process

Duality for topological abelian group stacks and T -duality 2351.5.5 The initial motivation of the present paper was a third choice for the additionalstructure of the pair which came to live in a discussion with T. Pantev in spring 2006.It was motivated by the analogy with some constructions in algebraic geometry, seee.g. [DP].The starting point is the observation that a T -dual exists if and only if the restrictionof the gerbe G ! E to Ej B .1/, the restriction of E to a one-skeleton of B, is trivial.In particular, the restriction of G to the fibres of E has to be trivial. Of course, theT n -bundle E ! B is locally trivial on B, too. Therefore, locally on the base B, thesequence of maps G ! E ! B is equivalent to.BT T n / jB ! T n jB ! B jB ;where we identify spaces over B with the corresponding sheaves. The stack .BT T n / jB is a Picard stack.Our proposal for the additional structure on G ! E ! B is that of a torsor overthe Picard stack .BT T n / jB .1.5.6 In order to avoid the definition of a torsor over a group object in the two-categorialworld of stacks we use the following trick (see 6.2 for details). Note that a torsor X overan abelian group G can equivalently be described as an extension of abelian groups0 ! G ! U ! p Z ! 0so that X Š p 1 .1/. We use the same trick to describe a torsor over the Picard stack.BT T n / jB as an extensionof Picard stacks..BT T n / jB ! U ! Z jB1.5.7 The sheaf of sections of E ! B is a torsor over T n . Let0 ! T n jB ! E ! Z jB ! 0be the corresponding extension of sheaves of abelian groups. The filtration (1) of thePicard stack U has the formBT jB ! U ! E;and locally on BU Š .BT T n Z/ jB : (5)1.5.8 The proposal 1.5.5 was motivated by the hope that the Pontrjagin dual D.U / ofU determines the T -dual pair. In fact, this can not be true directly since the structureof D.U / (this uses admissibility of T n and Z n ) is given locally on B byD.U / Š .BT BZ n Z/ jB :Here the factor Z in (5) gives rise to BT, the factor BT yields Z, and T n yields BZ naccording to the rules (2) and (3). The problematic factor is BZ n in a place wherewe expect a factor T n . The way out is to interpret the gerbe BZ n as the gerbe ofR n -reductions of the trivial principal bundle T n B ! B.