Isoparametric surfaces and the tt*-equations

In this paper we provide a link between tt*-bundles, which are solutions of a general version of the equations of topological-antitopological fusion considered by Cecotti-Vafa, Dubrovin and Hertling and isoparametric spherical surfaces, i.e. spherical surfaces with constant principal curvatures. More precisely, we construct a tt*-bundle from a pluriharmonic map into the hermitian symmetric space SO(l+2)/SO(2)×SO(l) which can be seen as the Grassmannian Gr(2,n+2) of oriented two-planes in Rn+2. Applying the above construction to the Gauss map of an isoparametric spherical surface we associate to a given isoparametric spherical surface a new solution to the tt∗-equations.