Nets of Subfactors

Abstract

A subtheory of a quantum field theory specifies von Neumann subalgebras A(O) (the 'observables' in the space-time region O) of the von Neumann algebras B(O) (the 'fields' localized in O). Every local algebra being a (type III 1 ) factor, the inclusion A(O) ⊂ B(O) is a subfactor. The assignment of these local subfactors to the space-time regions is called a 'net of subfactors'. The theory of subfactors is applied to such nets. In order to characterize the 'relative position' of the subtheory, and in particular to control the restriction and induction of superselection sectors, the canonical endomorphism is studied. The crucial observation is this: the canonical endomorphism of a local subfactor extends to an endomorphism of the field net, which in turn restricts to a localized endomorphism of the observable net. The method allows to characterize, and reconstruct, local extensions B of a given theory A in terms of the observables. Various non-trivial examples are given. Several results go beyond the quantum field theoretical application.

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

“…B is a Möbius covariant net R ⊃ I → B(I ) on its vacuum Hilbert space H B 0 such that for each I the inclusion A(I ) ⊂ B(I ) holds and is an irreducible subfactor, which has automatically finite Jones index [13] equal to the statistical dimension of the (reducible) representation of A on H B 0 [17]. The net B may be non-local, but is required to be relatively local w.r.t.…”

“…Namely, B (I 1 ) is generated by ι (A(I 1 )) and v 1 = ι(u) · v, where v ∈ B(I 0 ) is the canonical charged intertwiner v ∈ Hom(ι, ιθ ) for the canonical DHR endomorphism θ localized in I 0 [17] (see also App. B), and θ 1 is an equivalent DHR endomorphism localized in I 1 .…”

“…Here, ε is the global conditional expectation B → A, which preserves the vacuum state [17]. In particular, ε(t) ∈ Hom(τ, σ ).…”

“…(5.12)] a linear condition on ϕ =ῑ(ψ) ∈ Hom(θ, θ στ ) was given which guarantees that ϕ commutes withῑ(B(K )). Hereῑ : B → A is a homomorphism conjugate to the injection ι : A → B, such that γ = ιῑ on B(K ) is a canonical endomorphism for A(K ) ⊂ B(K ) and θ =ῑι is the dual canonical endomorphism, which is a DHR endomorphism of A localized in K [17]. Unfortunately, the condition displayed in [18] does not take into account that ϕ belongs toῑ(B(L)) (i.e., is in the range ofῑ).…”

Quantum Field Theory: Where We Are

“…Conformally invariant theories in two dimensions have been constructed rigorously (and partially classified [8]) by methods of operator algebras, especially the theory of finite index subfactors [9]. It is here crucial that a "germ" of the theory is given, such as the subtheory of the stress-energy tensor field, and is verified to share certain algebraic features.…”