process

Duality for topological abelian group stacks and T -duality 341By an inspection of the definitions one checks that the natural factorization u r;s existsif r D s. Furthermore one checks that.w; u r;s /W H r ˝Er B R sz y H s ! .E r B R s / BT jBis an isomorphism of gerbes with band T jEr B R sif and only if r D s D 1. Bothstatements can be checked already when restricting to a point, and therefore becomeclear when considering the argument in the proof of Lemma 6.19.We define the map (70) byuH H y ˝EB yE op H z1y ˝E1 ByR 1H 1u 1;1BT jBE ByE.id;can/ E1 ByR 1(note that R 1 D R, E 1 D E and H 1 D H ).Lemma 6.19. The triple ‰.P/ D ..E; H /; . yE; yH/;u/ constructed above is a T -duality triple.Proof. It remains to show that the isomorphism of gerbes u satisfies the condition 6.2,2 in the version of 6.1.11.By naturality of the construction ‰ in the base B and the fact that condition 6.2, 2can be checked at a single point b 2 B, we can assume without loss of generality thatB is a point. We can further assume that E D Z T n and P D BT Z T n .Inthis caseD.P / D D.BT/ D.Z/ D.T n / Š Z BT BZ n :We haveH z yH Š .BT T n / .BT BZ n /:The restriction of the evaluation map H z yH ! H ˝ zyH ! BT is the composition.BT T n / .BT BZ n / Š BT .BT/ T n BZ nididev! BT BT BT † ! BT:We are interested in the contribution ev W T n BZ n ! BT n .It suffices to see that in the case n D 1 we haveev .z/ D x ˝ y;where x 2 H 1 .TI Z/, y 2 H 1 .BZI Z/, and z 2 H 2 .BT/ Š Z are the canonicalgenerators. In fact this implies via the Künneth formula that ev .z/ D P niD1 x i ˝ y i ,