcal{T}}_n$ and ${Inf}_n$. We can derive a homotopy equivalent version of the $\Omega$–spectrum $inf$ in terms of “field theory type” super semigroup homomorphisms. Tracing back our connectivity condition to the functorial language of field theories provides a candidate for connective $1|1$–dimensional Euclidean field theories, $eft$, and might result in a more general criterion for instance for a connective version of $2|1$–dimensional such theories (which are conjectured to yield a spectrum for $TMF$).