Integrable QFT(2) Encoded on Products of Dynkin Diagrams

A large class of Thermodynamic Bethe Ansatz equations governing the Renormalization Group evolution of the Casimir energy of the vacuum on the cylinder for an integrable two-dimensional field theory, can often be encoded on a tensor product of two graphs. We demonstrate here that in this case the two graphs can only be of \(ADE\) type. We also give strong numerical evidence for a new large set of Dilogarithm sum Rules connected to \(ADE\times ADE\) and a simple formula for the ultraviolet perturbing operator conformal dimensions only in terms of rank and Coxeter numbers of \(ADE\times ADE\). We conclude with some remarks on the curious case \(ADE\times D\). [Talk given by F.R. at the Cargese Workshop "New Developments in String Theory, Conformal Models and Topological Field Theory" (May 1993)]