On Flatness of Ocneanu’s Connections on the Dynkin Diagrams and Classification of Subfactors

Abstract

We will give a proof of Ocneanu’s announced classification of subfactors of the AFD type II1 factor with the principal graphs An, Dn, E7, the Dynkin diagrams, and give a single explicit of exp π √ -1/24 and exp π √-1/60 for each of E6 and E8 such that its validity is equivalent to the existence of two (and only two) subfactors for these principal graphs. Our main tool is the flatness of connections on finite graphs, which is the key notion of Ocneanu’s paragroup theory. We give the difference between the diagrams D2n and D2n+1 a meaning as a Z/2Z-obstruction for flatness arising in orbifold construction, which is an analogue of orbifold models in solvable lattice models. © 1995 Acdemic Press, Inc.