Skip to main content
SpringerLink
Log in

Kac-Moody and Virasoro Algebras in Local Quantum Physics

  • Chapter
Fundamental Problems of Gauge Field Theory

Part of the book series: NATO ASI Series ((NSSB,volume 141))

Abstract

These notes are based on lectures given at the Erice International School in Mathematical Physics in July 1985, and treat the Sugawara construction of the Virasoro algebra in terms of a given Kac-Moody algebra, thereby complementing previous lecture notes. Deductions are made and applied to the string theory of unified particle interactions, conformally invariant two dimensional quantum field theories and the phase transitions of two dimensional lattice systems. The ‘quark model’ representation of Kac-Moody algebras is also treated. A symmetric space criterion is derived for the quantum equivalence of a chiral model with Wess-Zumino term to a free fermion theory.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. ADLER S. AND DASHEN R. : Current Algebras and applications to particle physics (Benjamin, New York 1968 ).

    MATH  Google Scholar 

  2. ALTSCHULER D. Critical Exponents from Infinite - Dimensional Symplectic Algebras, University of Geneva preprint UGVA-DPT 1985/06–466.

    Google Scholar 

  3. BARDAKCI K. AND HALPERN M. Phys. Rev. D3 (1971), 2493.

    MathSciNet  Google Scholar 

  4. BELAVIN A.A., POLYAKOV A.M. AND ZAMOLODCHIKOV A.B. Nucl. Phys. B241 (1984) 333.

    Article  MathSciNet  MATH  Google Scholar 

  5. BROWER R. Phys. Rev. D6 (1972) 1655.

    Google Scholar 

  6. CALLAN C.G., DASHEN R.F. AND SHARP D.H. Phys. Rev. 165 (1968) 1883

    Article  Google Scholar 

  7. COLEMAN S. Phys. Rev. D11 (1975) 2088.

    Google Scholar 

  8. COLEMAN S., GROSS D. AND JACKIW R. Phys Rev. 180 (1969) 1359.

    Article  Google Scholar 

  9. DASHEN R. AND FRISHMAN Y.: Phys. Rev. D11 (1975) 278.

    Google Scholar 

  10. DEL GIUDICE E., DI VECCHIA P. AND FUBINI S. Annals of Phys. 70 (1972) 378.

    Article  Google Scholar 

  11. DELL’ANTONIO G.F., FRISHMAN Y. AND ZWANZIGER D. Phys. Rev. D6 (1972) 988.

    Google Scholar 

  12. DI VECCHIA P., KNIZHNIK V.G., PETERSEN J.L. AND ROSSI P. Nucl. Phys. B253 (1985) 701.

    Article  Google Scholar 

  13. DIRAC P.A.M. Nuovo Cimento (10) 1 (1955) 16;

    MathSciNet  MATH  Google Scholar 

  14. DIRAC P.A.M. Rev. Mod. Physics 34 (1962) 592.

    Article  MathSciNet  Google Scholar 

  15. DYNKIN E.B. Transi. Amer. Math. Soc. (1) 9 (1962) 328

    Google Scholar 

  16. DYNKIN E.B. Amer. Math. Soc. (2) 6 (1957) 111.

    MATH  Google Scholar 

  17. DYSON F.J.: Phys. Rev. 75 (1949) 486.

    Article  MathSciNet  MATH  Google Scholar 

  18. FERRARA S., GRILLO A.F. AND GATTO R. Nuovo Cimento 12A (1972) 959.

    MathSciNet  Google Scholar 

  19. FRENKEL I.B. Proc. Natl. Acad. Sci. U.S.A. 77 (1980) 6303;

    Article  MathSciNet  MATH  Google Scholar 

  20. FRENKEL I.B. J. Funct. Anal. 44 (1981) 259.

    Article  MathSciNet  MATH  Google Scholar 

  21. FRENKEL I.B. AND KAC V.G. Inv. Math. 62 (1980) 23.

    Article  MathSciNet  MATH  Google Scholar 

  22. FREUND P.G., OH P. AND WHEELER J.T. Nucl. Phys. B246 (1984) 371.

    Article  MathSciNet  Google Scholar 

  23. FRIEDAN D.: 1982 Les Houches Sumner School; ed. J.B. Zuber and R. Stora: Les Houches Session XXXIX, Recent Advances in Field Theory and Statistical Mechanics (North Holland 1984 ).

