Exactly solvable orbifold models and subfactors

Abstract

§1 IntroductionSince the pioneering work of V. F. R. Jones [J], the theory of subfactors has had deep and unexpected relations to 3-dimensional topology, conformal field theory, quantum groups, etc. Here we present a relation between paragroup theory of Ocneanu on subfactors and exactly solvable lattice models in statistical mechanics, and in particular, show usefulness of the notion of orbifold lattice models in subfactor theory.Classification of subfactors of the approximately finite dimensional (AFD) factor of type II 1 is one of the most important and challenging problems in the theory of operator algebras. This is a study of inclusions of certain infinite dimensional simple algebras of bounded linear operators on a Hilbert space. The AFD type II 1 factor, the operator algebra we work on, is the most natural infinite dimensional analogue of finite dimensional matrix algebras M n (C) in a sense. (As a general reference on operator algebras, see [T], and as a basic reference on subfactor theory, we cite [GHJ]. All the basic notions are found there.) Here, by "a subfactor N ⊂ M" we mean that N and M are both AFD type II 1 factor and N is a subalgebra of M. V. Jones studied the Jones index [J], a real-valued invariant for the inclusion N ⊂ M, which measures the relative size of M with respect to N , roughly speaking.A classification approach based on higher relative commutants were studied by several people such as Jones, Ocneanu, Pimsner and Popa. In this approach, the classification problem can be divided into the following 3 steps.(1) Prove that the subfactor can be approximated by certain increasing sequence of finite dimensional algebras called higher relative commutants.(2) Characterize the higher relative commutants in an axiomatic way. (3) Work on the axioms to classify higher relative commutants. Part (1) is quite functional analytic, but parts (2) and (3) are rather algebraic and combinatorial. We are concerned mainly with (2) and (3) in this paper.V. Jones noticed that some graph called "principal graph" appears naturally from the higher relative commutants. A. Ocneanu [O1] first claimed that Step (1) is possible in the case with certain finiteness condition called "finite depth," which means that the principal graph is finite. As to Step (2), he further obtained a complete combinatorial characterization of higher relative commutants for the finite depth case, found a new algebraic structure, and named it paragroup. It was known that if the Jones

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

“…(After the sumission of this paper, A. Ocneanu showed to the author the full proof along this suggested line in October, 1991. We will present his original proof in the appendix of [12]. )…”