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.Comment: Typos corrected and a few minor changes, 37 pages, AMS LaTeX, epic,
eepic, doc-class conm-p-l.cl
