Multi-Interval Subfactors and Modularity¶of Representations in Conformal Field Theory (original) (raw)

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.

Extensions of Conformal Nets¶and Superselection Structures

Communications in Mathematical Physics, 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 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.

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.

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

1997

Modular structure and duality in conformal quantum field theory

Communications in Mathematical Physics, 1993

Making use of a recent result of Borchers, an algebraic version of the Bisognano-Wichmann theorem is given for conformal quantum field theories, i.e. the Tomita-Takesaki modular group associated with the von Neumann algebra of a wedge region and the vacuum vector concides with the evolution given by the rescaled pure Lorentz transformations preserving the wedge. A similar geometric description is valid for the algebras associated with double cones. Moreover essential duality holds on the Minkowski space M , and Haag duality for double cones holds provided the net of local algebras is extended to a pre-cosheaf on the superworldM , i.e. the universal covering of the Dirac-Weyl compactification of M . As a consequence a PCT symmetry exists for any algebraic conformal field theory in even spacetime dimension. Analogous results hold for a Poincaré covariant theory provided the modular groups corresponding to wedge algebras have the expected geometrical meaning and the split property is satisfied. In particular the Poincaré representation is unique in this case. * Supported in part by Ministero della Ricerca Scientifica and CNR-GNAFA. • Supported in part by INFN, sez. Napoli.

Operator algebras and conformal field theory

Communications in Mathematical Physics, 1993

We define and study two-dimensional, chiral conformal field theory by the methods of algebraic field theory. We start by characterizing the vacuum sectors of such theories and show that, under very general hypotheses, their algebras of local observables are isomorphic to the unique hyperfinite type III 1 factor. The conformal net determined by the algebras of local observables is proven to satisfy Haag duality. The representation of the Moebius group (and presumably of the entire Virasoro algebra) on the vacuum sector of a conformal field theory is uniquely determined by the Tomita-Takesaki modular operators associated with its vacuum state and its conformal net. We then develop the theory of Moebius covariant representations of a conformal net, using methods of Doplicher, Haag and Roberts. We apply our results to the representation theory of loop groups. Our analysis is motivated by the desire to find a "background-independent" formulation of conformal field theories. Contents

Topological Sectors and a Dichotomy in Conformal Field Theory

Communications in Mathematical Physics, 2004

Let A be a local conformal net of factors on S 1 with the split property. We provide a topological construction of soliton representations of the n-fold tensor product A ⊗ · · · ⊗ A, that restrict to true representations of the cyclic orbifold (A ⊗ · · · ⊗ A) Zn . We prove a quantum index theorem for our sectors relating the Jones index to a topological degree. Then A is not completely rational iff the symmetrized tensor product (A ⊗ A) flip has an irreducible representation with infinite index. This implies the following dichotomy: if all irreducible sectors of A have a conjugate sector then either A is completely rational or A has uncountably many different irreducible sectors. Thus A is rational iff A is completely rational. In particular, if the µ-index of A is finite then A turns out to be strongly additive. By , if A is rational then the tensor category of representations of A is automatically modular, namely the braiding symmetry is non-degenerate. In interesting cases, we compute the fusion rules of the topological solitons and show that they determine all twisted sectors of the cyclic orbifold. * Supported in part by GNAMPA-INDAM and MIUR. † Supported in part by NSF.

Modular theory for the von Neumann algebras of local quantum physics

Contemporary Mathematics, 2011

In the first part, the second quantization procedure and the free Bosonic scalar field will be introduced, and the axioms for quantum fields and nets of observable algebras will be discussed. The second part is mainly devoted to an illustration of the Bisognano-Wichmann theorem for Wightman fields and in the algebraic setting, with a discussion on the physical meaning of this result. In the third part some reconstruction theorems based on modular groups will be described, in particular the possibility of constructing an action of the symmetry group of a given theory via modular groups, and the construction of free field algebras via representations of the symmetry group.

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.

Tensor categories of endomorphisms and inclusions of von Neumann algebras

Q-systems describe "extensions" of an infinite von Neumann factor NNN, i.e., finite-index unital inclusions of NNN into another von Neumann algebra MMM. They are (special cases of) Frobenius algebras in the C* tensor category of endomorphisms of NNN. 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)subsetB(O)A(O)\subset B(O)A(O)subsetB(O) of local nets. These applications, notably in quantum field theories with boundaries, are discussed in a separate paper [arXiv:1405.7863].

Multi-Interval Subfactors and Modularity¶of Representations in Conformal Field Theory (original) (raw)

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