Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 6 (2010), 099, 44 pages      arXiv:1007.0721      https://doi.org/10.3842/SIGMA.2010.099

Notes on TQFT Wire Models and Coherence Equations for SU(3) Triangular Cells

Robert Coquereaux a, Esteban Isasi b and Gil Schieber a
a) Centre de Physique Théorique (CPT) Luminy, Marseille, France
b) Departamento de Física, Universidad Simón Bolívar, Caracas, Venezuela

Received July 09, 2010, in final form December 16, 2010; Published online December 28, 2010

Abstract
After a summary of the TQFT wire model formalism we bridge the gap from Kuperberg equations for SU(3) spiders to Ocneanu coherence equations for systems of triangular cells on fusion graphs that describe modules associated with the fusion category of SU(3) at level k. We show how to solve these equations in a number of examples.

Key words: quantum symmetries; module-categories; conformal field theories; 6j symbols.

pdf (805 Kb)   ps (899 Kb)   tex (960 Kb)

References

  1. Böhm G., Szlachányi K., A coassociative C*-quantum group with non-integral dimensions, Lett. Math. Phys. 38 (1996), 437-456, math.QA/9509008.
  2. Cappelli A., Itzykson C., Zuber J.-B., The ADE classification of minimal and A1(1) conformal invariant theories, Comm. Math. Phys. 13 (1987), 1-26.
  3. Carter J.S., Flath D.E., Saito M., The classical and quantum 6j-symbols, Mathematical Notes, Vol. 43, Princeton University Press, Princeton, NJ, 1995.
  4. Coquereaux R., Hammaoui D., Schieber G., Tahri E.H., Comments about quantum symmetries of SU(3) graphs, J. Geom. Phys. 57 (2006), 269-292, math-ph/0508002.
  5. Coquereaux R., Racah-Wigner quantum 6j symbols, Ocneanu cells for AN diagrams and quantum groupoids, J. Geom. Phys. 57 (2007), 387-434, hep-th/0511293.
  6. Coquereaux R., Schieber G., Orders and dimensions for sl(2) or sl(3) module categories and boundary conformal field theories on a torus, J. Math. Phys. 48 (2007), 043511, 17 pages, math-ph/0610073.
  7. Coquereaux R., Schieber G., Quantum symmetries for exceptional SU(4) modular invariants associated with conformal embeddings, SIGMA 5 (2009), 044, 31 pages, arXiv:0805.4678.
  8. Di Francesco P., Matthieu P., Sénéchal D., Conformal field theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York, 1997.
  9. Di Francesco P., Zuber J.-B., SU(N) Lattice integrable models associated with graphs, Nuclear Phys. B 338 (1990), 602-646.
    Di Francesco P., Zuber J.-B., SU(N) lattice integrable models and modular invariance, in Recents Developments in Conformal Field Theories (Trieste, 1989), Editors S. Randjbar-Daemi, E. Sezgin and J.-B. Zuber, World Sci. Publ., River Edge, NJ, 1990, 179-215.
  10. Di Francesco P., Meander determinants, Comm. Math. Phys. 191 (1998), 543-583, hep-th/9612026.
  11. Evans D.E., Kawahigashi Y., Quantum symmetries on operator algebras, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998.
  12. Evans D.E., Pugh M., Ocneanu cells and Boltzmann weights for the SU(3) ADE graphs, Müsnter J. Math. 2 (2009), 95-142, arXiv:0906.4307.
  13. Evans D.E., Pugh M., SU(3)-Goodman-de la Harpe-Jones subfactors and the realisation of SU(3) modular invariants, Rev. Math. Phys. 21 (2009), 877-928, arXiv:0906.4252.
  14. Ewen H., Ogievetsky O., Classification of the GL(3) quantum matrix groups, q-alg/9412009.
  15. Gannon T., The classification of affine SU(3) modular invariants, Comm. Math. Phys. 161 (1994), 233-263, hep-th/9212060.
  16. Gel'fand I.M., Ponomarev V.A., Model algebras and representations of graphs, Funktsional. Anal. i Prilozhen. 13 (1979), no. 3, 1-12 (in Russian).
  17. Hammaoui D., Schieber G., Tahri E.H., Higher Coxeter graphs associated to affine su(3) modular invariants, J. Phys. A: Math. Gen. 38 (2005), 8259-8286, hep-th/0412102.
  18. Hammaoui D., Géométrie quantique d'Ocneanu des graphes de Di Francesco-Zuber associés aux modèles conformes de type su(3), PhD Thesis,, LPTP, Université Mohammed I, Oujda, Maroc, 2007.
