  Using nonlinear pde techniques, we construct a new family of globally smooth
tt* structures. This includes tt* structures associated to the (orbifold)
quantum cohomology of a finite number of complex projective spaces and weighted
projective spaces. The existence of such "magical solutions" of the tt*
equations, namely smooth solutions characterized by asymptotic boundary
conditions, was predicted by Cecotti and Vafa. In our situation, the tt*
equations belong to a class of equations which we call the tt*-Toda lattice.
Solutions of the tt*-Toda lattice are harmonic maps which have dual
interpretations as Frobenius structures or variations of (semi-infinite) Hodge
structures.
