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Graphs and reflection groups

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Abstract

It is shown that graphs that generalize the ADE Dynkin diagrams and have appeared in various contexts of two-dimensional field theory may be regarded in a natural way as encoding the geometry of a root system. After recalling what are the conditions satisfied by these graphs, we define a bilinear form on a root system in terms of the adjacency matrices of these graphs and undertake the study of the group generated by the reflections in the hyperplanes orthogonal to these roots. Some “non-integrally laced” graphs are shown to be associated with subgroups of these reflection groups. The empirical relevance of these graphs in the classification of conformal field theories or in the construction of integrable lattice models is recalled, and the connections with recent developments in the context of\(\mathcal{N} = 2\) supersymmetric theories and topological field theories are discussed. A. Claude

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References

  1. Zuber, J.-B.: Conformal, Integrable and Topological Theories, Graphs and Coxeter Groups. In: Proceedings of the XIth International Conference of Mathematical Physics, Paris July 1994, D. Iagolnitzer ed., Boston: International Press, 1995

    Google Scholar 

  2. Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge: Cambridge Univ. Pr., 1990; Bourbaki, N.: Groupes et Algèbres de Lie. Chaps. 4–6, Paris: Masson, 1981

    Google Scholar 

  3. Coxeter, H.S.M.: Ann. Math.35, 588 (1934)

    MathSciNet  Google Scholar 

  4. Goodman, F.M., de la Harpe, P., Jones, V.F.R.: Coxeter Graphs and Towers of Algebras. Berlin, Heidelberg, New York: Springer, 1989

    Google Scholar 

  5. Verlinde, E.: Nucl. Phys.300, [FS22], 360 (1988)

    Article  Google Scholar 

  6. Kac, V.: Funct. Anal. Appl.3, 252 (1969); Moody, R.V., Patera, J.: SIAM J. Alg. Disc. Meth.5, 359 (1984) and further references therein

    Article  Google Scholar 

  7. Gepner, D.: Commun. Math. Phys.141, 381 (1991)

    Google Scholar 

  8. Di Francesco, P., Zuber, J.-B.: Nucl. Phys.B338, 602 (1990)

    Article  Google Scholar 

  9. Coxeter, H.S.M.: Duke Math. J.18, 765 (1951)

    Article  Google Scholar 

  10. Kazama, Y., Suzuki, H.: Phys. Lett.B216, 112 (1989); Nucl. Phys.B321, 232 (1989)

    Article  Google Scholar 

  11. Nemeschansky, D., Warner, N.P.: Nucl. Phys.B380, 241 (1992); Di Francesco, P., Lesage, F., Zuber, J.-B.: Nucl. Phys.B408, 600 (1993)

    Article  Google Scholar 

  12. Coxeter, H.S.M., Moser, W.O.J.: Generators and Relations for Discrete Groups. Berlin, Heidelberg, New York: Springer, 1957

    Google Scholar 

  13. Fuchs, J.: Fortschr. Phys.42, 1 (1994) and further references therein

    Google Scholar 

  14. Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of differentiable maps. Basel: Birkhäuser, 1985

    Google Scholar 

  15. A'Campo, N.: Math. Ann.213, 1 (1975)

    Article  Google Scholar 

  16. Warner, N.: Private communication

  17. Zuber, J.-B.: Mod. Phys. Lett.A8, 749 (1994)

    Article  Google Scholar 

  18. Pasquier, V.: J. Phys.A20, 5707 (1987)

    Google Scholar 

  19. Ocneanu, A.: In: Operator Algebras and Applications. Vol. 2, London Math. Soc. Lecture Notes Series, Vol. 136, London: Cambridge Univ. Press, 1988, pp. 119–172

    Google Scholar 

  20. Shcherbak, O.P.: Russ. Math. Surveys43:3, 149 (1988); Moody, R.V., Patera, J.: J. Phys. A26, 2829 (1993)

    Google Scholar 

  21. Christe, P., Ravanini, F.: Int. J. Mod. Phys.A4, 897 (1989)

    Article  Google Scholar 

  22. Pasquier, V.: Nucl. Phys.B285 [FS19], 162 (1987)

    Article  Google Scholar 

  23. Di Francesco, P., Zuber, J.-B.: In: Recent Developments in Conformal Field Theories. Trieste Conference, 1989, S. Randjbar-Daemi, E. Sezgin, J.-B. Zuber eds., Singapore: World Scientific 1990; Di Francesco, P.: Int. J. Mod. Phys.A7, 407–500 (1992)

