Ungauging Schemes and Coulomb Branches of Non-simply Laced Quiver Theories

Abstract

Three-dimensional Coulomb branches have a prominent role in the study of moduli spaces of supersymmetric gauge theories with $8$ supercharges in $3,4,5$, and $6$ dimensions. Inspired by simply laced $3$d $\mathcal{N}=4$ supersymmetric quiver gauge theories, we consider Coulomb branches constructed from non-simply laced quivers with edge multiplicity $k$ and no flavor nodes. In a computation of the Coulomb branch as the space of dressed monopole operators, a center-of-mass $U(1)$ symmetry needs to be ungauged. Typically, for a simply laced theory, all choices of the ungauged $U(1)$ (i.e. all choices of ungauging schemes) are equivalent and the Coulomb branch is unique. In this note, we study various ungauging schemes and their effect on the resulting Coulomb branch variety. It is shown that, for a non-simply laced quiver, inequivalent ungauging schemes exist which correspond to inequivalent Coulomb branch varieties. Ungauging on any of the long nodes of a non-simply laced quiver yields the same Coulomb branch $\mathcal{C}$. For choices of ungauging the $U(1)$ on a short node of rank higher than $1$, the GNO dual magnetic lattice deforms such that it no longer corresponds to a Lie group, and therefore, the monopole formula yields a non-valid Coulomb branch. However, if the ungauging is performed on a short node of rank $1$, the one-dimensional magnetic lattice is rescaled conformally along its single direction and the corresponding Coulomb branch is an orbifold of the form $\mathcal{C}/\mathbb{Z}_k$. Ungauging schemes of $3$d Coulomb branches provide a particularly interesting and intuitive description of a subset of actions on the nilpotent orbits studied by Kostant and Brylinski arXiv:math/9204227. The ungauging scheme analysis is carried out for minimally unbalanced $C_n$, affine $F_4$, affine $G_2$, and twisted affine $D_4^{(3)}$ quivers, respectively.