Physics of Particles and Nuclei

, Volume 48, Issue 4, pp 509–563 | Cite as

Quantum groups as generalized gauge symmetries in WZNW models. Part I. The classical model

  • L. Hadjiivanov
  • P. Furlan


Wess–Zumino–Novikov–Witten (WZNW) models over compact Lie groups G constitute the best studied class of (two dimensional, 2D) rational conformal field theories (RCFTs). A WZNW chiral state space is a finite direct sum of integrable representations of the corresponding affine (current) algebra, and the correlation functions of primary fields are monodromy invariant combinations of left times right sector conformal blocks solving the Knizhnik–Zamolodchikov equation. However, even in this very well understood case of 2D RCFT, the “internal” (gauge) symmetry that governs the ensuing fusion rules remains unclear. On the other hand, the canonical approach to the classical chiral WZNW theory developed by Faddeev, Alekseev, Shatashvili, Gawedzki and Falceto reveals its Poisson–Lie symmetry. After a covariant quantization, the latter gives rise to an associated quantum group symmetry which naturally requires an extension of the state space. This paper contains a review of earlier work on the subject with a special emphasis, in the case G = SU(n), on the emerging chiral “WZNW zero modes” which provide an adequate algebraic description of the internal symmetry structure of the model. Combining further left and right zero modes, one obtains a specific dynamical quantum group, the structure of its Fock representation resembling the axiomatic approach to gauge theories in which a “restricted” quantum group plays the role of a generalized gauge symmetry.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Yu. Alekseev and L. D. Faddeev, “(T* G)t: A Toy model for conformal field theory”, Commun. Math. Phys. 141, 413–422 (1991).ADSMathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    A. Yu. Alekseev and L. D. Faddeev, “An involution and dynamics for the -deformed top”, Zap. Nauchn. Sem. POMI, 1992, Vol. 200, pp. 3–16 [in Russian]Google Scholar
  3. 2a.
    A. Yu. Alekseev and L. D. Faddeev, J. Math. Sci. 77, 3137–3145 (1995); arXiv:hep-th/9406196).MathSciNetCrossRefGoogle Scholar
  4. 3.
    A. Yu. Alekseev, L. D. Faddeev, and M. A. Semenov-Tian-Shansky, “Hidden quantum groups inside Kac–Moody algebras”, Commun. Math. Phys. 149, 335–345 (1992).ADSMathSciNetMATHCrossRefGoogle Scholar
  5. 4.
    A. Alekseev and S. Shatashvili, “From geometric quantization to conformal field theory”, Commun. Math. Phys. 128, 197–212 (1990).ADSMathSciNetMATHCrossRefGoogle Scholar
  6. 5.
    A. Alekseev and S. Shatashvili, “Quantum groups and WZNW models”, Commun. Math. Phys. 133, 353–368 (1990).ADSMathSciNetMATHCrossRefGoogle Scholar
  7. 6.
    A. Yu. Alekseev and I. T. Todorov, “Quadratic brackets from symplectic forms”, Nucl. Phys. B 421, 413–428 (1994), arXiv:hep-th/9307026.ADSMathSciNetMATHCrossRefGoogle Scholar
  8. 7.
    L. Alvarez-Gaumé, C. Gomez, and G. Sierra, “Hidden quantum symmetries in rational conformal field theories”, Nucl. Phys. B 319, 155–186 (1989)ADSMathSciNetMATHCrossRefGoogle Scholar
  9. 7a.
    L. Alvarez-Gaumé, C. Gomez, and G. Sierra, “Quantum group interpretation of some conformal field theories”, Phys. Lett. B 220, 142–152 (1989)ADSMathSciNetMATHCrossRefGoogle Scholar
  10. 7b.
    L. Alvarez-Gaumé, C. Gomez, and G. Sierra, “Duality and quantum groups”, Nucl. Phys. B 330, 347–398 (1990).ADSMathSciNetMATHCrossRefGoogle Scholar
  11. 8.
    H. H. Andersen, “Tensor products of quantized tilting modules”, Commun. Math. Phys. 149, 149–159 (1992).ADSMathSciNetMATHCrossRefGoogle Scholar
  12. 9.
    H. H. Andersen and J. Paradowski, “Fusion categories arising from semisimple Lie algebras”, Commun. Math. Phys. 169, 563–588 (1995).ADSMathSciNetMATHCrossRefGoogle Scholar
  13. 10.
    H. Araki, “Indecomposable representations with invariant inner product. A theory of the Gupta-Bleuler triplet”, Commun. Math. Phys. 97, 149–159 (1985).ADSMathSciNetMATHCrossRefGoogle Scholar
  14. 11.
    D. Arnaudon, “Composition of kinetic momenta: The case”, Commun. Math. Phys. 159, 175–194 (1994), arXiv:hep-th/9212067.ADSMathSciNetMATHCrossRefGoogle Scholar
  15. 12.
    V. I. Arnold, “On the braids of algebraic functions and the cohomologies of swallow tails”, Uspekhi Mat. Nauk 23 (4), 247–248 (1968).MathSciNetGoogle Scholar
  16. 12a.
    V. I. Arnold, “The cohomology ring of the colored braid group”, Math. Notes 5, 138–140 (1969).CrossRefGoogle Scholar
  17. 12b.
    V. I. Arnold, “On some topological invariants of algebraic functions”, Trans. Moscow Math. Soc. 21, 30–52 (1970).Google Scholar
  18. 13.
    E. Artin, “Theorie der Zöpfe”, Abh. Math. Sem. Univ. Hamburg 4, 47–72 (1925).MathSciNetMATHCrossRefGoogle Scholar
  19. 13a.
    E. Artin, “Theory of braids”, Ann. Math., Part 2 48, 101–126 (1947).MathSciNetMATHCrossRefGoogle Scholar
  20. 14.
    L. Atanasova, P. Furlan, and L. Hadjiivanov, “Zero modes of the SU(2)k Wess–Zumino–Novikov–Witten model in Euler angles parametrization”, J. Phys. A 37, 5329–5339 (2004), arXiv:hep-th/0311170.ADSMathSciNetMATHCrossRefGoogle Scholar
  21. 15.
    O. Babelon, “Extended conformal algebra and the Yang-Baxter equation”, Phys. Lett. B 215, 523–529 (1988).ADSMathSciNetCrossRefGoogle Scholar
  22. 16.
    O. Babelon, “Universal exchange algebra for Bloch waves and Liouville theory”, Commun. Math. Phys. 139, 619–649 (1991).ADSMathSciNetMATHCrossRefGoogle Scholar
  23. 17.
    B. Bakalov and A. Kirillov, “Lectures on tensor categories and modular functors”, in University Lecture, Ser. V (AMS, Providence, RI, 2001), Vol. 21.Google Scholar
  24. 18.
    J. Balog, L. Dabrowski, and L. Fehér, “Classical r-matrix and exchange algebras in WZNW and Toda theories”, Phys. Lett. B 244, 227–234 (1990).ADSMathSciNetMATHCrossRefGoogle Scholar
  25. 19.
    J. Balog, L. Fehér, and L. Palla, “The chiral WZNW phase space and its Poisson-Lie groupoid”, Phys. Lett. B 463, 83–92 (1999), arXiv:hep-th/9907050.ADSMathSciNetMATHCrossRefGoogle Scholar
  26. 20.
    J. Balog, L. Fehér, and L. Palla, “Chiral extensions of the WZNW phase space, Poisson–Lie symmetries and groupoids”, Nucl. Phys. B 568, 503–542 (2000), arXiv:hep-th/9910046.ADSMathSciNetMATHCrossRefGoogle Scholar
  27. 21.
    J. Balog, L. Fehér, and L. Palla, “Classical Wakimoto realizations of chiral WZNW Bloch waves”, J. Phys. A 33, 945–956 (2000), arXiv:hep-th/9910112.ADSMathSciNetMATHCrossRefGoogle Scholar
  28. 22.
    J. Balog, L. Fehér, and L. Palla, “The chiral WZNW phase space as a quasi-Poisson space”, Phys. Lett. A 277, 107–114, arXiv:hep-th/0007045.Google Scholar
  29. 23.
    V. Bargmann, “On the representations of the rotation group”, Rev. Mod. Phys. 34, 829–845 (1962).ADSMathSciNetMATHCrossRefGoogle Scholar
  30. 24.
