Towards a Classification of Fusion Rule Algebras in Rational Conformal Field Theories

Abstract

We review the main topics concerning Fusion Rule Algebras (FRA) of Rational Conformal Field Theories. After an exposition of their general properties, we examine known results on the complete classification for low number of fields (≤ 4). We then turn our attention to FRA's generated polynomially by one (real) fundamental field, for which a classification is known. Attempting to generalize this result, we describe some connections between FRA's and Graph Theory. The possibility to get new results on the subject following this "graph" approach is briefly discussed.

Nonsemisimple Fusion Algebras and the Verlinde Formula

“…The category of representations of W(p) in the (1, p) model decomposes into linkage classes of representations, which are full subcategories of the representation category. 3 The representation category of the algebra W(p) associated with the (1, p) model has p + 1 linkage classes; we denote them as LC, LC ′ , and LC(s) with 1 ≤ s ≤ p−1. The indecomposable representations in each linkage class are as follows.…”

“…The indecomposable representations in each linkage class are as follows. The classes LC and LC ′ contain only a single indecomposable (hence, irreducible) 3 The term "linkage class" is borrowed from the theory of finite-dimensional Lie algebras. The linkage classes of an additive category C are additive full subcategories C i such that there are no (nonzero) morphisms between objects in two distinct linkage classes, every object of C is a direct sum of objects of the linkage classes, and none of the C i can be split further in the same manner.…”

Fusion potentials. I

“…The general study of these fusion algebras [3] and their classification have been the object of much work [4]. In particular, the possibility that they may be represented by sets of polynomials has been considered [4].…”

PATH SPACES AND $\mathcal W$ FUSION IN MINIMAL MODELS

“…2 , the Chebychev polynomials of the second kind (defined by p 0 = 1, p 1 (x) = x, p n+2 (x) = xp n+1 (x) − p n (x)) evaluated at the adjacency matrix T p ′ −1 2 of the tadpole graph A p ′ −1 /Z Z 2 [30]. Moreover these polynomials are generators of the fusion ring of the (2, p ′ )-Virasoro minimal model.…”