Extensions of Conformal Nets¶and Superselection Structures

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 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.

How to Remove the Boundary in CFT – An Operator Algebraic Procedure

“…In [12] it was shown that a unitary representation of the Möbius group Möb is generated by the modular groups of a "halfsided modular triple", i.e., three von Neumann algebras A i (i = 0, 1, 2) with a joint cyclic and separating vector Ψ such that if σ i t is the modular group for (A i , Ψ ), then σ i t (A i+1 ) ⊂ A i+1 for t 0. (Here, i + 1 is understood mod 3.)…”

On the Representation Theory of Virasoro Nets

“…Finally if d(c) = 1 then, for every I ∈ I, π (λ,q) (A (Vir,c) (I)) = A U(1) (I) which is impossible since A U(1) is strongly additive (see [5,26]) while A (Vir,c) it is not. Now let G be a simply connected compact Lie group with simple Lie algebra Lie(G) and let k be a positive integer.…”

On the Uniqueness of Diffeomorphism Symmetry in Conformal Field Theory

“…All these properties are indeed "additional": there are Möbius covariant nets which are not even 3-regular (see the examples in [12]). …”

“…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.…”