Quantum Double Actions on Operator Algebras and Orbifold Quantum Field Theories

Abstract

Starting from a local quantum field theory with an unbroken compact symmetry group G in 1 + 1-dimensional spacetime we construct disorder fields implementing gauge transformations on the fields (order variables) localized in a wedge region. Enlarging the local algebras by these disorder fields we obtain a nonlocal field theory, the fixpoint algebras of which under the appropriately extended action of the group G are shown to satisfy Haag duality in every simple sector. The specifically 1+1 dimensional phenomenon of violation of Haag duality of fixpoint nets is thereby clarified. In the case of a finite group G the extended theory is acted upon in a completely canonical way by the quantum double D(G) and satisfies R-matrix commutation relations as well as a Verlinde algebra. Furthermore, our methods are suitable for a concise and transparent approach to bosonization. The main technical ingredient is a strengthened version of the split property which is expected to hold in all reasonable massive theories. In the appendices (part of) the results are extended to arbitrary locally compact groups and our methods are adapted to chiral theories on the circle.

Construction of Kink Sectors for Two-Dimensional Quantum Field Theory Models – An Algebraic Approach

“…Note that a Dirac spinor field contributes to the field content of the Yukawa 2 and Wess-Zumino models, and the nets of Cauchy data fulfill twisted duality instead of Haag duality [58]. According to recent results which have been established by M. Müger [48], our results remain true for these cases also.…”

“…Unfortunately, the operators U I (g) are not contained in C * (M) and the term α (t,x) (U I (g)) has no well defined mathematical meaning. To get a well defined solution for γ (g,I) , we construct an extension of the net M which contains the operators U I (g) (compare also [48]). (b) The automorphisms α t commute with the automorphisms χ g , i.e.…”

On revolutionizing quantum field theory with Tomita’s modular theory

“…Models with a breakdown of Haag duality and the construction of the associated dual net were discussed in Refs. [152][153][154]. Sufficient conditions to reconstruct, using modular theory, 55 a chiral net with conformal symmetry and spectrum condition from a single half-sided modular inclusion of von Neumann algebras were formulated in Refs.…”

Tensor Product Representations of the Quantum Double of a Compact Group

“…* email: thk@wins.uva.nl † email: bais@phys.uva.nl ‡ email: nmuller@phys.uva.nl rigorous definitions for the quantum double or its dual have been proposed, see in particular Majid [17] and Podles' and Woronowicz' [20].An important mathematical application of the Drinfel'd double is a rather simple construction of the 'ordinary' quasi-triangular quantum groups (i.e. q-deformations of universal enveloping algebras of semisimple Lie algebras and of algebras of functions on the corresponding groups), see for example [8] and [17].In physics the quantum double has shown up in various places: in integrable field theories [6], in algebraic quantum field theory [18], and in lattice quantum field theories. For a short summary of these applications, see [12].…”