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Two-Dimensional Conformal Field Theory and Three-Dimensional Topology

  • J. Fröhlich
  • C. King
Part of the NATO ASI Series book series (NSSB, volume 234)

Summary

We present a survey of two-dimensional conformal field theory and show how the mathematical structures underlying conformal field theory can be used to construct invariants of links imbedded in a general class of three-dimensional manifolds. After a general introduction, we discuss chiral algebras and their representation theory. Chiral vertices are introduced as analogues of Clebsch-Gordan operators in group theory. Braiding and fusing of chiral vertices is analyzed, and it is sketched how to define conformal field theory on arbitrary Riemann surfaces by a sewing procedure. We then show how to construct link invariants from the data provided by a conformal field theory and sketch connections with quantum group theory.

Keywords

Riemann Surface Quantum Group Conformal Block Braid Group Conformal Field Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Basic references on conformal field theory:Google Scholar
  2. A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Nucl. Phys. B241 (1984) 333;MathSciNetADSMATHCrossRefGoogle Scholar
  3. D. Friedan, Z. Qiu and S. Shenker, Phys. Rev. Lett. 52. (1984) 455;MathSciNetCrossRefGoogle Scholar
  4. D. Friedan, Z. Qiu and S. Shenker, Colman. Math. Phys. 107 (1986) 535.MathSciNetADSMATHCrossRefGoogle Scholar
  5. Monodromy of conformal blocks and braid group representations:Google Scholar
  6. VI.S. Dotsenko and V. A. Fateev, Nucl. Phys. 240 [FS12] (1984) 312;Nucl. Phys. B251 [FS13] (1985) 691; Phys. Lett. B413 (1985) 291;Google Scholar
  7. V. G. Knizhnik and A. B. Zamolodchikov, Nucl. Phys. B247 (1984) 83;MathSciNetADSMATHCrossRefGoogle Scholar
  8. A. Tsuchiya and Y. Kanie, Lett. Math. Phys. 13 (1987) 303;MathSciNetADSMATHCrossRefGoogle Scholar
  9. J. Fröhlich, “Statistics of fields, the Yang-Baxter equation, and the theory of knots and links”, 1987 Cargèse lectures, in Non-Perturbative Quantum Field Theory, eds. G’t Hooft et al. ( Plenum Press, New York, 1988 );Google Scholar
  10. G. Moore and N. Seiberg, Phys. Leu. B212 (1988) 451.MathSciNetADSGoogle Scholar
  11. 2.
    The revolution in knot theory was initiated in:Google Scholar
  12. V.F.R. Jones, Bull. AMS 12 (1985) 103;MATHCrossRefGoogle Scholar
  13. V.F.R. Jones, Ann. Math. 126 (1987) 335.MATHCrossRefGoogle Scholar
  14. Background from the theory of operator algebras can be found in:Google Scholar
  15. V. F. R. Jones, Invent. Math. 72 (1983) 1; Braid Groups, Hecke Algebras and type II c Factors, Proceedings of Japan-US Conf. on Operator Algebras, 1983.Google Scholar
  16. General texts on knot theory:Google Scholar
  17. D. Rolfsen, Knots and Links, Publish or Perish, Math. Lecture Series, 1976;MATHGoogle Scholar
  18. G. Burde and H. Zieschang, Knots, de Gruyter Studies in Mathematics 5.Google Scholar
  19. Recent papers on link-invariants:Google Scholar
  20. P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millet and A. Ocneanu, Bull. AMS 12 (1985) 239;MATHCrossRefGoogle Scholar
  21. L. H. Kauffman, Topology 26 (1987) 395.MathSciNetMATHCrossRefGoogle Scholar
  22. Y. Akutsu and M. Wadati, J. Phys. Soc. Japan 56 (1987) 839, 3039;MathSciNetADSGoogle Scholar
  23. Y. Akutsu, T. Deguchi and M. Wadati, J. Phys. Soc. Japan 56 (1987) 3464; 57 (1988) 757, 1905;Google Scholar
