Conformal field theory and torsion elements of the Bloch group. (English) Zbl 1193.81092

Nahm, Werner

Cartier, Pierre (ed.) et al., Frontiers in number theory, physics, and geometry II. On conformal field theories, discrete groups and renormalization. Papers from the meeting, Les Houches, France, March 9–21, 2003. Berlin: Springer (ISBN 978-3-540-30307-7/hbk). 67-132 (2007).

Summary: We argue that rational conformally invariant quantum field theories in two dimensions are closely related to torsion elements of the algebraic \(K\)-theory group \(K_3(\mathbb C)\). 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 \(K_3(\mathbb C)\). The algebraic equations are solved for integrable models given by arbitrary pairs of A-type Cartan matrices. The paper should be readable by mathematicians.
For the entire collection see [Zbl 1104.11003].

MSC:

81T40	Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
11G55	Polylogarithms and relations with \(K\)-theory
11Z05	Miscellaneous applications of number theory
19D45	Higher symbols, Milnor \(K\)-theory
33B30	Higher logarithm functions
33D80	Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics