Real bundle gerbes, orientifolds and twisted KR-homology

TL;DR

This paper develops a geometric and algebraic framework for Real bundle gerbes and twisted KR-homology on manifolds with involution, extending previous theories and providing tools for orientifold classification in string theory.

Contribution

It introduces a classification of Real bundle gerbes via Real Dixmier-Douady classes, constructs geometric cycles for twisted KR-homology, and establishes an isomorphism with analytic twisted KR-homology, extending prior work to the Real setting.

Findings

Real bundle gerbes classified by Real Dixmier-Douady classes

Isomorphism between Grothendieck group of modules and twisted KR-theory

Geometric cycles generate a real-oriented homology theory dual to twisted KR-theory

Abstract

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 and Baum-Carey-Wang 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.