Karl-Henning Rehren

Superselection sectors with braid group statistics and exchange algebras. I. General theory

The theory of superselection sectors is generalized to situations in which normal statistics has to be replaced by braid group statistics. The essential role of the positive Markov trace of algebraic quantum field theory for this analysis is explained, and the relation to exchange algebras is established.

Einstein causality and Artin braids

Abstract Within the restricted context of conformal QFTh 2 we present a systematic analysis of the exchange algebras of light-cone fields which result from the previously studied global decomposition theory of Einstein-causal fields. Although certain aspects of the representation theory of exchange algebras with their Artin braid structure-constants appear in our illustrative examples (minimal and WZW models), our main interest are algebraic aspects. We view the present work as a new non-lagrangian (non-hamiltonian) approach to non-perturbative QFTh.

Superselection sectors with braid group statistics and exchange algebras. 2. Geometric aspects and conformal covariance

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 Poincare 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.

Local quantum observables in the anti-de-Sitter–conformal QFT correspondence

Abstract Quantum field theory on ( d +1)-dimensional anti-de-Sitter space–time admits a reinterpretation as a quantum field theory with conformal symmetry on d -dimensional Minkowski space–time. This conjecture originally emerged from string theory considerations. Here, it is proven in a general framework by an explicit identification between the local observables of the two corresponding theories.

Space-time fields and exchange fields

We derive discrete symmetries of braid group statistics related to charge conjugation and outer automorphisms of the local algebra. The structure of the latter (which are abelian superselection charges) is analyzed in some detail. We use the results to study in great generality a phenomenon recently observed in conformal quantum field theories: the existence of two-dimensional space-time fields with conventional (local, fermionic, dual) commutation relations, expressible as bilinear sums over light-cone fields with exchange algebra commutation relations.

Canonical Tensor Product Subfactors

Abstract:Canonical tensor product subfactors (CTPSs) describe, among other things, the embedding of chiral observables in two-dimensional conformal quantum field theories. A new class of CTPSs is constructed some of which are associated with certain modular invariants, thereby establishing the expected existence of the corresponding two-dimensional theories.

Chiral Observables and Modular Invariants

Abstract:Various definitions of chiral observables in a given Möbius covariant two-dimensional (2D) theory are shown to be equivalent. Their representation theory in the vacuum Hilbert space of the 2D theory is studied. It shares the general characteristics of modular invariant partition functions, although SL(2, ℤ) transformation properties are not assumed. First steps towards a classification are made.

Locality of Conformal Fields in Two Dimensions: Exchange Algebra on the Light-Cone

We discuss the exchange algebra of light-cone operators as the fundamental structure of two-dimensional conformal quantum field theory. It is necessary in order to account for the locality properties of Wightman functions of conformal fields. We discuss the consistency requirements of this new type of algebra, and obtain a classification containing the well known “minimal models”.

Generalized free fields and the AdS-CFT correspondence

Abstract. 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.

A Comment on the Dual Field in the AdS–CFT Correspondence

In the perturbative AdS–CFT correspondence, the dual field whose source are the prescribed boundary values of a bulk field in the functional integral, and the boundary limit of the quantized bulk field are the same thing. This statement is due to the fact that Witten graphs are boundary limits of the corresponding Feynman graphs for the bulk fields, and hence the dual conformal correlation functions are limits of bulk correlation functions. This manifestation of duality is analyzed in terms of the underlying functional integrals of different structure.