Advances in Theoretical and Mathematical Physics

Volume 23 (2019)

Number 8

Real bundle gerbes, orientifolds and twisted $KR$-homology

Pages: 2093 – 2159



Pedram Hekmati (Department of Mathematics, University of Auckland, New Zealand)

Michael K. Murray (School of Mathematical Sciences, University of Adelaide, SA, Australia)

Richard J. Szabo (Dept. of Mathematics, Maxwell Institute, and Higgs Centre for Theoretical Physics, Heriot-Watt University, Edinburgh, United Kingdom; and Centro de Matemática, Computacão e Cognicão, Universidade de Federal do A.B.C., Santo André, SP, Brazil)

Raymond F. Vozzo (School of Mathematical Sciences, University of Adelaide, SA, Australia)


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.

The authors acknowledge support under the Australian Research Council’s Discovery Projects funding scheme (project numbers DP120100106 and DP130102578), the Consolidated Grant ST/L000334/1 from the UK Science and Technology Facilities Council, the Action MP1405 QSPACE from the European Cooperation in Science and Technology (COST), and the Visiting Researcher Program Grant 2016/04341–5 from the Fundacão de Amparo á Pesquisa do Estado de São Paulo (FAPESP, Brazil). Report no.: EMPG–16–15.