We proved that the integration of the Chern-Weil forms on CY moduli are always rational numbers. This result follows from a more general one: the integration of the Chern-Weil forms of the Hodge bundles on any coarse moduli spaces are rationa numbers. When the dimension of the moduli space is one, this was a result of Zucker and Peters. For the fisrt Chern class, this was proved by Kollar.

We will also discuss the applications in string theory. This is joint with M. Douglas.