process

234 U. Bunke, T. Schick, M. Spitzweck, and A. Thom(a) G satisfies the two-three condition,(b) G admits an open subgroup of the form C R n with C compact such that G=C R nis finitely generated(c) the connected component of the identity of G is locally topologically divisible.Then G is admissible over S lc .The whole Section 4 is devoted to the proof of this theorem. This section is verylong and technical. A reader who is interested in the extension of Pontrjagin duality totopological abelian group stacks and applications to T -duality is advised to skip thissection in a first reading and to take Theorem 4.8 for granted.1.5 T -duality1.5.1 The aim of topological T -duality is to model the underlying topology of mirrorsymmetry in algebraic geometry and T -duality in string theory. For a detailed motivationwe refer to [BRS]. The objects of topological T -duality over a space B are pairs.E; G/ of a T n -principal bundle E ! B and a gerbe G ! E with band T. A gerbewith band T over a space E is a map of stacks G ! E which is locally isomorphicto BT jE ! E. Topological T -duality associates to .E; G/ dual pairs . yE; yG/. Forprecise definitions we refer to [BRS] and 6.1.1.5.2 The case n D 1 (n is the dimension of the fibre of E ! B) is quite easyto understand (see [BS05]). In this case every pair admits a unique T -dual (up toisomorphism, of course). The higher dimensional case is more complicated since onthe one hand not every pair admits a T -dual, and on the other hand, in general a T -dualis not unique, compare [BS05], [BRS].1.5.3 The general idea is that the construction of a T -dual pair of .E; G/ needs thechoice of an additional structure. This structure might not exist, and in this case thereis no T -dual. On the other hand, if the additional structure exists, then it might not beunique, so that the T -dual is not unique, too.1.5.4 The additional structure in [BRS] was the extension of the pair to a T -dualitytriple. We review this notion in 6.1.One can also interpret the approach to T -duality via non-commutative topology[MR05], [MR06] in this way. The gerbe G ! E determines an isomorphism class ofa bundle of compact operators (by equality of the Dixmier–Douady classes). The additionalstructure which determines a T -dual is an R n -action on this bundle of compactoperators which lifts the T n -action on E. Let us call a bundle of algebras of compactoperators on E together with such a R n -action a dynamical triple.The precise relationship between T -duality triples and dynamical triples is studiedin the thesis of Ansgar Schneider [Sch07].