    C. Becchi, A. Rouet, and R. Stora, “The Abelian Higgs–Kibble model. Unitarity of the S-operator”, Phys. Lett. B 52, 344–346 (1974).ADSCrossRefGoogle Scholar
  31. 25.
    C. Becchi, A. Rouet, and R. Stora, “Renormalization of gauge theories”, Ann. Phys. 98, 287–321 (1976).ADSMathSciNetCrossRefGoogle Scholar
  32. 26.
    A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, “Infinite conformal symmetry in two-dimensional quantum field theory”, Nucl. Phys. B 241, 333–380 (1984).ADSMathSciNetMATHCrossRefGoogle Scholar
  33. 27.
    L. Biedenharn, E. Lieb, B. Simon, and F. Wilczek, “The ancestry of the “anyon”, Letter, Physics Today, Part 1, p. 90 (1990).CrossRefGoogle Scholar
  34. 28.
    W. Bishara, “Non-abelian quantum Hall states and fractional statistics”, Caltech PhD Thesis, ID Code: 1759 (2009). thesis.pdf.Google Scholar
  35. 29.
    B. Blok, “Classical exchange algebra in the Wess–Zumino–Witten model”, Phys. Lett. B 233, 359–362 (1989).ADSMathSciNetMATHCrossRefGoogle Scholar
  36. 30.
    J. Blom and E. Langmann, “Finding and solving Calogero–Moser type systems using Yang–Mills gauge theories”, Nucl. Phys. B 563, 506–532 (1999), arXiv:math-ph/9909019.ADSMathSciNetMATHCrossRefGoogle Scholar
  37. 31.
    N. N. Bogoliubov, A. A. Logunov, A. I. Oksak, and I. T. Todorov, General Principles of Quantum Field Theory (Kluwer, Dordrecht, 1990).MATHCrossRefGoogle Scholar
  38. 32.
    R. E. Borcherds, “Vertex algebras, Kac–Moody algebras, and the monster”, Proc. Natl. Acad. Sci. USA 83, 3068–3071 (1986).ADSMathSciNetMATHCrossRefGoogle Scholar
  39. 33.
    G. Böhm and K. Szlachányi, “A coassociative -quantum group with non-integral dimensions”, Lett. Math. Phys. 38, 437–456 (1996), arXiv:qalg/ 9509008MathSciNetMATHCrossRefGoogle Scholar
  40. 33a.
    G. Böhm, F. Nill, and K. Szlachányi, “Weak Hopf algebras, I: Integral theory and -structure”, J. Algebra 221, 385–438 (1999), arXiv:math.QA/9805116.MathSciNetMATHCrossRefGoogle Scholar
  41. 33b.
    G. Böhm and K. Szlachányi, “Weak Hopf algebras, II: Representation theory, dimensions and the Markov trace”, J. Algebra 223, 156–212 (2000), arXiv:math.QA/9906045.MathSciNetMATHCrossRefGoogle Scholar
  42. 34.
    D. Buchholz, “Algebraic quantum theory: A status report”, in XIIIth ICMP (London, 2000), arXiv:mathph/ 0011044.Google Scholar
  43. 35.
    D. Buchholz and R. Haag, “The quest for understanding in relativistic quantum physics”, J. Math. Phys. 41, 3674–3697 (2000), arXiv:hep-th/9910243.ADSMathSciNetMATHCrossRefGoogle Scholar
  44. 36.
    D. Buchholz and J. E. Roberts, “New light on infrared problems: sectors, statistics, symmetries and spectrum”, Commun. Math. Phys. 330, 935–972 (2014), arXiv:1304.2794[math-ph].ADSMathSciNetMATHCrossRefGoogle Scholar
  45. 37.
    A. G. Bytsko and L. D. Faddeev, “(T*B)q q-analogue of model space and the CGC generating matrices”, J. Math. Phys. 37, 6324–6348 (1996), arXiv:qalg/ 9508022.ADSMathSciNetMATHCrossRefGoogle Scholar
  46. 38.
    M. Cahen, S. Gutt, and J. Rawnsley, “Some remarks on the classification of Poisson-Lie groups”, Contemp. Math. 179, 1–16 (1994).MathSciNetMATHCrossRefGoogle Scholar
  47. 39.
    F. E. Camino, W. Zhou, and V. J. Goldman, “Observation of Aharonov–Bohm superperiod in a Laughlin quasiparticle interferometer”, Phys. Rev. Lett. 95, 246802 (2005), arXiv:cond-mat/0504341.ADSCrossRefGoogle Scholar
  48. 40.
    L. Caneschi and M. Lysiansky, “Chiral quantization of the WZW SU(n) model”, Nucl. Phys. B 505, 701–726 (1997), arXiv:hep-th/9605099.ADSMathSciNetMATHCrossRefGoogle Scholar
  49. 41.
    A. Cappelli, L. S. Georgiev, and I. T. Todorov, “A unified conformal field theory description of paired quantum Hall states”, Commun. Math. Phys. 205, 657–689 (1999), arXiv:hep-th/9810105.ADSMathSciNetMATHCrossRefGoogle Scholar
  50. 42.
    A. Cappelli, L. S. Georgiev, and I. T. Todorov, “Parafermion Hall states from coset projections of Abelian conformal theories”, Nucl. Phys. B 599, 499–530 (2001), arXiv:hep-th/0009229.ADSMathSciNetMATHCrossRefGoogle Scholar
  51. 43.
    A. Cappelli, C. Itzykson, and J.-B. Zuber, “Modular invariant partition functions in two dimensions”, Nucl. Phys. B 280[FS], 445–465 (1987).ADSMathSciNetMATHCrossRefGoogle Scholar
  52. 44.
    A. Cappelli, C. Itzykson, and J.-B. Zuber, “The classification of minimal and A 1(1) conformal invariant theories”, Commum. Math. Phys. 113, 1–26 (1987).ADSMathSciNetMATHCrossRefGoogle Scholar
  53. 45.
    S. Carpi, Y. Kawahigashi, R. Longo, and M. Weiner, “From vertex operator algebras to conformal nets and back”, arXiv:1503.01260[math.OA].Google Scholar
  54. 46.
    V. Chari and A. Pressley, A Guide to Quantum Groups (Cambridge University Press, 1994).MATHGoogle Scholar
  55. 47.
    P. Christe and R. Flume, “The four-point correlations of all primary operators of the d = 2 conformally invariant SU(2) sigma model with Wess–Zumino term”, Nucl. Phys. B 282, 466–494 (1987).ADSCrossRefGoogle Scholar
  56. 48.
    M. Chu, P. Goddard, I. Halliday, D. Olive, and A. Schwimmer, “Quantisation of the Wess-Zumino-Novikov-Witten model on a circle”, Phys. Lett. B 266, 71–81 (1991).ADSMathSciNetMATHCrossRefGoogle Scholar
  57. 48a.
    M. Chu and P. Goddard, “Quantisation of the SU(n) WZW model at level k”, Nucl. Phys. B 445, 145–168 (1995), arXiv:hep-th/9407116.ADSMathSciNetMATHCrossRefGoogle Scholar
  58. 49.
    S. Coleman, “Quantum sine-Gordon equation as the massive Thirring model”, Phys. Rev. D 11, 2088–2097 (1975).ADSCrossRefGoogle Scholar
  59. 50.
    R. Coquereaux and R. Trinchero, “On quantum symmetries of ADE graphs”, Adv. Theor. Math. Phys. 8, 189–216 (2004).MathSciNetMATHCrossRefGoogle Scholar
  60. 51.
    C. Crnkovic and E. Witten, “Covariant description of canonical formalism in geometrical theories”, in Three Hundred Years of Gravitation, Ed. by S. W. Hawking and W. Israel (Cambridge Univ. Press, Cambridge, 1987), pp. 676–684.Google Scholar
  61. 52.
    P. Dedecker, “Calcul des variations, formes différentielles et champs géodésiques”, Colloque International de Géometrie Différentielle (Publications C.N.R.S., Strasbourg, 1953).MATHGoogle Scholar
  62. 53.
