Graded Lie Derivatives and Short Distance Expansions in Two Dimensions

  • Rainer Dick
Part of the NATO ASI Series book series (NSSB, volume 315)


The exploration of algebraic structures in two-dimensional field theories underwent a rapid development in recent years, particularly through the investigation of W-algebras. It was demonstrated by Zamolodchikov that there exist algebras which close nonlinearly into finitely many (quasi-)primary fields1. For recent work on these W N -algebras see ref. 2–5 and references therein. A certain N → ∞ limit of these algebras, denoted meanwhile commonly as w 1+∞, has been studied by Bakas6 (see also ref. 7 for studies of homology complexes of a subalgebra of w 1+∞), and Pope, Romans and Shen succeeded8 in the construction of a unique W -algebra with the property to admit central terms for all integer weights ≥ 2. For a discussion of field theoretic realisations of these algebras and their relations see ref. 9 and references therein. Both w 1+∞ and the related algebra w are subalgebras of the algebra of area-preserving diffeomorphisms in two dimensions, and it has been pointed out by Witten that these algebras are realized as symmetries of the ground ring of two-dimensional string theory10, implying in particular the existence of operator representations of these algebras in the framework of Liouville theory coupled to c = 1 matter fields. A very nice account of this has also been given by Eguchi, who studied the equivalence of Liouville theory coupled to matter fields and SU(2)/U(1) coset theory11.


Central Charge Central Extension Conformal Weight Primary Field Liouville Theory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A.B. Zamolodchikov, Theor. Math. Phys. 65 (1985) 1205.MathSciNetCrossRefGoogle Scholar
  2. [2]
    R. Blumenhagen, M. Flohr, A. Kliem, W. Nahm, A. Recknagel and R. Varnhagen, Nucl. Phys. B361 (1991) 255.MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    L. Fehér, L. O’Raifeartaigh, P. Ruelle, I. Tsutsui and A. Wipf, Ann. Phys. 213 (1992) 1.ADSMATHCrossRefGoogle Scholar
  4. [4]
    L. Frappat, E. Ragoucy and P. Sorba, ENSLAPP-AL-391/92 (July 1992).Google Scholar
  5. [5]
    T. Tjin, A geometric construction of W-algebras, contribution to this workshop.Google Scholar
  6. [6]
    I. Bakas, Phys. Lett. B228 (1989) 57.MathSciNetADSGoogle Scholar
  7. [7]
    D.B. Fuks, Funct. Anal Appl. 19 (1985) 305.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    C.N. Pope, L.J. Romans and X. Shen, Phys. Lett. B236 (1990) 173.MathSciNetADSGoogle Scholar
  9. C.N. Pope, L.J. Romans and X. Shen, Nucl. Phys. B339 (1990) 191.MathSciNetADSCrossRefGoogle Scholar
  10. [9]
    E. Bergshoeff, P.S. Howe, C.N. Pope, E. Sezgin, X. Shen and K.S. Stelle, Nucl. Phys. B363 (1991) 163.MathSciNetADSCrossRefGoogle Scholar
  11. [10]
    E. Witten, Nucl Phys. B373 (1992) 187.MathSciNetADSCrossRefGoogle Scholar
  12. [11]
    T. Eguchi, w -algebra in 2D black holes, contribution to this workshop.Google Scholar
  13. [12]
    R. Dick, in: “Nonperturbative Methods in Low Dimensional Quantum Field Theories”, ed. G. Domokos et al., World Scientific, Singapore 1991, p. 455.Google Scholar
  14. [13]
    R. Dick, Fortschr. Phys. 40 (1992) 519.MathSciNetCrossRefGoogle Scholar
  15. [14]
    R. Dick and M. Weigt, LMU-TPW 92-11 (June 1992).Google Scholar
  16. [15]
    D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B271 (1986) 93.MathSciNetADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Rainer Dick
    • 1
  1. 1.Sektion PhysikUniversität MünchenMünchen 2Germany

Personalised recommendations