Presentations of Wess-Zumino-Witten Fusion Rings

Abstract

The fusion rings of Wess-Zumino-Witten models are re-examined. Attention is drawn to the difference between fusion rings over Z (which are often of greater importance in applications) and fusion algebras over C. Complete proofs are given characterising the fusion algebras (over C) of the SU(r+1) and Sp(2r) models in terms of the fusion potentials, and it is shown that the analagous potentials cannot describe the fusion algebras of the other models. This explains why no other representation-theoretic fusion potentials have been found. Instead, explicit generators are then constructed for general WZW fusion rings (over Z). The Jacobi-Trudy identity and its Sp(2r) analogue are used to derive the known fusion potentials. This formalism is then extended to the WZW models over the spin groups of odd rank, and explicit presentations of the corresponding fusion rings are given. The analogues of the Jacobi-Trudy identity for the spinor representations (for all ranks) are derived for this purpose, and may be of independent interest.

arXiv comment32 pages, 3 figures, added references, minor additions to text. To be published in Rev. Math. Phys

References in corpus (7)

• Dirichlet-Branes and Ramond-Ramond Charges
• Fusion rules in conformal field theory
• Topological Landau-Ginzburg Matter from Sp(N)_{K} Fusion Rings
• Generalized Fusion Potentials
• Symplectic fusion rings and their metric
• Fusion Potentials for G_k and Handle Squashing
• Integrable N = 2 Landau-Ginzburg Theories from Quotients of Fusion Rings

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• Twisted D-branes of the SU(N)_K WZW model and level-rank duality
• A new approach to twisted K-theory of compact Lie groups
• Affine Pieri rule for periodic Macdonald spherical functions and fusion rings
• Spectral Sequence Computation of Higher Twisted $K$-Groups of $SU(n)$