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

SIGMA 7 (2011), 102, 29 pages      arXiv:1104.4630      https://doi.org/10.3842/SIGMA.2011.102

Classical and Quantum Dilogarithm Identities

Rinat M. Kashaev a and Tomoki Nakanishi b
a) Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, Case postale 64, 1211 Genève 4, Switzerland
b) Graduate School of Mathematics, Nagoya University, Nagoya, 464-8604, Japan

Received May 03, 2011, in final form October 26, 2011; Published online November 01, 2011

Abstract
Using the quantum cluster algebra formalism of Fock and Goncharov, we present several forms of quantum dilogarithm identities associated with periodicities in quantum cluster algebras, namely, the tropical, universal, and local forms. We then demonstrate how classical dilogarithm identities naturally emerge from quantum dilogarithm identities in local form in the semiclassical limit by applying the saddle point method.

Key words: dilogarithm; quantum dilogarithm; cluster algebra.

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References

  1. Baseilhac S., Benedetti R., Quantum hyperbolic invariants of 3-manifolds with PSL(2,C)-characters, Topology 43 (2004), 1373-1423, math.GT/0306280.
  2. Bazhanov V.V., Baxter R.J., Star-triangle relation for a three-dimensional model, J. Statist. Phys. 71 (1993), 839-864, hep-th/9212050.
  3. Bazhanov V.V., Mangazeev V.V., Sergeev S.M., Faddeev-Volkov solution of the Yang-Baxter equation and discrete conformal symmetry, Nuclear Phys. B 784 (2007), 234-258, hep-th/0703041.
  4. Bazhanov V.V., Mangazeev V.V., Sergeev S.M., Quantum geometry of 3-dimensional lattices, J. Stat. Mech. Theory Exp. 2008 (2008), no. 7, P07004, 27 pages, arXiv:0801.0129.
  5. Bazhanov V.V., Reshetikhin N.Yu., Restricted solid-on-solid models connected with simply laced algebras and conformal field theory, J. Phys. A: Math. Gen. 23 (1990), 1477-1492.
  6. Bazhanov V.V., Reshetikhin N.Yu., Remarks on the quantum dilogarithm, J. Phys. A: Math. Gen. 28 (1995), 2217-2226.
  7. Berenstein A., Zelevinsky A., Quantum cluster algebras, Adv. Math. 195 (2005), 405-455, math.QA/0404446.
  8. Berezin F.A., The method of second quantization, Pure and Applied Physics, Vol. 24, Academic Press, New York - London, 1966.
  9. Cecotti S., Neitzke A., Vafa C., R-twisting and 4d/2d correspondences, arXiv:1006.3435.
  10. Chapoton F., Functional identities for the Rogers dilogarithm associated to cluster Y-systems, Bull. London Math. Soc. 37 (2005), 755-760.
  11. 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.
  12. Dirac P.M.A., The principles of quantum mechanics, 4th ed., Clarendon Press, Oxford, 1958.
  13. Faddeev L.D., Discrete Hisenberg-Weyl group and modular group, Lett. Math. Phys. 34 (1995), 249-254, hep-th/9504111.
  14. Faddeev L.D., Current-like variables in massive and massless integrable models, in Quantum Groups and their Applications in Physics (Varenna, 1994), Editors L. Castellani and J. Wess, IOS, Amsterdam, 1996, 117-135, hep-th/9408041.
  15. Faddeev L.D., Kashaev R.M., Quantum dilogarithm, Modern Phys. Lett. A 9 (1994), 427-434, hep-th/9310070.
  16. Faddeev L.D., Kashaev R.M., Volkov A.Yu., Strongly coupled quantum discrete Liouville theory. I. Algebraic approach and duality, Comm. Math. Phys. 219 (2001), 199-219, hep-th/0006156.
  17. Faddeev L.D., Volkov A.Yu., Abelian current algebra and the Virasoro algebra on the lattice, Phys. Lett. 315 (1993), 311-318, hep-th/9307048.
  18. Fock V.V., Goncharov A.B., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 865-930, math.AG/0311245.
  19. Fock V.V., Goncharov A.B., Cluster ensembles, quantization and the dilogarithm. II. The intertwiner, in Algebra, Arithmetic, and Geometry: in Honor of Yu.I. Manin, Vol. I, Progr. Math., Vol. 269, Birkhäuser Boston, Inc., Boston, MA, 2009, 655-673, math.QA/0702398.
  20. Fock V.V., Goncharov A.B., The quantum dilogarithm and representations of quantum cluster varieties, Invent. Math. 172 (2009), 223-286, math.QA/0702397.
  21. Fomin S., Zelevinsky A., Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), 497-529, math.RT/0104151.
  22. Fomin S., Zelevinsky A., Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), 63-121, math.RA/0208229.
  23. Fomin S., Zelevinsky A., Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), 977-1018, hep-th/0111053.
  24. Fomin S., Zelevinsky A., Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), 112-164, math.RT/0602259.
  25. Frenkel E., Szenes A., Thermodynamic Bethe ansatz and dilogarithm identities. I, Math. Res. Lett. 2 (1995), 677-693, hep-th/9506215.
  26. Gaiotto D., Moore G.W., Neitzke A., Four-dimensional wall-crossing via three-dimensional field theory, arXiv:0807.4723.
  27. Gaiotto D., Moore G.W., Neitzke A., Framed BPS states, arXiv:1006.0146.
