Skip to main content
SpringerLink
Log in

A Bosonization of \({U_q(\widehat{sl}(M|N))}\)

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

Abstract

A bosonization of the quantum affine superalgebra \({U_q(\widehat{sl}(M|N))}\) is presented for an arbitrary level \({k \in {\bf C}}\). Screening operators that commute with \({U_q(\widehat{sl}(M|N))}\) are presented for the level \({k \neq -M+N}\).

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. Feigin B.L., Fuchs D.B.: Representations of the Virasoro algebra, representation of Lie groups and related topics. Adv. Stud. Contemp. Math. 7, 465–554 (1990)

    Google Scholar 

  2. Jimbo, M., Miwa, T.: Algebraic analysis of solvable lattice model. In: CBMS Regional Conference Series in Mathematics, vol. 85, AMS (1984)

  3. Frenkel I.B., Jing N.: Vertex representations of quantum affine algebras. Proc. Natl. Acad. Sci. 85, 9373–9377 (1988)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. Bernard D.: Vertex operator representations of quantum affine algebra \({U_q(B_r^{(1)})}\). Lett. Math. Phys. 17, 239–245 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  5. Jing N., Koyama Y., Misra K.: Level one representations of quantum affine algebra \({U_q(C_n^{(1)})}\). Sel. Math. 5(2), 243–255 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Jing N.: Twisted vertex representations of quantum affine algebras. Invent. Math. 102, 663–690 (1990)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Jing N.: Level one representations of \({U_q(G_2^{(1)})}\). Proc. Am. Math. Soc. 127(1), 21–27 (1999)

    Article  MATH  Google Scholar 

  8. Kimura K., Shiraishi J., Uchiyama J.: A level-one representation of the quantum affine superalgebra \({U_q(\widehat{sl}(M+1|N+1))}\). Commun. Math. Phys. 188(2), 367–378 (1997)

    Article  ADS  MATH  Google Scholar 

  9. Zhang Y.-Z.: Level-one representation and vertex operators of quantum affine superalgebra \({U_q(\widehat{gl}(N|N))}\). J. Math. Phys. 40, 6110–6124 (1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Yang W.-L., Zhang Y.-Z.: Vertex operators of \({U_q(\widehat{gl}(N|N))}\) and highest weight representations of \({U_q(\widehat{gl}(2|2))}\). J. Math. Phys. 41, 2460–2481 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Yang W.-L., Zhang Y.-Z.: Drinfeld basis and free boson representation of twisted quantum affine superalgebra \({U_q(osp(2|2)^{(2)})}\). Phys. Lett. A 261, 252–258 (1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. Wakimoto M.: Fock representations of the affine Lie algebra \({A_1^{(1)}}\). Commun. Math. Phys. 104(4), 605–609 (1986)

    Article  ADS  MATH  Google Scholar 

  13. Feigin, B.L., Frenkel, E.V.: Representations of affine Kac-Moody algebras and bosonization. In: Physics and Mathematics of Strings. World Scientific, Singapore, pp. 271–316 (1990)

  14. Iohara K., Koga Y.: Wakimoto modules for the affine Lie superalgebras \({A(m-1, n-1)^{(1)}}\) and \({D(2,1, a)^{(1)}}\). Math. Proc. Camb. Phil. Soc. 132, 419–433 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Matsuo A.: A q-deformation of Wakimoto modules, primary fields and screening operators. Commun. Math. Phys. 160, 33–48 (1994)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. Shiraishi J.: Free boson representation. Phys. Lett. A 171, 243–248 (1992)

    Article  MathSciNet  Google Scholar 

  17. Konno H.: BRST cohomology in quantum affine algebra \({U_q(\widehat{sl}_2)}\). Mod. Phys. Lett. A 9, 1253–1265 (1994)

    Article  ADS  MATH  Google Scholar 

  18. Awata H., Odake S., Shiraishi J.: Free boson realization of \({U_q(\widehat{sl}_N)}\). Commun. Math. Phys. 162(1), 61–83 (1994)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  19. Awata H., Odake S., Shiraishi J.: q-difference realization of \({U_q({\rm sl}(M\vert N))}\) and its application to free boson realization of \({U_q(\widehat{\rm sl}(2\vert 1))}\). Lett. Math. Phys. 42(3), 271–279 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kojima T.: Free field realization of quantum affine superalgebra \({U_q(\widehat{sl}(N|1))}\). J. Math. Phys. 53, 013515 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. Zhang Y.-Z., Gould M.D.: \({U_q(\widehat{sl}(2|1))}\) Vertex operators, screening currents and correlation functions at arbitrary level. J. Math. Phys. 41, 5577–5291 (2000)

