Topological \( K \)-theory has many variants which have been developed and exploited for geometric purposes. There are real or quaternionic versions, “real” \( K \)-theory in the sense of, equivariant \( K \)-theory and combinations of all these.
In recent years \( K \)-theory has found unexpected application in the physics of string theories and all variants of \( K \)-theory that had previously been developed appear to be needed. There are even variants, needed for the physics, which had previously escaped attention, and it is one such variant that is the subject of this paper.
This variant, denoted by \( K_{\pm}(X) \), was introduced by Witten in relation to “orientifolds”. The geometric situation concerns a manifold \( X \) with an involution \( \tau \) having a fixed sub-manifold \( Y \). On \( X \) one wants to study a pair of complex vector bundles \( (E^+, E^-) \) with the property that \( \tau \) interchanges them. If we think of the virtual vector bundle \( E^+ - E^- \), then \( \tau \) takes this into its negative, and \( K_{\pm}(X) \) is meant to be the appropriate K-theory of this situation.
In physics, \( X \) is a 10-dimensional Lorentzian manifold and maps \( \Sigma \to X \) of a surface \( \Sigma \) describe the world-sheet of strings. The symmetry requirements for the appropriate Feynman integral impose conditions that the putative \( K \)-theory \( K_{\pm}(X) \) has to satisfy.
The second author proposed a precise topological definition of \( K_{\pm}(X) \) which appears to meet the physics requirements, but it was not entirely clear how to link the physics with the geometry.
