Conformal Field Theory and Torsion Elements of the Bloch Group

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(C). If such a theory has an integrable matrix perturbation with purely elastic scattering matrix, then the partition function has a canonical sum representation. Its asymptotic behaviour is given in terms of the solution of an algebraic equation which can be read off from the scattering matrix. The solutions yield torsion elements of an extension of the Bloch group which seems to be equal to K_3(C). These algebraic equations are solved for integrable models given by arbitrary pairs of equations are solved for integrable models given by arbitrary pairs of A-type Cartan matrices. The paper should be readable by mathematicians.