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Conformal Field Theory and Torsion Elements of the Bloch Group

  • Werner Nahm
Chapter

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.

Keywords

Central Charge Irreducible Representation Operator Product Expansion Conformal Dimension Dynkin Diagram 
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References

  1. E. Ardonne, P. Bouwknegt, and P. Dawson, K-matrices for 2D conformal field theories, Nucl.Phys. B660 (2003), 473–531Google Scholar
  2. G.E. Andrews, The theory of partitions, in: Encyclopedia of Mathematics and its Applications, Vol. 2, Addison Wesley, 1976Google Scholar
  3. A. Berkovich, B.M. McCoy, The universal chiral partition function for exclusion statistics, hep-th/9808013Google Scholar
  4. P. Dorey, Exact S-matrices, hep-th/9810026Google Scholar
  5. A. Coste, T. Gannon, Remarks on Galois symmetry in rational conformal field theories, Phys.Lett. B323 (1994), 316–321Google Scholar
  6. P. Dorey, R. Tateo, Excited states in some simple perturbed conformal field theories, Nucl.Phys. B515 (1998), 575–623CrossRefGoogle Scholar
  7. E. Frenkel, A. Szenes, Crystal bases, dilogarithm identities and torsion in algebraic K-groups, J. Amer. Math. Soc. 8 (1995), 629–664, hep-th/9304118zbMATHCrossRefGoogle Scholar
  8. E. Frenkel, A. Szenes, Thermodynamic Bethe Ansatz and Dilogarithm Identities I, Math. Res. Lett. 2 (1995), 677–693, hep-th/9506215zbMATHGoogle Scholar
  9. 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, 2000Google Scholar
  10. F. Gliozzi, R. Tateo, Thermodynamic Bethe Ansatz and Threefold Triangulations, Int.J.Mod.Phys. A11 (1996), 4051–4064Google Scholar
  11. F. Gliozzi, R. Tateo, ADE functional dilogarithm identities and integrable models, Phys.Lett. B348 (1995), 84–88Google Scholar
  12. R. Hirota, Discrete analogue of a generalized Toda equation, J.Phys. Soc. Jpn. 50 (1981) 3785–3791CrossRefGoogle Scholar
  13. 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), 263–270Google Scholar
  14. A.N. Kirillov, Identities for the Rogers Dilogarithm Function Connected with Simple Lie Algebras, J. Soviet Mathematics 47 (1989), 2450–2459CrossRefGoogle Scholar
  15. 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), 3156–3164CrossRefGoogle Scholar
  16. T. Klassen, E. Melzer, Purely Elastic Scattering Theories And Their Ultraviolet Limits, Nucl.Phys. B338 (1990), 485–528CrossRefGoogle Scholar
  17. A. Kuniba, T. Nakanishi, Spectra in Conformal Field Theories from the Rogers Dilogarithm, Mod.Phys.Lett. A7 (1992), 3487–3494Google Scholar
  18. A. Kuniba, T. Nakanishi, J. Suzuki, Characters in Conformal Field Theories from Thermodynamic Bethe Ansatz, Mod.Phys.Lett. A8 (1993), 1649–1660Google Scholar
  19. M. Lässig, M.J. Martins, Finite-Size Effects in Theories with Factorizable S-Matrices, Nucl.Phys. B354 (1991), 666–688CrossRefGoogle Scholar
  20. T. Miwa, On Hirota's difference equations, Proc. Japan Acad. A58 (1982) 9–12.Google Scholar
  21. M.J. Martins, Complex Excitations in the Thermodynamic Bethe Ansatz, Phys.Rev.Lett. 67 (1991), 419–421zbMATHCrossRefGoogle Scholar
  22. 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 ScientificGoogle Scholar
  23. W. Nahm, A. Recknagel, and M. Terhoeven, Dilogarithm Identities in Conformal Field Theory, Mod.Phys.Lett. A8 (1993), 1835–1848Google Scholar
  24. W.D. Neumann, Extended Bloch Group and the Cheeger-Chern-Simons Class, math.GT/0307092Google Scholar
  25. W.D. Neumann, D. Zagier, Volumes of hyperbolic 3-manifolds, Topology 24 (1985), 307–332zbMATHCrossRefGoogle Scholar
  26. 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’Google Scholar
  27. V. Kac, Infinite Dimensional Lie Algebras, Third edition, Cambridge University Press, Cambridge 1990zbMATHGoogle Scholar
  28. A.A. Suslin, Algebraic K-theory of fields, ICM, Berkeley, I (1986) 222–244Google Scholar
  29. C.H. Sah, Homology of Classical Lie Groups made Discrete, III, J. Pure Appl. Algebra 56 (1989) 313–318CrossRefGoogle Scholar
  30. 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.Google Scholar
  31. S. Weinzierl, Algebraic Algorithms in Perturbative Calculations, hep-th/0305260, this volumeGoogle Scholar
  32. D. Zagier, Polylogarithms, Dedekind zeta functions and the algebraic K-theory of fields, in: Progr. Math. 89 (1991), Birkhäuser Boston, Boston MA, 391–430Google Scholar
  33. A.B. Zamolodchikov, Integrable field theory from conformal field theory, Adv. Stud. in Pure Math. 19 (1989), 641Google Scholar
  34. A.B. Zamolodchikov, On The Thermodynamic Bethe Ansatz Equations For REffectionless Ade Scattering Theories, Phys.Lett. B253 (1991), 391–394.Google Scholar
  35. D. Zagier, The Dilogarithm Function, this volumeGoogle Scholar

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  • Werner Nahm

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