Skip to main content
SpringerLink
Log in

Quantum Reduction for Affine Superalgebras

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We extend the homological method of quantization of generalized Drinfeld–Sokolov reductions to affine superalgebras. This leads, in particular, to a unified representation theory of superconformal algebras.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bakalov, B., Kac, V. G.: Field algebras. IMRN 3, 123–159 (2003) QA/ 0204282

    Article  MATH  Google Scholar 

  2. Bershadsky, M.: Conformal field theory via Hamiltonian reduction. Commun. Math. Phys. 139, 71–82 (1991)

    MathSciNet  MATH  Google Scholar 

  3. Bershadsky, M.: Phys. Lett. 174B, 285 (1986); V.G. Knizhnik, Theor. Math. Phys. 66, 68 (1986)

    Article  MathSciNet  Google Scholar 

  4. de Boer, J., Tjin, T.: The relation between quantum W-algebras and Lie algebras. Commun. Math. Phys. 160, 317–332 (1994).

    MATH  Google Scholar 

  5. Bouwknegt, P., Schoutens, K.: W-symmetry. Advanced ser Math. Phys., Vol~22, Singapore: World Scientific, 1995

  6. Dobrev, V. K.: Characters of the unitarizable highest weight modules over N=2 superconformal algebras. Phys. Lett. B 186, 43–51 (1987)

    Article  MathSciNet  Google Scholar 

  7. Feigin, B.L., Frenkel, E.: Quantization of Drinfeld-Sokolov reduction. Phys. Lett. B 246, 75–81 (1990)

    Article  MathSciNet  Google Scholar 

  8. Feigin, B.L., Frenkel, E.: Representations of affine Kac-Moody algebras, bozonization and resolutions. Lett. Math. Phys. 19, 307–317 (1990)

    MathSciNet  MATH  Google Scholar 

  9. Fiebig, P.: The combinatorics of category ᵊA for symmetrizable Kac–Moody algebras. 2002 preprint

  10. Fradkin, E.S., Linetsky, V. Ya.: Classification of superconformal and quasisuperconformal algebras in two dimensions. Phys. Lett. B 291, 71–76 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Frenkel, E., Kac, V., Wakimoto, M.: Characters and fusion rules for W-algebras via quantized Drinfeld-Sokolov reduction. Commun. Math. Phys. 147, 295–328 (1992)

    MathSciNet  MATH  Google Scholar 

  12. Goddard, P., Schwimmer, A.: Factoring out free fermions and superconformal algebras. Phys. Lett. 214B, 209–214 (1988)

    Article  MathSciNet  Google Scholar 

  13. Kac, V.G.: Lie superalgebras. Adv. Math. 26, 8–96 (1977)

    MATH  Google Scholar 

  14. Kac, V.G.: Infinite-dimensional algebras, Dedekind’s η-function, classical Möbius function and the very strange formula. Adv. Math. 30, 85–136 (1978)

    MathSciNet  MATH  Google Scholar 

  15. Kac, V.G.: Infinite-dimensional Lie algebras. 3rd edition. Cambridge: Cambridge University Press, 1990

  16. Kac, V.G.: Vertex algebras for beginners. Providence: AMS, University Lecture Series, Vol. 10, 1996. Second edition, 1998

    Google Scholar 

  17. Kac, V.G.: Classification of supersymmetries. ICM talk, August 2002

  18. Kac, V.G., Wakimoto, M.: Modular invariant representations of infinite-dimensional Lie algebras and superalgebras. Proc. Natl. Acad. Sci. USA 85, 4956–4960 (1988)

    MATH  Google Scholar 

  19. Kac, V.G., Wakimoto, M.: Classification of modular invariant representations of affine algebras. In: Infinite-dimensional Lie algebras and groups. Advanced Ser. Math. Phys. Vol. 7, Singapore: World Scientific, 1989, pp. 138–177

  20. Kac, V.G., Wakimoto, M.: Integrable highest weight modules over affine superalgebras and number theory. Progress in Math., 123, Boston: Birkhäuser, 1994, pp. 415–456

  21. Kac, V.G., Wakimoto, M.: Integrable highest weight modules over affine superalgebras and Appell’s function. Commun. Math, Phys. 215, 631–682 (2001)

    Google Scholar 

  22. Kac, V.G., Wakimoto, M.: Quantum reduction and representation theory of superconformal algebras. math-ph/0304011

  23. Khovanova, T.: Super K dV equation related to the Neveu–Schwarz-2 Lie superalgebra of string theory. Teor. Mat. Phys. 72, 306–312 (1987)

    MATH  Google Scholar 

  24. Kiritsis, E.B.: Character formulae and the structure of the presentations of the N=1, N=2 superconformal algebras. Int. J. Mod. Phys. A. 3, 1871–1906 (1988)

    MathSciNet  Google Scholar 

  25. Matsuo, Y.: Character formula of C<1 unitary representation of N=2 superconformal algebra. Prog. Theor. Phys. 77, 793–797 (1987)

    MathSciNet  Google Scholar 

  26. Ravanini, F., Yang, S-K.: Modular invariance in N=2 superconformal field theories. Phys. Lett. B195, 202–208 (1987)

    Google Scholar 

  27. Roan, S.S.: Heisenberg and modular invariance of N=2 conformal field theory, Intern. J. Mod. Phys. A. 15, 3065–3094 (2000), hep-th/9902198

    Article  MathSciNet  MATH  Google Scholar 

  28. Wakimoto, M.: Lectures on infinite-dimensional Lie algebra. Singapore: World Scientific, 2001

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, M. I. T., Cambridge, MA 02139, USA

    Victor Kac

  2. Institute of Mathematics, Academia Sinica, Taipei, Taiwan. R.O.C

    Shi-Shyr Roan

  3. Graduate School of Mathematics, Kyushu University, Fukuoka, 812-8581, Japan

    Minoru Wakimoto

Authors
  1. Victor Kac
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shi-Shyr Roan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Minoru Wakimoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Victor Kac.

Additional information

Communicated by L. Takhtajan

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kac, V., Roan, SS. & Wakimoto, M. Quantum Reduction for Affine Superalgebras. Commun. Math. Phys. 241, 307–342 (2003). https://doi.org/10.1007/s00220-003-0926-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-003-0926-1

Keywords

Search

Navigation