    Google Scholar 

  24. FRIEDAN D., QIU Z. AND SHENKER S.: Phys. Rev. Lett. 52 (1984) 1575; Vertex Operators in Mathematics and Physics; MSRI publication No. 3 (Springer 1984 ), p491.

    Google Scholar 

  25. FRIFDAN D. AND SHENKER S.: Proceedings of the Santa Fe meeting, edited T. Goldman and M.N. Nieto (World Scientific 1985 ), p437.

    Google Scholar 

  26. FUBINI S., HANSON A.J. AND JACKIW R.: Phys. Rev. D7 (1973) 1732.

    Google Scholar 

  27. FUBINI S. AND VENEZIANO G.: Nuovo Cimento 67A (1970) 29.

    Google Scholar 

  28. GFLL-MANN M.: Phys. Rev. 125 (1962) 1067;

    Article  MathSciNet  Google Scholar 

  29. GFLL-MANN M.: Phys. Letters 8 (1964) 214.

    Article  Google Scholar 

  30. GFLL-MANN M. AND NE’FMAN Y.: The Eightfold Way (Benjamin, New York 1964 ).

    Google Scholar 

  31. GERVAIS J-L. AND SAKITA B.: Nucl. Phys. B34 (1971) 477.

    Article  MathSciNet  Google Scholar 

  32. GLIOZZI F., OLIVE D. AND SCHERK J.: Nuclear Phys. B122 (1977) 253.

    Google Scholar 

  33. GODDARD P.: Kac-Moody and Virasoro Algebras: Representations and Applications, DAMPP 85/7.

    Google Scholar 

  34. GODDARD P., KENT A. AND OLIVE D. 1985a: Phys. Lett. 152B(1985) 88.

    MathSciNet  Google Scholar 

  35. GODDARD P., KENT A. AND OLIVE D. 1985b: Unitary Representations of the Virasoro and Super Virasoro Algebras, DAMPT preprint 85–21.

    Google Scholar 

  36. GODDARD P., NUYTS J. AND OLIVE D.: Nucl. Phys. B125 (1977) 1.

    Article  MathSciNet  Google Scholar 

  37. GODDARD P., NAHM W. AND OLIVE D.: Phys. Lett. 160B (1985) 111.

    MathSciNet  Google Scholar 

  38. GODDARD P. AND OLIVE D. 1984: Vertex Operators in Mathematics and Physics, MSRI Publication No. 3 (Springer 1984) p51.

    Google Scholar 

  39. GODDARD P. AND OLIVE D. 1985: Nuclear Phys. B257 [FS14] (1985) 226.

    MathSciNet  Google Scholar 

  40. GODDARD P., OLIVE D. AND SCHWIMMER A.: Phys. Lett. 157B (1985) 393.

    MathSciNet  Google Scholar 

  41. GODDARD P. AND THORN C. Phys. Letters 408 (1972) 235.

    Google Scholar 

  42. GOODMAN R. AND WALLACH N.R. J. Reine. Angew. Math. 347 (1984) 69.

    MathSciNet  MATH  Google Scholar 

  43. GREEN M. AND SCHWARZ J. 1981 Nuclear Phys. B181 (1981) 502.

    Google Scholar 

  44. GREEN M. AND SCHWARZ J. 1984 Physics Lett. 149B (1984) 117.

    MathSciNet  Google Scholar 

  45. GROSS D., HARVEY J.A., MARTINEC E. AND ROHM R. Phys. Rev. Lett. 54 (1985) 502;

    Article  MathSciNet  Google Scholar 

  46. GROSS D., HARVEY J.A., MARTINEC E. AND ROHM R. Nucl. Phys. B256 (1985) 253.

    Article  MathSciNet  Google Scholar 

  47. HELGASON S. Differential Geometry, Lie Groups, and Symmetric Spaces, ( Academic Press 1978 ).

    Google Scholar 

  48. HUMPHREYS J.E. Introduction to Lie Algebras and Representation Theory, (Springer 1972 ).

    Google Scholar 

  49. JACOB M. (Editor) Dual Theory, (North Holland 1974 ).

    Google Scholar 

  50. JAIN S., SHANKAR P. AND WADIA S. Conformal Invariance and String Theory in Compact Space: Bosons, Tata Institute preprint TIFR/PH/85–3.

    Google Scholar 

  51. KAC V.G. 1978: Proceedings of the International Congress of Mathematicians, Helsinki 1978 and Lecture Notes in Physics 94 (1979)

    Google Scholar 

  52. For the proof see B.L. Feigin and D.B. Fuchs: Functs. Anal. Prilozhen. 16 (1982) 47 [Funct. Anal. and App. 16 (1982) 114].

