Graph rings and integrable perturbations of N = 2 superconformal theories

We show that the connection between certain integrable perturbations of N = 2 superconformal theories and graphs found by Lerche and Warner extends to a broader class. These perturbations are such that the generators of the perturbed chiral ring may be diagonalized in an orthonormal basis. This allows one to define a dual ring, whose generators are labelled by the ground states of the theory and are encoded in a graph of set of graphs, that reproduce the pattern of the ground states and interpolating solitons. All known perturbations of the ADE potentials and some others are shown to satisfy this criterion. This suggests a test of integrability.