Abstract
We argue that rational conformally invariant quantum field theories in two dimensions are closely related to torsion elements of the algebraic K-theory group K3(C). If such a field theory has an integrable perturbation with purely elastic scattering matrix, then its partition function has a canonical sum representation. The corresponding asymptotic behaviour of the density of states is given in terms of the solutions of an algebraic equation which can be read off from the scattering matrix. These solutions yield torsion elements of an extension of the Bloch group which seems to be equal to K3(C). The algebraic equations are solved for integrable models given by arbitrary pairs of A-type Cartan matrices. The paper should be readable by mathematicians.
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References
E. Ardonne, P. Bouwknegt, and P. Dawson, K-matrices for 2D conformal field theories, Nucl.Phys. B660 (2003), 473–531
G.E. Andrews, The theory of partitions, in: Encyclopedia of Mathematics and its Applications, Vol. 2, Addison Wesley, 1976
A. Berkovich, B.M. McCoy, The universal chiral partition function for exclusion statistics, hep-th/9808013
P. Dorey, Exact S-matrices, hep-th/9810026
A. Coste, T. Gannon, Remarks on Galois symmetry in rational conformal field theories, Phys.Lett. B323 (1994), 316–321
P. Dorey, R. Tateo, Excited states in some simple perturbed conformal field theories, Nucl.Phys. B515 (1998), 575–623
E. Frenkel, A. Szenes, Crystal bases, dilogarithm identities and torsion in algebraic K-groups, J. Amer. Math. Soc. 8 (1995), 629–664, hep-th/9304118
E. Frenkel, A. Szenes, Thermodynamic Bethe Ansatz and Dilogarithm Identities I, Math. Res. Lett. 2 (1995), 677–693, hep-th/9506215
H. Gangl, D. Zagier, Classical and Elliptic Polylogarithms and Special Values of L-Series, in: B.B. Gordon et al. eds., The Arithmetic and Geometry of Algebraic Cycles, 2000
F. Gliozzi, R. Tateo, Thermodynamic Bethe Ansatz and Threefold Triangulations, Int.J.Mod.Phys. A11 (1996), 4051–4064
F. Gliozzi, R. Tateo, ADE functional dilogarithm identities and integrable models, Phys.Lett. B348 (1995), 84–88
R. Hirota, Discrete analogue of a generalized Toda equation, J.Phys. Soc. Jpn. 50 (1981) 3785–3791
R. Kedem, T.R. Klassen, B.M. McCoy, E. Melzer, Fermionic Quasiparticle Representations for Characters of G1 (1) × G1 (1)/G2 (1), Phys.Lett. B304 (1993), 263–270
A.N. Kirillov, Identities for the Rogers Dilogarithm Function Connected with Simple Lie Algebras, J. Soviet Mathematics 47 (1989), 2450–2459
A.N. Kirillov, N.Yu. Reshetikhin, Representations of Yangians and Multiplicities of Occurrence of the Irreducible Components of the Tensor Product of Representations of Simple Lie Algebras, J. Soviet Mathematics 52 (1990), 3156–3164
T. Klassen, E. Melzer, Purely Elastic Scattering Theories And Their Ultraviolet Limits, Nucl.Phys. B338 (1990), 485–528
A. Kuniba, T. Nakanishi, Spectra in Conformal Field Theories from the Rogers Dilogarithm, Mod.Phys.Lett. A7 (1992), 3487–3494
A. Kuniba, T. Nakanishi, J. Suzuki, Characters in Conformal Field Theories from Thermodynamic Bethe Ansatz, Mod.Phys.Lett. A8 (1993), 1649–1660
M. Lässig, M.J. Martins, Finite-Size Effects in Theories with Factorizable S-Matrices, Nucl.Phys. B354 (1991), 666–688
T. Miwa, On Hirota's difference equations, Proc. Japan Acad. A58 (1982) 9–12.
M.J. Martins, Complex Excitations in the Thermodynamic Bethe Ansatz, Phys.Rev.Lett. 67 (1991), 419–421
W. Nahm, Conformal Field Theory, Dilogarithms, and Three Dimensional Manifolds, in: Interface between physics and mathematics, Proceedings, Hangzhou 1993, W. Nahm and J.M. Shen eds., World Scientific
W. Nahm, A. Recknagel, and M. Terhoeven, Dilogarithm Identities in Conformal Field Theory, Mod.Phys.Lett. A8 (1993), 1835–1848
W.D. Neumann, Extended Bloch Group and the Cheeger-Chern-Simons Class, math.GT/0307092
W.D. Neumann, D. Zagier, Volumes of hyperbolic 3-manifolds, Topology 24 (1985), 307–332
A. Ocneanu, Paths on Coxeter Diagrams: From Platonic Solids and Singularities to Minimal Models and Subfactors, AMS Fields Institute Monographs no. 13, (1999), B.V. Rajarama Bhat, G, A, Elliott, P.A. Fillmore eds., vol. ‘Lectures on Operator Theory’
V. Kac, Infinite Dimensional Lie Algebras, Third edition, Cambridge University Press, Cambridge 1990
A.A. Suslin, Algebraic K-theory of fields, ICM, Berkeley, I (1986) 222–244
C.H. Sah, Homology of Classical Lie Groups made Discrete, III, J. Pure Appl. Algebra 56 (1989) 313–318
S.O. Warnaar and P.A. Pearce, Exceptional structure of the dilute A 3 model: E8 and E7 Rogers-Ramanujan identities J.Phys. A27 (1994) L891-L898.
S. Weinzierl, Algebraic Algorithms in Perturbative Calculations, hep-th/0305260, this volume
D. Zagier, Polylogarithms, Dedekind zeta functions and the algebraic K-theory of fields, in: Progr. Math. 89 (1991), Birkhäuser Boston, Boston MA, 391–430
A.B. Zamolodchikov, Integrable field theory from conformal field theory, Adv. Stud. in Pure Math. 19 (1989), 641
A.B. Zamolodchikov, On The Thermodynamic Bethe Ansatz Equations For REffectionless Ade Scattering Theories, Phys.Lett. B253 (1991), 391–394.
D. Zagier, The Dilogarithm Function, this volume
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Nahm, W. (2007). Conformal Field Theory and Torsion Elements of the Bloch Group. In: Cartier, P., Moussa, P., Julia, B., Vanhove, P. (eds) Frontiers in Number Theory, Physics, and Geometry II. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30308-4_2
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DOI: https://doi.org/10.1007/978-3-540-30308-4_2
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