ADE Quiver Duality Proposals: Graph Classification and Nodewise Realizations

Abstract

We use the large-$N$ index strategy to find new quiver dualities in four-dimensional $\mathcal N=1$ quiver gauge theories. We study gauge quivers with or without an adjoint field at each gauge node, and bifundamental matter between selected nodes with fixed $R$-charges. Imposing the finite-meson condition restricts the allowed gauge graphs to finite ADE diagrams and leads to a new simultaneous duality for ADE quivers. We present an explicit superpotential that produces a uniform classical meson spectrum consistent with the large-$N$ index approach's prediction. We then relate the simultaneous rank transformation to the longest element of the ADE Weyl group, whose reduced expressions organize nodewise duality sequences realizing the simultaneous ADE duality. Realizing the simultaneous ADE duality through such nodewise sequences offers a finite-$N$ consistency check beyond the large-$N$ index construction. Moreover, we realize the simultaneous ADE duality as a sequence of subquiver simultaneous dualities.