Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Communications in Mathematical Physics

Conformal field algebras with quantum symmetry from the theory of superselection sectors

G. Mack, Volker Schomerus

Published Nov 1, 1990• 86 citations
Publisher
DOI
10.1007/bf02102093

Related papers

Quantum group interpretation of some conformal field theories

Luis Álvarez Gaumé, César Gómez, Germán Sierra

Physics Letters B•Mar 1, 1989•284 citations

BRAID STATISTICS IN LOCAL QUANTUM THEORY

Jürg Martin Fröhlich, Fabrizio Gabbiani

We present details of a mathematical theory of superselection sectors and their statistics in local quantum theory over (two- and) three-dimensional space-time. The framework for our analysis is algebraic quantum field theory. Statistics of superselection sectors in three-dimensional local quantum theory with charges not localizable in bounded space-time regions and in two-dimensional chiral theories is described in terms of unitary representations of the braid groups generated by certain Yang-Baxter matrices. We describe the beginnings of a systematic classification of those representations. Our analysis makes contact with the classification theory of subfactors initiated by Jones. We prove a general theorem on the connection between spin and statistics in theories with braid statistics. We also show that every theory with braid statistics gives rise to a “Verlinde algebra”. It determines a projective representation of SL(2, ℤ) and, presumably, of the mapping class group of any Riemann surface, even if the theory does not display conformal symmetry.

Reviews in Mathematical Physics•Jan 1, 1990•233 citations

Infinite conformal symmetry in two-dimensional quantum field theory

Alexander A. Belavin, Andrey Polyakov, Al.B. Zamolodchikov

Nuclear Physics B•Jul 1, 1984•4,830 citations

CHIRAL DECOMPOSITION AS A SOURCE OF QUANTUM SYMMETRY IN THE ISING MODEL

Kornél Szlachányi

The superselection sectors of three closely related models are studied and compared. I. The full Ising model with observable algebra [Formula: see text], the universal algebra of local even CAR algebras on the circle. [Formula: see text] has four sectors with Z(2) × Z(2) symmetry. II. The chiral Ising model with observable algebra [Formula: see text] (c = L or R), which is the universal algebra of local even Majorana algebras, has three sectors with quantum symmetry Gc ≅ M1⊕M1⊕M2. The Mack-Schomerus endomorphism creating the non-Abelian sector of [Formula: see text], respectively of [Formula: see text] is shown to coincide with the restriction [Formula: see text], respectively [Formula: see text] of a Z (2) charge creating automorphism ρ of [Formula: see text]. III. As an intermediate step the superselection sectors of the algebra [Formula: see text] — which can be interpreted as the observable algebra of the conformal Ising model — are found to have ordinary group symmetry described by the dihedral group D4. The relation between the sectors of [Formula: see text] and [Formula: see text] is explained in terms of a strange 'symmetry breaking': symmetry enhancement in the Neveu-Schwarz sector and symmetry breaking in the Ramond. Covariant charged fields are constructed in all three cases and the truncation in Gc is shown to arise from the failure of the Cuntz algebra relations for the chiral charged fields.

Reviews in Mathematical Physics•Aug 1, 1994•7 citations

Entanglement entropy and superselection sectors. Part I. Global symmetries

Horacio Casini, Marina Huerta, Javier M. Magán +1 more

Abstract Some quantum field theories show, in a fundamental or an effective manner, an alternative between a loss of duality for algebras of operators corresponding to complementary regions, or a loss of additivity. In this latter case, the algebra contains some operator that is not generated locally, in the former, the entropies of complementary regions do not coincide. Typically, these features are related to the incompleteness of the operator content of the theory, or, in other words, to the existence of superselection sectors. We review some aspects of the mathematical literature on superselection sectors aiming attention to the physical picture and focusing on the consequences for entanglement entropy (EE). For purposes of clarity, the whole discussion is divided into two parts according to the superselection sectors classification: the present part I is devoted to superselection sectors arising from global symmetries, and the forthcoming part II will consider those arising from local symmetries. Under this perspective, here restricted to global symmetries, we study in detail different cases such as models with finite and Lie group symmetry as well as with spontaneous symmetry breaking or excited states. We illustrate the general results with simple examples. As an important application, we argue the features of holographic entanglement entropy correspond to a picture of an sub-theory with a large number of superselection sectors and suggest some ways in which this identification could be made more precise.

arXiv:1905.10487•Journal of High Energy Physics•Feb 1, 2020•87 citations

Quantum group approach to conformal field theory

Luis Álvarez-Gaumé

Nuclear Physics B - Proceedings Supplements•Aug 1, 1990•1 citation
We present details of a mathematical theory of superselection sectors and their statistics in local quantum theory over (two- and) three-dimensional space-time. The framework for our analysis is algebraic quantum field theory. Statistics of superselection sectors in three-dimensional local quantum theory with charges not localizable in bounded space-time regions and in two-dimensional chiral theories is described in terms of unitary representations of the braid groups generated by certain Yang-Baxter matrices. We describe the beginnings of a systematic classification of those representations. Our analysis makes contact with the classification theory of subfactors initiated by Jones. We prove a general theorem on the connection between spin and statistics in theories with braid statistics. We also show that every theory with braid statistics gives rise to a “Verlinde algebra”. It determines a projective representation of SL(2, ℤ) and, presumably, of the mapping class group of any Riemann surface, even if the theory does not display conformal symmetry.
The superselection sectors of three closely related models are studied and compared. I. The full Ising model with observable algebra [Formula: see text], the universal algebra of local even CAR algebras on the circle. [Formula: see text] has four sectors with Z(2) × Z(2) symmetry. II. The chiral Ising model with observable algebra [Formula: see text] (c = L or R), which is the universal algebra of local even Majorana algebras, has three sectors with quantum symmetry Gc ≅ M1⊕M1⊕M2. The Mack-Schomerus endomorphism creating the non-Abelian sector of [Formula: see text], respectively of [Formula: see text] is shown to coincide with the restriction [Formula: see text], respectively [Formula: see text] of a Z (2) charge creating automorphism ρ of [Formula: see text]. III. As an intermediate step the superselection sectors of the algebra [Formula: see text] — which can be interpreted as the observable algebra of the conformal Ising model — are found to have ordinary group symmetry described by the dihedral group D4. The relation between the sectors of [Formula: see text] and [Formula: see text] is explained in terms of a strange 'symmetry breaking': symmetry enhancement in the Neveu-Schwarz sector and symmetry breaking in the Ramond. Covariant charged fields are constructed in all three cases and the truncation in Gc is shown to arise from the failure of the Cuntz algebra relations for the chiral charged fields.
Abstract Some quantum field theories show, in a fundamental or an effective manner, an alternative between a loss of duality for algebras of operators corresponding to complementary regions, or a loss of additivity. In this latter case, the algebra contains some operator that is not generated locally, in the former, the entropies of complementary regions do not coincide. Typically, these features are related to the incompleteness of the operator content of the theory, or, in other words, to the existence of superselection sectors. We review some aspects of the mathematical literature on superselection sectors aiming attention to the physical picture and focusing on the consequences for entanglement entropy (EE). For purposes of clarity, the whole discussion is divided into two parts according to the superselection sectors classification: the present part I is devoted to superselection sectors arising from global symmetries, and the forthcoming part II will consider those arising from local symmetries. Under this perspective, here restricted to global symmetries, we study in detail different cases such as models with finite and Lie group symmetry as well as with spontaneous symmetry breaking or excited states. We illustrate the general results with simple examples. As an important application, we argue the features of holographic entanglement entropy correspond to a picture of an sub-theory with a large number of superselection sectors and suggest some ways in which this identification could be made more precise.