  We present a derivation of the chiral ring relations, arising in ${\cal N}=1$
gauge theories in the presence of (anti-)self-dual background gravitational
field $G_{\alpha\beta\gamma}$ and graviphoton field strength $F_{\alpha\beta}$.
These were previously considered in the literature in order to prove the
relation between gravitational F-terms in the gauge theory and coefficients of
the topological expansion of the related matrix integral. We consider the
spontaneous breaking of ${\cal N} =2$ to ${\cal N} =1$ supergravity coupled to
vector- and hyper-multiplets, and take a rigid limit which keeps a non-trivial
$G_{\alpha\beta\gamma}$ and $F_{\alpha\beta}$ with a finite supersymmetry
breaking scale. We derive the resulting effective, global, ${\cal N}=1$ theory
and show that the chiral ring relations are just a consequence of the standard
${\cal N}=2$ supergravity Bianchi identities . We can also obtain models with
matter in different representations and in particular quiver theories. We also
show that, in the presence of non-trivial $F_{\alpha\beta}$, consistency of the
Konishi-anomaly loop equations with the chiral ring relations, demands that the
gauge kinetic function and the superpotential, a priori unrelated for an ${\cal
N}=1$ theory, should be derived from a prepotential, indicating an underlying
${\cal N}=2$ structure.
