Gravitational F-terms through anomaly equations and deformed chiral rings

TL;DR

This paper investigates gravitational F-terms in supersymmetric gauge theories with supergravity backgrounds, deriving anomaly equations and showing their equivalence to matrix model expansions, thus connecting gauge theory, supergravity, and string theory insights.

Contribution

It introduces generalized anomaly equations for gravitational F-terms and demonstrates their equivalence to matrix model genus expansions, extending previous results to more complex backgrounds.

Findings

Derived anomaly equations for gravitational F-terms.

Solved for one-point functions to all orders in F^2.

Established equivalence between gauge theory results and matrix model expansions.

Abstract

We study effective gravitational F-terms, obtained by integrating an $U(N)$ adjoint chiral superfield $\Phi$ coupled to the ${\cal N}=1$ gauge chiral superfield $W_\alpha$ and supergravity, to arbitrary orders in the gravitational background. The latter includes in addition to the ${\cal N}=1$ Weyl superfield $G_{\alpha\beta\gamma}$, the self-dual graviphoton field strength $F_{\alpha\beta}$ of the parent, broken ${\cal N}=2$ 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 ${\cal N}=2$ 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 $\langle {\rm Tr} W^2/(z-\Phi)\rangle$ to all orders in $F^2$. 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.