Partial N=2 → N=1 supersymmetry breaking and gravity deformed chiral rings -: art. no. 041

We present a derivation of the chiral ring relations, arising in N = 1 gauge theories in the presence of (anti-) self-dual background gravitational field G(alphabetagamma) and graviphoton field strength F-alphabeta. 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 N = 2 to N = 1 supergravity coupled to vector- and hyper-multiplets, and take a rigid limit which keeps a non-trivial G(alphabetagamma) and F-alphabeta with a finite supersymmetry breaking scale. We derive the resulting effective, global, N = 1 theory and show that the chiral ring relations are just a consequence of the standard 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-alphabeta, 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 N = 1 theory, should be derived from a prepotential, indicating an underlying N = 2 structure.