On Superselection Theory of Quantum Fields in Low Dimensions

Müger, Michael

doi:10.1142/9789814304634_0041

Cited by 15 publications

(12 citation statements)

References 27 publications

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“…where S 1 , S 2 are the S-matrices of C 1 , C 2 , respectively and T 1 , T 2 are the T -matrices of C 1 , C 2 , respectively. This fact was conjectured by Rehren [23] and proved by Müger [21] and Kawahigashi-Longo independently. We say that the Q-system is Lagrangian when this modular invariance property holds.…”

Section: Relative Tensor Products Of Heterotic Full Conformal Field Theories Yasuyuki Kawahigashimentioning

confidence: 69%

Recent Mathematical Developments in Quantum Field Theory

Abdesselam¹,

Hollands²,

Kopper³

et al. 2017

Oberwolfach Rep.

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Section: Relative Tensor Products Of Heterotic Full Conformal Field Theories Yasuyuki Kawahigashimentioning

confidence: 69%

Recent Mathematical Developments in Quantum Field Theory

Abdesselam¹,

Hollands²,

Kopper³

et al. 2017

Oberwolfach Rep.

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“…(Cf. [54] and [71] for proof sketches.) This fact implies that we have [Rep B] = [Rep A] for any finite extension B ⊃ A of rational chiral CFTs.…”

Section: Definitionmentioning

confidence: 99%

“…In view of Theorem 8.2, it is clear that a modular invariant for (C 1 , C 2 ) exists if and only if C 1 and C 2 are Witt equivalent. But more can be said (stated in [71] and proven in [24]): 8.5 Proposition If C 1 , C 2 are modular, there is a bijection (modulo natural equivalence relations) between modular invariants (A 1 , A 2 , E) for (C 1 , C 2 ) and connectedétale algebras A ∈ C 1 ⊠ C 2 such that…”

Section: Definitionmentioning

confidence: 99%

Modular Categories

MÜGER¹

2013

Quantum Physics and Linguistics

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We limit ourselves to recalling some basic definitions. A category is called an Ab-category if it is enriched over (the symmetric tensor category Ab of) abelian groups. I.e., all hom-sets come with abelian group structures and the composition • of morphisms is a homomorphism w.r.t. both arguments. An Ab-category is called additive if it has a zero object and every pair of objects has a direct sum. If k is a field (or more generally, a commutative unital ring) then a category C is called k-linear if all hom-sets are k-vector spaces (or k-modules) and • is bilinear. An object X is called simple if every monic morphism Y ֒→ X is an isomorphism and absolutely simple if End X ∼ = kid X . If k is an algebraically closed field, as we will mostly assume, the two notions coincide. A k-linear category is semisimple if every object is a finite direct sum of simple objects. A semisimple category is called finite if the number of isomorphism classes of simple objects is finite. (There is a notion of finiteness for non-semisimple categories, cf.[34], but we will not need it.) A positive * -operation on a C-linear category C is a contravariant endofunctor of C that acts like the identity on objects, is involutive ( * * = id) and anti-linear.(Dropping the C-linearity, one arrives at the notion of a dagger-category.) A * -operation is called positive if s * • s = 0 implies s = 0. A category equipped with a (positive) * -operation is called hermitian (unitary). A unitary category with finite-dimensional hom-spaces and splitting idempotents is semisimple (since finite dimensional algebras with positive * -operation are semisimple, thus multi-matrix algebras).

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“…α-induction relates the tensor category associated with a net A to that associated with an extension B, see [22] and references there. We also mention the work of [27] for a categorical relation corresponding to inclusions of nets.…”

Section: Introductionmentioning

confidence: 99%

A Rigidity Result for Extensions of Braided Tensor C*–Categories Derived from Compact Matrix Quantum Groups

Pinzari

Roberts

2011

Commun. Math. Phys.

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Let G be a classical compact Lie group and Gµ the associated compact matrix quantum group deformed by a positive parameter µ (or µ ∈ R \ {0} in the type A case). It is well known that the category of unitary representations of Gµ is a braided tensor C * -category. We show that any braided tensor * -functor ρ : Rep(Gµ) → M to another braided tensor C * -category with irreducible tensor unit is full if |µ| = 1. In particular, the functor of restriction RepGµ → Rep(K) to a proper compact quantum subgroup K cannot be made into a braided functor. Our result also shows that the Temperley-Lieb category T ±d for d > 2 can not be embedded properly into a larger category with the same objects as a braided tensor C * -subcategory.

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On Superselection Theory of Quantum Fields in Low Dimensions

Abstract: We discuss finite local extensions of quantum field theories in low space time dimensions in connection with categorical structures and the question of modular invariants in conformal field theory, also touching upon purely mathematical ramifications.

Cited by 15 publications

References 27 publications

Recent Mathematical Developments in Quantum Field Theory

Recent Mathematical Developments in Quantum Field Theory

Modular Categories

A Rigidity Result for Extensions of Braided Tensor C*–Categories Derived from Compact Matrix Quantum Groups

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