Abstract
We classify the automorphisms of the (chiral) level-k affineSU(3) fusion rules, for any value ofk, by looking for all permutations that commute with the modular matricesS andT. This can be done by using the arithmetic of the cyclotomic extensions where the problem is naturally posed. Whenk is divisible by 3, the automorphism group (∼Z 2) is generated by the charge conjugationC. Ifk is not divisible by 3, the automorphism group (∼Z 2×Z 2) is generated byC and the Altschüler-Lacki-Zaugg automorphism. Although the combinatorial analysis can become more involved, the techniques used here forSU(3) can be applied to other algebras.
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References
Cardy, J.: Operator content of two-diemnsional conformally invariant theories. Nucl. Phys. B270 [FS16], 186–204 (1986)
Moore, G., Seiberg, N.: Naturality in conformal field theory. Nucl. Phys.B313, 16–40 (1989)
Verlinde, E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys.B300 [FS22], 360–376 (1988)
Capelli, A., Itzykson, C., Zuber, J.B.: The A-D-E Classification of Minimal andA (1)1 Conformal Invariant Theories. Commun. Math. Phys.113, 1–26 (1987)
Capelli, A.: Modular invariant partition functions of superconformal theories. Phys. Lett.185B, 82–88 (1987)
Gepner, D., Qiu, Z.: Modular invariant partition functions for parafermionic theories. Nucl. Phys.B285 [FS19], 423–453 (1987)
Ginsparg, P.: Curiosities atc=1. Nucl. Phys.B295 [FS21], 153–170 (1988); Kiritsis, E.: Proof of the completeness of the classification of rational conformal theories withc=1. Phys. Lett.217B, 427–430 (1989)
Ruelle, P., Thiran, E., Weyers, J.: Modular Invariant Partition Functions for Affine\(\widehat{SU(3)}\) Theories at Prime Heights. Commun. Math. Phys.133, 305–322 (1990)
Ruelle, P., Thiran, E., Weyers, J.: Implications of an arithmetical symmetry of the commutant for modular invariants. Nucl. Phys.B402, 693–708 (1993)
Kac, V.: Infinite Dimensional Lie Algebras. Boston, Basel, Stuttgart: Birkhäuser 1983
Bégin, L., Mathieu, P., Walton, M.:\(\widehat{su(3)_k }\) fusion coefficients. Mod. Phys. Lett.A7, 3255–3266 (1992)
Bernard, D.: String characters from Kac-Moody automorphisms. Nucl. Phys.B288, 628–648 (1987)
Altschüler, D., Lacki, J., Zaugg, P.: The affine Weyl group and modular invariant partition functions. Phys. Lett.205B, 281–284 (1988)
Verstegen, D.: New exceptional modular invariant partition functions for simple Kac-Moody algebras. Nucl. Phys.B346, 349–386 (1990)
Washington, L.: Introduction to Cyclotomic Fields. Graduate Texts in Mathematics, Vol.83. Berlin, Heidelberg, New York: Springer 1982
Fuchs, J.: Simple WZW Currents. Commun. Math. Phys.136, 345–356 (1991)
Gannon, T.: The Classification of AffineSU(3) Modular Invariant Partition Functions. 1992
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Communicated by G. Felder
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Ruelle, P. Automorphisms of the affineSU(3) fusion rules. Commun.Math. Phys. 160, 475–492 (1994). https://doi.org/10.1007/BF02173425
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DOI: https://doi.org/10.1007/BF02173425