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

Elke K Markert

Algebraic & Geometric Topology 10 (2010) 1187–1219

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 Tn 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 Tn are homotopy equivalent to KOn and gave the idea for the connection between Tn 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).

$K$–theory, connective, field theory, Euclidean field theory
Mathematical Subject Classification 2000
Primary: 19L41, 55N15, 81Q60
Secondary: 81T60, 81T08
Received: 24 May 2007
Revised: 25 November 2009
Accepted: 6 December 2009
Published: 23 May 2010
Elke K Markert
School of Natural Sciences
Institute for Advanced Study
1 Einstein Drive
Princeton NJ 08540
United States