    P. Dedecker, “On the generalization of symplectic geometry to multiple integrals in the calculus of variations”, in Differential Geometrical Methods in Mathematical Physics, Proceedings of the Symposium, Bonn, July 1–4, 1975, Lect. Notes Math. 570, 395–456 (1977).MathSciNetMATHGoogle Scholar
  63. 54.
    R. De-Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, and D. Mahalu, “Direct observation of a fractional charge”, Nature 389, 162–164 (1997).ADSCrossRefGoogle Scholar
  64. 55.
    P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer, New York, 1997).MATHCrossRefGoogle Scholar
  65. 56.
    P. A. M. Dirac, “Generalized Hamiltonian dynamics”, Canad. J. Math. 2, 129–148 (1950).MathSciNetMATHCrossRefGoogle Scholar
  66. 57.
    S. Doplicher, R. Haag, and J. E. Roberts, “Local observables and particle statistics, I”, Commun. Math. Phys. 23, 199–230 (1971).ADSMathSciNetCrossRefGoogle Scholar
  67. 57a.
    S. Doplicher, R. Haag, and J. E. Roberts, “Local observables and particle statistics, II”, Commun. Math. Phys. 35, 49–85 (1974).ADSMathSciNetCrossRefGoogle Scholar
  68. 58.
    S. Doplicher and J. E. Roberts, “A new duality theory for compact groups”, Inv. Math. 98, 157–218 (1989).ADSMathSciNetMATHCrossRefGoogle Scholar
  69. 58a.
    S. Doplicher and J. E. Roberts, “Endomorphisms of -algebras, cross products and duality for compact groups”, Ann. Math. 130, 75–119 (1989).MathSciNetMATHCrossRefGoogle Scholar
  70. 58b.
    S. Doplicher and J. E. Roberts, “Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics”, Commun. Math. Phys. 131, 51–107 (1990).ADSMathSciNetMATHCrossRefGoogle Scholar
  71. 59.
    V. S. Dotsenko and V. A. Fateev, “Conformal algebra and multipoint correlation functions 2D in statistical model”, Nucl. Phys. B 240, 312–348 (1984).ADSCrossRefGoogle Scholar
  72. 60.
    S. Doty, “New versions of Schur-Weyl duality,” in Proceedings of the Gainesville Conference on Finite Groups, March 6–12, 2003, Ed. by Chat Yin Ho et al., De Gruyter Proceedings in Mathematics (Walter de Gruyter, Berlin, New York, 2004), pp. 59–71; arXiv:0704.1877[math.RT].Google Scholar
  73. 61.
    V. G. Drinfeld, “Hamiltonian structures on Lie groups, Lie bialgebras, and the geometric meaning of the Yang–Baxter equation”, Sov. Math. Dokl. 27, 68–70 (1983).Google Scholar
  74. 62.
    V. G. Drinfeld, “Hopf algebra and the quantum Yang–Baxter equation”, Dokl. Akad. Nauk SSSR 283, 1060–1064 (1985).MathSciNetGoogle Scholar
  75. 62a.
    V. G. Drinfeld, “Quantum Groups”, in Proc. of the International Congress of Mathematicians (Academic Press, Berkeley, 1986), Vol. 1, pp. 798–820.Google Scholar
  76. 63.
    V. G. Drinfeld, “Quasi-Hopf algebras”, Leningrad Math. J. 1, 1419–1457 (1990).MathSciNetGoogle Scholar
  77. 64.
    M. Dubois-Violette, P. Furlan, L. K. Hadjiivanov, A. P. Isaev, P. N. Pyatov, and I. T. Todorov, “A finite dimensional gauge problem in the WZNW model”, in Quantum Theory and Symmetries, Proc. International Symp. Goslar (Germany, 1999), Ed. by H.-D. Doebner and V. Dobrev (World Scientific, Singapore, 2000), pp. 331–349, arXiv:hep-th/9910206.Google Scholar
  78. 65.
    M. Dubois-Violette and I. T. Todorov, “Generalized cohomologies and the physical subspace of the SU(2) WZNW model”, Lett. Math. Phys. 42, 183–192 (1997), arXiv:hep-th/9704069MathSciNetMATHCrossRefGoogle Scholar
  79. 65a.
    M. Dubois-Violette and I. T. Todorov, “Generalized homologies for the zero modes of the WZNW model”, Lett. Math. Phys. 48, 323–338 (1999), arXiv:math.QA/9905071.MathSciNetMATHCrossRefGoogle Scholar
  80. 66.
    P. Etingof and D. Nikshych, “Dynamical quantum groups at roots of 1”, Duke Math J. 108, 135–168 (2001), arXiv:math.QA/0003221.MathSciNetMATHCrossRefGoogle Scholar
  81. 67.
    P. Etingof, D. Nikshych, and V. Ostrik, “On fusion categories”, arXiv:math.QA/0203060.Google Scholar
  82. 68.
    P. Etingof and V. Ostrik, “Finite tensor categories”, Moscow Math. J. 4 (3), 627–654 (2004), arXiv:math.QA/0301027.MathSciNetMATHGoogle Scholar
  83. 69.
    P. Etingof and A. Varchenko, “Geometry and classification of solutions of the classical dynamical Yang–Baxter equation”, Commun. Math. Phys. 192, 77–120 (1998), arXiv:q-alg/9703040.ADSMathSciNetMATHCrossRefGoogle Scholar
  84. 70.
    P. Etingof and A. Varchenko, “Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups”, Commun. Math. Phys. 196, 591–640 (1998), arXiv:q-alg/9708015.ADSMathSciNetMATHCrossRefGoogle Scholar
  85. 71.
    P. Etingof and A. Varchenko, “Exchange dynamical quantum groups”, Commun. Math. Phys. 205, 19–52 (1999), arXiv:q-alg/9801135.ADSMathSciNetMATHCrossRefGoogle Scholar
  86. 72.
    L. D. Faddeev, “On the exchange matrix for WZNW model”, Commun. Math. Phys. 132, 131–138 (1990).ADSMathSciNetMATHCrossRefGoogle Scholar
  87. 73.
    L. D. Faddeev, “Quantum symmetry in conformal field theory by Hamiltonian methods”, in New Symmetry Principles in Quantum Field Theory, Proceedings (Cargèse,1991), Ed. by J. Fröhlich et al. (Plenum Press, New York, 1992), pp. 159–175.CrossRefGoogle Scholar
  88. 74.
    L. D. Faddeev, N. Yu. Reshetikhin, and L. A. Takhtajan, “Quantization of Lie groups and Lie algebras”, Leningrad Math. J. 1, 193–225 (1990).MathSciNetMATHGoogle Scholar
  89. 75.
    F. Falceto and K. Gawedzki, Quantum Group Symmetries in WZW Models (Bures-sur-Yvette, I.H.E.S., 1991)MATHGoogle Scholar
  90. 75a.
    F. Falceto and K. Gawedzki, “On quantum group symmetries in conformal field theories”, in XXth International Congress on Differential Geometric Methods in Theoretical Physics (New York, 1991), arXiv:hep-th/9109023.Google Scholar
  91. 76.
    F. Falceto and K. Gawedzki, “Lattice Wess–Zumino–Witten model and quantum groups”, J. Geom. Phys. 11, 251–279 (1993), arXiv:hepth/ 9209076.ADSMathSciNetMATHCrossRefGoogle Scholar
  92. 77.
    L. Fehér and A. Gábor, “On interpretations and constructions of classical dynamical R-matrices”, in Quantum Theory and Symmetries, Ed. by E. Kapuscik et al. (World Scientific, Singapore, 2002), pp. 331–336, arXiv:hep-th/0111252.CrossRefGoogle Scholar
  93. 78.
    B. L. Feigin, A. M. Gainutdinov, A. M. Semikhatov, and I. Yu. Tipunin, “Modular group representations and fusion in LCFT and in the quantum group center”, Commun. Math. Phys. 265, 47–93 (2006), arXiv:hep-th/0504093.ADSMATHCrossRefGoogle Scholar
  94. 79.
    B. L. Feigin, A. M. Gainutdinov, A. M. Semikhatov, and I. Yu. Tipunin, “Kazhdan–Lusztig correspondence for the representation category of the triplet W-algebra in logarithmic CFT”, Theor. Math. Phys. 148, 1210–1235 (2006), arXiv:math.QA/0512621MATHCrossRefGoogle Scholar
  95. 80.
