Skip to main content
SpringerLink
Log in

Residues of $q$-Hypergeometric Integrals and Characters of Affine Lie Algebras

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

Abstract

We study certain subspaces of solutions to the $sl_2$ rational qKZ equation at level zero. Each subspace is specified by the vanishing of the residue at a certain divisor which stems from models in two dimensional integrable field theories. We determine the character of the subspace which is parametrized by the number of variables and the sl 2 weight of the solutions. The sum of all characters with a fixed weight gives rise to the branching functions of the irreducible representations of sl 2 in the level one integrable highest weight representations of \(\widehat{sl_2}\). It is written in the fermionic form.

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. Babelon, O., Bernard, D., Smirnov, F.: Null-vectors in integrable field theory. Commun. Math. Phys. 186, 601–648 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cardy, J., Mussardo, G.: Form factors of descendent operators in perturbed conformal field theories. Nucl. Phys. B340, 387–402 (1990)

  3. Christe, P.: Factorized characters and form factors of descendant operators in perturbed conformal systems. Int. J. Mod. Phys. A6, 5271–5286 (1991)

  4. Frenkel, I., Reshetikhin, N.: Quantum affine algebras and holonomic difference equations. Commun. Math. Phys. 146, 1–60 (1992)

    MathSciNet  MATH  Google Scholar 

  5. Kedem, R., McCoy, B., Melzer, E.: The sums of Rogers, Schur and Ramanujan and the Bose-Fermi correspondence in 1+1-dimensional quantum field theory. In: Recent Progress in Statistical Mechanics and Quantum Field Theory, P. Bouwknegt et al. (eds), Singapore: World Scientific, 1995, pp. 195–219

  6. Koubek, A.: The space of local operators in perturbed conformal field theories. Nucl. Phys. B. 435, 703–734 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kac, V.: Infinite Dimensional Lie Algebras. Third edition, Cambridge: Cambridge University Press, 1992

  8. Kirillov, A.N., Smirnov, F.: A representation of the current algebra connected with the su(2)-invariant Thirring model. Phys. Lett. B 198, 506–510 (1987)

    Article  Google Scholar 

  9. Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Second edition, Oxford: Oxford University Press, 1995

  10. Melzer, E.: The many faces of a character. Lett. Math. Phys. 31, 233–246 (1994)

    MathSciNet  MATH  Google Scholar 

  11. Nakayashiki, A., Pakuliak, S., Tarasov, V.: On solutions of the KZ and qKZ equations at level zero. Ann. Inst. Henri Poincaré 71, 459–496 (1999)

    MATH  Google Scholar 

  12. Nakayashiki, A., Takeyama, Y.: On form factors of SU(2) invariant Thirring model. In: MathPhys Odyssey 2001, Integrable Models and Beyond- in honor of Barry, M. McCoy, Kashiwara, M. Miwa, T., (eds), Progr. Math. Phys., Basel-Bostan: Birkhäuser, 2002, pp. 357–390

  13. Rocha-Caridi, A.: Vacuum vector representations of the Virasoro algebra. In: Vertex operators in Mathematics and Physics, Leowsky, J. et al. (eds), Berlin: Springer, 1985, pp. 451–473

  14. Smirnov, F.: Lectures on integrable massive models of quantum field theory. In: Nankai Lectures on Mathem. Physics, Ge,~M-L., Zhao,~B-H. (eds), Singapore: World Scientific, 1990, pp. 1–68

  15. Smirnov, F.: Dynamical symmetries of massive integrable models. Int. J. Mod. Phys. A 7, Suppl. 1B, 813–837 (1992)

  16. Smirnov, F.: Form factors in Completely Integrable Models of Quantum Field Theories. Singapore: World Scientific, 1992

  17. Smirnov, F.: Counting the local fields in SG theory. Nucl. Phys. B 453, 807–824 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tarasov, V.: Completeness of the hypergeometric solutions of the qKZ equations at level zero. Am. Math. Soc. Translations, Ser. 2 201, 309–321 (2000)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics, Kyushu University, Ropponmatsu 4-2-1, Fukuoka, 810-8560, Japan

    Atsushi Nakayashiki

Authors
  1. Atsushi Nakayashiki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Atsushi Nakayashiki.

Additional information

Communicated by L. Takhtajan

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nakayashiki, A. Residues of $q$-Hypergeometric Integrals and Characters of Affine Lie Algebras. Commun. Math. Phys. 240, 197–241 (2003). https://doi.org/10.1007/s00220-003-0893-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-003-0893-6

Keywords

Search

Navigation