Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 8 (2012), 086, 13 pages      arXiv:1208.0809      https://doi.org/10.3842/SIGMA.2012.086
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”

On Affine Fusion and the Phase Model

Mark A. Walton
Department of Physics and Astronomy, University of Lethbridge, Lethbridge, Alberta, T1K 3M4, Canada

Received August 01, 2012, in final form November 08, 2012; Published online November 15, 2012

Abstract
A brief review is given of the integrable realization of affine fusion discovered recently by Korff and Stroppel. They showed that the affine fusion of the su(n) Wess-Zumino-Novikov-Witten (WZNW) conformal field theories appears in a simple integrable system known as the phase model. The Yang-Baxter equation leads to the construction of commuting operators as Schur polynomials, with noncommuting hopping operators as arguments. The algebraic Bethe ansatz diagonalizes them, revealing a connection to the modular S matrix and fusion of the su(n) WZNW model. The noncommutative Schur polynomials play roles similar to those of the primary field operators in the corresponding WZNW model. In particular, their 3-point functions are the su(n) fusion multiplicities. We show here how the new phase model realization of affine fusion makes obvious the existence of threshold levels, and how it accommodates higher-genus fusion.

Key words: affine fusion; phase model; integrable system; conformal field theory; noncommutative Schur polynomials; threshold level; higher-genus Verlinde dimensions.

pdf (377 kb)   tex (21 kb)

References

  1. Altschüler D., Bauer M., Itzykson C., The branching rules of conformal embeddings, Comm. Math. Phys. 132 (1990), 349-364.
  2. Bogoliubov N.M., Izergin A.G., Kitanine N.A., Correlation functions for a strongly correlated boson system, Nuclear Phys. B 516 (1998), 501-528, solv-int/9710002.
  3. Cummins C.J., Mathieu P., Walton M.A., Generating functions for WZNW fusion rules, Phys. Lett. B 254 (1991), 386-390.
  4. Di Francesco P., Mathieu P., Sénéchal D., Conformal field theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York, 1997.
  5. Feingold A.J., Fredenhagen S., A new perspective on the Frenkel-Zhu fusion rule theorem, J. Algebra 320 (2008), 2079-2100, arXiv:0710.1620.
  6. Fomin S., Greene C., Noncommutative Schur functions and their applications, Discrete Math. 193 (1998), 179-200.
  7. Fulton W., Young tableaux. With applications to representation theory and geometry, London Mathematical Society Student Texts, Vol. 35, Cambridge University Press, Cambridge, 1997.
  8. Gepner D., Witten E., String theory on group manifolds, Nuclear Phys. B 278 (1986), 493-549.
  9. Goodman F.M., Nakanishi T., Fusion algebras in integrable systems in two dimensions, Phys. Lett. B 262 (1991), 259-264.
  10. Irvine S.E., Walton M.A., Schubert calculus and threshold polynomials of affine fusion, Nuclear Phys. B 584 (2000), 795-809, hep-th/0004055.
  11. Kac V.G., Peterson D.H., Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. Math. 53 (1984), 125-264.
  12. Kirillov A.N., Mathieu P., Sénéchal D., Walton M.A., Can fusion coefficients be calculated from the depth rule?, Nuclear Phys. B 391 (1993), 651-674, hep-th/9203004.
  13. Korepin V.E., Bogoliubov N.M., Izergin A.G., Quantum inverse scattering method and correlation functions, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1993.
  14. Korff C., Noncommutative Schur polynomials and the crystal limit of the Uqsl^(2)-vertex model, J. Phys. A: Math. Theor. 43 (2010), 434021, 20 pages, arXiv:1006.4710.
  15. Korff C., The su(n) WZNW fusion ring as integrable model: a new algorithm to compute fusion coefficients, in Infinite Analysis 2010 - Developments in Quantum Integrable Systems, RIMS Kôkyûroku Bessatsu, B28, Res. Inst. Math. Sci. (RIMS), Kyoto, 2011, 121-153, arXiv:1106.5342.
  16. Korff C., Stroppel C., The sl^(n)k-WZNW fusion ring: a combinatorial construction and a realisation as quotient of quantum cohomology, Adv. Math. 225 (2010), 200-268, arXiv:0909.2347.
  17. Nepomechie R.I., A spin chain primer, Internat. J. Modern Phys. B 13 (1999), 2973-2985, hep-th/9810032.
  18. Verlinde E., Fusion rules and modular transformations in 2D conformal field theory, Nuclear Phys. B 300 (1988), 360-376.
  19. Walton M.A., Tensor products and fusion rules, Canad. J. Phys. 72 (1994), 527-536.


Previous article  Next article   Contents of Volume 8 (2012)