  24. V. F. R. Jones, “On knot invariants related to some statistical mechanical models,” Berkeley Preprint, 1988;Google Scholar
  25. V. G. Turaev, “The Yang-Baxter equation and invariants of links”, LOMI Preprint E-3–87; N. Yu. Reshetikhin, “Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links” 1, II, LOMI Preprints E-4–87, E-17–87.Google Scholar
  26. 3.
    E. Witten, “Quantum field theory and the Jones polynomial”, Proc. LAMP Conf. 1988, to appear in Int’l J. Mod. Phys. A.Google Scholar
  27. 4.
    M. F. Atiyah, “New invariants of three-and four-dimensional manifolds” in: “Che Mathematical Heritage of Hermann Weyl, ed. R. Wells, Providence R. I.: ( AMS Publ., 1988 ).Google Scholar
  28. 5.
    J. Fröhlich and C. King, “The Chern-Simons theory and knot polynomials”, Preprint ETH-TH/89–10, to appear in Commun. Math. Phys.Google Scholar
  29. 6.
    G. Felder, J. Fröhlich and G. Keller, Commun. Math. Phys. in press; G. Moore and N. Seiberg, “Naturality in conformal field theory”, Preprint IAS SNS-HEP-88/31; J. Fröhlich, unpublished.Google Scholar
  30. 7.
    A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Nucl. Phys. B241 (1984) 333.MathSciNetADSMATHCrossRefGoogle Scholar
  31. 8.
    V. G. Knizhnik and A. B. Zamolodchikov, Nucl. Phys. B247 (1984) 83;MathSciNetADSMATHCrossRefGoogle Scholar
  32. P. Goddard, A. Kent and D. Olive, Phys. Lett. 152B (1985) 88;MathSciNetMATHCrossRefGoogle Scholar
  33. P. Goddard, A. Kent and D. Olive, Commun. Math. Phys. 103 (1986) 105.MathSciNetADSMATHCrossRefGoogle Scholar
  34. 9.
    A. B. Zamolodchikov, Theor. Math. Phys. 65 (1985) 1205;Google Scholar
  35. H. Eichenherr, Phys. Leu. 1518 (1985) 26;MathSciNetADSGoogle Scholar
  36. A. B. Zamolodchikov and V. A. Fateev, Soy. Phys. JET? 62 (1985) 215; 63 (1985) 913.Google Scholar
  37. 10.
    J. Fröhlich, “Statistics of fields, the Yang-Baxter equation and the theory of knots and links”, 1987 Cargèse lectures, in: Non-Perturbative Quantum Field Theory, eds. G. ‘t Ilooft et al. ( Plenum Press, New York, 1988 ).Google Scholar
  38. 11.
    G. Felder, J. Fröhlich and G. Keller, “On the structure of unitary conformal field theory 1: existence of conformal blocks”, Commun. Math. Phys. in press, and “On the Structure… 11: Representation Theoretic Approach”, to appear in Colman. Math. Phys.Google Scholar
  39. 12.
    G. Moore and N. Seiberg, “Classical and quantum conformal field theory”, Preprint IAS SNS-HEP-88/39, to appear in Commun. Math. Phys.Google Scholar
  40. 13.
    D. Frieda, Z. Qiu, and S. Shenker, Puys. Rev. Leu. 52 (1984) 455; see also Ref. 11.Google Scholar
  41. 14.
    G. Felder, K. Gawçdzki, and A. Kupiainen, Nucl. Phys. B299 (1988) 355;ADSCrossRefGoogle Scholar
  42. G. Felder, K. Gawçdzki, and A. Kupiainen, Commun. Math. Phys. 117 (1988) 127; and references given there.Google Scholar
  43. 15.
    G. Felder, “BRST approach to minimal models”, Nucl. Phys. B in press.Google Scholar
  44. 16.
    J. Birman, “Braids, links and mapping class groups”, Ann. Math. Studies 82, ( Princeton University Press, Princeton; 1974 ).Google Scholar
  45. 17.
    VI. S. Dotsenko and V. A. Fateev, Nucl. Phys. B 240 [FS12] (1984) 312; 8251 [FS13] (1985) 691; Phys. Lett. 154B (1985) 291.Google Scholar
  46. 18.
    G. Felder, J. Fröhlich and G. Keller, “Braid Matrices and structure constants for minimal models”, to appear in Commun. Math. Phys.Google Scholar
  47. 19.
    D. Bernard and G. Felder, ETH-preprint 1989.Google Scholar
  48. 20.
    M. Jimbo, Lett. Math. Phys. 10 (1985) 63;MathSciNetADSMATHCrossRefGoogle Scholar
