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Subfactors and Conformal Field Theory

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Quantum and Non-Commutative Analysis

Part of the book series: Mathematical Physics Studies ((MPST,volume 16))

Abstract

We discuss relations between the combinatorial structure of subfactors, solvable lattice models, (rational) conformal field theory, and topological quantum field theory.

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References

  1. J. W. Alexandar, The combinatorial theory of complexes, Ann. Math. (2) 31 294–322, (1930).

    Google Scholar 

  2. G. E. Andrews, R. J. Baxter, & P. J. Forrester, Eight vertex SOS model and generalized Rogers-Ramanujan type identities, J. Stat. Phys. 35 193–266, (1984).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. H. Araki D. E. Evans, On a C’-algebra approach to phase transition in the two dimensional Ising model, Comm. Math. Phys. 106 241–266, (1986).

    Article  MathSciNet  Google Scholar 

  4. J. Bion-Nadal, An example of a subfactor of the hyperfinite IIi factor whose principal graph invariant is the Coxeter graph E6, in “Current Topics in Operator Algebras”,World Scientific Publishing, pp. 104–113, (1991).

    Google Scholar 

  5. B. Blackadar, Symmetries of the CAR algebra, Ann. Math. 131 589–623, (1990).

    Article  MathSciNet  MATH  Google Scholar 

  6. O. Bratteli, G. A. Elliott, D. E. Evans, & A. Kishimoto, Non-commutative spheres, Inter. J. Math. 2 139–166, (1991).

    Google Scholar 

  7. O. Bratteli, G. A. Elliott, D. E. Evans, & A. Kishimoto, Finite group actions on AF algebras obtained by folding the interval, (to appear in K-theory).

    Google Scholar 

  8. O. Bratteli, G. A. Elliott, D. E. Evans, & A. Kishimoto, Non-commutative spheres II: rational rotations, (to appear in J. Operator Theory).

    Google Scholar 

  9. O. Bratteli, D. E. Evans, & A. Kishimoto, Crossed products of totally disconnected spaces by Z2 *; Z2, preprint, Swansea, (1991).

    Google Scholar 

  10. O. Bratteli & A. Kishimoto, Non-commutative spheres III: irrational rotations, preprint, Trondheim, (1991).

    Google Scholar 

  11. J. de Boer & J. Goeree, Markov traces and IIi factors in conformal field theory, Comm. Math. Phys. 139 267–304, (1991).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. A. Cappelli, C. Itzykson, & J.-B. Zuber, The A-D-E classification of minimal and Al(1) conformal invariant theories, Comm. Math. Phys. 113 1–26, (1987).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. A. L. Carey & D. E. Evans, The operator algebras of the two dimensional Ising model, in “Braids” eds. J. Birman, A. Libgober, Comternp. Math. 78 117–165, (1988).

    Google Scholar 

  14. M. Choda, Duality for finite bipartite graphs, to appear in Pac. J. Math.

    Google Scholar 

  15. M. Choda & H. Kosaki, Strongly outer actions for inclusion of factors, preprint, (1992).

    Google Scholar 

  16. A. Connes, Sur la classification des facteurs de type II C. R. Acad. Sc. Paris 281 13–15, (1975).

    Google Scholar 

  17. A. Connes, Outer conjugacy classes of automorphisms of factors, Ann. Sci. École Norm. Sup. 8 383–419, (1975).

    Google Scholar 

  18. A. Connes, On the classification of von Neumann algebras and their automorphisms, Symposia Math. XX 435–478, (1976).

    Google Scholar 

  19. A. Connes D. E. Evans, Embeddings of U(1)-curient algebras in non-commutative algebras of classical statistical mechanics, Comm. Math. Phys. 121 507–525, (1989).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. A. Connes M. Takesaki, The flow of weights on factors of type III, Tohoku Math. J. 29, 473–555, (1977).

    Google Scholar 

  21. E. Date, M. Jimbo, T. Miwa, & M. Okado, Solvable lattice models, in “Theta functions — Bowdoin 1987, Part 1,” Proc. Sympos. Pure Math. Vol. 49, Amer. Math. Soc., Providence, R.I., pp. 295–332.

