Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 523and these lie in the component of Ind(r − s).Since the F r,s ⊂ F are invariant under the action of U(H), it followsthat they can be defined fibrewise and this shows that the classes c r,s canbe defined for K α (X). However the classes for r = s (and so of index zero)are not sufficient to generate all Chern classes. It is a not unreasonableconjecture that the c r,r are the only integral characteristic classes for thetwisted K−theories [Atiyah and Anderson (1967); Atiyah (2000)].While the use of Hilbert spaces H and the corresponding projectivespaces P (H) may not come naturally to a topologist, these are perfectlynatural in physics. Recall that P (H) is the space of quantum states. Bundlesof such arise naturally in quantum field theory.4.5.7 Twisted K−Theory and the Verlinde AlgebraTwistings of cohomology theories are most familiar for ordinary cohomology[Freed (2001); Freed et. al. (2003)]. Let M be a smooth manifold.Then a flat real vector bundle E → M determines twisted real cohomologygroups H • (M; E). In differential geometry these cohomology groups aredefined by extending the de Rham complex to forms with coefficients in Eusing the flat connection. The sorts of twistings of K−theory we considerare 1D, so analogous to the case when E is a line bundle. There are also1D twistings of integral cohomology, determined by a local system Z → M.This is a bundle of groups isomorphic to Z, so is determined up to isomorphismby an element of H 1 (M; Aut(Z)) ∼ = H 1 (M; Z mod 2), since the onlynontrivial automorphism of Z is multiplication by −1. The twisted integralcohomology H • (M; Z) may be thought of as sheaf cohomology, or definedusing a cochain complex. We give a Čech description as follows. Let {U i}be an open covering of M andg ij : U i ∩ U j −→ {±1} (4.28)a cocycle defining the local system Z. Then an element of H q (M; Z) isrepresented by a collection of q−cochains a i ∈ Z q (U i ) which satisfya j = g ij a i on U ij = U i ∩ U j . (4.29)We can use any model of co–chains, since the group Aut(Z) ∼ = {±1} alwaysacts. In place of co–chains we represent integral cohomology classesby maps to an Eilenberg–MacLane space K(Z, q). The cohomology group isthe set of homotopy classes of maps, but here we use honest maps as representatives.The group Aut(Z) acts on K(Z, q). One model of K(Z, 0) is the