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.

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The general theory of superselection sectors is shown to provide almost all the structure observed in two-dimensional conformal field theories. Its application to two-dimensional conformally covariant and three-dimensional Poincaré covariant theories yields a general spin-statistics connection previously encountered in more special situations. CPT symmetry can be shown also in the absence of local (anti-) commutation relations, if the braid group statistics is expressed in the form of an exchange algebra.

Q-systems describe "extensions" of an infinite von Neumann factor N , i.e., finite-index unital inclusions of N into another von Neumann algebra M . They are (special cases of) Frobenius algebras in the C* tensor category of endomorphisms of N . 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) ⊂ B(O) of local nets. These applications, notably in conformal quantum field theories with boundaries, are briefly exposed, and are discussed in more detail in two separate papers [4,5].

A rigorous (and simple) proof is given that there is a one-to-one correspondence between causal anti-deSitter covariant quantum field theories on anti-deSitter space and causal conformally covariant quantum field theories on its conformal boundary. The correspondence is given by the explicit identification of observables localized in wedge regions in anti-deSitter space and observables localized in double-cone regions in its boundary. It takes vacuum states into vacuum states, and positive-energy representations into positive-energy representations.Comment: 16 pages, 1 figure, v3: new material added in response to referees' reports, v4: a hasty conclusion in v3 rectified + more cosmetic change

Dedicated to Detlev Buchholz on the occasion of his 60th birthdayConformal quantum field theory on the half-space x > 0 of Minkowski space-time ("boundary CFT") is analyzed from an algebraic point of view, clarifying in particular the algebraic structure of local algebras and the bi-localized charge structure of local fields. The field content and the admissible boundary conditions are characterized in terms of a non-local chiral field algebra.It follows that the components T 10 = T 01 , T 11 = T 00 of the stress-energy tensor are of the formi.e. bi-local expressions in terms of the chiral field T (cf. Fig. 1). 909 Rev. Math. Phys. 2004.16:909-960. Downloaded from www.worldscientific.com by UNIVERSITY OF SASKATCHEWAN on 02/03/15. For personal use only. Rev. Math. Phys. 2004.16:909-960. Downloaded from www.worldscientific.com by UNIVERSITY OF SASKATCHEWAN on 02/03/15. For personal use only. Local Fields in Boundary Conformal QFT 911has additional contributions at t 1 +x 1 = t 2 −x 2 and at t 1 −x 1 = t 2 +x 2 . But within a wedge region M + ⊃ W : x > |t| (⇔ t − x < 0 < t + x), the latter contributions are ineffective. The same holds for any time translate of W . A slightly stronger version of this algebraic indistinguishability is the following. It has been shown that the chiral stress-energy tensor satisfies the split property: namely for every pair of intervals J < I which do not touch (thus allowing to smooth out the UV singularities), there exists a state ϕ in the vacuum Hilbert space H 0 of T (depending on I and J; in particular not the vacuum state) which has no correlation between T (u 1 ) and T (u 2 ) when u 1 ∈ I and u 2 ∈ J. In other words, ϕ factorizes on products of T (u i ) with u i ∈ I ∪ J according to( 1.6) This implies that, for every double-cone O not touching the boundary (hence t − x and t + x belong to non-touching intervals as before), there is a state ϕ such that products of T µν (t, x) given by (1.2) in the boundary CFT with (t, x) ∈ O have the same expectation values in the state ϕ as the same products of T µν (t, x) given by (1.5) in the 2D Minkowski space CFT have in the state ϕ ⊗ ϕ. This property exhibits the local "decoupling" of left and right chiral components. Exactly as the split property fails when the intervals I and J touch, the decoupling of left-and right-movers breaks down at the boundary in BCFT. We shall assume the split property for all chiral fields of a boundary CFT. This property is known to be related to phase space properties of the CFT (existence of Tr exp(−βL 0 )) [9, 1], and it has been established for large classes of chiral models ([45] and references therein).Our aim in the present article is to understand the structure of local fields in boundary CFT which do not decompose in the manner of (1.2) or (1.4). These nonchiral fields have to satisfy local commutativity with the chiral fields and with each other, and transform covariantly under the conformal (Möbius) group generated by the chiral stress-energy tensor T . The crucial observation will be that non-chiral...

Dedicated to Detlev Buchholz on the occasion of his 70th birthday. AbstractWe study the structure of local algebras in relativistic conformal quantum field theory with phase boundaries. Phase boundaries are instances of a more general notion of boundaries that give rise to a variety of algebraic structures. These can be formulated in a common framework originating in Algebraic QFT, with the principle of Einstein Causality playing a prominent role. We classify the phase boundary conditions by the centre of a certain universal construction, which produces a reducible representation in which all possible boundary conditions are realized. For a large class of models, the classification reproduces results obtained in a different approach by Fuchs et al. before.

Motivated by structural issues in the AdS-CFT correspondence, the theory of generalized free fields is reconsidered. A stress-energy tensor for the generalized free field is constructed as a limit of Wightman fields. Although this limit is singular, it fulfills the requirements of a conserved local density for the Poincaré generators. An explicit "holographic" formula relating the Klein-Gordon field on AdS to generalized free fields on Minkowski space-time is provided, and contrasted with the "algebraic" notion of holography. A simple relation between the singular stress-energy tensor and the canonical AdS stress-energy tensor is exhibited.