process

Duality for topological abelian group stacks and T -duality 327After a choice of ! BT in order to define the map s below we obtain a mapr W T n T n ! BT by the following diagram.yH opb. Ov;v/ H b.T n BT op / .T n BT/s yH opb ˝Eu bb yE bH b Eb yE b BT prBT BTrT n T nThe condition 6.2, 2. is now equivalent to the condition that we can choose the isomorphismsv; Ov in (50) such thatr .z/ DnXpr 1;i x [ pr 2;i x;iD1where z 2 H 2 .BTI Z/ and x 2 H 1 .TI Z/ are the canonical generators, andpr k;i ; W T n T n pr k! T n pr i! T, k D 1; 2, i D 1;:::;n are projections onto thefactors.6.1.12 Important topological invariants of T -duality triples are the Chern classes ofthe underlying T n -principal bundles. For a triple t D ..E; H /; . yE; yH/;u/we definec.t/ WD c.E/;Oc.t/ WD c. yE/:These classes belong to H 2 .BI Z n /.6.2 Torus bundles, torsors and gerbes6.2.1 In this subsection we review various interpretations of the notion of a T n -principalbundle.6.2.2 Let G be a topological group. Let us start with giving a precise definition of aG-principal bundle.Definition 6.8. A G-principal bundle E over a space B consists of a map of spaces W E ! B which admits local sections together with a fibrewise right action E G !E such that the natural mapE G ! E B E;.e; t/ 7! .e; et/is an homeomorphism. An isomorphism of G-principal bundles E ! E 0 is a G-equivariantmap E ! E 0 of spaces over B.