  19. Isasi E., Schieber G., From modular invariants to graphs: the modular splitting method, J. Phys. A: Math. Theor. 40 (2007), 6513-6537, math-ph/0609064.
  20. Jones V.F.R., Index of subfactors, Invent. Math. 72 (1983), 1-25.
  21. Kauffman L.H., State models and the Jones polynomial, Topology 26 (1987), 395-407.
  22. Kauffman L.H., Lins S.L., Temperley-Lieb recoupling theory and invariant of 3-manifolds, Annals of Mathematics Studies, Vol. 134, Princeton University Press, Princeton, NJ, 1994.
  23. Kuperberg G., Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 (1996), 109-151, q-alg/9712003.
  24. Lickorish W.B.R., Calculations with the Temperley-Lieb algebra, Comment. Math. Helv. 67 (1992), 571-591.
  25. Malkin A., Ostrik V., Vybornov M., Quiver varieties and Lusztig's algebra, Adv. Math. 203 (2006), 514-536, math.RT/0403222.
  26. Ocneanu A., Quantized group string algebras and Galois theory for algebras, in Operator Algebras and Applications, Vol. 2 (Warwick, 1987), London Math. Soc. Lecture Note Ser., Vol. 136, Cambridge Univ. Press, Cambridge, 1988, 119-172.
  27. Ocneanu A., Quantum symmetry, differential geometry of finite graphs and classification of subfactors (notes by Y. Kawahigashi), University of Tokyo Seminary Notes, Vol. 45, University of Tokyo, 1991.
  28. Ocneanu A., Quantum cohomology, quantum groupoïds and subfactors, Talk at First Caribean School of Mathematics and Theoretical Physics, Saint François, Guadeloupe, 1993.
  29. Ocneanu A., Talks given in various institutions between 1995 and 1999.
  30. Ocneanu A., Paths on Coxeter diagrams: from Platonic solids and singularities to minimal models and subfactors (notes by S. Goto), Fields Institute Monographs, Amer. Math. Soc., Providence, RI, 1999.
  31. Ocneanu A., The classification of subgroups of quantum SU(N), in Quantum Symmetries in Theoretical Physics and Mathematics (Bariloche, 2000), Editors R. Coquereaux, A. García and R. Trinchero, Contemp. Math., Vol. 294, Amer. Math. Soc., Providence, RI, 2002, 133-159.
  32. Ocneanu A., Private communication.
  33. Ogievetsky O., Uses of quantum spaces, in Quantum Symmetries in Theoretical Physics and Mathematics (Bariloche, 2000), Editors R. Coquereaux, A. García and R. Trinchero, Contemp. Math., Vol. 294, Amer. Math. Soc., Providence, RI, 2002, 161-232, hep-th/0006151.
  34. Ohtsuki T., Yamada S., Quantum SU(3) invariant of 3-manifolds via linear skein theory, J. Knot Theory Ramifications 6 (1997), 373-404.
  35. Ostrik V., Module categories, weak Hopf algebras and modular invariants, Transform. Groups 8 (2003), 177-206, math.QA/0111139.
  36. Petkova V.B., Zuber J.-B., The many faces of Ocneanu cells, Nuclear Phys. B 603 (2001), 449-496, hep-th/0101151.
  37. Racah G., Theory of complex spectra. II, Phys. Rev. 62 (1942), 438-462.
  38. Reshetikhin N.Yu., Turaev V.G., Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), 1-26.
  39. Rumer G., Teller E., Weyl H., Eine für die Valenztheorie geeignete Basis der binären Vektorinvarianten, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. I (1932), 499-504.
  40. Schieber G., L'algèbre des symétries quantiques d'Ocneanu et la classification des systèmes conformes à 2D, PhD Thesis, UP (Marseille) and UFRJ (Rio de Janeiro), 2003.
  41. Suciu L., The SU(3) wire model, PhD Thesis, Pennsylvania State University, 1997.
  42. Temperley H.N.V., Lieb E.H., Relations between the "percolation" and "colouring" problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the "percolation" problem, Proc. Roy. Soc. London Ser. A 322 (1971), 251-280.
  43. Trinchero R., Quantum symmetries of face models and the double triangle algebra, Adv. Theor. Math. Phys. 10 (2006), 49-75, hep-th/0501140.
  44. Wenzl H., Hecke algebras of type An and subfactors, Invent. Math. 92 (1988), 349-383.

Previous article   Next article   Contents of Volume 6 (2010)