  28. Gliozzi F., Tateo R., ADE functional dilogarithm identities and integrable models, Phys. Lett. B 348 (1995), 84-88, hep-th/9411203.
  29. Gliozzi F., Tateo R., Thermodynamic Bethe ansatz and three-fold triangulations, Internat. J. Modern Phys. A 11 (1996), 4051-4064, hep-th/9505102.
  30. Goncharov A.B., Pentagon relation for the quantum dilogarithm and quantized M0,5cyc, in Geometry and Dynamics of Groups and Spaces, Progr. Math., Vol. 265, Birkhäuser, Basel, 2008, 415-428, arXiv:0706.4054.
  31. Inoue R., Iyama O., Keller B., Kuniba A., Nakanishi T., Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras I: type Br, Publ. Res. Inst. Math. Sci., to appear, arXiv:1001.1880.
  32. 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, Publ. Res. Inst. Math. Sci., to appear, arXiv:1001.1881.
  33. Kashaev R.M., Quantum dilogarithm as a 6j-symbol, Modern Phys. Lett. A 9 (1994), 3757-3768, hep-th/9411147.
  34. Kashaev R.M., The hyperbolic volume of knots from the quantum dilogarithm, Lett. Math. Phys. 39 (1997), 269-275, q-alg/9601025.
  35. Kashaev R.M., Quantization of Teichmüller spaces and quantum dilogarithm, Lett. Math. Phys. 43 (1998), 105-115, q-alg/9705021.
  36. Kashaev R.M., The q-binomial formula and the Rogers dilogarithm identity, math.QA/0407078.
  37. Kashaev R.M., Discrete Liouville equation and Teichmüller theory, arXiv:0810.4352.
  38. Keller B., The periodicity conjecture for pairs of Dynkin diagrams, arXiv:1001.1531.
  39. Keller B., On cluster theory and quantum dilogarithm identities, in Representations of Algebras and Related Topics, Editors A. Skowronski and K. Yamagata, EMS Series of Congress Reports, European Mathematical Society, 2011, 85-11, arXiv:1102.4148.
  40. Kirillov A.N., Identities for the Rogers dilogarithm function connected with simple Lie algebras, J. Soviet Math. 47 (1989), 2450-2458.
  41. Kirillov A.N., Dilogarithm identities, Progr. Theoret. Phys. Suppl. 118 (1995), no. 118, 61-142, hep-th/9408113.
  42. Kirillov A.N., Reshetikhin N.Yu., 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.
  43. Kontsevich M., Soibelman Y., Stability structures, Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435.
  44. Kontsevich M., Soibelman Y., Motivic Donaldson-Thomas invariants: summary of results, arXiv:0910.4315.
  45. Kontsevich M., Soibelman Y., Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, Commun. Number Theory Phys. 5 (2011), 231-352, arXiv:1006.2706.
  46. Kuniba A., Thermodynamics of the Uq(Xr(1)) Bethe ansatz system with q a root of unity, Nuclear Phys. B 389 (1993), 209-244.
  47. Lewin L., Polylogarithms and associated functions, North-Holland Publishing Co., New York - Amsterdam, 1981.
  48. Nagao K., Donaldson-Thomas theory and cluster algebras, arXiv:1002.4884.
  49. Nagao K., Quantum dilogarithm identities, RIMS Kôkyûroku Bessatsu B 28 (2011), 165-170.
  50. Nagao K., Wall-crossing of the motivic Donaldson-Thomas invariants, arXiv:1103.2922.
  51. Nakanishi T., Dilogarithm identities for conformal field theories and cluster algebras: simply laced case, Nagoya Math. J. 202 (2011), 23-43, arXiv:0909.5480.
  52. Nakanishi T., Periodicities in cluster algebras and dilogarithm identities, in Representations of Algebras and Related Topics, Editors A. Skowronski and K. Yamagata, EMS Series of Congress Reports, European Mathematical Society, 2011, 407-444, arXiv:1006.0632.
  53. Nakanishi T., Tateo R., Dilogarithm identities for sine-Gordon and reduced sine-Gordon Y-systems, SIGMA 6 (2010), 085, 34 pages, arXiv:1005.4199.
  54. Nakanishi T., Zelevinsky A., On tropical dualities in cluster algebras, arXiv:1101.3736.
  55. Plamondon P., Cluster algebras via cluster categories with infinite-dimensional morphism spaces, arXiv:1004.0830.
  56. Reineke M., Poisson automorphisms and quiver moduli, J. Inst. Math. Jussieu 9 (2009), 653-667, arXiv:0804.3214.
  57. Ruijsenaars S.N.M., First order analytic difference equations and integrable quantum systems, J. Math. Phys. 38 (1997), 1069-1146.
  58. Schützenberger M.P., Une interprétation de certaines solutions de l'équation fonctionnelle F(x+y)=F(x)F(y), C. R. Acad. Sci. Paris 236 (1953), 352-353.
  59. Takhtajan L.A., Quantum mechanics for mathematicians, Graduate Studies in Mathematics, Vol. 95, American Mathematical Society, Providence, RI, 2008.
  60. Tran T., F-polynomials in quantum cluster algebras, Algebr. Represent. Theory 14 (2011), 1025-1061, arXiv:0904.3291.
  61. Volkov A.Yu., Noncommutative hypergeometry, Comm. Math. Phys. 258 (2005), 257-273, math.QA/0312084.
  62. Volkov A.Yu., In preparation.
  63. Volkov A.Yu., Pentagon identity revisited I, arXiv:1104.2267.
  64. Woronowicz S.L., Quantum exponential function, Rev. Math. Phys. 12 (2000), 873-920.
  65. Zagier D., The dilogarithm function, in Frontiers in Number Theory, Physics, and Geometry II, Springer, Berlin, 2007, 3-65.

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