The A-D-E classification of minimal and A1(1) conformal invariant theories

We present a detailed and complete proof of our earlier conjecture on the classification of minimal conformal invariant theories. This is based on an exhaustive construction of all modular invariant sequilinear forms, with positive integral coefficients, in the characters of the Virasoro or of the A1(1) Kac-Moody algebras, which describe the corresponding partition functions on a torus. A remarkable correspondence emerges with simply laced Lie algebras. (orig.)