    Google Scholar 

  24. Cappelli, A., Itzykson, C., Zuber, J.-B.: Nucl. Phys.B280, [FS], 445 (1987); Commun. Math. Phys.113, 1 (1987); Kato, A.: Mod. Phys. Lett.A2, 585 (1987)

    Article  Google Scholar 

  25. Itzykson, C.: Nucl. Phys. Proc. Suppl.58, 150 (1988)

    Article  Google Scholar 

  26. Gannon, T.: Commun. Math. Phys.161, 233 (1994); The classification ofSU(3) modular invariants revisited. hep-th 9404185, Ann. I.H.P., to appear

    Google Scholar 

  27. Petkova, V., Zuber, J.-B.: From CFT's to graphs. To appear in Nucl. Phys. B 463 (1996)

  28. Petkova, V., Zuber, J.-B.: Nucl. Phys. B.B438, 347 (1995)

    Article  Google Scholar 

  29. Bannai, E., Ito, T.: Algebraic Combinatorics I: Association schemes. Reading, MA: Benjamin-Cummings, 1984

    Google Scholar 

  30. Ocneanu, A.: Communication at the Workshop Low Dimensional Topology, Statistical Mechanics and Quantum Field Theory, Fields Institute, Waterloo, Ontario, April 26–30, 1995

  31. Martinec, E.: Phys. Lett.B217, 431 (1989); Criticality, catastrophes and compactifications. In: Physics and mathematics of strings. V.G. Knizhnik memorial volume, L. Brink, D. Friedan, A.M. Polyakov eds., Singapore: World Scientific, 1990; Vafa, C., Warner, N.P.: Phys. Lett.B218, 51 (1989); Lerche, W., Vafa, C., Warner, N.P.: Nucl. Phys.B324, 427 (1989)

    Article  Google Scholar 

  32. Cecotti, S., Vafa, C.: Commun. Math. Phys.158, 569 (1993)

    Google Scholar 

  33. Cecotti, S., Vafa, C.: Nucl. Phys.B367, 359 (1991); Dubrovin, B.: Commun. Math. Phys.152, 539 (1993)

    Article  Google Scholar 

  34. Fendley, P., Intriligator, K.: Nucl. Phys.B372, 533 (1992)

    Article  Google Scholar 

  35. Lerche, W., Warner, N.P.: In: Strings and Symmetries, 1991. N. Berkovits, H. Itoyama et al. (eds.) Singapore: World Scientific 1992; Intriligator, K.: Mod. Phys. Lett.A6, 3543 (1991)

    Google Scholar 

  36. Witten, E.: Nucl. Phys.B340, 281 (1990); Dijkgraaf, R., Verlinde, E., Verlinde, H.: Nucl. Phys.B352, 59 (1991); In: String Theory and Quantum Gravity. Proceedings of the 1990 Trieste Spring School, M. Green et al. (eds.) Singapore: World Sci. 1991

    Article  Google Scholar 

  37. Dubrovin, B.: Nucl. Phys.B379, 627 (1992); Differential Geometry of the space of orbits of a reflection group. hep-th/9303152

    Article  Google Scholar 

  38. Eguchi, T., Yamada, Y., Yang, S.-K.: On the genus expansion in topological string theory. Rev. Math. Phys.7, 279 (1995)

    Article  Google Scholar 

  39. Dubrovin, B.: Geometry of 2D Topological Field Theories. hep-th/9407018

  40. Vafa, C.: Mod. Phys. Lett.A6, 337 (1991)

    Article  Google Scholar 

  41. Dorey, P.: Int. J. Mod. Phys.8A, 193 (1993)

    Google Scholar 

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Authors and Affiliations

  1. Service de Physique Théorique de Saclay, CEA, F-91191, Gif sur Yvette Cedex, France

    J. -B. Zuber

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  1. J. -B. Zuber
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Communicated by R.H. Dijkgraaf

Dedicated to: A. Claude

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Zuber, J.B. Graphs and reflection groups. Commun.Math. Phys. 179, 265–293 (1996). https://doi.org/10.1007/BF02102590

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