Disorder Operators, Quantum Doubles, and Haag Duality in 1 + 1 Dimensions

Abstract

Since the notion of the ‘quantum double’ was coined by Drinfel’d in his famous ICM lecture [8] there have been several attempts aimed at a clarification of its relevance to two dimensional quantum field theory. The quantum double appears implicitly in the work [3] on orbifold constructions in conformai field theory, where conformal quantum field theories (CQFTs) are considered whose operators are fixpoints under the action of a symmetry group on another CQFT. Whereas the authors emphasize that ‘the fusion algebra of the holomorphic G-orbifold theory naturally combines both the representation and class algebra of the group G’ the relevance of the double is fully recognized only in [4]. The quantum double also appears in the context of integrable quantum field theories, e.g. [1], as well as in certain lattice models (e.g. [18]). Common to these works is the role of disorder operators or ‘twist fields’ which are ‘local with respect to A up to the action of an element g ∈ G’ [3].