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Modular invariants from subfactors

Böckenhauer, Jens and Evans, David Emrys 2000. Modular invariants from subfactors. Contemporary Mathematics , pp. 95-131.

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Abstract

In these lectures we explain the intimate relationship between modular invariants in conformal field theory and braided subfactors in operator algebras. A subfactor with a braiding determines a matrix $Z$ which is obtained as a coupling matrix comparing two kinds of braided sector induction ("alpha-induction"). It has non-negative integer entries, is normalized and commutes with the S- and T-matrices arising from the braiding. Thus it is a physical modular invariant in the usual sense of rational conformal field theory. The algebraic treatment of conformal field theory models, e.g. $SU(n)_k$ models, produces subfactors which realize their known modular invariants. Several properties of modular invariants have so far been noticed empirically and considered mysterious such as their intimate relationship to graphs, as for example the A-D-E classification for $SU(2)_k$. In the subfactor context these properties can be rigorously derived in a very general setting. Moreover the fusion rule isomorphism for maximally extended chiral algebras due to Moore-Seiberg, Dijkgraaf-Verlinde finds a clear and very general proof and interpretation through intermediate subfactors, not even referring to modularity of $S$ and $T$. Finally we give an overview on the current state of affairs concerning the relations between the classifications of braided subfactors and two-dimensional conformal field theories. We demonstrate in particular how to realize twisted (type II) descendant modular invariants of conformal inclusions from subfactors and illustrate the method by new examples.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Additional Information: Quantum Symmetries in Theoretical Physics and Mathematics (Bariloche, 2000). Pdf uploaded in accordance with publisher's policy at http://www.sherpa.ac.uk/romeo/issn/0271-4132/ (accessed 24/02/2014)
Publisher: American Mathematical Society
ISSN: 0271-4132
Date of First Compliant Deposit: 30 March 2016
Last Modified: 04 Jun 2017 04:09
URI: http://orca-mwe.cf.ac.uk/id/eprint/33657

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