    Google Scholar 

  22. P. Di Francesco & J.-B. Zuber SU(N) lattice integrable models associated with graphs, Nucl. Phys. B338, 602–646, (1990).

    Google Scholar 

  23. P. Di Francesco J.-B. Zuber, SU(N) lattice integrable models and modular invariance, (to appear in Proc. Trieste Conf. Recent Developments in Conformal Field Theories, Trieste, 1989 ).

    Google Scholar 

  24. R. Dijkgraaf & E. Verlinde, Modular invariance and the fusion algebra, Proceedings of the Annecy Conference on Conformal Field Theory. Nucl. Phys. B(Proc. Suppl.) 5B 87–97, (1988).

    Google Scholar 

  25. R. Dijkgraaf & E. Witten, Topological gauge theories and group cohomology, Comm. Math. Phys. 129, 393–429, (1990).

    Google Scholar 

  26. L. Dixon, J. A. Harvey, C. Vafa, & E. Witten, Strings on orbifolds, Nucl. Phys. B261, 678–686, (1985); B274 285–314, (1986).

    Google Scholar 

  27. B. Durhuus, H. P. Jakobsen, & R. Nest, Topological quantum field theories from generalized 6j-symbols, preprint, (1991).

    Google Scholar 

  28. D. E. Evans, The C-algebras of topological Markov chains, Tokyo Metropolitan University Lecture Notes, (1984).

    Google Scholar 

  29. D. E. Evans, Quasi-product states on C*-algebras, in “Operator algebras and their connections with topology and ergodic theory”, Springer Lecture Notes in Math., 1132 129–151, (1985).

    Chapter  Google Scholar 

  30. D. E. Evans, Ce-algebraic methods in statistical mechanics and field theory, Inter. J. Mod. Phys. 4 1069–1118, (1990).

    Google Scholar 

  31. D. E. Evans & J. D. Gould, Embeddings and dimension groups of non-commutative AF algebras associated to models in classical statistical mechanics, preprint, Swansea.

    Google Scholar 

  32. D. E. Evans & Y. Kawahigashi, Orbifold subfactors from Hecke algebras, preprint, (1992).

    Google Scholar 

  33. D. E. Evans & Y. Kawahigashi, From subfactors to 3-dimensional topological quantum field theories and back, preprint, (1992).

    Google Scholar 

  34. D. E. Evans & Y. Kawahigashi, The E7 commuting squares produce D10 as principal graph, preprint, (1992).

    Google Scholar 

  35. D. E. Evans & J. T. Lewis, On a C*-algebra approach to phase transition in the two-dimensional Ising model II, Comm. Math. Phys. 102 521–535, (1986).

    Article  MathSciNet  Google Scholar 

  36. P. Fendley, New exactly solvable orbifold models, J. Phys. A22 4633–4642, (1989).

    MathSciNet  ADS  MATH  Google Scholar 

  37. P. Fendley & P. Ginsparg, Non-critical orbifolds, Nucl. Phys. B324 549–580, (1989).

    Article  Google Scholar 

  38. F. Goodman, & P. de la Harpe, V. F. R. Jones, “Coxeter graphs and towers of algebras”, MSRI publications 14, Springer, (1989).

    Google Scholar 

  39. M. Gross & S. Varsted, Elementary moves and ergodicity in D-dimensional simplicial quantum gravity, Nucl. Phys. B378 367–380, (1992).

    Article  ADS  Google Scholar 

  40. U. Haagerup, in preparation.

    Google Scholar 

  41. U. Haagerup E. Stormer, Pointwise inner automorphisms of von Neumann algebras (with an appendix by C. Sutherland), J. Funct. Anal. 92 177–201, (1990).

    Article  MathSciNet  Google Scholar 

  42. T. Hayashi, Quantum group symmetry of partition functions of IRF models and their application to Jones’ index theory, preprint, (1992).

    Google Scholar 

  43. M. Izumi, Application of fusion rules to classification of subfactors, Publ. RIMS Kyoto Univ. 27 953–994, (1991).

    Article  MathSciNet  MATH  Google Scholar 

  44. M. Izumi, On flatness of the Coxeter graph Es,to appear in Pac. J. Math.

    Google Scholar 

  45. M. Izumi & Y. Kawahigashi, Classification of subfactors with the principal graph D (1)n appear in J. Funct. Anal.

    Google Scholar 

  46. M. Jimbo, T, M.wa, & M. Okado, Solvable lattice models whose states are dominant integral weights of A(7P1, Lett. Math. Phys. 14 123–131, (1987).