    B. L. Feigin, A. M. Gainutdinov, A. M. Semikhatov, and I. Yu. Tipunin, “Logarithmic extensions of minimal models: characters and modular transformations”, Nucl. Phys. B 757, 303–343 (2006), arXiv:hep-th/0606196.ADSMathSciNetMATHCrossRefGoogle Scholar
  96. 81.
    B. L. Feigin, A. M. Gainutdinov, A. M. Semikhatov, and I. Yu. Tipunin, “Kazhdan–Lusztig-dual quantum group for logarithmic extensions of Virasoro minimal models”, J. Math. Phys. 48, 032303 (2007), arXiv:math.QA/0606506.ADSMathSciNetMATHCrossRefGoogle Scholar
  97. 82.
    G. Felder, “Conformal field theory and integrable systems associated to elliptic curves,” in Proceedings of the International Congress of Mathematicians, Zürich, Switzerland, 1994, Ed. by S. D. Chatterji (Birkhäuser, Basel, 1995), pp. 1247–1255; arXiv:hep-th/9407154Google Scholar
  98. 82a.
    G. Felder, “Elliptic quantum groups,” in Proceedings of the 11th International Congress of Mathematical Physics, Paris, France, July 18–23, 1994, Ed. by D. Iagolnitzer (International, Cambridge, MA, 1995), pp. 211–218; arXiv:hep-th/9412207.Google Scholar
  99. 83.
    R. Ferrari, L. E. Picasso, and F. Strocchi, “Some remarks on local operators in quantum electrodynamics”, Commun. Math. Phys. 35, 25–38 (1974).ADSMathSciNetCrossRefGoogle Scholar
  100. 84.
    M. Fierz, “Über die relativistische Theorie kräftefreier Teilchen mit beliebigem Spin”, Helv. Phys. Acta 12, 3–37 (1939).MATHCrossRefGoogle Scholar
  101. 85.
    M. Finkelberg, “Fusion categories”, Ph.D. Thesis (Harvard University, 1993).MATHGoogle Scholar
  102. 86.
    M. Finkelberg, “An equivalence of fusion categories”, Geom. Funct. Anal. 6, 249–267 (1996).MathSciNetMATHCrossRefGoogle Scholar
  103. 86a.
    M. Finkelberg, “An equivalence of fusion categories”, Geom. Funct. Anal. 23(E), 810–811 (2013).MathSciNetCrossRefGoogle Scholar
  104. 87.
    D. Finkelstein and J. Rubinstein, “Connection between spin, statistics, and kinks”, J. Math. Phys. 9, 1762–1779 (1968).ADSMathSciNetMATHCrossRefGoogle Scholar
  105. 88.
    M. A. I. Flohr, “On modular invariant partition functions of conformal field theories with logarithmic operators”, Int. J. Mod. Phys. A 11, 4147–4172 (1996), arXiv:hep-th/9509166.ADSMathSciNetMATHCrossRefGoogle Scholar
  106. 89.
    M. A. I. Flohr, “On fusion rules in logarithmic conformal field theories”, Int. J. Mod. Phys. A 12, 1943–1958 (1997), arXiv:hep-th/9605151.ADSMathSciNetMATHCrossRefGoogle Scholar
  107. 90.
    K. Fredenhagen, K.-H. Rehren, and B. Schroer, “Superselection sectors with braid group statistics and exchange algebras, I: General theory”, Commun. Math. Phys. 125, 201–226 (1989)ADSMathSciNetMATHCrossRefGoogle Scholar
  108. 90a.
    K. Fredenhagen, K.-H. Rehren, and B. Schroer, “Superselection sectors with braid group statistics and exchange algebras, II: Geometric aspects and conformal covariance”, Rev. Math. Phys., Spec. Issue S, 113–157 (1992).MathSciNetMATHCrossRefGoogle Scholar
  109. 91.
    D. Friedan and S. Shenker, “The analytic geometry of two-dimensional conformal field theory”, Nucl. Phys. B 281, 509–545 (1987).ADSMathSciNetCrossRefGoogle Scholar
  110. 92.
    J. Fröhlich, “Statistics of fields, the Yang-Baxter equation, and the theory of knots and links”, in Nonperturbative Quantum Field Theory, Ed. by G.’ t Hooft et al., NATO ASI Ser. 185, 71–100 (1988).MathSciNetCrossRefGoogle Scholar
  111. 92a.
    J. Fröhlich, “On the structure of (unitary) rational conformal field theory”, Nucl. Phys. B: Proc. Suppl. 5, 110–118 (1988).ADSMathSciNetMATHCrossRefGoogle Scholar
  112. 93.
    J. Fröhlich and F. Gabbiani, “Braid statistics in local quantum theory”, Rev. Math. Phys. 2, 251–353 (1990).MathSciNetMATHCrossRefGoogle Scholar
  113. 94.
    J. Fröhlich and T. Kerler, “Quantum groups, quantum categories and quantum field theory”, in Lecture Notes in Mathematics, Vol. 1542 (Springer, Berlin, 1993).Google Scholar
  114. 95.
    J. Fuchs, “On non-semisimple fusion rules and tensor categories”, arXiv:hep-th/0602051.Google Scholar
  115. 96.
    J. Fuchs, S. Hwang, A. M. Semikhatov, and I. Yu. Tipunin, “Nonsemisimple fusion algebras and the Verlinde formula”, Commun. Math. Phys. 247, 713–742 (2004), arXiv:hep-th/0306274.ADSMathSciNetMATHCrossRefGoogle Scholar
  116. 97.
    J. Fuchs and C. Schweigert, Symmetries, Lie Algebras and Representations (Cambridge University Press, 1997).MATHGoogle Scholar
  117. 98.
    J. Fuchs and C. Schweigert, “Hopf algebras and finite tensor categories in conformal field theory”, Rev. Union Mat. Argentina 51, 43–90 (2010), arXiv:1004.3405[hep-th].MathSciNetMATHGoogle Scholar
  118. 99.
    W. Fulton, Young Tableaux with Applications to Representation Theory and Geometry (Cambridge University Press, 1997).MATHGoogle Scholar
  119. 100.
    W. Fulton and J. Harris, Representation Theory, A First Course (Springer, New York, 1997).MATHGoogle Scholar
  120. 101.
    P. Furlan and L. Hadjiivanov, “Quantum su(n)k monodromy matrices”, J. Phys. A 45, 165202 (2012), arXiv:1111.2037[math-ph].ADSMathSciNetMATHCrossRefGoogle Scholar
  121. 102.
    P. Furlan, L. Hadjiivanov, A. P. Isaev, O. V. Ogievetsky, P. N. Pyatov, and I. Todorov, “Quantum matrix algebra for the SU(n) WZNW model”, J. Phys. A 36, 5497–5530 (2003), arXiv:hep-th/0003210.ADSMathSciNetMATHCrossRefGoogle Scholar
  122. 103.
    P. Furlan, L. K. Hadjiivanov, and I. T. Todorov, “Canonical approach to the quantum WZNW model”, Preprint IC/95/74, ESI 234 (ICTP Trieste and ESI Vienna, 1995).MATHGoogle Scholar
  123. 104.
    P. Furlan, L. K. Hadjiivanov, and I. T. Todorov, “Operator realization of the SU(2) WZNW model”, Nucl. Phys. B 474, 497–511 (1996), arXiv:hepth/ 9602101.ADSMathSciNetMATHCrossRefGoogle Scholar
  124. 105.
    P. Furlan, L. Hadjiivanov, and I. Todorov, “A quantum gauge approach to the 2DSU(n) WZNW model”, Int. J. Mod. Phys. A 12, 23–32 (1997), arXiv:hepth/ 9610202.ADSMATHCrossRefGoogle Scholar
  125. 106.
    P. Furlan, L. K. Hadjiivanov, and I. T. Todorov, “Chiral zero modes of the SU(n) Wess–Zumino–Novikov–Witten model”, J. Phys. A 36, 3855–3875 (2003), arXiv:hep-th/0211154.ADSMathSciNetMATHCrossRefGoogle Scholar
  126. 107.
