Twistings of KR for Real groupoids

Equivariant Cohomotopy implies orientifold tadpole cancellation

“…In more detail, D-brane charge on orbifolds is traditionally expected [Wi98, 5.1][dBD + 02, 4.5.2][GC99] to be in equivariant K-theory KU G , KO G (see [Gr05]). Hence orientifolds are expected to have charge quantization in a combination of these aspects in some Real equivariant orbifold K-theory [Mou11]…”

String Theory on Elliptic Curve Orientifolds and KR-Theory

“…In this section, we will briefly review KR-theory and KR-theory with a sign choice, as well as certain twisted variants. All these twistings of KR-theory were discussed and classified by Moutuou [24,23,26], though this may not be readily apparent because of the great generality of Moutuou's framework. (Moutuou deals with Z 2 -graded algebras over Real groupoids, but here we only need the case where the grading is trivial and the groupoid reduces to a Real space.…”

“…Twistings and sign choices in KR-theory have been unified in work of Moutuou [24,23]. He constructs and computes a graded Brauer group [26] of graded real continuous-trace algebras over a Real space (X, ι).…”

“…We then record all the groups that occur for the various possible involutions and twistings. Most of these calculations are taken from [12], but we also relate the results to earlier calculations made in [28] and [18] and to classifications of twistings of KR by Moutuou [24,23].…”

Algebraic K-theory and derived equivalences suggested by T-duality for torus orientifolds

“…In these theories, D-brane charges lie in twisted KR-groups of (X, ι), where X is the spacetime manifold and ι is the involution on X defining the orientifold structure. (That D-brane charges for orientifolds are classified by KR-theory was pointed out in [46, §5.2], [19], and [18], but twisting (as defined in [25,24,26] and [12]) may arise due to the B-field, as in [47], and/or the charges of the O-planes, as explained in [12].) These orientifold theories were found in [13] to split up into a number of T-duality groupings, with the theories in each grouping all related to one another by various T-dualities.…”