Modular invariants from subfactors
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 ("alphainduction"). It has nonnegative integer entries, is normalized and commutes with the S and Tmatrices 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 ADE 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 MooreSeiberg, DijkgraafVerlinde 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 twodimensional 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2000
 arXiv:
 arXiv:math/0006114
 Bibcode:
 2000math......6114B
 Keywords:

 Mathematics  Operator Algebras;
 Mathematical Physics;
 Mathematics  Mathematical Physics;
 Mathematics  Quantum Algebra;
 High Energy Physics  Theory;
 81T40 (Primary) 46L37;
 46L60;
 81T05;
 81R10;
 22E67;
 82B23;
 18D10 (Secondary)
 EPrint:
 Typos corrected and a few minor changes, 37 pages, AMS LaTeX, epic, eepic, docclass conmpl.cls