Phase Boundaries in Algebraic Conformal QFT

Bischoff, Marcel; Kawahigashi, Yasuyuki; Longo, Roberto; Rehren, Karl-Henning

doi:10.1007/s00220-015-2560-0

Cited by 22 publications

(117 citation statements)

References 41 publications

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“…In order to show that the braided product M A × ± ε M B actually contains M A and M B as subalgebras (see Proposition 4.5 below) we need to consider generalized Q-systems of intertwiners with an additional property, which is a weaker version of the unit property in ordinary Q-systems, namely θ(w * )x = 1, cf. [BKLR16,Prop. 4.12].…”

Section: Braided Productsmentioning

confidence: 99%

“…In this section we apply the braided product construction to nets of von Neumann algebras and show that it enjoys some remarkable properties, in analogy to the finite index case, which allows one to extract boundary quantum field theories as in [BKLR16]. Such field theories with transparent boundaries will be discussed in the next section.…”

Section: Braided Product Of Netsmentioning

confidence: 99%

“…Lastly, the analysis of extensions can be used to construct new examples of QFTs. Starting from some extensions [LR95] and of theories with defects and boundaries [BKLR16] cover the finite index case only, both being based on the notion of Q-system (which is tightly connected to the existence of conjugate morphismsῑ of the inclusion morphism ι : N ֒→ M for a subfactor N ⊂ M, hence to the finiteness of the dimension of ι in the sense of [LR97]).…”

Section: Introductionmentioning

confidence: 99%

“…An advantage of using generalized Q-systems (in the finite index case as well) is that no factoriality or irreducibility assumption on the inclusion is needed along the way. This enhanced flexibility is particularly desirable in the study of boundary conditions, see comments after [BKLR16,Thm. 4.4], where non-irreducible, non-factorial extensions necessarily appear.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, we leave open the questions about universality of the braided product construction and the classification of boundary conditions in the infinite index case, cf. [BKLR16,Sec. 5].…”

Section: Introductionmentioning

confidence: 99%

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Infinite index extensions of local nets and defects

Vecchio

Giorgetti

2018

Rev. Math. Phys.

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Subfactor theory provides a tool to analyze and construct extensions of Quantum Field Theories, once the latter are formulated as local nets of von Neumann algebras. We generalize some of the results of [LR95] to the case of extensions with infinite Jones index. This case naturally arises in physics, the canonical examples are given by global gauge theories with respect to a compact (non-finite) group of internal symmetries. Building on the works of Izumi, Longo, Popa [ILP98] and Fidaleo, Isola [FI99], we consider generalized Q-systems (of intertwiners) for a semidiscrete inclusion of properly infinite von Neumann algebras, which generalize ordinary Q-systems introduced by Longo [Lon94] to the infinite index case. We characterize inclusions which admit generalized Q-systems of intertwiners and define a braided product among the latter, hence we construct examples of QFTs with defects (phase boundaries) of infinite index, extending the family of boundaries in the grasp of [BKLR16].Comment: 50 page

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Section: Braided Productsmentioning

confidence: 99%

Section: Braided Product Of Netsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Infinite index extensions of local nets and defects

Vecchio

Giorgetti

2018

Rev. Math. Phys.

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Tensor Categories and Endomorphisms of von Neumann Algebras

Bischoff

Kawahigashi

Longo

et al. 2015

SpringerBriefs in Mathematical Physics

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167

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Q-systems describe "extensions" of an infinite von Neumann factor N , i.e., finite-index unital inclusions of N into another von Neumann algebra M . They are (special cases of) Frobenius algebras in the C* tensor category of endomorphisms of N . We review the relation between Q-systems, their modules and bimodules as structures in a category on one side, and homomorphisms between von Neumann algebras on the other side. We then elaborate basic operations with Q-systems (various decompositions in the general case, and the centre, the full centre, and the braided product in braided categories), and illuminate their meaning in the von Neumann algebra setting. The main applications are in local quantum field theory, where Q-systems in the subcategory of DHR endomorphisms of a local algebra encode extensions A(O) ⊂ B(O) of local nets. These applications, notably in conformal quantum field theories with boundaries, are briefly exposed, and are discussed in more detail in two separate papers [4,5].

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