Modular Invariants, Graphs and α-Induction for Nets of Subfactors. II

Abstract

We apply the theory of α-induction of sectors which we elaborated in our previous paper to several nets of subfactors arising from conformal field theory. The main application are conformal embeddings and orbifold inclusions of SU (n) WZW models. For the latter, we construct the extended net of factors by hand. Developing further some ideas of F. Xu, our treatment leads canonically to certain fusion graphs, and in all our examples we rediscover the graphs Di Francesco, Petkova and Zuber associated empirically to the corresponding SU (n) modular invariants. We establish a connection between exponents of these graphs and the appearance of characters in the block-diagonal modular invariants, provided that the extended modular S-matrices diagonalize the endomorphism fusion rules of the extended theories. This is proven for many cases, and our results cover all the block-diagonal SU (2) modular invariants, thus provide some explanation of the A-D-E classification.

Torus structure on graphs and twisted partition functions for minimal and affine models

“…It is also well known that the obtained minimal model is unitary if and only if the Coxeter numbers κ 1 and κ 2 of the two diagrams G (1) and G (2) just differ by one unit. The usual situation for minimal models corresponds therefore to the choice of the two trivial torus structures for the graphs G (1) and G (2) ; the possibility of replacing these two torus structures by more general ones (i.e., matrices W (1) 0,0 and W (2) 0,0 by matrices W (1) x 1 ,y 1 and W (2) x 2 ,y 2 ) leads to a natural classification of twisted partition functions for minimal models.…”

“…Our discussion of twisted partition functions for minimal models can be summarized as follows : to a pair (G (1) , G (2) ) of ADE Dynkin diagrams one can associate six types of sesquilinear forms on the space of Virasoro characters. These forms can be interpreted, in terms of minimal models, as partition functions in boundary conformal field theory with defects.…”

“…These forms can be interpreted, in terms of minimal models, as partition functions in boundary conformal field theory with defects. This classification rests on the possibility of introducing several "torus structures" for the two diagrams G (1) and G (2) . Torus structures are parametrized by elements of a particular base in the Ocneanu algebra of quantum symmetries; a torus structure may have a single twist, two twists, or no twist at all.…”

“…In our set-up, this affirmation can be precisely formulated as follows: the partition function of a minimal model of type (G (1) , G (2) ) can be obtained as the sesquilinear form associated with the matrix W (1) 0,0 ⊗W (2) 0,0 where these two matrices respectively describe the undeformed torus structures of diagrams G (1) and G (2) . It is also well known that the obtained minimal model is unitary if and only if the Coxeter numbers κ 1 and κ 2 of the two diagrams G (1) and G (2) just differ by one unit.…”

“…Analogues of minimal models for general Coxeter-Dynkin systems The general case of minimal models corresponds to the choice of two graphs of type SU (2) (i.e., two arbitrary Dynkin diagrams of type ADE) but one can also replace the two ADE diagrams G (1) and G (2) by members of an higher Coxeter-Dynkin system (for example the Di Francesco -Zuber diagrams of type SU (3)) and obtain in this way similar classifications. Here the notion of "minimal model" is generalized and the corresponding partition functions, twisted or not, can be interpreted in terms of minimal models for W n algebras (in particular W 3 for the Di Francesco -Zuber diagrams).…”