Vacuum ward identities for higher genera

The minimal models of two-dimensional conformal field theory are considered on surfaces with nontrivial topology. Due to degeneration of the vacuum module in these models, the stress tensor components satisfy special equations of motion - the vacuum Ward identities. It is shown that these identities can be written in the form of partial differential equations on the moduli space, satisfied by the partition function of the theory. Some examples are written down explicitly in the case of torus and g = 2 surface, represented as a double-fold covering of a sphere. For the simplest minimal theory M( 2 5), equations are closed on hyperelliptic surface of any genus and the situation is governed by the other minimal model M( 3 10).