N=1 Superpotentials from Multi-Instanton Calculus

TL;DR

We develop and apply localization on the multi-instanton moduli space to compute gaugino and scalar condensates in ${\cal N}=1$ gauge theories, including mass deformations of ${\cal N}=2$ and ${\cal N}=2^*$ to ${\cal N}=1$ and ${\cal N}=1^*$. Observables in the ${\cal N}=1$ theory are related to their ${\cal N}=2$ counterparts via chiral insertions, with vacua fixed by minimizing the quantum potential; the fixed-point sums over Young tableaux yield explicit condensates and generating functions that satisfy Konishi-anomaly–driven chiral ring relations. The paper analyzes two phases (completely broken ${\rm U}(1)^N$ and unbroken ${\rm U}(N)$) in the ${\cal N}=1$ limit, performs multi-instanton tests (including the $U(1)$ and $SU(2)$ cases), and extends the analysis to ${\cal N}=1^*$ with exact results for abelian and non-abelian groups, connecting to elliptic functions and integrable models (Calogero-Moser). The results demonstrate exact nonperturbative control over chiral correlators and reveal the deep interplay between localization, Seiberg–Witten geometry, and integrable systems in ${\cal N}=1$ dynamics. These findings provide a robust framework for computing nonperturbative superpotentials and chiral rings in supersymmetric gauge theories.

Abstract

In this paper we compute gaugino and scalar condensates in N=1 supersymmetric gauge theories with and without massive adjoint matter, using localization formulae over the multi--instanton moduli space. Furthermore we compute the chiral ring relations among the correlators of the $N=1^*$ theory and check this result against the multi-instanton computation finding agreement.