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数理解析研究所講究録

New subfactors associated with closed systems of sectors (Progress in Operator Algebras)

Karl-Henning Rehren

Published Feb 1, 2000• 0 citations
OpenAlex

Abstract

A theorem is derived which (i) provides a new class of subfactors which may be interpreted as generalized asymptotic subfactors, and which (ii) ensures the existence of two- dimensional local quantum field theories associated with certain modular invariant matrices.

Related papers

New subfactors associated with closed systems of sectors

Karl-Henning Rehren

A theorem is derived which (i) provides a new class of subfactors which may be interpreted as generalized asymptotic subfactors, and which (ii) ensures the existence of two-dimensional local quantum field theories associated with certain modular invariant matrices.

arXiv:math/9911148•Nov 19, 1999•1 citation

NETS OF SUBFACTORS

R. LONGO, Karl-Henning Rehren

A subtheory of a quantum field theory specifies von Neumann subalgebras [Formula: see text] (the ‘observables’ in the space-time region [Formula: see text]) of the von Neumann algebras [Formula: see text] (the 'field' localized in [Formula: see text]). Every local algebra being a (type III1) factor, the inclusion [Formula: see text] 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 single 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 one to characterize, and reconstruct, local extensions ℬ of a given theory [Formula: see text] in terms of the observables. Various non-trivial examples are given. Several results go beyond the quantum field theoretical application.

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Modular Invariants from Subfactors:¶Type I Coupling Matrices and Intermediate Subfactors

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Critical Phenomena, modular invariants and operator algebras

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We review the framework subfactors provide for understanding modular invariants. We discuss the structure of a generalized Longo-Rehren subfactor and the relationship between the coupling matrices of such subfactors, modular invariance and local extensions. We relate results of Kostant, in the context of the McKay correspondence for finite subgroups of SU(2), to subfactors. A direct proof of how alpha-induction produces modular invariants is presented.

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Yasuyuki Kawahigashi, Roberto Longo, Michael Müger

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A subtheory of a quantum field theory specifies von Neumann subalgebras [Formula: see text] (the ‘observables’ in the space-time region [Formula: see text]) of the von Neumann algebras [Formula: see text] (the 'field' localized in [Formula: see text]). Every local algebra being a (type III1) factor, the inclusion [Formula: see text] 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 single 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 one to characterize, and reconstruct, local extensions ℬ of a given theory [Formula: see text] in terms of the observables. Various non-trivial examples are given. Several results go beyond the quantum field theoretical application.