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

Sign Choices for Orientifolds

“…There are two natural Z 2 -sheaves on the orientifold which in [13] we denoted by U (1) and U (1). They are the equivariant sheaves of smooth functions into U (1), where the first is endowed with a trivial Z 2 -action on U (1) and the second has the Z 2 -action induced by complex conjugation.…”

“…This will be useful when considering more geometric constructions of the sign choice map below. We showed in [13] that there exist long exact sequences relating the equivariant sheaf cohomology of U 0 and U 1 . For ∈ Z 2 we have…”

“…We begin by recalling some basic facts about Real bundle gerbes, referring the reader to [13] for more details.…”

“…In the sequel we will suppress the subscripts on τ P and τ Y . In [13] we proved that Real bundle gerbes are classified up to Real stable isomorphism by their Real Dixmier-Douady class in the degree two equivariant sheaf cohomology group: [17] that if (P, Y ) is a bundle gerbe, a connection A on the U (1)-bundle P → Y [2] is called a bundle gerbe connection if it respects the bundle gerbe multiplication. If A is a bundle gerbe connection, then the curvature F A ∈ 2 (Y [2] ) satisfies δ(F A ) = 0.…”

“…In these cases the K R-theory classes are twisted by a Real bundle gerbe with trivial complex Dixmier-Douady class, whereby the possible Real structures on the bundle gerbe are given by equivariant line bundles (it is a torsor under H 1 (M; Z 2 , U 0 )). The spectrum of O-plane charges is then determined by the fact that, on the connected components of the fixed point set of the orientifold involution, the K R-theory becomes KO-theory when the restriction of the bundle gerbe has sign +1 and so is Real trivial, or KSp-theory when the restriction of the bundle gerbe has 2-torsion Dixmier-Douady class with sign −1 corresponding to non-trivial torsion H -flux [13]. However, in general there are sign choices that cannot be achieved in such a way unless the complex Dixmier-Douady class (or H -flux of the string background) is non-trivial, 1 in which case the more general geometric realisation of twisted K R-theory from [13] contains the appropriate sign choices.…”