    P. Furlan, L. Hadjiivanov, and I. Todorov, “Zero modes’ fusion ring and braid group representations of the extended chiral WZNW model”, Lett. Math. Phys. 82, 117–151 (2007), arXiv:0710.1063v3[hep-th].ADSMathSciNetMATHCrossRefGoogle Scholar
  127. 108.
    P. Furlan, L. Hadjiivanov, and I. Todorov, “Canonical approach to the WZNW model”, arXiv:1410.7228[hepth].Google Scholar
  128. 109.
    P. Furlan, G. M. Sotkov, and I. T. Todorov, “Twodimensional conformal field theory”, Riv. Nuovo Cim. 12 (6), 1–202 (1989).CrossRefGoogle Scholar
  129. 110.
    M. R. Gaberdiel and H. G. Kausch, “A rational logarithmic conformal field theory”, Phys. Lett. B 386, 131–137 (1996), arXiv:hep-th/9606050.ADSMathSciNetCrossRefGoogle Scholar
  130. 111.
    M. R. Gaberdiel and H. G. Kausch, “Indecomposable fusion products”, Nucl. Phys. B 477, 293–318 (1996), arXiv:hep-th/9604026.ADSMathSciNetCrossRefGoogle Scholar
  131. 112.
    A. Gainutdinov, D. Ridout, and I. Runkel, “Special issue on logarithmic conformal field theory”, J. Phys. A: Math. Theor. 46 (49), 490301; 494001–494015 (2013).MATHCrossRefGoogle Scholar
  132. 113.
    A. Ch. Ganchev and V. B. Petkova, “U q(sL(2)) invariant operators and minimal theories fusion matrices”, Phys. Lett. B 233, 374–382 (1989).ADSMathSciNetMATHCrossRefGoogle Scholar
  133. 114.
    K. Gawedzki, “Classical origin of quantum group symmetries in Wess–Zumino–Witten conformal field theory”, Commun. Math. Phys. 139, 201–213 (1991).ADSMathSciNetMATHCrossRefGoogle Scholar
  134. 115.
    K. Gawedzki and N. Reis, “WZW branes and gerbes”, Rev. Math. Phys. 14, 1281–1334 (2002), arXiv:hepth/ 0205233.MathSciNetMATHCrossRefGoogle Scholar
  135. 116.
    D. Gepner and E. Witten, “String theory on group manifolds”, Nucl. Phys. B 278, 493–549 (1986).ADSMathSciNetCrossRefGoogle Scholar
  136. 117.
    J.-L. Gervais and A. Neveu, “Novel triangle relation and absence of tachions in Liouville theory”, Nucl. Phys. B 238, 125–141 (1984).ADSCrossRefGoogle Scholar
  137. 118.
    P. Goddard and D. Olive, “Kac-Moody and Virasoro algebras in relation to quantum physics”, Int. J. Mod. Phys. 1, 303–414 (1986).ADSMathSciNetMATHCrossRefGoogle Scholar
  138. 119.
    G. A. Goldin, R. Menikoff, and D. H. Sharp, “Particle statistics from induced representations of a local current group”, J. Math. Phys. 21, 650–664 (1980).ADSMathSciNetMATHCrossRefGoogle Scholar
  139. 119a.
    G. A. Goldin, R. Menikoff, and D. H. Sharp, “Representations of a local current algebra in nonsimply connected space and the Aharonov-Bohm effect”, J. Math. Phys. 22, 1664–1668 (1981).ADSMathSciNetCrossRefGoogle Scholar
  140. 120.
    H. Goldschmidt and S. Sternberg, “The Hamilton- Cartan formalism in the calculus of variations”, Ann. Inst. Fourier. Grenoble 23, 203–267 (1973).MathSciNetMATHCrossRefGoogle Scholar
  141. 121.
    C. Gómez and G. Sierra, “A brief history of hidden quantum symmetries in conformal field theories”, Proc. XXI DGMTP Conference (Tianjin, China, 1992), pp. 66–85, arXiv:hep-th/9211068.Google Scholar
  142. 122.
    F. M. Goodman, P. de la Harpe, and V. F. R. Jones, Coxeter Graphs and Towers of Algebras (Springer, Berlin, New York, 1989).MATHCrossRefGoogle Scholar
  143. 123.
    S. Goto, “On Ocneanu theory of double triangle algebras for subfactors and classification of irreducible connections on the Dynkin diagrams”, Exp. Math. 28, 218–253 (2010).MathSciNetMATHCrossRefGoogle Scholar
  144. 124.
    G. S. Guralnik and C. R. Hagen, “Where have all the Goldstone bosons gone?”, arXiv:1401.6924[hep-th].Google Scholar
  145. 125.
    M. B. Green and J. H. Schwarz, “Anomaly cancellations in supersymmetric D = 10 gauge theory and superstring theory”, Phys. Lett. B 149, 117–122 (1984).ADSMathSciNetCrossRefGoogle Scholar
  146. 126.
    M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory (Cambridge University Press, 1987).MATHGoogle Scholar
  147. 127.
    V. Gurarie, “Logarithmic operators in conformal field theory”, Nucl. Phys. B 410, 535–549 (1993), arXiv:hep-th/9303160.ADSMathSciNetMATHCrossRefGoogle Scholar
  148. 128.
    R. Haag, Local Quantum Physics. Fields, Particles, Algebras, 2nd ed. (Springer, Berlin, 1996).MATHCrossRefGoogle Scholar
  149. 129.
    R. Haag and D. Kastler, “An algebraic approach to quantum field theory”, J. Math. Phys. 5, 848–861 (1964).ADSMathSciNetMATHCrossRefGoogle Scholar
  150. 130.
    L. Hadjiivanov and P. Furlan, “Extended su(n)k and restricted U q sl(2)”, in Proc. VII International Workshop “Lie Theory and its Applications in Physics” (Varna, Bulgaria, 2007), Ed. by H.-D. Doebner and V. K. Dobrev (Heron Press, Sofia, 2008), pp. 151–160, arXiv:0712.2158[hep-th].Google Scholar
  151. 131.
    L. Hadjiivanov and P. Furlan, “On quantum WZNW monodromy matrix–factorization, diagonalization, and determinant”, in Proc. IX International Workshop “Lie Theory and Its Applications in Physics” (Varna, Bulgaria, 2011), Ed. by V. Dobrev (Series: Springer Proceedings in Mathematics and Statistics, 2013), Vol. 36, pp. 287–297, arXiv:1112.6274[math-ph].Google Scholar
  152. 132.
    L. Hadjiivanov and P. Furlan, “SU(n) WZNW fusion and a -algebra”, Bulg. J. Phys. 40 (2), 141–146 (2013).MathSciNetMATHGoogle Scholar
  153. 133.
    L. Hadjiivanov and P. Furlan, “On the 2D zero modes’ algebra of the SU(n) WZNW model”, in Proc. X International Workshop “Lie Theory and Its Applications in Physics” (Varna, Bulgaria,2013), Ed. by V. Dobrev (Series: Springer Proceedings in Mathematics and Statistics, 2014), Vol. 111, pp. 381–391, arXiv:1401.4394[math-ph].Google Scholar
  154. 134.
    L. Hadjiivanov and P. Furlan, ““Spread” restricted Young diagrams from a WZNW dynamical quantum group”, in Proc. XI International Workshop “Lie Theory and Its Applications in Physics” (Varna, Bulgaria, 2015), Ed. by V. Dobrev, arXiv:1512.09031[math-ph].Google Scholar
  155. 135.
    L. K. Hadjiivanov, A. P. Isaev, O. V. Ogievetsky, P. N. Pyatov, and I. T. Todorov, “Hecke algebraic properties of dynamical R-matrices. Application to related quantum matrix algebras”, J. Math. Phys. 40, 427–448 (1999), arXiv:q-alg/9712026.ADSMathSciNetMATHCrossRefGoogle Scholar
  156. 136.
    L. K. Hadjiivanov, R. R. Paunov, and I. T. Todorov, “Extended chiral conformal theories with a quantum symmetry”, Nucl. Phys. B, Proc. Suppl. 18, 141–165 (1990).ADSMathSciNetMATHCrossRefGoogle Scholar
  157. 136a.
    L. K. Hadjiivanov, R. R. Paunov, and I. T. Todorov, Lecture Notes in Mathematics (Springer, Berlin, 1992), Vol. 1510, pp. 277–302.Google Scholar
  158. 137.
