process

Duality for topological abelian group stacks and T -duality 3236 T -duality of twisted torus bundles6.1 Pairs and topological T -duality6.1.1 The goal of this subsection is to introduce the main objects of topological T -dualityand review the structure of the theory. Let B be a topological space.Definition 6.1. A pair .E; H / over B consists of a locally trivial principal T n -bundleE ! B and a gerbe H ! E with band T jE . An isomorphism of pairs is a diagramH H0EB E0Bconsisting of an isomorphism of T n -principal bundles and an isomorphism ofT-banded gerbes . By P.B/ we denote the set of isomorphism classes of pairsover B. Iff W B ! B 0 is a continuous map, then pull-back induces a functorial mapP.f/W P.B 0 / ! P.B/.6.1.2 The functorP W TOP op ! Setshas been introduced in [BS05] for n D 1 and in [BRS] for n 1 in connection withthe study of topological T -duality.The main result of [BS05] in the case n D 1 is the construction of an involutivenatural isomorphism T W P ! P , the T -duality isomorphism, which associates to eachisomorphism class of pairs t 2 P.B/ the class of the T -dual pair Ot WD T.t/2 P.B/.In the higher-dimensional case n 2 the situation is more complicated. First ofall, not every pair t 2 P.B/ admits a T -dual. Furthermore, a T -dual, if it exists, maynot be unique. The results of [BRS] are based on the notion of a T -duality triple. Inthe following we recall the definition of a T -duality triple.6.1.3 As a preparation we recall the following two facts. Let W E ! B be aT n -principal bundle. Associated to the decomposition of the global section functor.E;:::/ as composition .E;:::/ D .B;:::/ı W Sh Ab S=E ! Ab there is adecreasing filtrationF n H .EI Z/ F n 1 H .EI Z/ F 0 H .EI Z/of the cohomology groups H .EI Z/ Š R .EI Z/ and a Serre spectral sequencewith second term E i;j2Š H i .BI R j Z/ which converges to GrH .EI Z/.