Affine su(3) and su(4) fusion multiplicities as polytope volumes

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

Section: Resultsmentioning

confidence: 84%

“…Affine fusion rules give the number of integrable representationsν that appear in the product of two integrable representationsλ andμ for a given affine algebra g at fixed level k (see e.g., [1] Chapter 16). Fusions are in fact truncated finite Lie algebra tensor products, with the degree of truncation fixed solely by the level.…”

Section: Introductionmentioning

confidence: 99%

Fusion bases as facets of polytopes

Bégin

Cummins

Lapointe

et al. 2002

“…It is known how to do that for su(Nр4) 14,15 and has been explored further in Ref. 16. To the BZ osp͑1͉2͒ triangle ͑11͒ we may assign the threshold level…”

Section: ͑40͒mentioning

confidence: 99%

“…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.…”

Section: Aϩbϩcϫ1р2k ͑43͒mentioning

confidence: 99%

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

Rasmussen

2003

Articles you may be interested inMinimal unitary representation of S U ( 2 , 2 ) and its deformations as massless conformal fields and their supersymmetric extensions J. Math. Phys. 51, 082301 (2010) We study affine osp͑1͉2͒ fusion, the fusion in osp͑1͉2͒ conformal field theory, for example. Higher-point and higher-genus fusion is discussed. The fusion multiplicities are characterized as discretized volumes of certain convex polytopes, and are written explicitly as multiple sums measuring those volumes. We extend recent methods developed to treat affine su͑2͒ fusion. They are based on the concept of generalized Berenstein-Zelevinsky triangles and virtual couplings. Higher-point tensor products of finite-dimensional irreducible osp͑1͉2͒ representations are also considered. The associated multiplicities are computed and written as multiple sums.

“…The problem of finding this direct sum decomposition is important in many parts of physics. 15 These tensor products are also strongly related to affine sû͑3͒ k fusions, 5,13 which originate from conformal field theory. 14,16 This decomposition could also be used to find decompositions in the asymptotic limit, 10,11 which are used in the study of the noncompact rigid rotor algebra.…”

Section: Introductionmentioning

confidence: 99%

A geometric description of tensor product decompositions in su(3)

Wesslén

2008

The direct sum decomposition of tensor products for su͑3͒ has many applications in physics, and the problem has been studied extensively. This has resulted in many decomposition methods, each with its advantages and disadvantages. The description given here is geometric in nature and it describes both the constituents of the direct sum and their multiplicities. In addition to providing decompositions of specific tensor products, this approach is very well suited to studying tensor products as the parameters vary and helping draw general conclusions. After a description and proof of the method, several consequences are discussed and proved. In particular, questions regarding multiplicities are considered.