PartialN=2 toN=1 supersymmetry breaking and gravity deformed chiral rings.

We present a derivation of the chiral ring relations, arising in ${\cal N}=1$gauge theories in the presence of (anti-)self-dual background gravitationalfield $G_{\alpha\beta\gamma}$ and graviphoton field strength $F_{\alpha\beta}$.These were previously considered in the literature in order to prove therelation between gravitational F-terms in the gauge theory and coefficients ofthe topological expansion of the related matrix integral. We consider thespontaneous breaking of ${\cal N} =2$ to ${\cal N} =1$ supergravity coupled tovector- and hyper-multiplets, and take a rigid limit which keeps a non-trivial$G_{\alpha\beta\gamma}$ and $F_{\alpha\beta}$ with a finite supersymmetrybreaking scale. We derive the resulting effective, global, ${\cal N}=1$ theoryand show that the chiral ring relations are just a consequence of the standard${\cal N}=2$ supergravity Bianchi identities . We can also obtain models withmatter in different representations and in particular quiver theories. We alsoshow that, in the presence of non-trivial $F_{\alpha\beta}$, consistency of theKonishi-anomaly loop equations with the chiral ring relations, demands that thegauge kinetic function and the superpotential, a priori unrelated for an ${\calN}=1$ theory, should be derived from a prepotential, indicating an underlying${\cal N}=2$ structure.Comment: 42 pages, uses JHEP.cls;v2: typos corrected and references adde