    L. K. Hadjiivanov, R. R. Paunov, and I. T. Todorov, “Quantum group extended chiral p-models”, Nucl. Phys. B 356, 387–438 (1991).ADSMathSciNetCrossRefGoogle Scholar
  159. 138.
    L. Hadjiivanov and T. Popov, “On the rational solutions of the Knizhnik–Zamolodchikov equation”, Eur. Phys. J. B 29, 183–187 (2002), arXiv:hepth/ 0109219.ADSMathSciNetCrossRefGoogle Scholar
  160. 139.
    L. K. Hadjiivanov, Ya. S. Stanev, and I. T. Todorov, “Regular basis and R-matrices for the Knizhnik-su(n)k Zamolodchikov equation”, Lett. Math. Phys. 54, 137–155 (2000), arXiv:hep-th/0007187.MathSciNetMATHCrossRefGoogle Scholar
  161. 140.
    T. Hayashi, “A canonical Tannaka duality for finite semisimple tensor categories,” arXiv:math.QA/9904073.Google Scholar
  162. 141.
    Y.-Z. Huang and J. Lepowsky, “Tensor categories and the mathematics of rational and logarithmic conformal field theory”, J. Phys., A: Math. Theor. 46 (49), 494009 (2013), arXiv:1304.7556[hep-th].MathSciNetMATHCrossRefGoogle Scholar
  163. 142.
    J. E. Humphreys, “Introduction to Lie algebras and representation theory”, in Graduate Texts in Mathematics, Vol. 9 (Springer, New York, 1972), reprinted in 1997.Google Scholar
  164. 143.
    A. Hurwitz, “Über Riemann’sche Flächen mit gegebenen Verzweigungspunkten“, Mathematische Annalen 39 (1), 1–60 (1891).Google Scholar
  165. 144.
    A. P. Isaev, “Twisted Yang-Baxter equations for linear quantum (super)groups”, J. Phys. A 29, 6903–6910 (1996), arXiv:q-alg/9511006.ADSMathSciNetMATHCrossRefGoogle Scholar
  166. 145.
    M. Jimbo, “A q-difference analogue of U(g) and the Yang-Baxter equation”, Lett. Math. Phys. 10, 63–69 (1985).ADSMathSciNetMATHCrossRefGoogle Scholar
  167. 145a.
    M. Jimbo, “A q-analogue of the U(gl(N + 1)), Hecke algebra and the Yang-Baxter equation”, Lett. Math. Phys. 11, 247–252 (1986).ADSMathSciNetMATHCrossRefGoogle Scholar
  168. 146.
    V. F. R. Jones, “Index for subfactors”, Inv. Math. 72, 1–25 (1983).ADSMathSciNetMATHCrossRefGoogle Scholar
  169. 147.
    V. F. R. Jones, “A polynomial invariant for knots via von Neumann algebras”, Bull. Am. Math. Soc. 12, 103–112 (1985).MathSciNetMATHCrossRefGoogle Scholar
  170. 148.
    B. Julia and S. Silva, “On covariant phase space methods”, Preprint LPT-ENS 01/28, AEI-2001-062 (ENS Paris and AEI Golm, 2001), arXiv:hep-th/0205072.Google Scholar
  171. 149.
    V. G. Kac, Infinite Dimensional Lie Algebras, 3rd ed. (Cambridge University Press, Cambridge, 1990).MATHCrossRefGoogle Scholar
  172. 150.
    V. G. Kac, “Vertex Algebras for Beginners”, in University Lecture Series, 2nd ed., Vol. 10 (AMS, Providence, RI, 1998).Google Scholar
  173. 151.
    V. G. Kac and A. K. Raina, “Bombay lectures on highest weight representations of infinite dimensional Lie algebras”, in Advanced Series in Mathematical Physics, Vol. 2 (World Scientific, Singapore, 1987).Google Scholar
  174. 152.
    C. Kassel, Quantum Groups (Springer, New York, 1995).MATHCrossRefGoogle Scholar
  175. 153.
    H. G. Kausch, “Extended conformal algebras generated by a multiplet of primary fields”, Phys. Lett. B 259, 448–455 (1991).ADSMathSciNetCrossRefGoogle Scholar
  176. 154.
    Y. Kawahigashi, “Conformal field theory, tensor categories and operator algebras”, arXiv:1503.05675[math-ph].Google Scholar
  177. 155.
    D. Kazhdan and G. Lusztig, “Tensor structures arising from affine Lie algebras, I”, J. Am. Math. Soc. 6, 905–947 (1993).MathSciNetMATHCrossRefGoogle Scholar
  178. 155a.
    D. Kazhdan and G. Lusztig, “Tensor structures arising from affine Lie algebras, II”, J. Am. Math. Soc. 6, 949–1011 (1993).MathSciNetMATHCrossRefGoogle Scholar
  179. 155b.
    D. Kazhdan and G. Lusztig, “Tensor structures arising from affine Lie algebras, III”, J. Am. Math. Soc 7, 335–381 (1994).MathSciNetMATHCrossRefGoogle Scholar
  180. 155c.
    D. Kazhdan and G. Lusztig, “Tensor structures arising from affine Lie algebras, IV”, J. Am. Math. Soc. 7, 383–453 (1994).MathSciNetMATHCrossRefGoogle Scholar
  181. 156.
    J. Kijowski, “A finite-dimensional canonical formalism in the classical field theory”, Commun. Math. Phys. 30, 99–128 (1973).ADSMathSciNetCrossRefGoogle Scholar
  182. 157.
    J. Kijowski and W. Szczyrba, “A canonical structure for classical field theories”, Commun. Math. Phys. 46, 183–206 (1976).ADSMathSciNetMATHCrossRefGoogle Scholar
  183. 158.
    J. Kijowski and W. M. Tulczyjew, A Symplectic Framework for Field Theories (Springer-Verlag, Berlin, 1979).MATHCrossRefGoogle Scholar
  184. 159.
    A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons”, Ann. Phys. 303, 2–30 (2003), arXiv:quant-ph/9707021.ADSMathSciNetMATHCrossRefGoogle Scholar
  185. 160.
    K. von Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the finestructure constant based on quantized Hall resistance”, Phys. Rev. Lett. 45, 494–497 (1980).ADSCrossRefGoogle Scholar
  186. 161.
    V. G. Knizhnik and A. B. Zamolodchikov, “Current algebra and Wess-Zumino model in two dimensions”, Nucl. Phys. B 247, 83–103 (1984).ADSMathSciNetMATHCrossRefGoogle Scholar
  187. 162.
    T. Kohno, “Monodromy representations of braid groups and Yang-Baxter equations”, Ann. Inst. Fourier 37, 139–160 (1987).MathSciNetMATHCrossRefGoogle Scholar
  188. 163.
    R. B. Laughlin, “Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations”, Phys. Rev. Lett. 50 (18), 1395–1398 (1983).ADSCrossRefGoogle Scholar
  189. 164.
    J. M. Leinaas and J. Myrheim, “On the theory of identical particles”, Nuovo Cim. B 37, 1–23 (1977).ADSCrossRefGoogle Scholar
  190. 165.
    R. Longo, “I: Index of subfactors and statistics of quantum fields”, Commun. Math. Phys. 126, 217–247 (1989).ADSMathSciNetMATHCrossRefGoogle Scholar
  191. 165a.
    R. Long, “II: Correspondences, braid group statistics and Jones polynomial”, Commun. Math. Phys. 130, 285–309 (1990).ADSCrossRefGoogle Scholar
  192. 166.
    J.-H. Lu and A. Weinstein, “Poisson Lie groups, dressing transformations, and Bruhat decompositions”, J. Diff. Geom. 31, 501–526 (1990).MathSciNetMATHCrossRefGoogle Scholar
  193. 167.
    G. Mack and V. Schomerus, “Quasi Hopf quantum symmetry in quantum theory”, Nucl. Phys. 370, 185–230 (1992).ADSMathSciNetMATHCrossRefGoogle Scholar
  194. 168.
    J. Mickelsson, Current Algebras and Groups (Plenum Press, New York, 1989).MATHCrossRefGoogle Scholar
  195. 169.
