Real Baum–Connes assembly and T-duality for torus orientifolds

Abstract

Abstract. We show that the real Baum-Connes conjecture for abelian groups, possibly twisted by a cocycle, explains the isomorphisms of (twisted) KR-groups that underlie all T-dualities of torus orientifold string theories.

T-Duality Simplifies Bulk-Boundary Correspondence

“…More general dualities exist amongst other variants of real K-theory. They are explained in the context of the real Baum-Connes conjecture, in analogy to (4.6), in [65,3].…”

Crystallographic T-duality

“…It turns out that S 1 triv andŜ 1 flip are T-dual in this context, in that there is a naturally defined isomorphism between KR • (S 1 triv ) ≡ KO • (S 1 triv ) and KR •−1 (Ŝ 1 flip ). Such dualities were studied in the context of orientifold string theories in [29,15] and in the context of topological insulators in [38], and are related to the Baum-Connes conjecture over the reals [44]. In the latter setting,Ŝ 1 flip can be thought of as a 1D Brillouin torus Z with the flip involution induced by complex conjugating characters.…”

“…For instance, a 3-dimensional lens space L(p) with k units of H-flux is T-dual to a generally non-homeomorphic lens space L(k) with p units of H-flux [36,11]. The desire to understand the general mechanism behind "topological T-dualities" of this kind led to a rekindling of interest in twisted K-theory, and a very fruitful C * -algebraic approach [43,37,44,15] even relates T-duality to the deep Baum-Connes isomorphisms [5].…”

T-Duality Simplifies Bulk–Boundary Correspondence: Some Higher Dimensional Cases

“…Quaternionic and Real K-theories are related by a degree shift of 4, and for notational convenience, we work mostly with KR-theory. Real T-duality was discussed in [60] in the context of the real Baum-Connes conjecture. It can be expressed as the real Baum-Connes assembly map following Poincaré duality.…”