The Potency of Nilpotence
Eric Bryan, Arvind Rajaraman, Yuri Shirman
TL;DR
The paper investigates Seiberg-like dualities in $ olinebreak N=1$ theories with adjoint and fundamental matter and ADE-type superpotentials, introducing nilpotent flat directions on the moduli space as stringent tests of the proposed dual descriptions. It provides a detailed review of the $A_k$ and $D_{k+2}$ dualities, including the electric/m magnetic pairs and the structure of the chiral ring, and then analyzes the dynamics along nilpotent directions by comparing two RG-flow routes: electric Higgsing versus magnetic meson branches. The main result is a clean separation: duality is robust for the $A_k$ family, with the Higgs-branch analyses reproducing the expected IR physics, while the $D_{k+2}$ theories fail to yield a consistent dual description along these directions, indicating the duality conjecture does not hold in that class. This failure highlights the subtle role of nilpotent directions in constraining dualities and motivates further exploration of ADE models, possible non-perturbative chiral-ring effects, and extensions to other matter representations or superpotentials.
Abstract
The dynamics of N=1 SUSY gauge theories with matter in adjoint and fundamental representations and the superpotentials given by Arnold's ADE singularities has been extensively studied in the literature. It was also conjectured that supersymmetric models with $W_{A_k}$, $W_{D_{k+2}}$ and $W_{E_7}$ superpotentials possess a dual description. In this paper we revisit the analysis of the moduli space of $A_k$ and $D_{k+2}$ models by considering the duality along nilpotent directions on the moduli space. While our analysis provides additional evidence for the duality conjecture in $W_{A_k}$ models, we show that the duality conjecture fails for the $W_{D_{k+2}}$ models.