    Google Scholar 

  22. Kojima, T.: The q-Wakimoto realization of the superalgebras \({U_q(\widehat{sl}(N|1))}\) and \({U_{q,p}(\widehat{sl}(N|1))}\). In: Springer Proceedings: Mathematics and Statistics, Lie Theory and its Application in Physics, vol. 111, pp. 263–276 (2013)

  23. Kojima T.: Screenings and vertex operators of quantum superalgebra \({U_q(\widehat{sl}(N|1))}\). J. Math. Phys. 53, 083503 (2012). doi:10.1063/1.4742125

    Article  ADS  MATH  MathSciNet  Google Scholar 

  24. Kac V.G.: Lie superalgebras. Adv. Math. 26(1), 8–96 (1977)

    Article  MATH  Google Scholar 

  25. Kac V.G.: A sketch of Lie superalgebra theory. Commun. Math. Phys. 53(1), 31–64 (1977)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  26. Kac V.G., Wakimoto M.: Integrable highest weight modules over affine superalgebras and number theory, Lie theory and geometry. Progr. Math. 123, 415–456 (1994)

    MATH  Google Scholar 

  27. Kac V.G.: Representation of classical Lie superalgebras. Lect. Notes. Math. 676, 597–626 (1978)

    Article  MathSciNet  Google Scholar 

  28. Yamane H.: On defining relations of the affine Lie superalgebras and their quantized universal enveloping superalgebras. Publ. Res. Inst. Math. Sci. 35, 321–390 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  29. Palev T.D., Tolstoy V.N.: Finite-dimensional irreducible representations of the quantum superalgebra \({U_q(gl(n|1))}\). Commun. Math. Phys. 141, 549–558 (1991)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  30. Drinfeld V.G.: A new realization of Yangians and quantized affine algebras. Sov. Math. Dokl. 36, 212–216 (1988)

    Google Scholar 

  31. Felder G.: BRST approach to minimal model. Nucl. Phys. B 317, 215–236 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  32. Bars I.: Free fields and new cosets of current algebras. Phys. Lett. B 255, 353–358 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  33. Isidro J.M., Ramallo A.V.: gl(N, N) current algebras and topological field theories. Nucl. Phys. B 414, 715–762 (1994)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  34. Bowcock P., Koktava R.-L.K., Taormina A.: Wakimoto modules for the affine superalgebra \({\widehat{sl}(2/1; {\bf C)})}\) and noncritical N = 2 strings. Phys. Lett. B 388, 303–308 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  35. Bowcock P., Taormina A.: Representation theory of the affine Lie superalgebra \({\widehat{sl}(2/1; {\bf C)})}\) at fractional level. Commun. Math. Phys. 185, 467–493 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  36. Bowcock P., Feigin B.L., Semikhatov A.M., Taormina A.: \({\widehat{sl}(2|1)}\) and \({\widehat{D}(2|1;\alpha )}\) as vertex operator extensions of dual affine sl(2) algebras. Commun. Math. Phys. 214, 495–545 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  37. Ding X.-M., Gould M.D., Zhang Y.-Z.: \({gl(2|2)}\) current superalgebra and non-unitary conformal field theory. Phys. Lett. A 318, 354–363 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  38. Yang W.-L., Zhang Y.-Z., Liu X.: \({gl(4|4)}\) current algebra: free field realization and screening current. Phys. Lett. B 641, 329–334 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  39. Yang W.-L., Zhang Y.-Z., Liu X.: Free-field representation of super affine \({gl(m|n)}\) currents at general level. J. Math. Phys. 48, 053514 (2007)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Physics, Faculty of Engineering, Yamagata University, Jonan 4-3-16, Yonezawa, 992-8510, Japan

    Takeo Kojima

Authors
  1. Takeo Kojima
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Takeo Kojima.

Additional information

Communicated by Y. Kawahigashi

Dedicated to Professor Michio Jimbo on the occasion of his 65th birthday

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kojima, T. A Bosonization of \({U_q(\widehat{sl}(M|N))}\) . Commun. Math. Phys. 355, 603–644 (2017). https://doi.org/10.1007/s00220-017-2957-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-017-2957-z

Search

Navigation