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arXiv:math/0204281

Critical Phenomena, modular invariants and operator algebras

David Evans

Published Apr 23, 2002• 17 citations
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Abstract

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.

DOI
10.48550/arxiv.math/0204281

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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.
A braided subfactor determines a coupling matrix Z which commutes with the S- and T-matrices arising from the braiding. Such a coupling matrix is not necessarily of “type I”, i.e. in general it does not have a block-diagonal structure which can be reinterpreted as the diagonal coupling matrix with respect to a suitable extension. We show that there are always two intermediate subfactors which correspond to left and right maximal extensions and which determine “parent ” coupling matrices Z ± of type I. Moreover it is shown that if the intermediate subfactors coincide, so that Z + = Z − , then Z is related to Z + by an automorphism of the extended fusion rules. The intertwining relations of chiral branching coefficients between original and extended S- and T-matrices are also clarified. None of our results depends on non-degeneracy of the braiding, i.e. the S- and T-matrices need not be modular. Examples from SO(n) current algebra models illustrate that the parents can be different, Z + ̸ = Z − , and that Z need not be related to a type I invariant by such an automorphism. 1
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.