process

Duality for topological abelian group stacks and T -duality 337it corresponds to a group stack P 2 PIC.S=B/ with H 0 .P / Š Z and H 1 .P / Š T.The group stack h.d/ 2 Q E is given by the pull-back (two-cartesian diagram)h.d/ EPZ.In particular we see that the gerbe H ! E of the pair u.h.d// DW .E; H / is given bya pull-backH EG B,where G 2 Gerbe.B/ is a gerbe with Dixmier–Douady class d.G/ D d. The compositionf ı hW H 3 .BI Z/ ! H 3 .EI Z/ is thus given by the pull-back along the mapp W E ! B, i.e. p D f ı h. By construction the composition H 3 .BI Z/ ! A !F 2 H 3 .EI Z/ is a certain factorization of p . This shows that the left square commutes.Now we show that the right square in (66) commutes. We start with an explicitdescription of Oc. Let P 2 Q E . The principal T n -bundle in G.0/ ! E.0/ ! B istrivial (see (61) for notation). Therefore the Serre spectral sequence .E r .0/; d r .0//degenerates at the second term. We already know by Lemma 6.16 that the Dixmier–Douady class of G.0/ ! E.0/ satisfies d 0 2 F 2 H 3 .E.0/I Z/. Its symbol can bewritten as P niD1 x i ˝Oc i .P / for a uniquely determined sequence Oc i .P / 2 H 2 .BI Z/.These classes constitute the components of the class Oc.P/ 2 H 2 .BI Z n /.We write the symbol of f.P/D d 1 as P i x i ˝ a i for a sequence a i 2 H 2 .BI Z/.As in the proof of Lemma 6.16 the equation (63) gives the identitynXnXnXpr 2 x i ˝ a i C pr 1 x i ˝Oc i .P / .pr 1 x i C pr 2 x i/ ˝ a iiD1iD1modulo the image of a second differential d 0;22 .1/. This relation is solved by a i WDOc i .P / and determines the image of the vector a WD .a 1 ;:::;a n / under H 2 .BI Z n / !H 2 .BI Z n /=im.d 0;22.1// DW B uniquely. Note that e ı f.P/is also represented by theimage of the vector a in B. This shows that the right square in (66) commutes.6.4 T -duality triples and group stacks6.4.1 Let R n be the classifying space of T -duality triples introduced in [BRS]. Itcarries a universal T -duality triple t univ WD ..E univ ;H univ /; . yE univ ; yH univ /; u univ /. Letc univ ; Oc univ 2 H 2 .R n I Z n / be the Chern classes of the bundles E univ ! R n , yE univ ! R n .They satisfy the relation c univ [Oc univ D 0. Let E univ be the extension of sheavesiD1