  We discuss the large $N$ limit of the supersymmetric $CP^N$ models as an
illustration of Cecotti and Vafa's $tt^*$ formalism. In this limit the `$tt^*$
equation' becomes the long wavelength limit of the $2D$ Toda lattice, an
equation first studied in the context of self-dual gravity. We show how simple
finite temperature and large $N$ techniques determine the relevant solution,
and verify analytically that it solves the $tt^*$ equation, using Legendre
transform techniques from self-dual gravity.
