Superselection sectors with braid group statistics and exchange algebras

Fredenhagen, Klaus; Rehren, Karl-Henning; Schroer, Bert

doi:10.1007/bf01217906

Cited by 358 publications

(449 citation statements)

References 21 publications

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“…In the latter case, we express the cocycles as z ρ (g) = U 0 (g)U ρ (g) * , and (4.5) because the intertwiners between DHR endomorphisms also intertwine the representations of the Möbius group [10]. By the spectrum condition, F(x) has a bounded analytic continuation to the lower complex halfplane.…”

Section: ) With Charged Intertwiners ψ (I) and A (I)mentioning

confidence: 99%

“…Here z ρ (g) ∈ Hom(ρ, α g ρα −1 g ) are the unitary cocycles z ρ (g) = U 0 (g)U ρ (g) * ∈ A [10,16], where U 0 and U ρ are the representations of the Möbius group in the vacuum representation and in the DHR representation ρ.…”

Section: Reconstruction Of the 2d Symmetrymentioning

confidence: 99%

“…If A is completely rational, the C* tensor category defined by its DHR superselection sectors is modular [14], i.e., the unitary S and T matrices defined by the statistics [10] generate a representation of the group SL(2, Z). By the Verlinde formula [22], these matrices also describe the modular transformation behavior of chiral partition functions ("characters").…”

Section: Haag Duality and Modular Invariancementioning

confidence: 99%

“…The group composition law follows from the cocycle properties [10,16] of z ρ . (iv) The intertwining property of µ is due to the fact that µ annihilates all charged intertwiners except the neutral one (σ =τ = id).…”

Section: Proposition 1 (I) the Maps βmentioning

confidence: 99%

See 3 more Smart Citations

How to Remove the Boundary in CFT – An Operator Algebraic Procedure

Longo

Rehren

2008

Commun. Math. Phys.

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Section: ) With Charged Intertwiners ψ (I) and A (I)mentioning

confidence: 99%

Section: Reconstruction Of the 2d Symmetrymentioning

confidence: 99%

Section: Haag Duality and Modular Invariancementioning

confidence: 99%

Section: Proposition 1 (I) the Maps βmentioning

confidence: 99%

See 2 more Smart Citations

How to Remove the Boundary in CFT – An Operator Algebraic Procedure

Longo

Rehren

2008

Commun. Math. Phys.

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“…The ±-sign here is due to the two possibilities of the statistics operators coming from the non-trivial space-time topology of the punctured circle (see [13,14]). Therefore the two statistics operators, corresponding to braiding and opposite braiding, are in general different but they may coincide for some λ and µ.…”

Section: Nets Of Subfactors and α-Inductionmentioning

confidence: 99%

Modular Invariants, Graphs and α-Induction¶for Nets of Subfactors. III

Böckenhauer¹,

Evans²

1999

Commun. Math. Phys.

113

321

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In this paper we further develop the theory of α-induction for nets of subfactors, in particular in view of the system of sectors obtained by mixing the two kinds of induction arising from the two choices of braiding. We construct a relative braiding between the irreducible subsectors of the two "chiral" induced systems, providing a proper braiding on their intersection. We also express the principal and dual principal graphs of the local subfactors in terms of the induced sector systems. This extended theory is again applied to conformal or orbifold embeddings of SU (n) WZW models. A simple formula for the corresponding modular invariant matrix is established in terms of the two inductions, and we show that it holds if and only if the sets of irreducible subsectors of the two chiral induced systems intersect minimally on the set of marked vertices, i.e. on the "physical spectrum" of the embedding theory, or if and only if the canonical endomorphism sector of the conformal or orbifold inclusion subfactor is in the full induced system. We can prove either condition for all simple current extensions of SU (n) and many conformal inclusions, covering in particular all type I modular invariants of SU (2) and SU (3), and we conjecture that it holds also for any other conformal inclusion of SU (n) as well. As a by-product of our calculations, the dual principal graph for the conformal inclusion SU (3) 5 ⊂ SU (6) 1 is computed for the first time.

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Dual Group Actions on C*-Algebras and Their Description by Hilbert Extensions

Baumgärtel

Lledó

2002

Math. Nachr.

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Given a C*‐algebra 𝒜, a discrete abelian group 𝒳 and a homomorphism Θ : 𝒳 → Out 𝒜, defining the dual action group Γ ⊂ aut 𝒜, the paper contains results on existence and characterization of Hilbert extensions of {𝒜, Γ}, where the action is given by $ \hat {\cal X} $. They are stated at the (abstract) C*‐level and can therefore be considered as a refinement of the extension results given for von Neumann algebras for example by V. F. R. Jones [18] or C. E. Sutherland [22, 23]. A Hilbert extension exists iff there is a generalized 2‐cocycle. These results generalize those in [12], which are formulated in the context of superselection theory, where it is assumed that the algebra 𝒜 has a trivial center, i.e. 𝒵 = ℂ1. In particular the well‐known “outer characterization” of the second cohomology H2(𝒳,𝒰(𝒵),α𝒳) can be reformulated: there is a bijection to the set of all 𝒜‐module isomorphy classes of Hilbert extensions. Finally, a Hilbert space representation (due to C. E. Sutherland [22, 23] in the von Neumann case) is mentioned. The C*‐norm of the Hilbert extension is expressed in terms of the norm of this representation and it is linked to the so‐called regular representation appearing in superselection theory.

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Superselection sectors with braid group statistics and exchange algebras

Cited by 358 publications

References 21 publications

How to Remove the Boundary in CFT – An Operator Algebraic Procedure

How to Remove the Boundary in CFT – An Operator Algebraic Procedure

Modular Invariants, Graphs and α-Induction¶for Nets of Subfactors. III

Dual Group Actions on C*-Algebras and Their Description by Hilbert Extensions

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