Volume 10, issue 2 (2010)

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

Elke K Markert

Algebraic & Geometric Topology 10 (2010) 1187–1219
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 ${\mathsc{ℰ}\mathsc{ℱ}\mathsc{T}}_{n}$ 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 ${\mathsc{ℰ}\mathsc{ℱ}\mathsc{T}}_{-n}$ are homotopy equivalent to ${KO}_{n}$ and gave the idea for the connection between ${\mathsc{ℰ}\mathsc{ℱ}\mathsc{T}}_{n}$ 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 $TMF$).

Keywords
$K$–theory, connective, field theory, Euclidean field theory
Mathematical Subject Classification 2000
Primary: 19L41, 55N15, 81Q60
Secondary: 81T60, 81T08