Equivariant K-theory

Topological K-theory [2] 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 [1], equivariant K-theory [14] and combinations of all these. In recent years K-theory has found unexpected application in the physics of string theories [6, 12, 13, 16] 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±(X), was introduced by Witten [16] in relation to “orientifolds”. The geometric situation concerns a manifold X with an involution τ leaving a fixed sub-manifold Y . On X one wants to study a pair of complex vector bundles (E, E) with the property that τ interchanges them. If we think of the virtual vector bundle E − E, then τ takes this into its negative, and K±(X) is meant to be the appropriate K-theory of this situation. In physics, X is a 10-dimensional Lorentzian manifold and maps Σ → X of a surface Σ describe the world-sheet of strings. The symmetry requirements for the appropriate Feynman integral impose conditions that the putative K-theory K±(X) has to satisfy. The second author proposed a precise topological definition of K±(X) which appears to meet the physics requirements, but it was not entirely clearly how to link the physics with the geometry. In this paper we elaborate on this definition and also a second (but equivalent) definition of K±(X). Hopefully this will bring the geometry and physics closer together, and in particular link it up with the analysis of Dirac operators. Although K±(X) is defined in the context of spaces with involution it is rather different from Real K-theory or equivariant K-theory (for G = Z2) although it has superficial resemblances to both. The differences will become clear as we proceed but at this stage it may be helpful to consider the analogy with cohomology. Equivariant cohomology can be defined (for any compact Lie group G), and this has relations with equivariant K-theory. But there is also “cohomology with local coefficients”, where the fundamental group π1(X) acts on the abelian coefficient group. In particular for integer coefficients Z the only such action is via a homomorphism π1(X) → Z2, i.e. by an element of H (X ;Z2) or equivalently a double-covering X̃ of X . This is familiar for an unoriented manifold with X̃ its oriented double-cover. In this situation, if say X is a compact n-dimensional manifold, then we do not have