From modular invariants to graphs: the modular splitting method

Abstract

We start with a given modular invariant M of a two dimensional su(n) k conformal field theory (CFT) and present a general method for solving the Ocneanu modular splitting equation and then determine, in a step-by-step explicit construction, 1) the generalized partition functions corresponding to the introduction of boundary conditions and defect lines; 2) the quantum symmetries of the higher ADE graph G associated to the initial modular invariant M. Notice that one does not suppose here that the graph G is already known, since it appears as a by-product of the calculations. We analyze several su(3) k exceptional cases at levels 5 and 9.

Conformal embeddings and quantum graphs with self-fusion

“…Therefore we fix K and look for "overgroups" G such that the pair (K, G) is conformal. The fact that each such pair gives rise to a quantum subgroup of K results from investigations carried out more recently (in the last ten years) but we should stress than few of them have been worked out explicitly: only the SU (N ) cases with N = 2, 3, 4 are described (their associated graphs and algebras of quantum symmetries are known) in the available literature [23,4,26,24,5,14,15,6,7]. We always assume that G is simple.…”

Racah–Wigner quantum 6J symbols, Ocneanu cells for AN diagrams and quantum groupoids

“…The algebra of characters Oc(G), associated with the product • on B is called the Ocneanu algebra of G, or algebra of quantum symmetries. Its structure is very much case dependent and it is not always commutative (see [32] and [12] for members of the SU (2) family and [33], [13], [17], [21] for other results). When G = A , the situation is simple, since G = A(G) = Oc(G).…”

Comments about quantum symmetries of SU(3) graphs

“…It is possible to use the modular splitting equation to determine the toric matrices. This was certainly the road followed by A. Ocneanu but a general method of resolution was first described in [11], many more details and examples can be found in [27]. One starts from the modular splitting equation (23).…”

“…It is described (for a particular example) in one section of [11]. A detailed study of this method together with several SU (3) examples will be given in [27].…”

“…In particular we give in most cases an algebraic realization of Oc(G) that allows one to perform calculations without having to use the graph of quantum symmetries. A detailed study of several cases has already been made available in the litterature [13,47] and details concerning the others will be published elsewhere [25,27,24]. Several graphs are displayed in figures 4 and 5.…”