Gravitational F-terms through anomaly equations and deformed chiral rings

We study effective gravitational F-terms, obtained by integrating an U(N) adjoint chiral superfield Φ coupled to the = 1 gauge chiral superfield W_α and supergravity, to arbitrary orders in the gravitational background. The latter includes in addition to the = 1 Weyl superfield G_αβγ, the self-dual graviphoton field strength F_αβ of the parent, broken = 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 = 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 ⟨TrW²/(z−Φ)⟩ to all orders in F². 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.