Communications in Mathematical Physics

, Volume 257, Issue 2, pp 395–423 | Cite as

A Monomial Basis for the Virasoro Minimal Series M(p,p′) : The Case 1<p′/p<2

  • B. FeiginEmail author
  • M. Jimbo
  • T. Miwa
  • E. Mukhin
  • Y. Takeyama


Quadratic relations are given explicitly in two cases of chiral conformal field theory, and monomial bases of the representation spaces are constructed by using the Fourier components of the intertwiners. The first case is the (2,1) primary fields for the (p,p′)-minimal series M r,s (1≤rp−1,1≤sp′−1) for the Virasoro algebra where 1<p′/p<2. We restrict ourselves to the case p≥3, for which the (2,1) primary field exists. The second case is the intertwiners corresponding to the two-dimensional representation for the level k integrable highest weight modules V(λ) (0≤λk) for the affine Lie algebra Open image in new window


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Di Francesco, P., Mathieu, P., Sénéchal, D.: Conformal Field Theory. Springer GTCP, New York: Springer, 1997Google Scholar
  2. 2.
    Feigin, B., Jimbo, M., Loktev, S., Miwa, T.: Two character formulas for Open image in new window spaces of coinvariants. Int. J. Mod. Phys. A 19 Suppl.02, 134–154 (2004)Google Scholar
  3. 3.
    Lepowsky, J., Wilson, R.: The structure of standard modules, I: Universal algebras and the Rogers-Ramanujan identities. Invent. Math. 77, 199–290 (1984); II: The case A(1)1, principal gradation, ibid. 79, 417–442 (1985)CrossRefGoogle Scholar
  4. 4.
    Lepowsky, J., Primc, M.: Structure of the standard modules for the affine Lie algebra A(1)1. Contemp. Math. 46, Providence, RI: Am. Math. Soc., 1985, pp. 1–84Google Scholar
  5. 5.
    Primc, M.: Vertex operator construction of standard modules for A(1)n. Pacific J. Math. 162, 143–187 (1994)Google Scholar
  6. 6.
    Meurman, A., Primc, M.: Annihilating fields of standard modules of Open image in new window and combinatorial identities. Mem. Am. Math. Soc. 137(652), 1999Google Scholar
  7. 7.
    Andrews, G., Baxter, R., Forrester, P.: Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities. J. Stat. Phys. 35, 193–266 (1984)CrossRefGoogle Scholar
  8. 8.
    Forrester, P., Baxter, R.: Further exact solutions of the eight-vertex SOS model and generalizations of the Rogers-Ramanujan-type identities. J. Stat. Phys. 38, 435–472 (1985)CrossRefGoogle Scholar
  9. 9.
    Huse, D.A.: Exact exponents for infinitely many new multi-critical points. Phys. Rev. B30, 3908–3915 (1984)Google Scholar
  10. 10.
    Feigin, B., Frenkel, E.: Coinvariants of Nilpotent Subalgebras of the Virasoro Algebra and Partition Identities. Adv. Soviet Math. 30, Part I 139–148 (1993)Google Scholar
  11. 11.
    Feigin, B., Jimbo, M., Miwa, T.: Vertex operator algebra arising from the minimal series M(3,p) and monomial basis. In: Proceedings of MathPhys Odessey 2001, Okayama, Basel-Boston: Birkhäuser, 2003, pp. 179–204Google Scholar
  12. 12.
    Foda, O., Lee, K.S.M., Pugai, Y., Welsh, T.A.: Path generating transforms. q-series from a contemporary perspective (South Hadley, MA, 1998). Contemp. Math. 254, Providence, RI: Amer. Math. Soc., 2000, pp. 157–186Google Scholar
  13. 13.
    Feigin, B., Miwa, T.: Extended vertex operator algebras and monomial bases. In: McGuire Festschrift, Statistical Physics on the Eve of the Twenty-First Century, M. Batchelor et al (eds.), Singapore: World Scientific, 1999Google Scholar
  14. 14.
    Huang, Y.-Z.: Generalized rationality and a “Jacobi identity” for intertwining operator algebras. Selecta Mat. 6, 225–267 (2000)Google Scholar
  15. 15.
    Feigin, B., Nakanishi, T., Ooguri, H.: The annihilating ideals of minimal models. In: Proceedings of the RIMS Research Project 1991, Infinite Analysis B, Advanced Series in Mathematical Physics Vol. 16, Singapore: World Scientific, 1992, pp. 217–238Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • B. Feigin
    • 1
    Email author
  • M. Jimbo
    • 2
  • T. Miwa
    • 3
  • E. Mukhin
    • 4
  • Y. Takeyama
    • 5
  1. 1.Landau Institute for Theoretical PhysicsChernogolovkaRussia
  2. 2.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan
  3. 3.Department of Mathematics, Graduate School of ScienceKyoto UniversityKyotoJapan
  4. 4.Department of MathematicsIndiana University-Purdue University-IndianapolisIndianapolisUSA
  5. 5.Institute of MathematicsUniversity of TsukubaTsukubaJapan

Personalised recommendations