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 3d $$ \mathcal{N} $$ 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 $$ \mathcal{C} $$ 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 $$ \mathcal{C} $$ 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 $$ {D}_4^{(3)} $$ D 4 3 quivers, respectively. The analysis is complemented with computations of the Highest Weight Generating functions.

S-fold magnetic quivers

“…A natural expectation from the F-theory setup is that the Higgs branch of S-fold theories can be interpreted as the moduli space of r G instantons on C 2 /Z since we have 7-branes probed by a stack of r D3 branes. As explained below, this correlates with the existence of discretely-gauged versions of S-fold theories and we find that this can be understood at the level of magnetic quivers by changing the ungauging scheme [46]. G, theories.…”

“…Discrete gauging. Since the magnetic quivers of tables 1 and 2 are non-simply laced one can change the ungauging scheme [46] and obtain new quivers, listed in tables 7 and 8. These quivers are proposed to be magnetic quivers for SCFTs obtained from discretely gauging the Z global symmetry of the S G predicted in [4] is beyond the reach of current magnetic quiver techniques.…”

“…The expectation is that the Higgs branch of the gauged theory (calledT (r) G, , where the circle symbolizes the cyclic gauging) is a Z quotient of the Higgs branch of the parent theory T (r) G, . Since the associated magnetic quivers are non simply-laced, we know there is a natural way to implement such a quotient -the ungauging scheme [46]: recall that for the quivers in table 2, the ungauging is on a long node. Furthermore, using the Crawley-Boevey trick [66], the ungauging on the short node can be made into an ungauging of a long node by replacing the flavor with a suitable long node.…”

“…A feature of all magnetic quivers coming from S-folds studied in the following is the appearance of non-simply laced edges, where the order of the non-simply laced edges corresponds to the order of the S-fold (the folding parameter). The 3d N = 4 Coulomb branches of non-simply laced quivers were studied in [44][45][46][47]. The non-simply laced quivers can be obtained from simply laced ones through an operation called folding [48][49][50].…”

Magnetic quivers from brane webs with O7+-planes

“…Therefore one denotes with the squircle the location at which the overall U(1) has to be decoupled. See [34] for more details on this point.…”