We describe a new –spectrum
for connective –theory
formed from spaces
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
stands between an older one by Segal, who uses nonsymmetric configurations without
Clifford-structure on spheres, and the well-known Atiyah–Singer model for
using
Clifford-linear Fredholm operators. Dropping the connectivity condition we obtain operator spaces
. These are homotopy
equivalent to the spaces
of –dimensional
supersymmetric Euclidean field theories of degree
which were defined by Stolz and Teichner (in terms of certain
homomorphisms of super semigroups). They showed that the
are homotopy equivalent
to and gave the idea for
the connection between
and .
We can derive a homotopy equivalent version of the
–spectrum
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
–dimensional Euclidean
field theories, ,
and might result in a more general criterion for instance for a connective version of
–dimensional
such theories (which are conjectured to yield a spectrum for
).
Keywords
$K$–theory, connective, field theory, Euclidean field
theory
