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Additional resources for Communications in Mathematical Physics - Volume 264
The H -field is usually linked to the equivariant Brauer group of a continuous trace C∗ -algebra A with spectrum E, on which a group G acts as automorphisms so that the dual action E agrees with the bundle structure. However, the equivariant Brauer group can be described entirely in terms of cohomology classes in H p (M, H 3−p (G, T)) for p = 1, 2, 3, , leaving no room for H -fields with a component in H 0 (M, H 3 (G, T)), (for example, any non-trivial H -field on G considered as a principal G-bundle over a point).
Let φ be the tricharacter of G constructed from H0 , uρ , ρ defined on 2 L2 (N) as in Sect. 4, and αρ = adρ. The algebra uρ -indG N (K(L (N))) has spectrum G/N, the G action on the spectrum is transitive with stabiliser N, and the Dixmier-Douady class is described by the 3-form H0 . Proof. We assume that φ is obtained from the class f = H0 by the procedure described in Sect. 2. Choose an open set F ⊆ G on which the projection π to G/N is one-one, and translates Fi = F xi whose projections give a cover of G/N.
Then we have vector bundles Ei over Vi such that P (Ei ) are locally trivial. We have the following morphisms: P (E1 ) ←−−−− P (E1 ) ×V1 f ∗ P (E2 ) −−−−→ P (E2 ). Note that P (E1 ) ×V1 f ∗ P (E2 ) is locally trivial. Hence we have the naturally defined morphism: HU∗ (dE HU∗ (dE 2) P (E2 ) −→ HU∗ (dE 1 )×U (dE2 ) 1) P (E1 ) ↓ P (E1 ) ×V1 f ∗ P (E2 ) In other words, we have the naturally defined morphism H ∗ (V2 ) −→ H ∗ (V1 ). 4. Thom isomorphism. Let V be an s-locally trivializable orbispace, and P (E1 ) is locally trivial for a vector bundle E1 over V .