    Google Scholar 

  47. V. F. R. Jones. Notes on Connes’ invariant x(M), unpublished.

    Google Scholar 

  48. V. F. R. Jones, Index for subfactors, Invent. Math. 72 1–15, (1983).

    MATH  Google Scholar 

  49. V. F. R. Jones, A polynomial invariant for knots via von Neumann algebras Bull. Amer. Math. Soc. 12 103–112, (1985).

    Article  MathSciNet  MATH  Google Scholar 

  50. A. Kato, Classification of modular invariant partition functions in two dimensions, Mod. Phys. Lett. A 2 585–600, (1987).

    Google Scholar 

  51. Y. Kawahigashi, Automorphisms commuting with a conditional expectation onto a subfactor with finite index, to appear in J. Operator Theory.

    Google Scholar 

  52. Y. Kawahigashi, On flatness of Ocneanu’s connections on the Dynkin diagrams and classification of subfactors, University of Tokyo, preprint, (1990).

    Google Scholar 

  53. Y. Kawahigashi, Exactly solvable orbifold models and subfactors, to appear in Proceedings of the Conference on Functional Analysis and Related Topics, Kyoto, 1991.

    Google Scholar 

  54. Y. Kawahigashi, Centrally trivial automorphisms and an analogue of Connes’ x(M) for subfactors, preprint, (1992).

    Google Scholar 

  55. Y. Kawahigashi, C. E. Sutherland, & M. Takesaki, The structure of the automorphism group of an injective factor and the cocycle conjugacy of discrete abelian grout actions, Acta Math. 169 105–130, (1992).

    Article  MathSciNet  MATH  Google Scholar 

  56. A. N. Kirillov & N. Yu. Reshetikhin, Representations of the algebra U 9 (s1 2 ), q-orthogonal polynomials and invariants for links, in “Infinite dimensional Lie algebras and groups” (V. G. Kac, ed.), Adv. Ser. in Math. Phys., vol. 7, pp. 285–339, (1988).

    Google Scholar 

  57. H. Kosaki, Automorphisms in irreducible decompositions of sectors, in this volume.

    Google Scholar 

  58. R. Kuik, On the q-state Potts model by means of non-commutative algebras, Thesis Groningen, (1986).

    Google Scholar 

  59. A. Kumjian, An involutive automorphism of the Bunce-Deddens algebra, CR Math. Rep. Acad. Sci. Canada 10 217–218, (1988).

    MathSciNet  MATH  Google Scholar 

  60. J. T. Lewis, P. N. M. Sisson, A C*-algebra of the 2-dimensional Ising model, Comm. Math. Phys. 44 279–292, (1975).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  61. P. H. Loi, On automorphisms of subfactors, preprint, (1990).

    Google Scholar 

  62. R. Longo, Simple injective subfactors, Adv. Math. 63 152–171, (1987).

    Article  MathSciNet  MATH  Google Scholar 

  63. G. Moore & N. Seiberg, Classical and quantum conformal field theory, Comm. Math. Phys. 123 177–254, (1989).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  64. A. Ocneanu, Quantized group, string algebras and Galois theory for algebras, in “Operator algebras and applications, Vol. 2 (Warwick, 1987),” London Math. Soc. Lect. Note Series Vol. 136, Cambridge University Press, pp. 119–172, (1988).

    Google Scholar 

  65. A. Ocneanu, “Graph geometry, quantized groups and nonamenable subfactors”, Lake Tahoe Lectures, June—July, (1989).

    Google Scholar 

  66. A. Ocneanu, “Quantum symmetry, differential geometry of finite graphs and classification of subfactors”, University of Tokyo Seminary Notes 45, (Notes recorded by Y. Kawahigashi), (1991).

    Google Scholar 

  67. A. Ocneanu, An invariant coupling between 3-manifolds and subfactors, with connections to topological and conformal quantum field theory, unpublished announcement, (1991).

