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

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