process

Duality for topological abelian group stacks and T -duality 3356.3.6 Recall Definition 6.4 of a dualizable pair.Lemma 6.16. If P 2 Q E , then up.P / 2 P.B/ is dualizable.Proof. Let .E; H / WD up.E/. In view of Theorem 6.5 we must show that d.H/ 2F 2 H 3 .EI Z/. Fork 2 Z we define G.k/ ! E.k/ by the two-cartesian diagramsG.k/ PE.k/E(61)The group structure of E induces maps ¹kº jBZ jB .W E.k/ B E.m/ ! E.k C m/ (62)On fibres appropriately identified with T n , this map is the usual group structure on T n .Since P is a Picard stack these multiplications are covered by OW G.k/ G.m/ !G.k C m/. The isomorphism class of G.k/ therefore must satisfypr E.k/ G.k/ ˝ pr E.m/ G.m/ Š G.k C m/ 2 Gerbe.E.k/ B E.m//: (63)We now write out this isomorphism in terms of Dixmier–Douady classes d k WDd.G.k// 2 H 3 .E.k/I Z/.We fix a generator of H 1 .TI Z/. This fixes a choice of generators of x i 2 H 1 .T n I Z/,i D 1;:::;nvia pullback along the coordinate projections. Let a W T n T n ! T n bethe group structure. Then we havea .x i / D pr 1 x i C pr 2 x i: (64)where pr i W T n T n ! T n , i D 1; 2, are the projections onto the factors.Let us for simplicity assume that B is connected. Let .E r .k/; d r .k// be the Serrespectral sequence of the composition E.k/ ! B !(see 6.1.7). Then we can identifyE 0;32 .k/ Š ƒ3 Z H 1 .T n I Z/. The class d k has a symbol in E 0;32.k/ which can be writtenas P i