Comments on the Links Between su(3) Modular Invariants, Simple Factors in the Jacobian of Fermat Curves, and Rational Triangular Billiards
Les rencontres physiciens-mathématiciens de Strasbourg -RCP25, Tome 48 (1997), Exposé no. 1, 56 p. 
@article{RCP25_1997__48__1_0,
     author = {Bauer, M. and Coste, A. and Itzykson, C. and Ruelle, P.},
     title = {Comments on the {Links} {Between} $su(3)$ {Modular} {Invariants,} {Simple} {Factors} in the {Jacobian} of {Fermat} {Curves,} and {Rational} {Triangular} {Billiards}},
     journal = {Les rencontres physiciens-math\'ematiciens de Strasbourg -RCP25},
     note = {talk:1},
     pages = {1--56},
     publisher = {Institut de Recherche Math\'ematique Avanc\'ee - Universit\'e Louis Pasteur},
     volume = {48},
     year = {1997},
     language = {en},
     url = {http://www.numdam.org/item/RCP25_1997__48__1_0/}
}
TY  - JOUR
AU  - Bauer, M.
AU  - Coste, A.
AU  - Itzykson, C.
AU  - Ruelle, P.
TI  - Comments on the Links Between $su(3)$ Modular Invariants, Simple Factors in the Jacobian of Fermat Curves, and Rational Triangular Billiards
JO  - Les rencontres physiciens-mathématiciens de Strasbourg -RCP25
N1  - talk:1
PY  - 1997
SP  - 1
EP  - 56
VL  - 48
PB  - Institut de Recherche Mathématique Avancée - Université Louis Pasteur
UR  - http://www.numdam.org/item/RCP25_1997__48__1_0/
LA  - en
ID  - RCP25_1997__48__1_0
ER  - 
%0 Journal Article
%A Bauer, M.
%A Coste, A.
%A Itzykson, C.
%A Ruelle, P.
%T Comments on the Links Between $su(3)$ Modular Invariants, Simple Factors in the Jacobian of Fermat Curves, and Rational Triangular Billiards
%J Les rencontres physiciens-mathématiciens de Strasbourg -RCP25
%Z talk:1
%D 1997
%P 1-56
%V 48
%I Institut de Recherche Mathématique Avancée - Université Louis Pasteur
%U http://www.numdam.org/item/RCP25_1997__48__1_0/
%G en
%F RCP25_1997__48__1_0
Bauer, M.; Coste, A.; Itzykson, C.; Ruelle, P. Comments on the Links Between $su(3)$ Modular Invariants, Simple Factors in the Jacobian of Fermat Curves, and Rational Triangular Billiards. Les rencontres physiciens-mathématiciens de Strasbourg -RCP25, Tome 48 (1997), Exposé no. 1, 56 p. http://www.numdam.org/item/RCP25_1997__48__1_0/

[1] A. Cappelli, C. Itzykson and J.-B. Zuber, The A-D-E classification of minimal and A 1 (1) conformal invariant theories, Commun. Math. Phys. 113 (1987) 1-26. | MR | Zbl

A. Kato, Classification of modular invariant partition functions in two dimensions, Mod. Phys. Lett. A2 (1987) 585-600. | MR

[2] T. Gannon, The Classification of affine su(3) modular invariant partition functions, Commun. Math. Phys. 161 (1994) 233-264. | MR | Zbl

[3] T. Gannon, WZW commutants, lattices, and level-one partition functions, Nucl. Phys. B396 (1993) 708-736. | MR

[4] P. Ruelle, E. Thiran and J. Weyers, Implications of an arithmetical symmetry of the commutant for modular invariants, Nucl. Phys. B402 (1993) 693-708. | MR | Zbl

[5] A. Coste and T. Gannon, Remarks on Galois symmetry in rational conformal field theories, Phys. Lett. B323 (1994) 316-321. | MR

[6] N. Koblitz and D. Rohrlich, Simple factors in the Jacobian of a Fermat curve, Can. J. Math. XXX (1978) 1183-1205. | MR | Zbl

[7] E. Aurell and C. Itzykson, Rational billiards and algebraic curves, J. Geom. and Phys. 5 (1988) 191-208. | MR | Zbl

[8] Contributions by P. Cohen, J. Wolfart, M. Bauer and C. Itzykson in The Grothendieck theory of dessins d'enfants, L. Schneps ed., LMSLNS 200, Cambridge Univ. Press.

[9] V.G. Kac, Infinite dimensional Lie algebras, 3rd edition, Cambridge University Press, Cambridge 1990. | MR | Zbl

[10] C. Itzykson and J.-M. Drouffe, Théorie statistique des champs, editions du CNRS, Paris 1989.

[11] V.G. Kac and M. Wakimoto, Modular and conformal invariance constraints in representation theory of affine algebras, Adv. Math. 70 (1988) 156-236. | MR | Zbl

[12] E. Witten, Non-Abelian bosonization in two dimensions, Commun. Math. Phys. 92 (1984) 455-472. | Zbl

[13] P. Goddard and D. Olive, Kac-Moody and Virasoro algebras in relation to quantum physics, Int. J. Mod. Phys. A1 (1983) 303-414. | MR | Zbl

[14] G. Moore and N. Seiberg, Naturality in conformal field theory, Nucl. Phys. B313 (1989) 16-40. | MR

[15] J. Cardy, The operator content of two-dimensional conformally invariant theories, Nucl. Phys. B270 (1986) 186-204. | MR | Zbl

