Skip to main content

Classical and quantum conformal field theory

Communications in Mathematical Physics volume 123, pages 177–254 (1989)Cite this article

Abstract

We define chiral vertex operators and duality matrices and review the fundamental identities they satisfy. In order to understand the meaning of these equations, and therefore of conformal field theory, we define the classical limit of a conformal field theory as a limit in which the conformal weights of all primary fields vanish. The classical limit of the equations for the duality matrices in rational field theory together with some results of category theory, suggest that (quantum) conformal field theory should be regarded as a generalization of group theory.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Belavin, A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two dimensional quantum field theory. Nucl. Phys. B241, 33 (1984)

    Google Scholar 

  2. Knizhnik, V.G., Zamolodchikov, A.B.: Current algebra and Wess-Zumino model in two dimensions. Nucl. Phys. B247, 83 (1984)

    Google Scholar 

  3. Zamolodchikov, A.B.: Infinite additional symmetries in two-dimensional conformal quantum field theory. Theor. Math. Phys.65, 1205 (1986)

    Google Scholar 

  4. Zamolodchikov, A.B., Fateev, V.A.: Parafermionic currents in the two-dimensional conformal quantum field theory and self dual critical points inZ(n) invariant statistical systems. Sov. Phys. JETP62, 215 (1985)

    Google Scholar 

  5. Friedan, D., Shenker, S.: The analytic geometry of two-dimensional conformal field theory. Nucl. Phys. B281, 509 (1987)

    Google Scholar 

  6. Friedan, D.: A new formulation of string theory. Physica Scripta T15, 72 (1987)

    Google Scholar 

  7. Friedan, D, Shenker, S.: Talks at Cargese and I.A.S. (unpublished) (1987)

  8. Kastor, D., Martinec, E., Qiu, Z.: Current algebra and conformal discrete series. Phys. Lett.200 B, 434 (1988)

    Google Scholar 

  9. Bagger, J., Nemeschansky, D., Yankielowicz, S.: Virasoro algebras with central chargec>1. Phys. Rev. Lett.60, 389 (1988)

    Google Scholar 

  10. Douglas, M.R.:G/H conformal field theory. CALT-68-1453

  11. Ravanini, F.: An infinite class of new conformal field theories with extended algebras. Nordita-87/56-P

  12. Harvey, J.A., Moore, G., Vafa, C.: Quasicrystalline compactification. Nucl. Phys. B304, 269 (1988)

    Google Scholar 

  13. Anderson, G., Moore, G.: Rationality in conformal field theory. Commun. Math. Phys.117, 441 (1988)

    Google Scholar 

  14. Verlinde, E.: Fusion rules and modular transformations in 2-D conformal field theory. Nucl. Phys. B300, 360 (1988)

    Google Scholar 

  15. Vafa, C.: Toward classification of conformal theories. Phys. Lett.206 B, 421 (1988)

    Google Scholar 

  16. Moore, G., Seiberg, N.: Polynomial equations for rational conformal field theories. Phys. Lett. (in press)

  17. Bais, F.A., Bouwknegt, P., Surridge, M., Schoutens, K.: Extensions of the Virasoro algebra constructed from Kac-Moody algebra using higher order casimir invariants. Nucl. Phys. B304, 348 (1988); Coset constructions for extended Virasoro algebras. Nucl. Phys. B304, 371 (1988)

    Google Scholar 

  18. Mathur, S.D., Mukhi, S., Sen, A.: Differential equations for correlators in arbitrary rational conformal field theories. TIFR/TH/88-32; On the classification of rational conformal field theories. TIFR/TH/88-39

  19. Bakas, I.: Higher spin fields and the Gelfand-Dicke Algebra, the Hamiltonian structure of the spin 4 operator algebra. University of Texas at Austin preprints

  20. Lukyanov, S.L., Fateev, V.A.: Additional symmetries in two dimensional conformal field theory and exactly solvable models, Parts I, II, III. (In Russian) Institute for Theoretical Physics (Kiev) preprint ITP-88-74P

  21. Brustein, R., Yankielowicz, S., Zuber, J.-B.: Factorization and selection rules of operator product algebras in conformal field theory. TAUP-1647-88

  22. Dijkgraaf, R., Verlinde, E.: Modular invariance and the fusion algebra. Presented at Annecy Conf. on conformal field theory

