Extensions of Conformal Nets¶and Superselection Structures (original) (raw)

2 1 1 Ju l 1 99 7 Extensions of Conformal Nets and Superselection Structures

1997

Starting with a conformal Quantum Field Theory on the real line, we show that the dual net is still conformal with respect to a new representation of the Möbius group. We infer from this that every conformal net is normal and conormal, namely the local von Neumann algebra associated with an interval coincides with its double relative commutant inside the local von Neumann algebra associated with any larger interval. The net and the dual net give together rise to an infinite dimensional symmetry group, of which we study a class of positive energy irreducible representations. We mention how superselsection sectors extend to the dual net and we illustrate by examples how, in general, this process generates solitonic sectors. We describe the free theories associated with the lowest weight n representations of PSL(2, R), showing that they violate 3-regularity for n > 2. When n ≥ 2, we obtain examples of non Möbius-covariant sectors of a 3-regular (non 4-regular) net. Supported in part...

Extensions of Conformal Nets and Superselection Structures

1998

Starting with a conformal Quantum Field Theory on the real line, we show that the dual net is still conformal with respect to a new representation of the Mobius group. We infer from this that every conformal net is normal and conormal, namely the local von Neumann algebra associated with an interval coincides with its double relative commutant inside the local von Neumann algebra associated with any larger interval. The net and the dual net give together rise to an infinite dimensional symmetry group, of which we study a class of positive energy irreducible representations. We mention how superselsection sectors extend to the dual net and we illustrate by examples how, in general, this process generates solitonic sectors. We describe the free theories associated with the lowest weight n representations of PSL(2; R), showing that they violate 3-regularity for n ? 2. When n 2, we obtain examples of non Mobius-covariant sectors of a 3-regular (non 4-regular) net.

Extension of conformal nets and superselection structures

1998

Abstract: Starting with a conformal Quantum Field Theory on the real line, we show that the dual net is still conformal with respect to a new representation of the Möbius group. We infer from this that every conformal net is normal and conormal, namely the local von Neumann algebra associated with an interval coincides with its double relative commutant inside the local von Neumann algebra associated with any larger interval. The net and the dual net give together rise to an infinite dimensional symmetry group, of which we study a class of positive energy irreducible representations. We mention how superselection sectors extend to the dual net and we illustrate by examples how, in general, this process generates solitonic sectors. We describe the free theories associated with the lowest weight n representations of PSL(2, R), showing that they violate 3-regularity for n > 2. When n ≥ 2, we obtain examples of non Möbius-covariant sectors of a 3-regular (non 4-regular) net.

Representations of conformal nets, universal C*-algebras

2016

We study the representation theory of a conformal net A on S 1 from a K-theoretical point of view using its universal C*-algebra C * (A). We prove that if A satisfies the split property then, for every representation π of A with finite statistical dimension, π(C * (A)) is weakly closed and hence a finite direct sum of type I ∞ factors. We define the more manageable locally normal universal C*-algebra C * ln (A) as the quotient of C * (A) by its largest ideal vanishing in all locally normal representations and we investigate its structure. In particular, if A is completely rational with n sectors, then C * ln (A) is a direct sum of n type I ∞ factors. Its ideal K A of compact operators has nontrivial K-theory, and we prove that the DHR endomorphisms of C * (A) with finite statistical dimension act on K A , giving rise to an action of the fusion semiring of DHR sectors on K 0 (K A). Moreover, we show that this action corresponds to the regular representation of the associated fusion algebra.

Representations of Conformal Nets, Universal C*-Algebras and K-Theory

Communications in Mathematical Physics, 2013

We study the representation theory of a conformal net A on S 1 from a K-theoretical point of view using its universal C*-algebra C * (A). We prove that if A satisfies the split property then, for every representation π of A with finite statistical dimension, π(C * (A)) is weakly closed and hence a finite direct sum of type I ∞ factors. We define the more manageable locally normal universal C*-algebra C * ln (A) as the quotient of C * (A) by its largest ideal vanishing in all locally normal representations and we investigate its structure. In particular, if A is completely rational with n sectors, then C * ln (A) is a direct sum of n type I ∞ factors. Its ideal K A of compact operators has nontrivial K-theory, and we prove that the DHR endomorphisms of C * (A) with finite statistical dimension act on K A , giving rise to an action of the fusion semiring of DHR sectors on K 0 (K A ). Moreover, we show that this action corresponds to the regular representation of the associated fusion algebra.

Multi-Interval Subfactors and Modularity¶of Representations in Conformal Field Theory

Communications in Mathematical Physics, 2001

We describe the structure of the inclusions of factors A(E) ⊂ A(E ′ ) ′ associated with multi-intervals E ⊂ R for a local irreducible net A of von Neumann algebras on the real line satisfying the split property and Haag duality. In particular, if the net is conformal and the subfactor has finite index, the inclusion associated with two separated intervals is isomorphic to the Longo-Rehren inclusion, which provides a quantum double construction of the tensor category of superselection sectors of A. As a consequence, the index of A(E) ⊂ A(E ′ ) ′ coincides with the global index associated with all irreducible sectors, the braiding symmetry associated with all sectors is non-degenerate, namely the representations of A form a modular tensor category, and every sector is a direct sum of sectors with finite dimension. The superselection structure is generated by local data. The same results hold true if conformal invariance is replaced by strong additivity and there exists a modular PCT symmetry.

Classification of local conformal nets. Case c < 1

Annals of Mathematics, 2004

We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge c less than 1. The irreducible ones are in bijective correspondence with the pairs of A-D 2n -E 6,8 Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1.

N= 2 superconformal nets

arXiv preprint arXiv:1207.2398, 2012

Abstract: We provide an operator algebraic approach to N= 2 chiral conformal field theory and set up the noncommutative geometric framework. Compared to the N= 1 case, the structure here is much richer. There are naturally associated nets of spectral triples and the JLO cocycles separate the Ramond sectors. We construct the N= 2 superconformal nets of von Neumann algebras in general, classify them in the discrete series c< 3, and study spectral flow. We prove the coset identification for the N= 2 super-Virasoro nets with c< 3, ...

Characterization of 2D rational local conformal nets and its boundary conditions: the maximal case

Let A be a completely rational local Möbius covariant net on S 1 , which describes a set of chiral observables. We show that local Möbius covariant nets B2 on 2D Minkowski space which contain the chiral theory A are in one-to-one correspondence with Morita equivalence classes of Q-systems in the unitary modular tensor category DHR(A). The Möbius covariant boundary conditions with symmetry A of such a net B2 are given by the Q-systems in the Morita equivalence class or by simple objects in the module category modulo automorphisms of the dual category. We generalize to reducible boundary conditions.

Local conformal nets arising from framed vertex operator algebras

Advances in Mathematics, 2006

We apply an idea of framed vertex operator algebras to a construction of local conformal nets of (injective type III 1 ) factors on the circle corresponding to various lattice vertex operator algebras and their twisted orbifolds. In particular, we give a local conformal net corresponding to the moonshine vertex operator algebras of Frenkel-Lepowsky-Meurman. Its central charge is 24, it has a trivial representation theory in the sense that the vacuum sector is the only irreducible DHR sector, its vacuum character is the modular invariant Jfunction and its automorphism group (the gauge group) is the Monster group. We use our previous tools such as α-induction and complete rationality to study extensions of local conformal nets. * Supported in part by JSPS. † Supported in part by GNAMPA and MIUR.