The ring structure for equivariant twisted K-theory

Tu, Jean Louis; Xu, Ping

doi:10.1515/crelle.2009.077

Cited by 14 publications

(29 citation statements)

References 43 publications

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“…, [27], [39]. It is an interesting problem to calculate the holonomy of the bundleG δ → LX in terms of the Dixmier-Douady class d ∈ H 3 (X; Z) of the gerbe G → X.…”

Section: A Description Of the Resultsmentioning

confidence: 99%

Inertia and delocalized twisted cohomology

Bunke

Schick

Spitzweck

2008

Homology, Homotopy and Applications

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Orbispaces are the analog of orbifolds, where the category of manifolds is replaced by topological spaces. We construct the loop orbispace LX of an orbispace X in the language of stacks in topological spaces. Furthermore, to a twist given by aWe use sheaf theory on topological stacks in order to define the delocalized twisted cohomology byis the pull-back of the gerbe G → X via the natural map LX → X, and L ∈ Sh Ab LX is the sheaf of sections of the C δ -bundle associated toG δ → LX. The same constructions can be applied in the case of orbifolds, and we show that the sheaf theoretic delocalized twisted cohomology is isomorphic to the twisted de Rham cohomology, where the isomorphism depends on the choice of a geometric structure on the gerbe G → X.

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“…, [27], [39]. It is an interesting problem to calculate the holonomy of the bundleG δ → LX in terms of the Dixmier-Douady class d ∈ H 3 (X; Z) of the gerbe G → X.…”

Section: A Description Of the Resultsmentioning

confidence: 99%

Inertia and delocalized twisted cohomology

Bunke

Schick

Spitzweck

2008

Homology, Homotopy and Applications

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“…These K-groups τ K G * (X), τ K * G (X) possess a natural R G -module structure, coming from the map X → pt [99] -as mentioned next section, K G 0 (pt) H 0 G (pt) R G . When the twist τ ∈ H 3 G (X; Z) is transgressed from H 4 (X; Z), the K-groups τ K G * (X) and τ K * G (X) carry graded ring structures [104], coming from the external Kasparov product in equivariant KK-theory.…”

Section: Dim(h) Hmentioning

confidence: 99%

“…For the finite groups the α ± are found in [32] (see also Section 1.4 of this paper). Tu and Xu [104] find the natural ring structure on the K-groups of twisted equivariant K-theoryit is essentially the external Kasparov product in equivariant KK-theory. In particular, for twisted equivariant K-homology (see Remark 4.30 of [104]), there will be a graded product τ K G i (X) × τ K G j (X) → τ K G i+j (X) (at least when the twist is transgressed).…”

Section: The K-homological Calculationsmentioning

confidence: 99%

“…Tu and Xu [104] find the natural ring structure on the K-groups of twisted equivariant K-theoryit is essentially the external Kasparov product in equivariant KK-theory. In particular, for twisted equivariant K-homology (see Remark 4.30 of [104]), there will be a graded product τ K G i (X) × τ K G j (X) → τ K G i+j (X) (at least when the twist is transgressed). This should agree with the algebra of the full system, and from this we can obtain the braiding, etc.…”

Section: The K-homological Calculationsmentioning

confidence: 99%

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Modular invariants and twisted equivariant K-theory

Evans

Gannon

2009

Communications in Number Theory and Physics

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Freed-Hopkins-Teleman expressed the Verlinde algebra as twisted equivariant K-theory. We study how to recover the full system (fusion algebra of defect lines), nimrep (cylindrical partition function), etc. of modular invariant partition functions of conformal field theories associated to loop groups. We work out several examples corresponding to conformal embeddings and orbifolds. We identify a new aspect of the A-D-E pattern of SU(2) modular invariants.

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“…(For nonsimply connected groups G, the existence of ring structures on the twisted K-homology is a much more subtle matter [40]. )…”

Section: 4mentioning

confidence: 99%

On the quantization of conjugacy classes

Meinrenken¹

2009

Enseign. Math.

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Abstract. Let G be a compact, simple, simply connected Lie group. A theorem of FreedHopkins-Teleman identifies the level k ≥ 0 fusion ring R k (G) of G with the twisted equivariant K-homology at level k + h ∨ , where h ∨ is the dual Coxeter number of G. In this paper, we will review this result using the language of Dixmier-Douady bundles. We show that the additive generators of the group R k (G) are obtained as K-homology push-forwards of the fundamental classes of pre-quantized conjugacy classes in G.

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The ring structure for equivariant twisted K-theory

Abstract: We prove, under some mild conditions, that the equivariant twisted K

Cited by 14 publications

References 43 publications

Inertia and delocalized twisted cohomology

Inertia and delocalized twisted cohomology

Modular invariants and twisted equivariant K-theory

On the quantization of conjugacy classes

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