Field theory configuration spaces for connective ko–theory

Abstract

We describe a new –spectrum for connective ko–theory formed from spaces infn 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 [20], who uses nonsymmetric configurations without Clifford-structure on spheres, and the well-known Atiyah–Singer model for KO using Clifford-linear Fredholm operators [1]. Dropping the connectivity condition we obtain operator spaces Infn . These are homotopy equivalent to the spaces EFT n of 1j1–dimensional supersymmetric Euclidean field theories of degree n which were defined by Stolz and Teichner in [22; 23] and with Hohnhold in [9] (in terms of certain homomorphisms of super semigroups). They showed that the EFT n are homotopy equivalent to KOn and gave the idea for the connection between EFT n and Infn . We can derive a homotopy equivalent version of the –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 1j1–dimensional Euclidean field theories, eft, and might result in a more general criterion for instance for a connective version of 2j1–dimensional such theories (which are conjectured to yield a spectrum for TMF).