Quantum Groups pp 277-302 | Cite as

Extended chiral conformal theories with a quantum symmetry

  • L. K. Hadjiivanov
  • R. R. Paunov
  • I. T. Todorov
II. Quantum Groups, Symmetries Of Dynamical Systems And Conformal Field Theory
Part of the Lecture Notes in Mathematics book series (LNM, volume 1510)


A properly extended chiral part of rational conformal field theory (RCFT) possesses an internal quantum symmetry. There exist self dual chiral Green functions defined up to an overall phase factor which transform under a 1-dimensional (unitary) representation of the braid group B n . A Bn-invariant inner product is constructed in the space of quantum group invariants which allows to reproduce 2-dimensional monodromy free euclidean Green functions by pairing the correlation functions of the left and right sectors.


Green Function Hopf Algebra Quantum Group Structure Constant Conformal Block 
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  1. [1]
    A.A. Belavin, A.M. Polyakov, A.B. Zamolodchikov, Nucl. Phys. B241 (1984), 333, this and other basis original papers are collected in: Conformal Invariance and Applications to Statistical Mechanics (CLASM) Eds. C.Itzykson, H.Saleur, J-B.Zuber (World Scientific, Singapore 1988).MathSciNetCrossRefGoogle Scholar
  2. [2]
    P. Furlan, G.M. Sotkov, I.T. Todorov, Riv. Nuovo. Cim. 12 no. 6 (1989), 1.MathSciNetCrossRefGoogle Scholar
  3. [3]
    G. Moore, N. Seiberg, Lectures on RCFT, Rutgers and Yale Univ. preprint RU-89-32 and YCTP-P13-89; see also Commun. Math. Phys. 123 (1989), 177.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    A. Tsuchiya, Y. Kanie, Conformal Field Theory and Solvable Lattice Models, Advanced Studies in Pure Mathematics 16 (1988), 297–372; Lett. Math. Phys. 13 (1987), 303.MathSciNetGoogle Scholar
  5. [5]
    K.H. Rehren, Commun. Math. Phys. 116 (1988), 675; Nucl. Phys. B312 (1989), 715.MathSciNetCrossRefGoogle Scholar
  6. [6]
    J.Fröhlich, Statistics of fields, the Yang-Baxter equation and the theory of knots and links, 1987 Cargèse lectures, Non-Perturbative Quantum Field Theory, Eds. G.'t Hooft et al., New York, Plenum Press (to appear).Google Scholar
  7. [7]
    G. Felder, J. Fröhlich, G. Keller, Commun. Math. Phys. 124 (1989), 647.CrossRefGoogle Scholar
  8. [8]
    J.-L. Gervais, A. Neveu, Nucl. Phys. B238 (1984), 125.MathSciNetCrossRefGoogle Scholar
  9. [9]
    V.G.Drinfeld, Quantum groups, Proc. ICM-86, Vol.1, Berkeley, CA, 1987, pp. 798–820; Algebra and Anal. 1 no. 2 (1989), 30. (Russian)Google Scholar
  10. [10]
    L.D. Faddeev, N.Yu. Reshetikhin, L.A. Takhtajan, Alg. Anal. 1 no. 1 (1989), 178. (Russian)MathSciNetGoogle Scholar
  11. [11]
    V. Pasquier, H. Saleur, Nucl. Phys. B330 (1990), 523.MathSciNetCrossRefGoogle Scholar
  12. [12]
    V.G. Drinfeld, Alg. Anal. 1 no. 6 (1989), 114; 2 no. 4 (1990), 149.MathSciNetGoogle Scholar
  13. [13]
    R.Dijkgraaf, V.Pasquier, R.Roche, Quasi-quantum groups related to orbifold models, Talk at Inter. Colloq. Modern Quantum Field Theory, Tata Inst. Fund. Research, Jan. 1990.Google Scholar
  14. [14]
    B.Schroer, Algebraic QFT as a framework for classification and model building. A heretic view of the New Kinematics, Annecy-le-Vieux preprint LAPP-TH-280/90, (and references therein).Google Scholar
  15. [15]
    L. Alvarez-Gaumé, C. Gomes, G. Sierra, Nucl. Phys. Lett. B319 (1989), 155; Phys. Lett. B220 (1989), 142; Nucl. Phys. B330 (1990), 347.CrossRefGoogle Scholar
  16. [16]
    A.Ch. Canchev, V.B. Petkova, Phys. Lett. B233 (1989), 374.CrossRefGoogle Scholar
  17. [17]
    G. Moore, N. Reshetikhin, Nucl. Phys. B328 (1989), 557.MathSciNetCrossRefGoogle Scholar
  18. [18]
    P.Furlan, A.Ch.Ganchev, V.B.Petkova, Quantum groups and fusion rules multiplicities, Trieste preprint INFN/AE-89/15.Google Scholar
