From minimal geodesics to supersymmetric field theories

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There are many models for the K-theory spectrum known today, each one having its own history and applications. The purpose of this note is to give an elementary description of eight such models (and certain completions of them) and to relate all of them by canonical maps, some of which are homeomorphisms (rather than just homotopy equivalences). Our survey begins with Raoul Bott’s iterated spaces of minimal geodesics in orthogonal groups, which he used to prove his famous periodicity theorem, and includes Milnor’s spaces of Clifford module structures as well as the Atyiah – Singer spaces of Fredholm operators. From these classical descriptions we move via spaces of unbounded operators and super-semigroups of operators to our most recent model, which is given by certain spaces of supersymmetric (1|1)-dimensional field theories. These spaces were introduced by the second two authors for the purpose of generalizing them to spaces of certain supersymmetric (2|1)dimensional Euclidean field theories that are conjectured to be related to the Hopkins –Miller spectrum TMF of topological modular forms.