Graded Brauer groups of a groupoid with involution

Abstract

International audienceWe define a group BrR(G) containing, in a sense, the graded complex and orthogonal Brauer groups of a locally compact groupoid G equipped with an involution. When the involution is trivial, we show that the new group naturally provides a generalisation of Donovan-Karoubi's graded orthogonal Brauer group GBrO. More generally, it is shown to be a direct summand of the well-known graded complex Brauer goup. In addition, we prove that BrR(G) identifies with a direct sum of a Real cohomology group and the abelian group ExtR(G, S 1) of Real graded S 1-central extensions. A cohomological picture is then given

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

“…and 2 Br E is exactly the group of "sign choices" discussed in [12] and [26]. Thus there is a unique Brauer class of Azumaya algebras A split over E(R) + and nonsplit over E(R) − , which can be chosen to be represented by the quaternion algebra over the function field R(E) given by the quaternion algebra symbol (−1, x − γ).…”

“…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.…”

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). The equivalence relation is Morita equivalence over X and the group operation is graded tensor product (over X).…”

Real Baum–Connes assembly and 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 [32, §5.2], [13], and [10], but twisting (as defined in [19,18,20] and [7]) may arise due to the B-field, as in [33], and/or the charges of the O-planes, as explained in [7].) These orientifold theories were found in [8] 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.…”