Foundations of Rational Quantum Field Theory, I

We study the connection between Rational Conformal Field Theory (RCFT), $N=2$ massive supersymmetric field theory, and solvable Interaction Round the Face (IRF) lattice models. Specifically, one identifies the fusion rings with the chiral rings. The theories so obtained are conjectured, and largely shown, to be integrable. A variety of examples and the structure of the metric in moduli space are given. The kink scattering theory is given by the Boltzmann weights of an IRF model, which is built entirely in terms of the conformal data of the original RCFT. This procedure produces all solvable IRF models in terms of projection operators of the RCFT. The soliton structure and their scattering amplitudes are described. A host of new rational conformal field theories is constructed generalizing most, if not all, of the known ones.