  23. Moore, G., Seiberg, N.: Naturality in conformal field theory. Nucl. Phys. (in press)

  24. Blok, B., Yankielowicz, S.: Extended algebras and the coset construction of conformal field theories. TAU-1661-88

  25. Witten, E.: Quantum field theory and the Jones polynomials. To appear in the proc. of the IAMP Congress, Swansea, July, 1988

  26. Segal, G.: Talks at IAS 1987

  27. Ishibashi, N., Matsuo, Y., Ooguri, H.: Soliton equations and free fermions on Riemann surfaces. UT-499-Tokyo

  28. Alvarez-Gaumé, L., Gomez, C., Reina, C.: Loop groups, Grassmannians and string theory. Phys. Lett.190 B, 55 (1987)

    Google Scholar 

  29. Vafa, C.: Operator formulation on Riemann surfaces. Phys. Lett.190 B, 47 (1987)

    Google Scholar 

  30. Also see, Alvarez-Gaumé, L., Gomez, C., Moore, G., Vafa, C.: Strings in the operator formalism. Nucl. Phys. B303, 455 (1988)

    Google Scholar 

  31. Alvarez-Gaumé, L., Gomez, C., Nelson, P., Sierra, G., Vafa, C.: Fermionic strings in the operator formalism. BUHEP-88-11, and refs. therein

  32. In the mathematical literature these have been discussed In: Borcherds, R.: Proc. Nat. Acad. Sci. USA,83, 3068 (1986)

    Google Scholar 

  33. Lepowsky, J.: Perspectives on the Monster. Lepowsky, J., Frenkel, I., Meurman, A.: Vertex operators and the Monster. Academic Press, New York to appear; Borcherds, R.: Berkeley preprints

  34. Goddard, P., Kent, A., Olive, D.: Unitary representations of the Virasoro and super Virasoro algebras. Commun. Math. Phys.103, 105 (1986)

    Google Scholar 

  35. Dixon, L., Harvey, J.A., Vafa, C., Witten, E.: Strings on orbifolds. Nulc. Phys. B261, 678 (1985); Strings on orbifolds. II. Nucl. Phys. B274, 285 (1986)

    Google Scholar 

  36. Goddard, P., Schwimmer, A.: Unitary constructions of extended conformal algebras. Phys. Lett.206 B, 62 (1988)

    Google Scholar 

  37. Schroer, B.: Quasiprimary fields: An approach to positivity of 2-D conformal quantum field theory. Nucl. Phys. B295, 4 (1988); Algebraic aspects of non-perturbative quantum field theories. Como lectures; Rehren, K.-H.: Locality of conformal fields in two-dimensions: Exchange algebras on the light cone. Commun. Math. Phys.116, 675 (1988); Fröhlich, J.: Statistics of fields, the Yang-Baxter equation, and the theory of knots and links. Lectures at Cargese 1987, to appear In: Nonperturbative quantum field theory, Plenum Press: New York. Felder, G., Fröhlich, J.: Unpublished lectures notes

    Google Scholar 

  38. Rehren, K.-H., Schroer, B.: Einstein causality and artin braids. FU preprint 88-0439

  39. Tsuchiya, A., Kanie, Y.: Vertex operators in the conformal field theory onP1 and monodromy representations of the braid group. In: Conformal field theory and solvable lattice models. Adv. Stud. Pure Math.16, 297 (1988); Vertex operators in the conformal field theory onP1 and monodromy representations of the braid group. Lett. Math. Phys.13, 303 (1987)

    Google Scholar 

  40. Vafa, C.: Conformal theories and punctured surfaces. Phys. Lett.199 B, 195 (1987)

    Google Scholar 

  41. Frenkel, I.: Talk at Canadian Society of Math. Vancouver, Nov. 1987 H. Sonoda. Nucl. Phys. B311, 417 (1988)

    Google Scholar 

  42. DiFrancesco, P.: Structure constants for rational conformal field theories. Saclay preprint, PhT-88/139

  43. Witten, E.: Non-Abelian bosonization. Commun. Math. Phys.92, 455 (1984)

    Google Scholar 

  44. Rose, M.E.: Elementary theory of angular momentum. New York: Wiley 1957

    Google Scholar 

  45. Pressley, N., Segal, G.: Loop groups. Oxford: Oxford Univ. Press 1986

    Google Scholar 

  46. Felder, G., Gawedzki, K., Kupiainen, A.: The spectrum of Wess-Zumino-Witten models. IHES/p/87/35

  47. Kirillov, A.A.: Elements of the theory of representations. Berlin, Heidelberg, New York: Springer 1976

    Google Scholar 

  48. Saavedra, N.: Catégories tannakiennes. Lecture Notes in Mathematics, Vol. 265. Berlin, Heidelberg, New York: Springer 1972

