Gravitational F-terms through anomaly equations and deformed chiral rings

Abstract

We study effective gravitational F-terms, obtained by integrating an U(N) adjoint chiral superfield Φ coupled to the Script N = 1 gauge chiral superfield Wα and supergravity, to arbitrary orders in the gravitational background. The latter includes in addition to the Script N = 1 Weyl superfield Gαβγ, the self-dual graviphoton field strength Fαβ of the parent, broken Script 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 Script 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 langTrW2/(z−Φ)rang 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.