  We classify the moduli spaces of the four-dimensional topological half-flat
gravity models by using the canonical bundle. For a K3-surface or
four-dimensional torus, they describe an equivalent class of a trio of the
Einstein-Kahler forms ( the hyperkahler forms ). We calculate the dimensions of
these moduli spaces by using the Atiyah-Singer Index theorem. We mention the
partition function and the possibility of the observables in the Witten-type
topological half-flat gravity model case.
