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Dilogarithm identities for conformal field theories and cluster algebras: Simply laced case

  • Tomoki Nakanishi (a1)

Abstract

The dilogarithm identities for the central charges of conformal field theories of simply laced type were conjectured by Bazhanov, Kirillov, and Reshetikhin. Their functional generalizations were conjectured by Gliozzi and Tateo. They have been partly proved by various authors. We prove these identities in full generality for any pair of Dynkin diagrams of simply laced type based on the cluster algebra formulation of the Y-systems.

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References

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[BR1] Bazhanov, V. V. and Reshetikhin, N. Y., Critical RSOS models and conformal field theory, Internat. J. Modern Phys. A 4 (1989), 115142.
[BR2] Bazhanov, V. V. and Reshetikhin, N. Y., Restricted solid-on-solid models connected with simply laced algebras and conformal field theory, J. Phys. A 23 (1990), 14771492.
[B] Bloch, S., “Applications of the dilogarithm function in algebraic K-theory and algebraic geometry” in Proceedings of the International Symposium on Algebraic Geometry (Kyoto, 1977), Kinokuniya, Tokyo, 1978, 103114.
[CGT] Caracciolo, R., Gliozzi, F., and Tateo, R., A topological invariant of RG flows in 2D integrable quantum field theories, Internat. J. Modern Phys. 13 (1999), 29272932.
[C] Chapoton, F., Functional identities for the Rogers dilogarithm associated to cluster Y-systems, Bull. Lond. Math. Soc. 37 (2005), 755760.
[DWZ] Derksen, H., Weyman, J., and Zelevinsky, A., Quivers with potentials and their representations II: Applications to cluster algebras, J. Amer. Math. Soc. 23 (2010), 749790.
[DS] Dupont, J. L. and Sah, C.-H., Dilogarithm identities in conformal field theory and group homology, Comm. Math. Phys. 161 (1994), 265282.
[FZ] Fateev, V. A. and Zamolodchikov, A. B., Nonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in ZN-symmetric statistical systems, J. Exp. Theor. Phys. 62 (1985), 215225.
[FG] Fock, V. V. and Goncharov, A. B., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4) 39 (2009), 865930.
[FZ1] Fomin, S. and Zelevinsky, A., Cluster algebras, I: Foundations, J. Amer. Math. Soc. 15 (2002), 497529.
[FZ2] Fomin, S. and Zelevinsky, A., Cluster algebras, II: Finite type classification, Invent. Math. 154 (2003), 63121.
[FZ3] Fomin, S. and Zelevinsky, A., Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), 9771018.
[FZ4] Fomin, S. and Zelevinsky, A., Cluster algebras, IV: Coefficients, Compos. Math. 143 (2007), 112164.
[FS] Frenkel, E. and Szenes, A., Thermodynamic Bethe ansatz and dilogarithm identities, I, Math. Res. Lett. 2 (1995), 677693.
[G] Gepner, D., New conformal field theories associated with Lie algebras and their partition functions, Nuclear Phys. B 290 (1987), 1024.
[GW] Gepner, D. and Witten, E., String theory on group manifolds, Nuclear Phys. B 278 (1986), 493549.
[GT1] Gliozzi, F. and Tateo, R., ADE functional dilogarithm identities and integrable models, Phys. Lett. B 348 (1995), 677693.
[GT2] Gliozzi, F. and Tateo, R., Thermodynamic Bethe ansatz and three-fold triangulations, Internat. J. Modern Phys. A 11 (1996), 40514064.
[HW] Hutchins, H. C. and Weinert, H. J., Homomorphisms and kernels of semifields, Period. Math. Hungar. 21 (1990), 113152.
[IIKKN1] Inoue, R., Iyama, O., Keller, B., Kuniba, A., and Nakanishi, T., Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras, I: Type Br , preprint, arXiv:1001.1880 [math.QA]
[IIKKN2] Inoue, R., Iyama, O., Keller, B., Kuniba, A., and Nakanishi, T., Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras, II: Types Cr, F4, and G2 , preprint, arXiv:1001.1881 [math.QA]
[IIKNS] Inoue, R., Iyama, O., Kuniba, A., Nakanishi, T., and Suzuki, J., Periodicities of T-systems and Y-systems, Nagoya Math. J. 197 (2010), 59174.
[Ke1] Keller, B., Cluster algebras, quiver representations and triangulated categories, preprint, arXiv:0807.1960 [math.RT]
[Ke2] Keller, B., The periodicity conjecture for pairs of Dynkin diagrams, preprint, arXiv:1001.1531 [math.RT]
[K1] Kirillov, A. N., Identities for the Rogers dilogarithm function connected with simple Lie algebras, J. Soviet Math. 47 (1989), 24502458.
[K2] Kirillov, A. N., Dilogarithm identities, Progr. Theoret. Phys. Suppl. 118 (1995), 61142.
[KR1] Kirillov, A. N. and Reshetikhin, N. Y., Exact solution of the Heisenberg XXZ model of spin s, J. Soviet Math. 35 (1986), 26272643.
[KR2] Kirillov, A. N. and 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), 31563164.
[KnZ] Knizhnik, V. G. and Zamolodchikov, A. B., Current algebra and Wess-Zumino model in two dimensions, Nuclear Phys. B 247 (1984), 83103.
[Ku] Kuniba, A., Thermodynamics of the Uq () Bethe ansatz system with q a root of unity, Nuclear Phys. B 389 (1993), 209244.
[KuN] Kuniba, A. and Nakanishi, T., Spectra in conformal field theories from the Rogers dilogarithm, Modern Phys. Lett. A 7 (1992), 34873494.
[KuNS] Kuniba, A., Nakanishi, T., and Suzuki, J., T-systems and Y-systems for quantum affinizations of quantum Kac-Moody algebras, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), 123.
[L] Lewin, L., Polylogarithms and Associated Functions, North-Holland, Amsterdam, 1981.
[N] Nahm, W., “Conformal field theory and torsion elements of the Bloch group” in Frontiers in Number Theory, Physics, and Geometry, II, Springer, Berlin, 2007, 67132
[NK] Nahm, W. and Keegan, S., Integrable deformations of CFTs and the discrete Hirota equations, preprint, arXiv.0905.3776 [hep-th]
[NRT] Nahm, W., Recknagel, A., and Terhoeven, M., Dilogarithm identities in conformal field theory, Modern Phys. Lett. A 8 (1993), 18351847.
[RTV] Ravanini, F., Tateo, R., and Valleriani, A., Dynkin TBA’s, Internat. J. Modern Phys. A 8 (1993), 17071727.
[RS] Richmond, B. and Szekeres, G., Some formulas related to dilogarithm, the zeta function and the Andrews-Gordon identities, J. Aust. Math. Soc. 31 (1981), 362373.
[S] Szenes, A., Periodicity of Y-systems and flat connections, Lett. Math. Phys. 89 (2009), 217230.
[V] Volkov, A. Y., On the periodicity conjecture for Y-systems, Comm. Math. Phys. 276 (2007), 509517.
[Zag1] Zagier, D., “Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields” in Arithmetic Algebraic Geometry, Progr. Math. 89, Birkhäuser, Boston, 1990, 391430.
[Zag2] Zagier, D., “The dilogarithm function” in Frontiers in Number Theory, Physics, and Geometry, II, Springer, Berlin, 2007, 365.
[Zam] Zamolodchikov, A. B., On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories, Phys. Lett. B 253 (1991), 391394.
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Dilogarithm identities for conformal field theories and cluster algebras: Simply laced case

  • Tomoki Nakanishi (a1)

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