    G. Moore and N. Read, “Nonabelions in the fractional quantum Hall effect”, Nucl. Phys. B 360, 362–396 (1991).ADSMathSciNetCrossRefGoogle Scholar
  196. 170.
    G. Moore and N. Reshetikhin, “A comment on quantum group symmetry in conformal field theory”, Nucl. Phys. B 328, 557–574 (1989).ADSMathSciNetCrossRefGoogle Scholar
  197. 171.
    G. Moore and N. Seiberg, “Polynomial equations for rational conformal field theories”, Phys. Lett. B 212, 451–460 (1988).ADSMathSciNetCrossRefGoogle Scholar
  198. 171a.
    G. Moore and N. Seiberg, “Naturality in conformal field theory”, Nucl. Phys. B 313, 16–40 (1989).ADSMathSciNetCrossRefGoogle Scholar
  199. 171b.
    G. Moore and N. Seiberg, “Classical and quantum conformal field theory”, Commun. Math. Phys. 123, 177–254 (1989).ADSMathSciNetMATHCrossRefGoogle Scholar
  200. 171c.
    G. Moore and N. Seiberg, “Taming the conformal zoo”, Phys. Lett. B 220, 422–430 (1989).ADSMathSciNetCrossRefGoogle Scholar
  201. 172.
    G. Moore and N. Seiberg, “Lectures on RCFT”, in Superstrings’ 89, Proceedings, Trieste Spring School and Workshop on Superstrings (Trieste, Italy,1989), Ed. by M. Green et al. (World Scientific, Singapore, 1990), pp.1–129; LecturesRCFT.pdf.Google Scholar
  202. 173.
    G. D. Mostow, “Braids, hypergeometric functions, and lattices”, Bull. Am. Math. Soc. 16, 225–246 (1987).MathSciNetMATHCrossRefGoogle Scholar
  203. 174.
    M. H. A. Newman, “On a string problem of Dirac”, J. London Math. Soc. 17, 173–177 (1942).MathSciNetMATHCrossRefGoogle Scholar
  204. 175.
    D. Nikshych and L. Vainerman, “Finite quantum groupoids and their applications”, New Directions in Hopf Algebras. MSRI Publications (Cambridge University Press, 2002), Vol. 43, pp. 211–262, arXiv:math.QA/0006057.MATHGoogle Scholar
  205. 176.
    S. P. Novikov, “The Hamiltonian formalism and a multivalued analogue of Morse theory”, Russ. Math. Surv. 37 (5), 1–56 (1982). 74.pdf.MathSciNetMATHCrossRefGoogle Scholar
  206. 177.
    K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films”, Science 306, 666–669 (2004).ADSCrossRefGoogle Scholar
  207. 178.
    A. Ocneanu, “Paths on Coxeter diagrams: From platonic solids and singularities to minimal models and subfactors,” in Lectures on Operator Theory, Ed. by S. Goto and Rajarama Bhat et al. (Fields Institute, Waterloo, Ontario, 1995; Fields Institute Monographs, AMS, 1999)Google Scholar
  208. 178a.
    A. Ocneanu, “Quantum symmetries for SU(3) CFT models,” in Quantum Symmetries in Theoretical Physics and Mathematics, Proceedings of the Bariloche School, Patagonia, Argentina, Jan. 10–21, 2000, Ed. by R. Coquereaux, A. Garcia, and R. Trinchero, AMS Contemp. Math. 294, 133–160 (2002).Google Scholar
  209. 179.
    L. Onsager, “Crystal statistics, I: A two-dimensional model with an order-disorder transition”, Phys. Rev., Ser. II 65 (3-4), 117–149 (1944).ADSMathSciNetMATHGoogle Scholar
  210. 180.
    V. Ostrik, “Module categories, weak Hopf algebras and modular invariants”, Transform. Groups 8 (2), 177–206 (2003), arXiv:math.QA/0111139.MathSciNetMATHCrossRefGoogle Scholar
  211. 181.
    V. Pasquier, “Continuum limit of lattice models built on quantum groups”, Nucl. Phys. B 295[FS21], 491–510 (1988).ADSMathSciNetCrossRefGoogle Scholar
  212. 181a.
    V. Pasquier, “Etiology of IRF models”, Commun. Math. Phys. 118, 355–364 (1988).ADSMathSciNetMATHCrossRefGoogle Scholar
  213. 182.
    V. Pasquier and H. Saleur, “Common structures between finite systems and conformal field theories through quantum groups”, Nucl. Phys. B 330, 523–556 (1990).ADSMathSciNetCrossRefGoogle Scholar
  214. 183.
    W. Pauli, “The connection between spin and statistics”, Phys. Rev. 58, 716–722 (1940).ADSMathSciNetMATHCrossRefGoogle Scholar
  215. 184.
    V. Petkova and J.-B. Zuber, “The many faces of Ocneanu cells”, Nucl. Phys. B 603, 449–496 (2001), arXiv:hep-th/0101151ADSMathSciNetMATHCrossRefGoogle Scholar
  216. 184a.
    V. Petkova and J.-B. Zuber, “Conformal field theories, graphs and quantum algebras”, in MathPhys Odyssey 2001, Integrable Models and Beyond (in honor of B. M. McCoy), Ed. by M. Kashiwara and T. Miwa, Ser.: Prog. Math. Phys. (Birkhäuser, Boston, 2002), Vol. 23, pp. 415–436, arXiv:hep-th/0108236.CrossRefGoogle Scholar
  217. 185.
    A. M. Polyakov, “Quantum geometry of bosonic strings”, Phys. Lett. B 103, 207–210 (1981)ADSMathSciNetCrossRefGoogle Scholar
  218. 185a.
    A. M. Polyakov, “Quantum geometry of fermionic strings”, Phys. Lett. B 103, 211–213 (1981).ADSMathSciNetCrossRefGoogle Scholar
  219. 186.
    W. Pusz and S. L. Woronowicz, “Twisted second quantization”, Rep. Math. Phys. 27, 231–257 (1989).ADSMathSciNetMATHCrossRefGoogle Scholar
  220. 187.
    N. Read and E. Rezayi, “Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Landau level”, Phys. Rev. B 59, 80–84 (1999).CrossRefGoogle Scholar
  221. 188.
    K.-H. Rehren, “Charges in quantum field theory”, in Proceedings of the 10th Congress on Mathematical Physics, Leipzig, Germany, July 30–Aug. 9, 1991, Ed. by K. Schmüdgen (Springer, Berlin,1992), pp. 388–392; Preprint DESY (DESY, Hamburg, 1991).Google Scholar
  222. 189.
    K.-H. Rehren and B. Schroer, “Einstein causality and Artin braids”, Nucl. Phys. B 312, 715–750 (1989).ADSMathSciNetCrossRefGoogle Scholar
  223. 190.
    E. S. Reich, “Phosphorene excites materials scientists”, Nature 506, 19 (2014).ADSCrossRefGoogle Scholar
  224. 191.
    N. Y. Reshetikhin and M. A. Semenov-Tian-Shansky, “Quantum R-matrices and factorization problems”, J. Geom. Phys. 5, 533–550 (1988). RADSMathSciNetMATHCrossRefGoogle Scholar
  225. 192.
    L. Rozansky and H. Saleur, “Quantum field theory for the multivariable Alexander-Conway polynomial”, Nucl. Phys. B 376, 461–509 (1992).ADSCrossRefGoogle Scholar
  226. 193.
    A. S. Schwarz, “Topology for physicists”, Grundlehren der mathematischen Wissenschaften, Ed. by S. Levy (Springer, Berlin, 1994), Vol. 308, p. 296.Google Scholar
  227. 194.
    J. Schwinger, On Angular Momentum: 1952 Preprint, Reprinted in Quantum Theory of Angular Momentum, Ed. by L. C. Biedernharn and H. van Dam (Academic Press, New York, 1965), pp. 229–279.Google Scholar
  228. 195.
    J. Schwinger, “Gauge invariance and mass, 2”, Phys. Rev. 128, 2425–2429 (1962).ADSMathSciNetMATHCrossRefGoogle Scholar
  229. 196.
    M. A. Semenov-Tian-Shansky, “What is a classical R-matrix?”, Funct. Anal. Appl. 17, 259–272 (1983).MATHCrossRefGoogle Scholar
  230. 196a.
