Solutions of the tt*-equations constructed from the SU(2)${}_k$-fusion ring, and Smyth potentials

Cecotti and Vafa introduced the tt*-equation (topological-antitopological fusion equation), whose solutions describe massive deformations of supersymmetric conformal field theories. We describe some solutions of the tt*-equation constructed from the SU(2)${}_k$-fusion algebra. The idea of the construction is due to Cecotti and Vafa, but we give a precise mathematical formulation and a description of the “holomorphic data” corresponding to the solutions by using the DPW method. Furthermore, we give a relation between the solutions and the representations of SU(2). As a special case, we consider the solutions corresponding to the supersymmetric A${}_k$-minimal model.