[16] W. Nahm, Lie group exponents and SU(2) current algebras, Commun. Math. Phys. 118 (1988) 171-176. | MR | Zbl

[17] P. Di Francesco and J.-B. Zuber, SU(N) lattice integrable models associated with graphs, Nucl. Phys. B338 (1990) 602-646. | MR

[18] T. Gannon, The level two and three modular invariants of SU(n), preprint (November 1995). | MR | Zbl

[19] T. Gannon, Towards a classification of su(2)⊕...⊕su(2) modular invariant partition functions, J. Math. Phys. 36 (1995) 675-706. | MR | Zbl

[20] T. Gannon, P. Ruelle and M. Walton, Automorphism modular invariants of current algebras, Commun. Math. Phys. 179 (1996) 121-156. | MR | Zbl

[21] T. Gannon, Kac-Peterson, Perron-Frobenius, and the classification of conformal field theories, preprint (q-alg 9510026).

[22] M. Bauer and C. Itzykson, Modular transformations of SU(N) affine characters and their commutant, Commun. Math. Phys. 127 (1990) 617-636. | MR | Zbl

[23] P. Ruelle, Dimension of the commutant for the SU(N) affine algebras, Commun.Math. Phys. 133 (1990) 181-196. | MR | Zbl

[24] E. Buffenoir, A. Coste, J. Lascoux, P. Degiovanni and A. Buhot, Precise study of some number fields and Galois actions occurring in conformal field theory, Ann. Inst. Poincaré, Theor. Phys. 63 (1995) 41-79. | EuDML | Numdam | MR | Zbl

[25] C. Vafa, Toward classification of conformal theories, Phys. Lett. 206B (1988) 421-426. | MR

G. Anderson and G. Moore, Rationality in conformal field theory, Commun. Math. Phys. 117 (1988) 441-450. | MR | Zbl

[26] J. De Boer and J. Goeree, Markov traces and II1 factors in conformal field theory, Commun. Math. Phys. 139 (1991) 267-304. | MR | Zbl

[27] P. Ruelle, E. Thiran and J. Weyers, Modular invariants for affine SU ^(3) theories at prime heights, Commun. Math. Phys. 133 (1990) 305-322. | MR | Zbl

[28] J. Fuchs, B. Gato-Rivera, B. Schellekens and C. Schweigert, Modular invariants and fusion rule automorphisms from Galois theory, Phys. Lett. B334 (1994) 113-120. | MR

[29] J. Fuchs, B. Schellekens and C. Schweigert, Galois modular invariants of WZW models, Nucl. Phys. B437 (1995) 667-694. | MR | Zbl

30] J.-B. Bost, Les Houches lectures "Introduction to compact Riemann surfaces, Jacobians and Abelian varieties", in From Number Theory to Physics, edited by M. Waldschmidt, P. Moussa, J.-M. Luck and C. Itzykson, Springer 1990. | MR | Zbl

[31] H.P.F. Swinnerton-Dyer, Analytic theory of Abelian varieties, Cambridge University Press, Cambridge 1974. | MR | Zbl

[32] S. Lang, Introduction to algebraic and Abelian functions, GTM 89, Springer 1982. | MR | Zbl

[33] N. Aoki, Simple factors of the Jacobian of a Fermat curve and the Picard number of a product of Fermat curves, Amer. J. Math. 113 (1991) 779-833. | MR | Zbl

[34] D. Rohrlich, appendix to B. Gross, On the periods of Abelian integrals and a formula of Chowla and Seiberg, Invent. Math. 45 (1978) 193-211. | EuDML | MR | Zbl

[35] A. Weil, Sur les périodes d'intégrales abéliennes, Commun. Pure and Appl. Math. XXIX (1976) 813-819. | MR | Zbl

[36] S. Lang, Complex multiplication, Springer 1983. | MR | Zbl

[37] G. Shimura and Y. Taniyama, Complex multiplication of Abelian varieties and its applications to number theory, Publ. Math. Soc. Jap., no. 6, 1961. | MR | Zbl

[38] N. Koblitz, Gamma function identities and elliptic differentials on Fermat curves, Duke Math. J. 45 (1978) 87-99. | MR | Zbl

[39] A. Grothendick, Esquisse d'un programme. | Zbl

[40] A.W. Knapp, Elliptic curves, Princeton Univ. Press, Princeton 1992. | MR | Zbl

[41] Arithmetic and Geometry of Fermat Curves Proceedings of the Algebraic Geometry Seminar, Singapore (1987).

[42] P.J. Richens and M.V. Berry, Pseudo-integrable systems in classical and quantum mechanics, Physica 2D (1981) 495-512. | MR | Zbl

[43] D. Gepner, E. Witten, String theory on group manifolds, Nucl. Phys. B278 (1986) 493-549. | MR

[44] R.C. Gunning, Lectures on modular forms Annals of Mathematics Studies 48, Princeton Univ. Press, Princeton 1992. | MR | Zbl

[45] G.A. Jones and D. Singerman, Complex functions, Cambridge University Press, Cambridge 1987. | MR | Zbl

[46] C.L. Siegel, Tropics in complex function theory, Vol. II, Wiley & Sons 1971. | MR | Zbl

[47] M. Bauer, unpublished.

[48] D.J. Benson, Polynomial invariants of finite groups, LMSLNS 190, Cambridge Univ. Press, Cambridge 1993. | MR | Zbl

[49] G.C. Shephard and J.A. Todd, Finite unitary reflection groups, Can. J. Math. VI (1954) 274-304. | MR | Zbl