Partial ${\cal N}=2 \to {\cal N}=1$ supersymmetry breaking and gravity deformed chiral rings

TL;DR

This paper derives chiral ring relations in ${ m extbf{N}=1}$ gauge theories influenced by gravitational and graviphoton backgrounds, revealing their origin from ${ m extbf{N}=2}$ supergravity identities and suggesting an underlying ${ m extbf{N}=2}$ structure.

Contribution

It demonstrates that chiral ring relations in ${ m extbf{N}=1}$ theories with gravitational backgrounds originate from ${ m extbf{N}=2}$ supergravity Bianchi identities, extending understanding of supersymmetry breaking effects.

Findings

Chiral ring relations are derived from ${ m extbf{N}=2}$ supergravity Bianchi identities.

Models with matter in various representations, including quiver theories, are constructed.

Consistency conditions imply gauge kinetic functions and superpotentials derive from a prepotential, indicating an ${ m extbf{N}=2}$ structure.

Abstract

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