    M. A. Semenov-Tian-Shansky, “Dressing transformations and Poisson group actions”, Publ. RIMS. Kyoto Univ. 21, 1237–1260 (1985).MathSciNetMATHCrossRefGoogle Scholar
  231. 197.
    A. M. Semikhatov, “Toward logarithmic extensions of sl(2)k conformal field models”, Theor. Math. Phys. 153, 1597–1642 (2007), arXiv:hep-th/0701279.MATHCrossRefGoogle Scholar
  232. 198.
    J.-P. Serre, Complex Semisimple Lie Algebras (Springer, New York, 1987).MATHCrossRefGoogle Scholar
  233. 199.
    B. Simon, “R(φ)2 Euclidean (quantum) field theory”, Princeton Series in Physics (Princeton Univ. Press, 1974).Google Scholar
  234. 200.
    E. Sklyanin, “Some algebraic structures connected with the Yang-Baxter equation”, Funct. Anal. Appl. 16, 263–270 (1982).MathSciNetMATHCrossRefGoogle Scholar
  235. 201.
    Ya. S. Stanev, I. T. Todorov, and L. K. Hadjiivanov, “Braid invariant rational conformal models with a quantum group symmetry”, Phys. Lett. B 276, 87–94 (1992).ADSMathSciNetCrossRefGoogle Scholar
  236. 202.
    R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That (Benjamin, New York, 1964).MATHGoogle Scholar
  237. 203.
    F. Strocchi, “Local and covariant gauge quantum field theories. Cluster property, superselection rules, and the infrared problem”, Phys. Rev. D 17, 2010–2021 (1978).ADSMathSciNetCrossRefGoogle Scholar
  238. 204.
    F. Strocchi, “Selected topics on the general properties of quantum field theory”, in Lecture Notes in Physics, Vol. 51 (World Scientific, Singapore, 1993).Google Scholar
  239. 205.
    H. Sugawara, “A field theory of currents”, Phys. Rev. 170, 1659–1662 (1968).ADSCrossRefGoogle Scholar
  240. 206.
    M. E. Sweedler, Hopf Algebras (Benjamin, New York, 1969).MATHGoogle Scholar
  241. 207.
    K. Szlachányi, “Finite quantum groupoids and inclusions of finite type,” in Proceedings of the Conference on Mathematical Physics in Mathematics and Physics Siena, June 20–25, 2000, Fields Inst. Commun. 30, 393–407 (2000); arXiv:math.QA/0011036.MathSciNetMATHGoogle Scholar
  242. 208.
    T. Tannaka, “Über den Dualitätssatz der nichtkommutativen topologischen Gruppen”, Tôhoku Math. J. 45, 1–12 (1939).MATHGoogle Scholar
  243. 209.
    W. Thirring, “A soluble relativistic field theory?”, Ann. Phys. 3, 91–112 (1958).ADSMathSciNetMATHCrossRefGoogle Scholar
  244. 210.
    I. T. Todorov, “Infinite Lie algebras in -dimensional conformal field theory”, in Proc. of the XIII International Conference on Differential Geometric Methods in Physics (Shumen, Bulgaria, 1984), Ed. by H.-D. Doebner and T. Palev, pp. 297–347Google Scholar
  245. 210a.
    I. T. Todorov, “Current algebra approach to conformal invariant two-dimensional models”, Phys. Lett. B 153, 77–81 (1985).ADSMathSciNetCrossRefGoogle Scholar
  246. 211.
    I. T. Todorov, “Quantum groups as symmetries of chiral conformal algebras,” in Quantum Groups, Proceedings of the 8th International Workshop on Mathematical Physics, Clausthal, FRG, 1989, Ed. by H.-D. Doebner and J.-D. Hennig, Lect. Notes Phys.370, 231–277 (1990).MathSciNetMATHGoogle Scholar
  247. 212.
    I. Todorov and L. Hadjiivanov, Quantum Groups and Braid Group Statistics in Conformal Current Algebra Models (Editora da Universidade Federal do Espirito Santo, Vitoria, Brazil, 2010), p. 163, ISBN 978-85-7772-045-3.Google Scholar
  248. 213.
    A. Tsuchiya and Y. Kanie, “Vertex operators in the conformal field theory on P1 and monodromy representations of the braid group”, Lett. Math. Phys. 13, 303–312 (1987)ADSMathSciNetMATHCrossRefGoogle Scholar
  249. 213a.
    A. Tsuchiya and Y. Kanie, “Vertex operators in conformal field theory on P1 and monodromy representations of braid group”, in Conformal Field Theory and Solvable Lattice Models, Ed. by M. Jimbo, T. Miwa, and A. Tsuchiya, Vol. 16 of Adv. Stud. Pure Math. (Academic, Boston, MA, 1988), pp. 297–372.CrossRefGoogle Scholar
  250. 214.
    D. C. Tsui, H. L. Störmer, and A. C. Gossard, “Twodimensional magnetotransport in the extreme quantum limit”, Phys. Rev. Lett. 48 (22), 1559–1562 (1982).ADSCrossRefGoogle Scholar
  251. 215.
    E. Verlinde, “Fusion rules and modular transformations in conformal field theory”, Nucl. Phys. B 300, 360–375 (1988).ADSMathSciNetMATHCrossRefGoogle Scholar
  252. 216.
    J. Wess and B. Zumino, “Consequences of anomalous Ward identities”, Phys. Lett. B 37, 95–97 (1971).ADSMathSciNetCrossRefGoogle Scholar
  253. 217.
    A. Weil, Elliptic Functions According to Eisenstein and Kronecker (Springer, Berlin, 1976).MATHCrossRefGoogle Scholar
  254. 218.
    X.-G. Wen, “Non-Abelian statistics in the fractional quantum Hall states”, Phys. Rev. Lett. 66, 802–805 (1991).ADSMathSciNetMATHCrossRefGoogle Scholar
  255. 219.
    G. C. Wick, A. S. Wightman, and E. P. Wigner, “The intrinsic parity of elementary particles”, Phys. Rev. 88, 101–105 (1952).ADSMathSciNetMATHCrossRefGoogle Scholar
  256. 220.
    A. S. Wightman, Problems in Relativistic Dynamics of Quantized Fields (Nauka, Moscow, 1968), p. 184 [in Russian].Google Scholar
  257. 221.
    F. Wilczek, “Quantum mechanics of fractional-spin particles”, Phys. Rev. Lett. 49, 957–959 (1982).ADSMathSciNetCrossRefGoogle Scholar
  258. 222.
    E. Witten, “Non-Abelian bosonization in two dimensions”, Commun. Math. Phys. 92, 455–472 (1984).ADSMATHCrossRefGoogle Scholar
  259. 223.
    E. Witten, “Quantum field theory and the Jones polynomial”, Commun. Math. Phys. 121, 351–399 (1989).ADSMathSciNetMATHCrossRefGoogle Scholar
  260. 224.
    S. L. Woronowicz, “Twisted SU(2) group. An example of noncommutative differential calculus”, in Publ. RIMS (Kyoto Univ., 1987), Vol. 23, pp. 117–181Google Scholar
  261. 224a.
    S. L. Woronowicz, “Compact matrix pseudogroups”, Commun. Math. Phys. 111, 613–665 (1987).ADSMathSciNetMATHCrossRefGoogle Scholar
  262. 225.
    Y.-L. Wu, B. Estienne, N. Regnault, and B. A. Bernevig, “Braiding non-Abelian quasiholes in fractional quantum Hall states”, Phys. Rev. Lett. 113, 116801 (2014), arXiv:1405.1720 [cond-mat.str-el].ADSCrossRefGoogle Scholar
  263. 226.
    A. B. Zamolodchikov and V. A. Fateev, “Operator algebra and correlation functions in the two-dimensional SU(2) × SU(2) chiral Wess–Zumino model”, Sov. J. Nucl. Phys. 43, 657–664 (1986).Google Scholar
  264. 227.
    D. P. Zhelobenko, “Compact Lie groups and their representations”, Translations of Math. Monographs (AMS, Providence, RI, 1973), Vol. 40.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Institute for Nuclear Research and Nuclear EnergyBulgarian Academy of SciencesSofiaBulgaria
  2. 2.Dipartimento di Fisica dell’ Università degli Studi di TriesteTriesteItaly

Personalised recommendations