Real bundle gerbes, orientifolds and twisted KR-homology

We consider Real bundle gerbes on manifolds equipped with an involution and prove that they are classified by their Real Dixmier–Douady class in Grothendieck’s equivariant sheaf cohomology. We show that the Grothendieck group of Real bundle gerbe modules is isomorphic to twisted KR-theory for a torsion Real Dixmier–Douady class. Using these modules as building blocks, we introduce geometric cycles for twisted KR-homology and prove that they generate a real-oriented generalised homology theory dual to twisted KR-theory for Real closed manifolds, and more generally for Real finite CW-complexes, for any Real Dixmier–Douady class. This is achieved by defining an explicit natural transformation to analytic twisted KR-homology and proving that it is an isomorphism. Our model both refines and extends previous results by Wang [55] and Baum–Carey–Wang [9] to the Real setting. Our constructions further provide a new framework for the classification of orientifolds in string theory, providing precise conditions for orientifold lifts of H-fluxes and for orientifold projections of open string states.