The Ramanujan Journal

, Volume 17, Issue 1, pp 33–52 | Cite as

Andrews–Gordon type identities from combinations of Virasoro characters

Article

Abstract

For p∈{3,4} and all p′>p, with p′ coprime to p, we obtain fermionic expressions for the combination χ1,sp,p+qΔχp−1,sp,p of Virasoro (W2) characters for various values of s, and particular choices of Δ. Equating these expressions with known product expressions, we obtain q-series identities which are akin to the Andrews–Gordon identities. For p=3, these identities were conjectured by Bytsko. For p=4, we obtain identities whose form is a variation on that of the p=3 cases. These identities appear to be new.

The case (p,p′)=(3,14) is particularly interesting because it relates not only to W2, but also to W3 characters, and offers W3 analogues of the original Andrews–Gordon identities. Our fermionic expressions for these characters differ from those of Andrews et al. which involve Gaussian polynomials.

Keywords

Andrews–Gordon identities q-Series identities Virasoro characters 

Mathematics Subject Classification (2000)

05A30 05A19 17B68 81T40 11P82 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andrews, G.E.: An analytic generalization of the Rogers–Ramanujan identities for odd moduli. Proc. Nat. Acad. Sci. USA 71, 4082–4085 (1974) MATHCrossRefGoogle Scholar
  2. 2.
    Andrews, G.E.: Multiple series Rogers–Ramanujan type identities. Pac. J. Math. 114, 267–283 (1984) MATHGoogle Scholar
  3. 3.
    Andrews, G.E., Schilling, A., Warnaar, S.O.: An A2 Bailey lemma and Rogers–Ramanujan-type identities. J. Am. Math. Soc. 12, 677–702 (1999) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bailey, W.N.: Some identities in combinatory analysis. Proc. Lond. Math. Soc. (2) 49, 421–435 (1947) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Berkovich, A., McCoy, B.M.: Continued fractions and fermionic representations for characters of M(p,p′) minimal models. Lett. Math. Phys. 37, 49–66 (1996) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Berkovich, A., McCoy, B.M., Schilling, A.: Rogers–Schur–Ramanujan type identities for the M(p,p′) minimal models of conformal field theory. Commun. Math. Phys. 191, 325–395 (1998) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bytsko, A.G.: Fermionic representations for characters of ℳ(3,t), ℳ(4,5), ℳ(5,6) and ℳ(6,7) minimal models and related dilogarithm and Rogers–Ramanujan-type identities. J. Phys. A Math. Gen. 32, 8045–8058 (1999) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bytsko, A.G., Fring, A.: Factorized combinations of Virasoro characters. Commun. Math. Phys. 209, 179–205 (2000) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Christe, P.: Factorized characters and form factors of descendant operators in perturbed conformal systems. Int. J. Mod. Phys. A 6, 5271–5286 (1991) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Fateev, V.A., Lykyanov, S.L.: The models of two-dimensional conformal quantum field theory with ℤn symmetry. Int. J. Mod. Phys. A 3, 507–520 (1988) CrossRefMathSciNetGoogle Scholar
  11. 11.
    Feigin, B.L., Fuchs, D.B.: Verma modules over the Virasoro algebra. Funct. Anal. Appl. 17, 241–242 (1983) CrossRefGoogle Scholar
  12. 12.
    Foda, O., Quano, Y.-H.: Virasoro character identities from the Andrews–Bailey construction. Int. J. Mod. Phys. A 12, 1651–1675 (1997) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications, vol. 35. Cambridge University Press, Cambridge (1990) MATHGoogle Scholar
  14. 14.
    Gordon, B.: A combinatorial generalization of the Rogers–Ramanujan identities. Am. J. Math. 83, 393–399 (1961) MATHCrossRefGoogle Scholar
  15. 15.
    Kedem, R., Klassen, T.R., McCoy, B.M., Melzer, E.: Fermionic sum representations for conformal field theory characters. Phys. Lett. B 307, 68–76 (1993) MATHMathSciNetGoogle Scholar
  16. 16.
    Rocha-Caridi, A.: Vacuum vector representations of the Virasoro algebra. In: Lepowsky, J., Mandelstam, S., Singer, I.M. (eds.) Vertex Operators in Mathematics and Physics. Springer, Berlin (1985) Google Scholar
  17. 17.
    Rogers, L.J.: Second memoir on the expansion of certain infinite products. Proc. Lond. Math. Soc. 25, 318–343 (1894) CrossRefGoogle Scholar
  18. 18.
    Rogers, L.J.: On two theorems of combinatory analysis and some allied identities. Proc. Lond. Math. Soc. (2) 16, 315–336 (1917) Google Scholar
  19. 19.
    Rogers, L.J., Ramanujan, S.: Proof of certain identities in combinatory analysis. Proc. Cambridge Philos. Soc. 19, 211–216 (1919) MATHGoogle Scholar
  20. 20.
    Slater, L.J.: Further identities of the Rogers–Ramanujan type. Proc. Lond. Math. Soc. (2) 54, 147–167 (1952) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Warnaar, S.O.: Hall–Littlewood functions and the A2 Rogers–Ramanujan identities. Preprint: math.CO/0410592 (2004) Google Scholar
  22. 22.
    Welsh, T.A.: Fermionic expressions for minimal model Virasoro characters. Mem. Am. Math. Soc. 175(827) (2005) Google Scholar
  23. 23.
    Zamolodchikov, A.B.: Infinite additional symmetries in two-dimensional conformal quantum field theory. Teor. Mat. Fiz. 65, 347–359 (1985) MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsMoscow regionRussia
  2. 2.Department of Mathematics and StatisticsUniversity of MelbourneVictoriaAustralia
  3. 3.School of MathematicsUniversity of SouthamptonSouthamptonUK

Personalised recommendations