    Google Scholar 

  68. S. Okamoto, Invariants for subfactors arising from Coxeter graphs, in “Current Topics in Operator Algebras,” World Scientific Publishing, pp. 84–103, (1991).

    Google Scholar 

  69. V. Pasquier, Operator content of the ADE lattice models, J. Phys. A. Math. Gen. 20 5701–5717, (1987).

    Article  ADS  Google Scholar 

  70. S. Pirogov, States associated with the two dimensional Ising model, Theor. Math. Phys. 11 (3) 614–617, (1972).

    Article  MathSciNet  Google Scholar 

  71. S. Popa, Classification of amenable subfactors of type II, IRES preprint, (1992).

    Google Scholar 

  72. S. Popa, Classification of actions of discrete amenable groups on amenable subfactors of of type II, IRES preprint, (1992).

    Google Scholar 

  73. Ph. Roche, Ocneanu cell calculus and integrable lattice models, Comm. Math. Phys. 127 395–424, (1990).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  74. J. Schou, Commuting squares and index for subfactors, Ph. D. Thesis, Odense University, 1990.

    Google Scholar 

  75. V. S. Sunder, A model for AF-algebras and a representation of the Jones projections, J. Operator Theory 18 289–301, (1987).

    MathSciNet  MATH  Google Scholar 

  76. V. S. Sunder & A. K. Vijayarajan, On the non-occurrence of the Coxeter graphs ß2n+1, E7, D2n+1 as principal graphs of an inclusion of II1 factors, (to appear in Pac. J. Math.).

    Google Scholar 

  77. C. E. Sutherl & M. Takesaki, Actions of discrete amenable groups on injective factors of type IIIa, A # 1, Pac. J. Math. 137 405–444, (1989).

    Article  MathSciNet  MATH  Google Scholar 

  78. V. G. Turaev & O. Y. Viro, State sum invariants of 3-manifolds and quantum 6j-symbols, preprint.

    Google Scholar 

  79. M. Wakui, On Dijkgraaf-Witten invariant for 3-manifolds, (to appear in Osaka J. Math.).

    Google Scholar 

  80. H. Wenzl, Hecke algebras of type A n and subfactors, Invent. Math. 92 345–383, (1988).

    Article  MathSciNet  ADS  Google Scholar 

  81. C. Winslow, Approximately inner automorphisms on inclusions of type IIIa factors, preprint, (1992).

    Google Scholar 

  82. E. Witten, Gauge theories and integrable lattice models, Nucl. Phys. B322 629–697, (1989).

    Article  MathSciNet  ADS  Google Scholar 

  83. F. Xu, Orbifold construction in subfactors, preprint, (1992).

    Google Scholar 

  84. S. Yamagami, A note on Ocneanu’s approach to Jones’ index theory, preprint, (1992).

    Google Scholar 

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Author information

Authors and Affiliations

  1. University of Wales, Swansea, SA2 8PP, Wales

    David E. Evans

  2. University of Tokyo, Tokyo, 113, Japan

    Yasuyuki Kawahigashi

Authors
  1. David E. Evans
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  2. Yasuyuki Kawahigashi
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Editor information

Editors and Affiliations

  1. Research Institute for Mathematical Sciences, Kyoto University, 606-01, Sakyo-ku, Kyoto, Japan

    Huzihiro Araki

  2. Department of Mathematics and Information, College of Liberal Arts, Kyoto University, Sakyo-ku, Yoshida Kyoto 606, Japan

    Keiichi R. Ito

  3. Department of Mathematics Faculty of Science, Hokkaido University, Sapporo 060, Japan

    Akitaka Kishimoto

  4. Research Institute for Mathematical Sciences, Kyoto University, 606-01, Kyoto, Japan

    Izumi Ojima

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© 1993 Springer Science+Business Media Dordrecht

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Evans, D.E., Kawahigashi, Y. (1993). Subfactors and Conformal Field Theory. In: Araki, H., Ito, K.R., Kishimoto, A., Ojima, I. (eds) Quantum and Non-Commutative Analysis. Mathematical Physics Studies, vol 16. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2823-2_28

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  • DOI: https://doi.org/10.1007/978-94-017-2823-2_28

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4334-4

  • Online ISBN: 978-94-017-2823-2

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