Communications in Mathematical Physics

, Volume 102, Issue 2, pp 337–347 | Cite as

Superconformal current algebras and their unitary representations

  • Victor G. Kac
  • Ivan T. Todorov
Article

Abstract

A natural supersymmetric extension\((\widehat{dG})_\kappa\) is defined of the current (= affine Kac-Moody Lie) algebra\(\widehat{dG}\); it corresponds to a superconformal and chiral invariant 2-dimensional quantum field theory (QFT), and hence appears as an ingredient in superstring models. All unitary irreducible positive energy representations of\((\widehat{dG})_\kappa\) are constructed. They extend to unitary representations of the semidirect sumSκ(G) of\((\widehat{dG})_\kappa\) with the superconformal algebra of Neveu-Schwarz, for\(\kappa = \frac{1}{2}\), or of Ramond, for κ=0.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Witten, E.: Non-abelian bosonization in two dimensions. Commun. Math. Phys.92, 455–472 (1984)Google Scholar
  2. 2.
    Knizhnik, V. G., Zamolodchikov, A. B.: Current algebra and Wess-Zumino model in two dimensions. Nucl. Phys.B247, 83–103 (1984)Google Scholar
  3. 3.
    Todorov, I. T.: Current algebra approach to conformal invariant two-dimensional models. Phys. Lett.153B, 77–81 (1985); Infinite Lie algebras in 2-dimensional conformal field theory, ISAS Trieste lecture notes 2/85/E.P.Google Scholar
  4. 4.
    Kac, V. G.: Contravariant form for infinite dimensional Lie algebras and superalgebras. Lecture Notes in Physics,94 Heidelberg, New York, Berlin: Springer, 1979 pp. 441–445; Frenkel, I. B., Kac, V. G.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math.62, 23–66 (1980); Frenkel, I. B.: Two constructions of affine Lie algebra representations and Boson-Fermion correspondence in quantum field theory. J. Funct. Anal.44, 259–327 (1981)Google Scholar
  5. 5.
    Kac, V. G.: Infinite dimensional Lie algebras: An introduction, Boston: Birkhäuser 1983Google Scholar
  6. 6.
    Kac, V. G., Peterson, D. H.: Spin and wedge representations of infinite dimensional Lie algebras and groups. Proc. Natl. Acad. Sci. USA78, 3308–3312 (1981)Google Scholar
  7. 7.
    Kac, V. G., Peterson, D. H.: Infinite dimensional Lie algebras, theta functions and modular forms. Adv. Math.53, 125–264 (1984); Goodman, R., Wallach, N. R.: Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle. J. Reine Angew. Math.347, 69–133 (1984), Erratum, ibid.352, 220 (1984)Google Scholar
  8. 8.
    Goddard, P., Olive, D.: Kac-Moody algebras, conformal symmetry and critical exponents, Nucl. Phys.B257 [FS 14], 226–240 (1985)Google Scholar
  9. 9.
    Goddard, P., Kent, A., Olive, D.: Virasoro algebras and coset space models. Phys. Lett.152B, 88–93 (1985)Google Scholar
  10. 10.
    Green, M. B., Schwarz, J. H.: Anomaly cancellation in supersymmetricD = 10 gauge theory and superstring theory. Phys. Lett.149B, 117–122 (1984); Infinity cancellations in SO(32) superstring theory. Phys. Lett.151B, 21–25 (1984)Google Scholar
  11. 11.
    Witten, E.: Some properties of O(32) superstrings. Phys. Lett.149B, 351–356 (1984)Google Scholar
  12. 12.
    Gross, D. J., Harvey, J. A., Martinec, E., Rohm, R.: Heterotic string. Phys. Rev. Lett.54, 502–505 (1985); Heterotic string theory I. The free heterotic string. Princeton: Princeton University Press, preprint (1985)Google Scholar
  13. 13.
    Eichenherr, H.: Minimal operator algebras in superconformal quantum field theory. Phys. Lett.151B, 26–30 (1985); Bershadsky, M. A., Knizhnik, V. G., Teitelman, M. G.: Superconformal symmetry in two dimensions. Phys. Lett.151B, 31–36 (1985)Google Scholar
  14. 14.
    Friedan, D., Qiu, Z. Shenker, S.: Superconformal invariance in two dimensions and the tricritical Ising model. Phys. Lett.151B, 37–43 (1985)Google Scholar
  15. 15.
    Belavin, A. A., Polyakov, A. M., Zamolodchikov, A. B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys.B241, 333–380 (1984); Infinite conformal symmetry of critical fluctuations in two dimensions. J. Stat. Phys.34, 763–774 (1984)Google Scholar
  16. 16.
    Neveu, A., Schwarz, J. H.: Factorizable dual model of pions. Nucl. Phys.B31, 86–112 (1971)Google Scholar
  17. 17.
    Ramond, P.: Dual theory for free fermions. Phys. Rev.D3, 2415–2418 (1971)Google Scholar
  18. 18.
    Ferrara, S., Gatto, R., Grillo, A.: Conformal algebra in two-space time dimensions and the Thirring model. Nuovo Cim.12A, 959–968 (1972); Mansuri, F., Nambu, Y.: Gauge conditions in dual resonance models. Phys. Lett.39B, 375–378 (1972); Fubini, S., Hanson, A., Jackiw, R.: New approach to field theory. Phys. Rev.D7, 1732–1760 (1973); Lüscher, M., Mack, G.: The energy momentum tensor of critical quantum field theory in 1 + 1 dimensions (Hamburg 1975) (unpublished)Google Scholar
  19. 19.
    Sugawara, H.: A field theory of currents. Phys. Rev.170, 1659–1662 (1968); Sommerfield, C.: Currents as dynamical variables. Phys. Rev.176, 2019–2025 (1968); Coleman, S., Gross, D., Jackiw, R.: Fermion avatars of the Sugawara model. Phys. Rev.180, 1359–1366 (1969); Bardakci, K., Halpern, M.: New dual quark models. Phys. Rev.D3, 2493–2506 (1971)Google Scholar
  20. 20.
    Kac, V. G.: Infinite dimensional Lie algebras and Dedekind's η-function. Funkts. Anal. Prilozh.8, 77–78 (1974) [Engl. transl: Funct. Anal. Appl.8, 68–70 (1974)]; Garland, H.: The arithmetic theory of loop algebras. J. Algebra53, 480–551 (1978); Kac, V. G., Peterson, D. H.: Unitary structure in representations of infinite-dimensional groups and a convexity theorem. Invent. Math.76, 1–14 (1984)Google Scholar
  21. 21.
    Kac, V. G.: Some problems on infinite-dimensional Lie algebras and their representations. In: Lecture Notes in Mathematics. Vol.933, pp. 117–126, Heidelberg, New York, Berlin: Springer 1982Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Victor G. Kac
    • 1
  • Ivan T. Todorov
    • 1
  1. 1.Department of MathematicsM.I.T.CambridgeUSA

Personalised recommendations