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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.
236 U. Bunke, T. Schick, M. Spitzweck, and A. Thom1.5.9 We have canonical isomorphisms H 0 .D.U // Š D.H 1 .U // Š D.T jB / ŠZ jB and let D.U / 1 D.U / be the pre-image of ¹1º jB Z jB under the natural mapD.U / ! H 0 .D.U //. The quotient D.U / 1 =.BT/ jB is a Z n -gerbe over B which weinterpret as the gerbe of R n -reductions of a T n -principal bundle yE ! B which iswell-defined up to unique isomorphism. The full structure of D.U / supplies the dualgerbe yG ! yE. The details of this construction are explained in 6.4.1.5.10 In this way we use Pontrjagin duality of Picard stacks in order to construct aT -dual pair to .E; G/. Schematically the picture is.E; G/ 1 Ý U2 Ý D.U / 3 Ý . yE; yG/;where the steps are as follows:1. choice of the structure on G of a torsor over .BT T n / jB2. Pontrjagin duality D.U / WD Hom PIC S .U; BT/3. Extraction of the dual pair from D.U / as explained in 1.5.9.1.5.11 Consider a pair .E; G/. In Subsection 6.4 we provide two constructions:1. The construction ˆ starts with the choice of a T -duality triple extending .E; G/and constructs the structure of a torsor over .BT T n / jB on G.2. The construction ‰ starts with the structure on G of a torsor over .BT T n / jBand constructs an extension of .E; G/ to a T -duality triple.Our main result Theorem 6.23 asserts that the constructions ‰ and ˆ are inverses toeach other. In other words, the theories of topological T -duality via Pontrjagin dualityof Picard stacks and via T -duality triples are equivalent.2 Sheaves of Picard categories2.1 Picard categories2.1.1 In a cartesian closed category one can define the notion of a group object in thestandard way. It is given by an object, a multiplication, an identity, and an inversionmorphism. The group axioms can be written as a collection of commutative diagramsinvolving these morphisms.Stacks on a site S form a two-cartesian closed two-category. A group object in atwo-cartesian closed two-category is again given by an object and the multiplication,identity and inversion morphisms. In addition each of the commutative diagrams fromthe category case is now filled by a two-morphisms. These two-morphisms must satisfyhigher associativity relations.Instead of writing out all these relations we will follow the exposition of DeligneSGA4, exposé XVIII [Del], which gives a rather effective way of working with group
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Editors:Guillermo CortiñasDepartam
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ContentsPreface....................
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IntroductionSince its inception 50
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Introductionxiwith D. Blecher, he c
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Program list of speakers and topics
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Categorical aspects of bivariant K-
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72 H. Emerson and R. MeyerIn this s
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74 H. Emerson and R. MeyerK-theoret
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76 H. Emerson and R. MeyerExample 4
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78 H. Emerson and R. Meyerand exact
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80 H. Emerson and R. Meyercondition
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82 H. Emerson and R. MeyerAs a resu
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84 H. Emerson and R. MeyerLet h and
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86 H. Emerson and R. MeyerCorollary
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88 H. Emerson and R. Meyer5 Dual-Di
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On K 1 of a Waldhausen categoryFern
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On K 1 of a Waldhausen category 93t
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On K 1 of a Waldhausen category 95(
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On K 1 of a Waldhausen category 992
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118 M. Karoubi[35], M. F. Atiyah an
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120 M. Karoubihere by GrK n .A/, ar
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122 M. Karoubi(resp. 1) in A. In th
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124 M. Karoubithe class ˛ of A in
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126 M. Karoubiested in the complex
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128 M. KaroubiThe same method may b
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130 M. KaroubiBy the usual excision
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132 M. KaroubiLet A be a graded twi
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134 M. Karoubithe same ideas as in
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136 M. KaroubiTheorem 6.2. 25 The (
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138 M. KaroubiIn order to show this
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140 M. KaroubiThe following theorem
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142 M. KaroubiWe would like to poin
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144 M. KaroubiZ=n identified with t
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146 M. Karoubiwhere n is the ring
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148 M. Karoubi[6] M. F. Atiyah, R.
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174 C. Voigtand using ˇ.x; y/ D .S
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176 C. Voigtalgebra A. The space Cn
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178 C. VoigtKK-theory [16]. Hence,
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A Schwartz type algebra for the tan
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A Schwartz type algebra for the tan
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Parshin’s conjecture revisitedTho
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428 C. WeibelGiven (0.4) and axiom
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430 C. WeibelAs pointed out in the
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434 C. WeibelWe need to see that th
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List of contributorsArthur Bartels,
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List of participantsNatalia Abad Sa
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