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 3d \( \mathcal{N} \) = 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 anisotropically 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 along its single direction i.e. isotropically and the corresponding Coulomb branch is an orbifold of the form \( \mathcal{C} \)/ℤk . Ungauging schemes of 3d Coulomb branches provide a particularly interesting and intuitive description of a subset of actions on the nilpotent orbits studied by Kostant and Brylinski [1]. The ungauging scheme analysis is carried out for minimally unbalanced Cn, affine F4, affine G2, and twisted affine \( {D}_4^{(3)} \) quivers, respectively. The analysis is complemented with computations of the Highest Weight Generating functions.
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Hanany, A., Zajac, A. Ungauging schemes and Coulomb branches of non-simply laced quiver theories. J. High Energ. Phys. 2020, 193 (2020). https://doi.org/10.1007/JHEP09(2020)193
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DOI: https://doi.org/10.1007/JHEP09(2020)193