    Google Scholar 

  53. KAC V.G. 1983: Infinite-dimensional Lie Algebras - An Introduction (Birkhauser 1983).

    Google Scholar 

  54. KAC V.G. AND PETERSON D.H. Proc. Math. Acad. Sci. U.S.A. 78 (1981) 3308.

    Article  MathSciNet  MATH  Google Scholar 

  55. KAC V.G. AND TODOROV I.T. Superconformal current algebras and their unitary representations, MIT preprint.

    Google Scholar 

  56. KNIZHNIK V.G. AND ZAMOLODCHIKOV A.R. Nucl. Phys. B247 (1984) 83.

    MathSciNet  Google Scholar 

  57. MACKAY W. AND PATERA J. Tables of dimensions, indices and branching ratios for representations of simple groups, (Dekker 1981 ).

    Google Scholar 

  58. MANDELSTAM S. 1973: Phys. Rev. D7 (1973) 3763 and 3777.

    Google Scholar 

  59. MANDELSTAM S. 1975: Phys. Rev. D11 (1975) 3026.

    MathSciNet  Google Scholar 

  60. NAMBU Y.: Lectures for the Copenhagen Symposium 1970 (Unpublished).

    Google Scholar 

  61. NEMESCHANSKY D. AND YANKIELOWICZ S. Phys. Rev. Lett. 54 (1985) 620.

    Article  MathSciNet  Google Scholar 

  62. NEVEU A. AND SCHWARZ J.H. Nucl. Phys. B31 (1971) 86;

    Article  Google Scholar 

  63. NEVEU A. AND SCHWARZ J.H. Phys. Rev. D4 (1971) 1109.

    Google Scholar 

  64. OLIVE D.I.: Kac—Moody Algebras: An Introduction for Physicists, Imperial/TP/84–85/14.

    Google Scholar 

  65. ONSAGER L. Phys. Rev. 65 (1944) 117.

    Article  MathSciNet  MATH  Google Scholar 

  66. RAMOND P.: Phys. Rev. D3 (1971) 2415.

    MathSciNet  Google Scholar 

  67. SCHWINGER J. 1959: Phys. Rev. Lett. 3 (1959) 296 and ADLER AND DASHEN p235.

    Google Scholar 

  68. SCHWINGER J. 1962: Phys. Rev. 127 (1962) 324, 130 (1963) 406, 130 (1963) 800.

    Google Scholar 

  69. SEGAL G.: Comm. Math. Phys. 81 (1981) 301.

    Article  Google Scholar 

  70. SKYRME T.H.R. Proc. Roy. Soc. A262 (1961) 237.

    MathSciNet  Google Scholar 

  71. SLANSKY R.: Phys. Reports 79 (1981) 1.

    Article  MathSciNet  Google Scholar 

  72. SOMMERFIELD C. Phys. Rev. 176 (1968) 2019.

    Article  Google Scholar 

  73. SUGAWARA H. Phys. Rev. 170 (1968) 1659.

    Article  Google Scholar 

  74. TODOROV I.T.: Phys. Letters 153B (1985) 77.

    Google Scholar 

  75. VIRASORO M.: Phys Rev. D1 (1969) 2933.

    MathSciNet  Google Scholar 

  76. WEIS J.: Unpublished.

    Google Scholar 

  77. WICK G.C.: Phys. Rev. 80 (1950) 268.

    Article  MathSciNet  MATH  Google Scholar 

  78. WITTEN E. : Commun. Math. Phys. 92 (1984) 455.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The Blackett Laboratory, Imperial College, London, SW7 2BZ, UK

    D. I. Olive

Authors
  1. D. I. Olive
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Bologna, Bologna, Italy

    G. Velo

  2. Joseph Henry Laboratories, Princeton, New Jersey, USA

    A. S. Wightman

Rights and permissions

Reprints and permissions

Copyright information

© 1986 Springer Science+Business Media New York

About this chapter

Cite this chapter

Olive, D.I. (1986). Kac-Moody and Virasoro Algebras in Local Quantum Physics. In: Velo, G., Wightman, A.S. (eds) Fundamental Problems of Gauge Field Theory. NATO ASI Series, vol 141. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-0363-4_3

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-0363-4_3

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4757-0365-8

  • Online ISBN: 978-1-4757-0363-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation