Field theory configuration spaces for connective ko–theory

Markert, Elke K¹

¹School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton NJ 08540, United States

Algebraic and Geometric Topology, Tome 10 (2010) no. 2, pp. 1187-1219

Résumé

We describe a new Ω–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, 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 Infn. These are homotopy equivalent to the spaces ℰℱTn 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 ℰℱT−n are homotopy equivalent to KOn and gave the idea for the connection between ℰℱTn and Infn. We can derive a homotopy equivalent version of the Ω–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).

Détail

DOI : 10.2140/agt.2010.10.1187

Keywords: $K$–theory, connective, field theory, Euclidean field theory

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