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

Abstract

We present a derivation of the chiral ring relations, arising in $\mathcal{N}=1$ 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 $\mathcal{N}=2$ to $\mathcal{N}=1$ 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, $\mathcal{N}=1$ theory and show that the chiral ring relations are just a consequence of the standard $\mathcal{N}=2$ 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 $\mathcal{N}=1$ theory, should be derived from a prepotential, indicating an underlying $\mathcal{N}=2$ structure.

42 pages, uses JHEP.cls;v2: typos corrected and references added

References in corpus (1)

• A Perturbative Window into Non-Perturbative Physics

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