    Google Scholar 

  49. Deligne, P., Milne, J.S.: Tannakian categories. In: Hodge cycles, motives and Shimura varieties. Lecture Notes in Mathematics, Vol. 900. Berlin, Heidelberg, New York: Springer 1982

    Google Scholar 

  50. Deligne, P.: Catégories tannakiennes,. IAS preprint

  51. MacLane, S.: Categories for the Working Mathematician. GTM 5

  52. Jones, V.F.R.: Invent. Math.72, 1 (1983)

    Google Scholar 

  53. Pasquier, V.: Operator content of the ADE lattice models. J. Phys. A20, 5707 (1987); Continuum limit of lattice models built on quantum groups. Nucl. Phys. B295 491 (1988); Etiology of IRF models. Saclay-SPhT/88/20

    Google Scholar 

  54. Kirillov, A.N., Reshetikhin, N.Y.: Representations of the algebraU q (sl(2)).q-orthogonal polynomials and invariants of links. LOMI preprint E-9-88

  55. See, for example, Drinfeld, V.: Quantum Groups. In: Proc. at the Intl. Cong. of Math. 1986, p. 798, and references therein

  56. Reshetikhin, N.Y.: Quantized Universal Enveloping Algebras. The Yang-Baxter Equation and Invariants of Links. LOMI preprint E-4-87, E-17-87

  57. Beilinson, A.A., Schechtman, V.V.: Determinant bundles and Virasoro algebras. Commun. Math. Phys.118, 651–701 (1988)

    Google Scholar 

  58. LeClair, A., Peskin, M., Preitschopf, C.: String field theory on the conformal plane. SLAC-PUB-4306; SLAC-PUB-4307

  59. MacLane, S.: Natural associativity and commutativity. Rice University Studies, Vol. 49, 4, 28 (1963)

    Google Scholar 

  60. Harer, J.: The second homology group of the mapping class group of an orientable surface. Invent. Math.72, 221 (1983)

    Google Scholar 

  61. Birman, J.: Braids, links, and mapping class groups. Ann. Math. Studies, Vol. 82. Princeton, NJ: Princeton University Press 1974

    Google Scholar 

  62. Birman, J.: On braid groups. Commun. Pure App. Math.22, 41 (1969); Mapping class groups and their relationship to braid groups. Commun. Pure App. Math.22, 213 (1969)

    Google Scholar 

  63. Wajnryb, B.: A simple presentation for the mapping class group of an orientable surface. Israel J. Math.45, 157 (1983); See also the review: Birman, J.: Mapping class group of surfaces. In: Proc. of “Braids” conference, Contemporary Math. (to appear)

    Google Scholar 

  64. DiFrancesco, P., Saleur, H., Zuber, J.-B.: Critical Ising correlations in the plane and on the torus. Nucl. Phys. B290, 527 (1987)

    Google Scholar 

  65. Eilenberg, S., MacLane, S.: Cohomology theory in abstract groups. I. Ann. Math.48, 51 (1947)

    Google Scholar 

  66. Brown, K.: Cohomology of groups. Berlin, Heidelberg, New York: Springer 1982

    Google Scholar 

  67. Jackiw, R.: Three cocycles in mathematics and physics. Phys. Rev. Lett.54, 159 (1985)

    Google Scholar 

  68. Grossman, B.: A three cocycle in quantum mechanics. Phys. Lett.152 B, 93 (1985)

    Google Scholar 

  69. See, e.g., Loo-Keng, Hua: Introduction to Number Theory, p. 162. Berlin, Heidelberg, New York: Springer 1982

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Advanced Study, 08540, Princeton, NJ, USA

    Gregory Moore & Nathan Seiberg

Authors
  1. Gregory Moore
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nathan Seiberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by L. Alvarez-Gaumé

On leave of absence from the Department of Physics, Weizmann Institute of Science, Rehovot 76100, Israel

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Moore, G., Seiberg, N. Classical and quantum conformal field theory. Commun.Math. Phys. 123, 177–254 (1989). https://doi.org/10.1007/BF01238857

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01238857

Keywords

  • Neural Network
  • Statistical Physic
  • Field Theory
  • Complex System
  • Nonlinear Dynamics