  We study effective gravitational F-terms, obtained by integrating an $U(N)$
adjoint chiral superfield $\Phi$ coupled to the ${\cal N}=1$ gauge chiral
superfield $W_\alpha$ and supergravity, to arbitrary orders in the
gravitational background. The latter includes in addition to the ${\cal N}=1$
Weyl superfield $G_{\alpha\beta\gamma}$, the self-dual graviphoton field
strength $F_{\alpha\beta}$ of the parent, broken ${\cal N}=2$ theory. We first
study the chiral ring relations resulting from the above non-standard
gravitational background and find agreement, for gauge invariant operators,
with those obtained from the dual closed string side via Bianchi identities for
${\cal N}=2$ supergravity coupled to vector multiplets. We then derive
generalized anomaly equations for connected correlators on the gauge theory
side, which allow us to solve for the basic one-point function $\langle {\rm
Tr} W^2/(z-\Phi)\rangle$ to all orders in $F^2$. By generalizing the matrix
model loop equation to the generating functional of connected correlators of
resolvents, we prove that the gauge theory result coincides with the genus
expansion of the associated matrix model, after identifying the expansion
parameters on the two sides.
