Communications in Mathematical Physics

, Volume 154, Issue 3, pp 471–508 | Cite as

New Jacobi-like identities for Z K parafermion characters

  • Philip C. Argyres
  • Keith R. Dienes
  • S. -H. Henry Tye


We state and prove various new identities involving theZ K parafermion characters (or level-K string functions)c n l for the casesK=4,K=8, andK=16. These identities fall into three classes: identities in the first class are generalizations of the famous Jacobi ϑ-function identity (which is theK=2 special case), identities in another class relate the levelK>2 characters to the Dedekind η-function, and identities in a third class relate theK>2 characters to the Jacobi ϑ-functions. These identities play a crucial role in the interpretation of fractional superstring spectra by indicating spacetime supersymmetry and aiding in the identification of the spacetime spin and statistics of fractional superstring states.


Neural Network Statistical Physic Crucial Role Complex System Nonlinear Dynamics 
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  1. 1.
    Zamolodchikov, A.B., Fateev, V.A.: Sov. Phys. J.E.T.P.,62, 215 (1985)Google Scholar
  2. 2.
    Argyres, P.C., Tye, S.-H.H.: Phys. Rev. Lett.67, 3339 (1991)CrossRefGoogle Scholar
  3. 3.
    See, e.g., Kač, V.G.: Infinite Dimensional Lie Algebras. (3rd ed.) Cambridge: Cambridge University Press, 1990Google Scholar
  4. 4.
    Gepner, D., Qiu, Z.: Nucl. Phys.B285, 423 (1987)CrossRefGoogle Scholar
  5. 5.
    Knizhnik, V.G., Zamolodchikov, A.B.: Nucl. Phys.B247, 83 (1984)CrossRefGoogle Scholar
  6. 6.
    Kač, V.G.: Adv. Math.35, 264 (1980); Kač, V.G., Peterson, D.: Bull. AMS 3, 1057 (1980); ibid. Kač, V.G., Peterson, D.: Adv. Math.53, 125 (1984)CrossRefGoogle Scholar
  7. 7.
    Distler, J., Qiu, Z.: Nucl. Phys.B336, 533 (1980)CrossRefGoogle Scholar
  8. 8.
    Dienes, K.R., Tye, S.-H.H.: Nucl. Phys.B376, 297 (1992)CrossRefGoogle Scholar
  9. 9.
    Kastor, D., Martinec, E., Qiu, Z.: Phys. Lett.200B, 434 (1988); Bagger, J. Nemeschansky, D., Yankielowicz, S.: Phys. Rev. Lett.60, 389 (1988); Ravanini, F.: Mod. Phys. Lett.A3, 397 (1988); Chung, S.-W. Lyman, E., Tye, S.-H.H.: Int. J. Mod. Phys.A7, 3339 (1992)Google Scholar
  10. 10.
    Argyres, P.C., Grochocinski, J., Tye, S.-H.H.:Nucl. Phys B367, 217 (1991); ibid., Argyres, P.C., Grochocinski, J., Tye, S.-H.H.: Nucl. Phys.B391, 409 (1993)CrossRefGoogle Scholar
  11. 11.
    Gliozzi, F., Scherk, J., Olive, D.: Nucl. PhysB122, 253 (1977). For a review see: Green, M.B., Schwarz, J.H., Witten, E.: Superstring Theory. Vols. 1 and 2 Cambridge: Cambridge University Press, 1987CrossRefGoogle Scholar
  12. 12.
    Aneziris, C., Balachandran, A.P., Kauffmann L., Srivastava, A.M.: Int. J. Mod. Phys.A6, 2519 (1991); Harvey, J.A., Liu, J.: Phys. Lett.B240, 369 (1990)CrossRefGoogle Scholar
  13. 13.
    Goldhaber, A.S.: Phys. Rev. Lett.36, 1122 (1976)CrossRefGoogle Scholar
  14. 14.
    Koblitz, N.: Introduction to Elliptic Curves and Modular Forms. Berlin, Heidelberg, New York: Springer 1984Google Scholar
  15. 15.
    Gunning, R.C.: Lectures on Modular Forms. Princeton, NJ: Princeton University Press 1962Google Scholar
  16. 16.
    Lang, S.: Introduction to Modular Forms. Berlin, Heidelberg, New York: Springer 1976; Lehner, J.: Lectures on Modular Forms. National Bureau of Standards Applied Mathematics Series No. 61, Washington, D.C.: U.S. GPO, 1969; Rankin R.A.: Modular Forms and Functions. Cambridge, England: Cambridge University Press, 1977; Serre, J.-P.: A Course in Arithmetic. Berlin, Heidelberg, New York: Springer 1973Google Scholar
  17. 17.
    Vafa, C., Witten, E.: Phys. Lett.159B, 265 (1985)Google Scholar
  18. 18.
    Argyres, P.C., Dienes, K.R.: Cornell preprint CLNS 92/1168, McGill preprint McGill/92-41 (to appear)Google Scholar
  19. 19.
    Dienes, K.R.: On the Internal Projections and Massive Sectors of the Fractional Superstring. McGill preprint McGill/93-01 (to appear)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Philip C. Argyres
    • 1
  • Keith R. Dienes
    • 2
  • S. -H. Henry Tye
    • 1
  1. 1.Newman Laboratory of Nuclear StudiesCornell UniversityIthacaUSA
  2. 2.Dept. of PhysicsMcGill UniversityMontréalCanada

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