Haag Duality in Conformal Quantum Field Theory

Buchholz, Detlev; Schulz-Mirbach, Hanns

doi:10.1142/s0129055x90000053

Cited by 92 publications

(188 citation statements)

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“…[8,13]. Moreover from the relations [G −r , G r ] = 2L 0 + c 3 (r 2 − 1 4 ), we find the energy bounds…”

Section: Super-virasoro Netsmentioning

confidence: 64%

“…Moreover, for f real, T B (f ) is essentially self-adjoint on H ∞ (cf. [8]) and we shall denote again T B (f ) its self-adjoint closure. Now let f be a smooth function on S 1 whose support do not contains −1.…”

Section: Super-virasoro Netsmentioning

confidence: 99%

“…(26) and Eq. (27) and the fact that Z commutes with L 0 one can apply the argument in [8,Sect. 2] to show that e iT B (f 1 ) and e iT F (f 1 ) commute with Ze iT B (f 2 ) Z * and Ze iT F (f 2 ) Z * .…”

Section: Super-virasoro Netsmentioning

confidence: 99%

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Structure and Classification of Superconformal Nets

Carpi

Kawahigashi

Longo

2008

Ann. Henri Poincaré

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We study the general structure of Fermi conformal nets of von Neumann algebras on S 1 , consider a class of topological representations, the general representations, that we characterize as Neveu-Schwarz or Ramond representations, in particular a Jones index can be associated with each of them. We then consider a supersymmetric general representation associated with a Fermi modular net and give a formula involving the Fredholm index of the supercharge operator and the Jones index. We then consider the net associated with the super-Virasoro algebra and discuss its structure. If the central charge c belongs to the discrete series, this net is modular by the work of F. Xu and we get an example where our setting is verified by considering the Ramond irreducible representation with lowest weight c/24. We classify all the irreducible Fermi extensions of any super-Virasoro net in the discrete series, thus providing a classification of all superconformal nets with central charge less than 3/2.

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“…[8,13]. Moreover from the relations [G −r , G r ] = 2L 0 + c 3 (r 2 − 1 4 ), we find the energy bounds…”

Section: Super-virasoro Netsmentioning

confidence: 64%

Section: Super-virasoro Netsmentioning

confidence: 99%

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Structure and Classification of Superconformal Nets

Carpi

Kawahigashi

Longo

2008

Ann. Henri Poincaré

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“…The latter class (see Sect. 2 for the definition) includes every strongly additive diffeomorphism covariant net on S 1 and hence every diffeomorphism covariant net which is completely rational in the sense of [18], the nets generated by chiral current algebras [1,12,27,30] and their orbifold subnets [31]. Since the Möbius symmetry of a given net on S 1 is completely determined by the vacuum vector [9,Theorem 2.19] our result shows that in the above cases the Diff + (S 1 ) symmetry of the net is also determined by this vector.…”

Section: Introductionmentioning

confidence: 99%

“…Firstly the uniqueness in the case of Virasoro nets implies that two Virasoro nets cannot be isomorphic as Möbius covariant nets on the circle if they have different central charges (Corollary 3.4), a fact that seems to be widely expected (see e.g. the introduction of [1]) but that has not been explicitly stated in the literature. Similarly two 4-regular diffeomorphism covariant nets cannot be isomorphic as Möbius covariant nets on S 1 if the corresponding representations of Diff + (S 1 ) are not unitarily equivalent and in particular if they have a different central charge (Corollary 5.6).…”

Section: Introductionmentioning

confidence: 99%

On the Uniqueness of Diffeomorphism Symmetry in Conformal Field Theory

Carpi

Weiner

2005

Commun. Math. Phys.

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A Möbius covariant net of von Neumann algebras on S 1 is diffeomorphism covariant if its Möbius symmetry extends to diffeomorphism symmetry. We prove that in case the net is either a Virasoro net or any at least 4-regular net such an extension is unique: the local algebras together with the Möbius symmetry (equivalently: the local algebras together with the vacuum vector) completely determine it. We draw the two following conclusions for such theories. (1) The value of the central charge c is an invariant and hence the Virasoro nets for different values of c are not isomorphic as Möbius covariant nets. (2) A vacuum preserving internal symmetry always commutes with the diffeomorphism symmetries. We further use our result to give a large class of new examples of nets (even strongly additive ones), which are not diffeomorphism covariant; i.e. which do not admit an extension of the symmetry to Diff + (S 1 ).

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On the Existence of Pointlike Localized Fields in Conformally Invariant Quantum Physics

Jörß

1993

NATO ASI Series

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In quantum field theory ~he existence of pointlike localizable objects called "fidds" is a preassumption. Since chll.J:ged fie]ds are iu general not observable this situation is unsatisfying from a quantum physics point of view. Indeed in any quantum theory the existence of fields should follow form deeper physical concepts and more natural first principles like stability, localit}•, causality and symmetry.In the framework of algebraic quantum field theory with Haag-Kastler nets of local observables this is presented for the case of conformal symmetry in 1+ 1 dimensions. Conformal fields are explicitly constructed as limits of observables localized in finite regions of space-time. These fields then allow to derive a geometric identification of modular operators, Haag duality in the VB.("UUm sector, the PCT-tbeorem and an equivalence theorem for fields and algebras.

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Haag Duality in Conformal Quantum Field Theory

Cited by 92 publications

References 0 publications

Structure and Classification of Superconformal Nets

Structure and Classification of Superconformal Nets

On the Uniqueness of Diffeomorphism Symmetry in Conformal Field Theory

On the Existence of Pointlike Localized Fields in Conformally Invariant Quantum Physics

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