process

Duality for topological abelian group stacks and T -duality 2311.3.2 Since we can consider sheaves of groups as Picard stacks in two different waysone can now ask how Pontrjagin duality is properly reflected in the language of Picardstacks. It turns out that the correct dualizing object is the stack BT 2 PIC S, and notT 2 PIC S as one might guess first. For a Picard stack A we define its dual byD.A/ WD HOM PIC S .A; BT/:We hope that using the same symbol for the dual sheaf and dual Picard stack does notintroduce to much confusion (see the two footnotes attached to the formulas (2) and(3) below). One can ask what the duals of the Picard stacks F and BF look like. Ingeneral we have (see 5.5)D.BF/Š D.F /: 1 (2)One could expect thatD.F / Š BD.F /; 2 (3)but this only holds under the condition that Ext 1 Sh Ab S .F; T/ D 0 (see 5.6). This conditionis not always satisfied, e.g.Ext 1 Sh Ab S .˚n2NZ=2Z; T/ 6D 0(see 4.29). But the main effort of the present paper is made to show thatExt 1 Sh Ab S .G; T/ D 0for a large class of locally compact abelian groups (this condition is part of admissibility,see Definition 1.5).1.3.3 Let A 2 PICS be a Picard stack. There is a natural evaluationev A W A ! D.D.A//:In analogy to 1.2 we make the following definition.Definition 1.4. We call a Picard stack dualizable if ev A W A ! D.D.A// is an equivalenceof Picard stacks.1.3.4 If G is a locally compact abelian group andthen the evaluation mapsExt 1 Sh Ab S .G; T/ Š Ext1 Sh Ab S . yG; T/ Š 0; (4)ev G W G ! D.D.G//;ev BG W BG ! D.D.BG//are isomorphisms by applying (2) and (3) twice, and using the sheaf-theoretic versionof Pontrjagin duality 1.3. In other words, under the condition (4) above G and BG aredualizable Picard stacks.1 On the right-hand side D.F / is the dual sheaf as in 1.1.5, considered as a Picard stack as in 1.2.8.2 The symbol D.F / on the left-hand side of this equation denotes the dual of the Picard stack given bythe sheaf F , while D.F / on the right-hand side is the dual sheaf.