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

Robert Coquereaux ; Esteban Isasi ; Gil Schieber

Symmetry, integrability and geometry: methods and applications, Tome 6 (2010)

Résumé

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.

Détail

Keywords: quantum symmetries; module-categories; conformal field theories; $6j$ symbols.

Bibliographie

[1] Böhm G., Szlachányi K., “A coassociative $C^*$-quantum group with non-integral dimensions”, Lett. Math. Phys., 38 (1996), 437–456, arXiv: math.QA/9509008 | DOI | MR | Zbl

[2] Cappelli A., Itzykson C., Zuber J.-B., “The ADE classification of minimal and $A_1^{(1)}$ conformal invariant theories”, Comm. Math. Phys., 13 (1987), 1–26 | DOI | MR

[3] Carter J. S., Flath D. E., Saito M., The classical and quantum $6j$-symbols, Mathematical Notes, 43, Princeton University Press, Princeton, NJ, 1995 | MR

[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, arXiv: math-ph/0508002 | DOI | MR | Zbl

[5] Coquereaux R., “Racah–Wigner quantum $6j$ symbols, Ocneanu cells for $A_N$ diagrams and quantum groupoids”, J. Geom. Phys., 57 (2007), 387–434, arXiv: hep-th/0511293 | DOI | MR | Zbl

[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 pp., arXiv: math-ph/0610073 | DOI | MR | Zbl

[7] Coquereaux R., Schieber G., “Quantum symmetries for exceptional SU(4) modular invariants associated with conformal embeddings”, SIGMA, 5 (2009), 044, 31 pp., arXiv: 0805.4678 | DOI | MR | Zbl

[8] Di Francesco P., Matthieu P., Sénéchal D., Conformal field theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York, 1997 | MR

[9] Di Francesco P., Zuber J.-B., $\mathrm{SU}(N)$ Lattice integrable models associated with graphs, Nuclear Phys. B, 338 (1990), 602–646 | DOI | MR

[10] Di Francesco P., “Meander determinants”, Comm. Math. Phys., 191 (1998), 543–583, arXiv: hep-th/9612026 | DOI | MR | Zbl

[11] Evans D. E., Kawahigashi Y., Quantum symmetries on operator algebras, Oxford Mathematical Monographs, The Clarendon Press, New York; Oxford University Press, 1998 | MR

[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 | MR | Zbl

[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 | DOI | MR | Zbl

[14] Ewen H., Ogievetsky O., Classification of the GL(3) quantum matrix groups, arXiv: q-alg/9412009

[15] Gannon T., “The classification of affine SU(3) modular invariants”, Comm. Math. Phys., 161 (1994), 233–263, arXiv: hep-th/9212060 | DOI | MR | Zbl

[16] Gel'fand I. M., Ponomarev V. A., “Model algebras and representations of graphs”, Funktsional. Anal. i Prilozhen, 13:3 (1979), 1–12 (in Russian) | MR

[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, arXiv: hep-th/0412102 | DOI | MR | Zbl

[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, arXiv: math-ph/0609064 | DOI | MR | Zbl

[20] Jones V. F. R., “Index of subfactors”, Invent. Math., 72 (1983), 1–25 | DOI | MR | Zbl

[21] Kauffman L. H., “State models and the Jones polynomial”, Topology, 26 (1987), 395–407 | DOI | MR | Zbl

[22] Kauffman L. H., Lins S. L., Temperley–Lieb recoupling theory and invariant of 3-manifolds, Annals of Mathematics Studies, 134, Princeton University Press, Princeton, NJ, 1994 | MR | Zbl

[23] Kuperberg G., “Spiders for rank 2 Lie algebras”, Comm. Math. Phys., 180 (1996), 109–151, arXiv: q-alg/9712003 | DOI | MR | Zbl

[24] Lickorish W. B. R., “Calculations with the Temperley–Lieb algebra”, Comment. Math. Helv., 67 (1992), 571–591 | DOI | MR | Zbl

[25] Malkin A., Ostrik V., Vybornov M., “Quiver varieties and Lusztig's algebra”, Adv. Math., 203 (2006), 514–536, arXiv: math.RT/0403222 | DOI | MR | Zbl

[26] Ocneanu A., “Quantized group string algebras and Galois theory for algebras”, Operator Algebras and Applications (Warwick, 1987), v. 2, London Math. Soc. Lecture Note Ser., 136, Cambridge Univ. Press, Cambridge, 1988, 119–172 | MR

[27] Ocneanu A., Quantum symmetry, differential geometry of finite graphs and classification of subfactors (notes by Y. Kawahigashi), University of Tokyo Seminary Notes, 45, University of Tokyo, 1991

[28] Ocneanu A., 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 $\mathrm{SU}(N)$”, Quantum Symmetries in Theoretical Physics and Mathematics (Bariloche, 2000), Contemp. Math., 294, eds. R. Coquereaux, A. Garc\'{ı}a and R. Trinchero, Amer. Math. Soc., Providence, RI, 2002, 133–159 | MR | Zbl

[32] Ocneanu A., Private communication

[33] Ogievetsky O., “Uses of quantum spaces”, Quantum Symmetries in Theoretical Physics and Mathematics (Bariloche, 2000), Contemp. Math., 294, eds. R. Coquereaux, A. Garc\'{ı}a and R. Trinchero, Amer. Math. Soc., Providence, RI, 2002, 161–232, arXiv: hep-th/0006151 | MR | Zbl

[34] Ohtsuki T., Yamada S., “Quantum SU(3) invariant of 3-manifolds via linear skein theory”, J. Knot Theory Ramifications, 6 (1997), 373–404 | DOI | MR | Zbl

[35] Ostrik V., “Module categories, weak Hopf algebras and modular invariants”, Transform. Groups, 8 (2003), 177–206, arXiv: math.QA/0111139 | DOI | MR | Zbl

[36] Petkova V. B., Zuber J.-B., “The many faces of Ocneanu cells”, Nuclear Phys. B, 603 (2001), 449–496, arXiv: hep-th/0101151 | DOI | MR | Zbl

[37] Racah G., “Theory of complex spectra. II”, Phys. Rev., 62 (1942), 438–462 | DOI

[38] Reshetikhin N. Yu., Turaev V. G., “Ribbon graphs and their invariants derived from quantum groups”, Comm. Math. Phys., 127 (1990), 1–26 | DOI | MR | Zbl

[39] Rumer G., Teller E., Weyl H., “Eine für die Valenztheorie geeignete Basis der binären Vektorinvarianten”, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. I, 29 (1932), 499–504 | Zbl

[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 | MR

[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 | DOI | MR | Zbl

[43] Trinchero R., “Quantum symmetries of face models and the double triangle algebra”, Adv. Theor. Math. Phys., 10 (2006), 49–75, arXiv: hep-th/0501140 | MR | Zbl

[44] Wenzl H., “Hecke algebras of type $A_n$ and subfactors”, Invent. Math., 92 (1988), 349–383 | DOI | MR | Zbl