process

Duality for topological abelian group stacks and T -duality 343Therefore we can consider D.P / 2 Ext PIC.S=B/ .Z jB ;D.Z/ jB / in a canonical way, and.P/ and .D.P// belong to the same group.Note thatd.G.P// D .P /; d.G.D.P /// D .D.P//under the isomorphism (71). It suffices to show that d.G.P// D d.G.D.P ///.In fact, as in 6.4.7 we have the following factorization of the evaluation map:G.P / ˝BG .D.P //P ˝ZjB D.P / P ZjB D.P / P B D.P /evT jBdiag¹1º jBZ jB Z jBZ jB Z jB .This represents G.D.P // as the dual gerbe G.P / op of G.P / in the sense of Definition6.6. The relation d.G.P// D d.G.D.P /// follows.6.4.11 We now start the actual proof of Lemma 6.21.In order to see that ‰ is H 3 .BI Z/-equivariant we will first describe the action ofH 3 .BI Z/ on the sets of isomorphism classes of T -duality triples with fixed underlyingT n -bundles E and yE on the one hand, and on the set of isomorphism classes of Picardstacks Q E , on the other.Consider g 2 H 3 .BI Z/. It classifies the isomorphism class of a gerbe G ! Bwith band T jB .Ift is represented by ..E; H /; . yE; yH/;u/, then g C t is representedby..E; H ˝ G/;. yE; yH ˝O G/;u ˝ id r G/;where the maps ; O;r are as in (49).Note the isomorphism H 3 .BI Z/ Š H 2 .BI T/ Š Ext 2 Sh Ab S=B .Z jB ; T jB /. Thereforethe class g also classifies an isomorphism class of Picard stacks with H 0 . zG/ Š Z jBand H 1 . zG/ Š T jB . From zG we can derive the gerbe G by the pull-backG zG ¹1º jBZ jB .6.4.12 Recall that we consider an extension E 2 Sh Ab S=B of the form0 ! T n jB ! E ! Z jB ! 0:We consider a Picard stack P 2 Ext PIC.S=B/ .E;D.Z/ jB /.Let furthermore zG 2 Ext PIC.S=B/ .Z jB ;D.Z/ jB /. Then we defineP ˝EzG 2 Ext PIC.S=B/ .E;D.Z/ jB /