Fusion bases as facets of polytopes

Abstract

A new way of constructing fusion bases (i.e., the set of inequalities governing fusion rules) out of fusion elementary couplings is presented. It relies on a polytope reinterpretation of the problem: the elementary couplings are associated to the vertices of the polytope while the inequalities defining the fusion basis are the facets. The symmetry group of the polytope associated to the lowest rank affine Lie algebras is found; it has order 24 for $\su(2)$, 432 for $\su(3)$ and quite surprisingly, it reduces to 36 for $\su(4)$, while it is only of order 4 for $\sp(4)$. This drastic reduction in the order of the symmetry group as the algebra gets more complicated is rooted in the presence of many linear relations between the elementary couplings that break most of the potential symmetries. For $\su(2)$ and $\su(3)$, it is shown that the fusion-basis defining inequalities can be generated from few (1 and 2 respectively) elementary ones. For $\su(3)$, new symmetries of the fusion coefficients are found.Comment: Harvmac, 31 pages; typos corrected, symmetry analysis made more precise, conclusion expanded, and references adde

𝒩-point and higher-genus osp(1|2) fusion

“…Recently, efforts have been made to characterize fusion multiplicities in terms of polytopes. Most results so far pertain to three-point fusion [16,17], but also higher-genus and higher-point su(2) fusions have been discussed [9]. Below we shall extend the latter results to osp(1|2).…”

“…Most results so far pertain to three-point fusion, 16,17 but also higher-genus and higher-point su͑2͒ fusions have been discussed. 9 Below we shall extend the latter results to osp͑1͉2͒.…”

Fusion rules and the Patera-Sharp generating-function method

“…The reconstruction of the facets of a polytope from its vertices is thus another way to generate the fusion basis. This method is described in [16].…”