THE FUSING RULE AND THE SCATTERING MATRIX OF AFFINE TODA THEORY

Affine Toda theory is an integrable theory with many interesting features. Classically, the presence of trilinear couplings is given by Dorey's "fusing rule", whatever the simple Lie algebra concerned. This paper discusses the structure of this rule, alternative solutions and formulations, and the relationship to the quantum conservation laws. This insight is applied to the conjectured scattering matrix of the quantum theory. The crossing and bootstrap properties are verified in a general way, valid for any Lie algebra, but the analyticity properties require the extra assumption that the algebra be simply laced. Various identities satisfied by a Coxeter element play a crucial role.