Notes on TQFT Wire Models and Coherence Equations for SU(3) Triangular Cells

Abstract

Abstract. After a summary of the TQFT wire model formalism we bridge the gap from Kuperberg equations for SU (3) spiders to Ocneanu coherence equations for systems of triangular cells on fusion graphs that describe modules associated with the fusion category of SU (3) at level k. We show how to solve these equations in a number of examples.Key words: quantum symmetries; module-categories; conformal field theories; 6j symbols 2010 Mathematics Subject Classification: 81R50; 81R10; 20C08; 18D10 ForewordStarting with the collection of irreducible integrable representations (irreps) of SU (3) at some level k (constructed in the framework of affine algebras or in the framework of quantum groups at roots of unity), the problem is to decide whether a graph encoding the action of these irreps actually defines a "healthy fusion graph" associated with a bona fide SU (3) nimrep, i.e. a module over a particular kind of fusion category. This is done by associating a complex number (a "triangular cell") with every elementary triangle of the given graph in such a way that their collection (a "self-connection") obeys a system of non trivial quadratic and quartic equations, called Ocneanu coherence equations, that can be themselves derived from another set of equations (sometimes called Kuperberg equations) describing relations between the intertwiners of the underlying fusion category. One issue is to describe and derive the coherence equations themselves. Another issue is to use them on the family of examples giving rise to SU (3) nimreps. Both problems are studied in this article.Our paper consists of two largely independent parts. The first is a set of notes dealing with TQFT graphical models (wire models, spiders, etc.). Our motivation for this part was to show how the Ocneanu coherence equations for triangular cells of fusions graphs could be deduced from the so-called Kuperberg relations for SU (3), something that does not seem to be explained in the literature. This section was then enlarged in order to set the discussion in the larger framework of graphical TQFT models and to discuss some not so well known features that show up when comparing the SU (2) and SU (3) situations. Despite its size, this first part should not be considered as a general presentation of the subject starting from first principles, this would require a book, not an article. The second part of the paper deals about the coherence equations themselves. In particular we show, on a selection of examples chosen among quantum subgroups (module-categories) of type SU (3), how to solve them in order to obtain a self-connection on the corresponding fusion graphs.

Drinfeld Doubles for Finite Subgroups of SU(2) and SU(3) Lie Groups

“…At this place we just illustrate on one example the importance of the faithfulness requirement: the group Σ 168 × Z 3 has 18 irreps, six of them, actually three pairs of complex conjugates, labelled 4, 5, 6, 7, 8, 9 are of dimension 3, but the irreps of the pair (4, 5) are not faithful whereas the two pairs (6, 7) and (8,9) are. As it happens the fusion graphs associated with the faithful representations (namely those labeled 6,7,8,9,11,12,14,15,17,18) are connected; this is not so for the others, in particular for the 3-dimensional irreps labelled 4 and 5. So the natural (or embedding) irreps, with respect to SU(3), are 6, 7, 8, 9 and we may choose to draw the fusion graph of N 6 for instance (see Fig.…”

“…To probe equation (7), we have to look at groups possessing complex representations. In the case of SU(2) subgroups, equation (7) holds, and this was easy to check since only the cyclic and binary tetrahedral subgroups have complex representations. It was then natural to look at subgroups of SU(3) and we found that (7) holds true for most subgroups of SU(3) but fails for some subgroups like F = Σ 72×3 or L = Σ 360×3 .…”

“…In the case of SU(2) subgroups, equation (7) holds, and this was easy to check since only the cyclic and binary tetrahedral subgroups have complex representations. It was then natural to look at subgroups of SU(3) and we found that (7) holds true for most subgroups of SU(3) but fails for some subgroups like F = Σ 72×3 or L = Σ 360×3 . The second property (8) does not make sense for a finite group since there is no invertible S matrix, and Verlinde formula cannot be used.…”

“…• For each of the examples that we consider later, we shall also give the integer d B = m ( n,p N p mn ) 2 whose interpretation as the dimension of a weak Hopf algebra B (or double triangle algebra [40]) will not be discussed in this paper, see also [7,27,43,45].…”

“…Defining the charge conjugation matrix C = S 2 and the path matrix X = i N i , it is a standard fact that C.X.C = X. Property (7 ) reads instead…”