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.