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

SIGMA 6 (2010), 085, 34 pages      arXiv:1005.4199

Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems

Tomoki Nakanishi a and Roberto Tateo b
a) Graduate School of Mathematics, Nagoya University, Nagoya, 464-8604, Japan
b) Dipartimento di Fisica Teorica and INFN, Universitè di Torino, Via P. Giuria 1, 10125 Torino, Italy

Received May 29, 2010, in final form October 16, 2010; Published online October 19, 2010

We study the family of Y-systems and T-systems associated with the sine-Gordon models and the reduced sine-Gordon models for the parameter of continued fractions with two terms. We formulate these systems by cluster algebras, which turn out to be of finite type, and prove their periodicities and the associated dilogarithm identities which have been conjectured earlier. In particular, this provides new examples of periodicities of seeds.

Key words: cluster algebras; quantum groups; integrable models.

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  1. Baxter R.J., Exactly solved models in statistical mechanics, Academic Press, Inc., London, 1982.
  2. Bazhanov V.V., Reshetikhin N., Restricted solid-on-solid models connected with simply laced algebras and conformal field theory, J. Phys. A: Math. Gen. 23 (1990), 1477-1492.
  3. Belavin A.A., Polyakov A.M., Zamolodchikov A.B., Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Phys. B 241 (1984), 333-380.
  4. Bernard D., LeClair A., Residual quantum symmetries of the restricted sine-Gordon theories, Nuclear Phys. B 340 (1990), 721-751.
  5. Chapoton F., Functional identities for the Rogers dilogarithm associated to cluster Y-systems, Bull. London Math. Soc. 37 (2005), 755-760.
  6. Derksen H., Weyman J., Zelevinsky A., Quivers with potentials and their representations. II. Applications to cluster algebras, J. Amer. Math. Soc. 23 (2010), 749-790, arXiv:0904.0676.
  7. Fomin S., Zelevinsky A., Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), 497-529, math.RT/0104151.
  8. Fomin S., Zelevinsky A., Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), 63-121, math.RA/0208229.
  9. Fomin S., Zelevinsky A., Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), 977-1018, hep-th/0111053.
  10. Fomin S., Zelevinsky A., Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), 112-164, math.RT/0602259.
  11. Frenkel E., Szenes A., Thermodynamic Bethe ansatz and dilogarithm identities. I, Math. Res. Lett. 2 (1995), 677-693, hep-th/9506215.
  12. Gliozzi F., Tateo R., ADE functional dilogarithm identities and integrable models, Phys. Lett. B 348 (1995), 677-693, hep-th/9411203.
  13. Inoue R., Iyama O., Keller B., Kuniba A., Nakanishi T., Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras. I. Type Br, arXiv:1001.1880.
  14. Inoue R., Iyama O., Keller B., Kuniba A., Nakanishi T., Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras. II. Types Cr, F4, and G2, arXiv:1001.1881.
  15. Inoue R., Iyama O., Kuniba A., Nakanishi T., Suzuki J., Periodicities of T and Y-systems, Nagoya Math. J. 197 (2010), 59-174, arXiv:0812.0667.
  16. Keller B., Cluster algebras, quiver representations and triangulated categories, in Triangulated Categories, Editors T. Holm, P. Jørgensen and R. Rouquier, Lecture Note Series, Vol. 375, London Mathematical Society, Cambridge University Press, 2010, 76-160, arXiv:0807.1960.
  17. Keller B., The periodicity conjecture for pairs of Dynkin diagrams, arXiv:1001.1531.
  18. Kirillov A.N., Reshetikhin N.Y., Exact solution of the Heisenberg XXZ model of spin s, J. Soviet Math. 35 (1986), 2627-2643.
  19. Kirillov A.N., Reshetikhin N.Y., Representations of Yangians and multiplicities of the inclusion of the irreducible components of the tensor product of representations of simple Lie algebras, J. Soviet Math. 52 (1990), 3156-3164.
  20. Klassen T.R., Melzer E., Purely elastic scattering theories and their ultraviolet limits, Nuclear Phys. B 338 (1990), 485-528.
  21. Klümper A., Pearce P.A., Conformal weights of RSOS lattice models and their fusion hierarchies, Phys. A 183 (1992), 304-350.
  22. Kuniba A., Thermodynamics of the Uq(Xr(1)) Bethe ansatz system with q a root of unity, Nuclear Phys. B 389 (1993), 209-244.
  23. Kuniba A., Nakanishi T., Spectra in conformal field theories from the Rogers dilogarithm, Modern Phys. Lett. A 7 (1992), 3487-3494, hep-th/9206034.
  24. Kuniba A., Nakanishi T., Suzuki J., Functional relations in solvable lattice models. I. Functional relations and representation theory, Internat. J. Modern Phys. A 9 (1994), 5215-5266, hep-th/9309137.
  25. Lewin L., Polylogarithms and associated functions, North-Holland, Amsterdam, 1981.
  26. Nakanishi T., Dilogarithm identities for conformal field theories and cluster algebras: simply laced case, Nagoya Math. J., to appear, arXiv:0909.5480.
  27. Onsager L., Crystal statistics. I. A two-dimensional model with an order disorder transition, Phys. Rev. 65 (1944), 117-149.
  28. Plamondon P., Cluster algebras via cluster categories with infinite-dimensional morphism spaces, arXiv:1004.0830.
  29. Plamondon P., Cluster characters for cluster categories with infinite-dimensional morphism spaces, arXiv:1002.4956.
  30. Ravanini R., Tateo R., Valleriani A., Dynkin TBA's, Internat. J. Modern Phys. A 8 (1993), 1707-1727, hep-th/9207040.
  31. Smirnov F.A., Reductions of the sine-Gordon model as a perturbation of minimal models of conformal field theory, Nuclear Phys. B 337 (1990), 156-180.
  32. Tateo R., New functional dilogarithm identities and sine-Gordon Y-systems, Phys. Lett. B 355 (1995), 157-164, hep-th/9505022.
  33. Zamolodchikov A.B., Integrable field theory from conformal field theory, in Integrable Systems in Quantum Field Theory and Statistical Mechanics, Adv. Stud. Pure Math., Vol. 19, Academic Press, Boston, MA, 1989, 641-674.
  34. Zamolodchikov A.B., Thermodynamic Bethe ansatz in relativistic models: scaling 3-state Potts and Lee-Yang models, Nuclear Phys. B 342 (1990), 695-720.
  35. Zamolodchikov A.B., Zamolodchikov A.B., Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models, Ann. Physics 120 (1979), 253-291.
  36. Zamolodchikov A.B., On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories, Phys. Lett. B 253 (1991), 391-394.

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