  49. M. Jimbo, Lett Math. Phys. 11 (1986) 247;MathSciNetADSMATHCrossRefGoogle Scholar
  50. V. G. Drinfel’d, “Quantum Groups”, in Proceedings of ICM Berkeley 1986, ed.Google Scholar
  51. A. M. Gleason, Providence R.I.: (AMS Publ., 1987); M. Rosso, Commun. Math. Phys. 117 (1988) 581;Google Scholar
  52. N. Yu. Reshetikhin, “Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links, 1, 11”, LOMI Preprints E-4–87, E-17–87.Google Scholar
  53. 21.
    G. Felder, J. Fröhlich and G. Keller, “On the structure…, 1I”; see Ref. 11.Google Scholar
  54. 22.
    K.-H. Rehren and B. Schroer, “Einstein causality and Arlin braids”, Berlin Preprint FU-88–0439.Google Scholar
  55. 23.
    G. Moore and N. Seiberg, Phys. Lett. B212 (1988) 451.MathSciNetCrossRefGoogle Scholar
  56. 24.
    Ref. 17; and P. Di Francesco, “Structure constants for rational conformal field theories”; Preprint, Saclay PhT-88/139; Ref. 18.Google Scholar
  57. 25.
    J. Fröhlich, Commun. Math. Phys. 47 (1976) 233.ADSCrossRefGoogle Scholar
  58. 26.
    D. Friedan and S. Shenker, Nucl. Phys. B281 (1987) 509;MathSciNetADSCrossRefGoogle Scholar
  59. T. Eguchi and H. Ooguri, Nucl. Phys. B282 (1987) 308;MathSciNetADSCrossRefGoogle Scholar
  60. C. Vafa, Phys. Lett. B190 (1987) 47;MathSciNetCrossRefGoogle Scholar
  61. L. Alvarez-Gaumé, C. Gomez, G. Moore and C. Vafa, Nucl. Phys. B303 (1988) 455;ADSCrossRefGoogle Scholar
  62. P. West, “A review of duality, string vertices, overlap identities and the group theoretical approach to string theory, Preprint CERN Th 4819/87.Google Scholar
  63. 27.
    H. Sonoda, “Sewing conformal field theories 1, II,” Preprints LBL-25140, LBL-25316, 1988.Google Scholar
  64. 28.
    J. Fröhlich, Notes for seminars at I.H.É.S. and at École Polytechnique (Palaiseau), 1988.Google Scholar
  65. 29.
    G. Felder and R. Silvotti, “Modular covariance of minimal model correlation functions”, Preprint IAS SM 1988.Google Scholar
  66. 30.
    N. Kuiper, Math. Ann. 278 (1987) 193–209.MathSciNetMATHCrossRefGoogle Scholar
  67. 31.
    D. Rolfsen, Knots and Links, Publish or Perish, Math. Lecture Series, 1976;MATHGoogle Scholar
  68. G. Burde and H. Zieschang, Knots de Gruyter Studies in Mathematics 5.Google Scholar
  69. 32.
    N. Yu. Reshetikhin, see Ref. 20.Google Scholar
  70. 33.
    V. G. Knizhnik and A. B. Zamolodchikov, see Ref. 8.Google Scholar
  71. 34.
    A. Tsuchiya and Y. Kanie, Leu. Math. Phys. 13 (1987) 303.MathSciNetADSMATHCrossRefGoogle Scholar
  72. 35.
    A. A. Belavin and V. G. Drinfel’d, Funct. Anal. Appl. 16 (1982) 1; 17 (1983) 69.Google Scholar
  73. 36.
    M. Jimbo, see Ref. 20.Google Scholar
  74. 37.
    M. Rosso, see Ref. 20.Google Scholar
  75. 38.
    V. Pasquier, Colman. Math. Phys. 118 (1988). 355.MathSciNetADSMATHCrossRefGoogle Scholar
  76. 39.
    S. Doplicher and J. Roberts, “C*-algebras and duality for compact groups…”, in Proceedings of Vlllth International Congress on Mathematical Physics ed. M. Mebkhout and R. Sénéor (World Scientific, 1989); see also: S. Doplicher, R. Haag and J. E. Roberts, Commun. Math. Phys. 23 (1971) 199; 35 (1974) 49.Google Scholar
  77. 40.
    S. Woronowicz, private communication (to J. F.), 1987.Google Scholar

Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • J. Fröhlich
    • 1
  • C. King
    • 2
  1. 1.Theoretical PhysicsETH-HönggerbergZürichSwitzerland
  2. 2.Department of MathematicsCornell UniversityIthacaUSA

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