  19. [19]
    E. Witten, Nucl. Phys. B330 (1990), 285.CrossRefGoogle Scholar
  20. [20]
    E. Guadagnini, M. Martellini, M. Mintchev, Phys. Lett. B235 (1990), 275.MathSciNetCrossRefGoogle Scholar
  21. [21]
    D.J. Smit, Commun. Math. Phys. 128 (1990), 1.MathSciNetCrossRefGoogle Scholar
  22. [22]
    N.Reshetikhin, F.Smirnov, Hidden quantum group symmetry and integrable perturbations of conformal field theories, LOMI-Harvard preprint (October 1989).Google Scholar
  23. [23]
    O. Babelon, Phys. Lett. B215 (1988), 523.MathSciNetCrossRefGoogle Scholar
  24. [23]a
    A. Alexeev, S. Shatashvili, Nucl. Phys. B323 (1989), 719.Google Scholar
  25. [23]b
    L.D. Faddeev, Commun. Math. Phys. 132 (1990), 131.MathSciNetCrossRefGoogle Scholar
  26. [24]
    J.-L. Gervais, Commun. Math. Phys. 130 (1990), 257.MathSciNetCrossRefGoogle Scholar
  27. [24]a
    E. Cremmer, J.-L. Gervais, Commun. Math. Phys. 134 (1990), 619.MathSciNetCrossRefGoogle Scholar
  28. [24]b
    J.-L.Gervais, Critical dimensions for non critical strings, Paris preprint LPTENS 90/4.Google Scholar
  29. [25]
    J. Fröhlich, C. King, Int. J. Mod. Phys. A4 (1989), 5321.CrossRefGoogle Scholar
  30. [26]
    C. Gomez, G. Sierra, Phys. Lett. B240 (1990), 149; The quantum symmetry of rational conformal field theories, Genève preprint UGVA-DPT 1990/04-669.MathSciNetCrossRefGoogle Scholar
  31. [27]
    J.Fröhlich, T.Kerler, On the role of quantum groups in low dimensional local quantum field theory, ETH, Zürich preprint (1990).Google Scholar
  32. [28]
    Algebraic Theory of Superselection Sectors: Introduction and Recent Results, Proc. of the Convegno Internazionale di Presentazione dell' Instituto Scientifico Internazionale G.B. Guccia “Algebraic Theory of Superselection Sectors and Field Theory” held in Palermo (Italy) 23–30 November 1989, Ed. D.Kastler, World Scientific, Singapore, 1990, (See, in particular, the introductory article by D.Kastler, M.Mebkhout and K.H.Rehren, as well as the contributions by D.Buchholz, G.Mack, I.T.Todorov; G.Mack, V.Schomerus; R.H.Rehren).Google Scholar
  33. [29]
    D. Buchholz, G. Mack, I.T. Todorov, Conformal Field Theory and Related Topics, Eds. P.Binetruy, P.Sorba; Nucl. Phys. B (proc. Supl.) 5B (1988), 20.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [30]
    K. Fredenhagen, R.H. Rehren, B. Schroer, Commun. Math. Phys. 125 (1989), 201.MathSciNetCrossRefGoogle Scholar
  35. [31]
    R. Longo, Commun. Math. Phys. 126 (1989), 217, and to be published.MathSciNetCrossRefGoogle Scholar
  36. [31]a
    L.S. Dotsenko, V.A. Fateev, Nucl. Phys. B240 [FS12] (1984), 312; CLASM [1], 214–250.MathSciNetCrossRefGoogle Scholar
  37. [32]
    I.T. Todorov, Lect. Notes Phys. 370 (1990), 231–277.CrossRefGoogle Scholar
  38. [33]
    A.N.Kirillov, N.Yu.Reshetikhin, Representations of the algebra U q (sl(2)), q-orthogonal polynomials and invariant of links, LOMI (Leningrad) preprint E-9-88.Google Scholar
  39. [34]
    K.Gawedzki, Geometry of Wess-Zumino-Witten models of conformal field theory, Proceedings of the 4th Annecy Meeting in Theoretical Physics, Spring 1990, (in honour of Raymond Stora).Google Scholar
  40. [35]
    I.-G.Koh, S.Ouvry, I.T.Todorov, Quantum dimensions and modular forms in chiral conformal theory, Orsay preprint IPNO/TH 90–17, Phys. Lett. B.Google Scholar
  41. [36]
    J. Kurchan, P. Lebeuf, M. Saraceno, Phys. Rev. A40 (1989), 6800.CrossRefGoogle Scholar
  42. [37]
    L.K.Hadjiivanov, R.R.Paunov, I.T.Todorov, Quantum group extended chiral p-models (to appear).Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • L. K. Hadjiivanov
    • 1
  • R. R. Paunov
    • 1
  • I. T. Todorov
    • 1
  1. 1.Institute for Nuclear Research and Nuclear EnergySofiaBulgaria

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