$\widehat{{\rm su}}\left( 3 \right)_k $ FUSION COEFFICIENTS

Bégin, Luc; Mathieu, Pierre; Walton, Mark A.

doi:10.1142/s0217732392002640

Cited by 36 publications

(46 citation statements)

References 2 publications

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“…Furthermore, our approach seems amenable to the treatment of higher rank su(r + 1) fusions, whereas an application of the Verlinde formula appears technically very complicated. We are currently considering such an extention of our approach based on previous results on the role of BZ triangles in affine su(3) and su(4) fusions [8,9]. A different approach to fusion based on the depth rule and the correspondence to three-point functions in Wess-Zumino-Witten conformal field theory may be found in our recent work [10,11].…”

Section: Commentsmentioning

confidence: 99%

Fusion multiplicities as polytope volumes: -point and higher-genus su(2) fusion

Rasmussen

Walton

2002

Nuclear Physics B

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We present the first polytope volume formulas for the multiplicities of affine fusion, the fusion in Wess-Zumino-Witten conformal field theories, for example. Thus, we characterise fusion multiplicities as discretised volumes of certain convex polytopes, and write them explicitly as multiple sums measuring those volumes. We focus on su(2), but discuss higher-point (N > 3) and higher-genus fusion in a general way. The method follows that of our previous work on tensor product multiplicities, and so is based on the concepts of generalised Berenstein-Zelevinsky diagrams, and virtual couplings. As a by-product, we also determine necessary and sufficient conditions for non-vanishing higher-point fusion multiplicities. In the limit of large level, these inequalities reduce to very simple non-vanishing conditions for the corresponding tensor product multiplicities. Finally, we find the minimum level at which the higher-point fusion and tensor product multiplicities coincide.

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Section: Commentsmentioning

confidence: 99%

Fusion multiplicities as polytope volumes: -point and higher-genus su(2) fusion

Rasmussen

Walton

2002

Nuclear Physics B

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“…to any given su(3) BZ triangle (7). This means that the fusion multiplicity may be described by supplementing the tensor product conditions (16) by the affine condition…”

Section: Su(3) Fusion Multiplicitiesmentioning

confidence: 99%

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

Rasmussen

Walton

2002

J. Phys. A: Math. Gen.

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Affine su(3) and su(4) fusion multiplicities as polytope volumes Jørgen Rasmussen 1 and Mark A. Walton 2 Physics Department, University of Lethbridge, Lethbridge, Alberta, Canada T1K 3M4 AbstractAffine su(3) and su(4) fusion multiplicities are characterised as discretised volumes of certain convex polytopes. The volumes are measured explicitly, resulting in multiple sum formulas. These are the first polytope-volume formulas for higher-rank fusion multiplicities. The associated threshold levels are also discussed. For any simple Lie algebra we derive an upper bound on the threshold levels using a refined version of the Gepner-Witten depth rule.1 rasmussj@cs.uleth.ca; supported in part by a PIMS Postdoctoral Fellowship and by NSERC 2 walton@uleth.ca; supported in part by NSERC

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“…While this manuscript was being completed, I received the preliminary version of a preprint by T. Gannon which contains the full classification of SU (3) modular invariant partition functions [17]. In particular, it also contains for SU (3), by using the results of [11], the classification of automorphisms shown here.…”

Section: Perspectivesmentioning

confidence: 99%

“…This is indeed possible for SU (3), by using their explicit expressions, obtained recently in [11]. It would however definitely confine us to SU (3) since the fusion coefficients for higher ranks are not known.…”

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confidence: 98%

Automorphisms of the affineSU(3) fusion rules

Ruelle

1994

Commun.Math. Phys.

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We classify the automorphisms of the (chiral) level-k affine SU (3) fusion rules, for any value of k, by looking for all permutations that commute with the modular matrices S and T . This can be done by using the arithmetic of the cyclotomic extensions where the problem is naturally posed. When k is divisible by 3, the automorphism group (∼ Z 2 ) is generated by the charge conjugation C. If k is not divisible by 3, the automorphism group (∼ Z 2 × Z 2 ) is generated by C and the Altschüler-Lacki-Zaugg automorphism. Although the combinatorial analysis can become more involved, the techniques used here for SU (3) can be applied to other algebras. * e-mail address: ruelle@stp.dias.ie 1. Introduction.

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$\widehat{{\rm su}}\left( 3 \right)_k $ FUSION COEFFICIENTS

Cited by 36 publications

References 2 publications

Fusion multiplicities as polytope volumes: -point and higher-genus su(2) fusion

Fusion multiplicities as polytope volumes: -point and higher-genus su(2) fusion

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

Automorphisms of the affineSU(3) fusion rules

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