process

Duality for topological abelian group stacks and T -duality 3456.4.14 We can now finish the argument that ‰ is H 3 .BI Z/-equivariant. We let‰.P/ D ..E; H /; . yE; yH/;u/ and ‰.g C P/ D ..E 0 ;H 0 /; . yE 0 ; yH 0 /; u 0 /. Note thatg C P D P ˝EzG. An inspection of the construction of the first entry of ‰ shows thatE 0 Š E and H 0 Š H ˝E prE ! BG. Proposition 6.22 shows that D.P ˝EzG/ ŠD.P /. This implies that yE 0 Š yE. Furthermore, if we restrict D.P ˝EzG/ along¹1º jB! Z jB we get by Proposition 6.22 and the proof of Lemma 6.21 a diagram ofstacks over B:yH op ˝ yE Gop z y H ˝RyERG opD.P ˝EzG/Š D.P / ˝D.P /D. zG/yEcan RyERD.P ˝EzG/Š D.P /¹1º jBZ jB .The map u 0 is induced by the evaluation .P ˝EzG/ D.P ˝EzG/ ! BT. Withthe given identifications using the “duality” between (73) and (72) we see that thisevaluation the induced by the product of the evaluations.P D.P // . zG D. zG// ! BT BT ! BT:After restriction to ¹1º jBwe see that u 0 D u ˝ v, where v W G ˝ G op ! BT is thecanonical pairing. This finishes the proof of the equivariance of ‰.Theorem 6.23. The maps ‰ and ˆ are inverse to each other.Proof. We first show the assertion under the additional assumption that H 3 .BI Z/ D 0.In this case an element P 2 Q E is determined uniquely by the class Oc.P/ 2 H 2 .BI Z n /.Similarly, a T -duality triple t 2 Triple.B/ with c.t/ D c.E/ is uniquely determined bythe class Oc.t/ D c. yE/. Since for P 2 Q E we have Oc.ˆı‰.P// (67)Lemma 6.18D Oc.‰.P// DLemma 6.18Oc.P/and Oc.‰ıˆ.t// D Oc.ˆ.t// (67)D Oc.t/this implies that ˆı‰ jQE D id QE and‰ ı ˆjs 1 .E/ D id js 1 .E/, where s W Triple ! P is as in 6.1.7. Note that H 3 .R n I Z/ D0. Therefore‰.ˆ.t univ // D t univ ; ˆ.‰.P univ // Š P univ :Now consider a general space B 2 S. We first show that ‰ ı ˆ D id. Lett 2 s 1 .E/ be classified by the map f t W B ! R n , i.e. ft tuniv D t. Then we have‰.ˆ.t// D ‰.ft Puniv/ D ft ‰.Puniv/ D ft tuniv D t, i.e.We consider the group‰ ı ˆ D id: (74) E WD .im.˛/ C im.s//=im.˛/ H 3 .BI Z/=im.˛/: