process

Duality for topological abelian group stacks and T -duality 325Definition 6.3. We let Triple.B/ denote the set of isomorphism classes of T -dualitytriples over B. Forf W B 0 ! B we let Triple.f /W Triple.B/ ! Triple.B 0 / be the mapinduced by the pull-back of T -duality triples.6.1.7 In this way we define a functorThis functor comes with specializationsTripleW TOP op ! Sets:s; Os W Triple ! Pgiven by s..E; H /; . yE; yH/;u/ WD .E; H / and Os..E; H /; . yE; yH/;u/ D . yE; yH/.Definition 6.4. A pair .E; H / is called dualizable if there exists a triple t 2 Triple.B/such that s.t/ Š .E; H /. The pair Os.t/ D . yE; yH/is called a T -dual of .E; H /.Thus the choice of a triple t 2 s 1 .E; H / encodes the necessary choices in orderto fix a T -dual. One of the main results of [BRS] is the following characterization ofdualizable pairs.Theorem 6.5. A pair .E; H / is dualizable in the sense of Definition 6.4 if and only ifd.H/ 2 F 2 H 3 .EI Z/.Further results of [BRS, Theorem 2.24] concern the classification of the set of dualsof a given pair .E; H /.6.1.8 For the purpose of the present paper it is more natural to interpret the isomorphismof gerbes uW Op yH ! p H in a T -duality triple ..E; H /; . yE; yH/;u/ in a different,but equivalent manner. To this end we introduce the notion of a dual gerbe.6.1.9 First we recall the definition of the tensor product w W H ˝X H 0 ! X of gerbesuW H ! X and u 0 W H 0 ! X with band T jX over X 2 S. Consider first the fibreproduct of stacks .u; u 0 /W H X H 0 ! X. Let T 2 S=X. An object s 2 H X H 0 .T /is given by a triple .t; t 0 ;/ of objects t 2 H.T/, t 0 2 H 0 .T / and an isomorphism W u.t/ ! u 0 .t 0 /. By the definition of a T jX -banded gerbe the group of automorphismsof t relative to u is the group Aut H.T/=rel.u/ .t/ Š T.T /. We thus have an isomorphismAut H X H 0 =rel..u;u 0 //.s/ Š T.T / T.T /. Similarly, for s 0 ;s 1 2 H X H 0 .T / the setHom H X H 0 =rel..u;u 0 //.s 0 ;s 1 / is a torsor over T.T / T.T /.We now define a prestack H ˝pX H 0 . By definition the groupoid H ˝pX H 0 .T / has thesame objects as H X H 0 .T /, but the morphism sets are factored by the anti-diagonalT.T / antidiag T.T / T.T /, i.e.Hom H ˝pX H 0 =rel.w/ .s 0;s 1 / D Hom H X H 0 =rel..u;u 0 //.s 0 ;s 1 /=antidiag.T.T //:The stack H ˝X H 0 is defined as the stackification of the prestack H ˝pX H 0 . Itisagainagerbe with band T jX . We furthermore have the following relation of Dixmier–Douadyclasses.d.H/ C d.H 0 / D d.H ˝X H 0 /: