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

Abstract

We relate quantum 6J symbols of various types (quantum versions of Wigner and Racah symbols) to Ocneanu cells associated with A N Dynkin diagrams. We check explicitly the algebraic structure of the associated quantum groupoids and analyze several examples. Some features relative to cells associated with more general ADE diagrams are also discussed.

Topological defects in open string field theory

“…This object however differs from the 6J symbol by a tetrahedral invariant normalization factor (see discussion in section 2.2.2), and resembles thus an object often called T ET in the literature, see e.g. [77,78]. Another difference would arise in the case of oriented defects, where the tetrahedral invariance of the defect network is broken.…”

“…What Coquearaux[78] calls geometrical Racah symbols, or Carter et al[77] call the 6j symbol are our F -matrices.…”

From modular invariants to graphs: the modular splitting method

“…General results have been published on this quantum groupoïd (see [18,7,10,17,21]). But we are not aware of any definite list of properties that the graphs G should satisfy to obtain the right classification.…”

“…Following the works of [18], it was shown that to every modular invariant of a 2d CFT one can associate a special kind of quantum groupoïd B(G), constructed from the combinatorial and modular data [13] of a graph G [23,7,26,28,10]. This quantum groupoïd B(G) plays a central role in the classification of 2d CFT, since it also encodes information on the theory when considered in various environments (not only on the bulk but also with boundary conditions and defect lines): the corresponding generalized partition functions are expressed in terms of a set of non-negative integer coefficients that can be determined from associative properties of structural maps of B(G) [1,30,24,26].…”

Alternative formulation for the operator algebra over the space of paths in a ADE $SU(3)$ graph

“…In one of these fields, conformal field theory, the relation is established via the well known ADE classification of the modular invariant partition functions of su(N ) WZW Conformal Field Theories (CFT). The formal structure that lies behind this classification has been developed over time and can be studied thoroughly in [1], [2], [3], [4], [5], [6], [7].…”