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We develop a classification of emph{minimally unbalanced} $3d~mathcal{N}=4$ quiver gauge theories. These gauge theories are important because the isometry group $G$ of their Coulomb branch contains a single factor, which is either a classical or an exceptional Lie group. Concurrently, this provides a classification of hyperkahler cones with isometry group $G$ which are obtainable by Coulomb branch constructions. HyperKahler cones such as Coulomb branches of $3d~mathcal{N}=4$ quivers are indispensable tools for describing Higgs branches of different theories in various dimensions. In particular, they are used to describe Higgs branches of $5d~mathcal{N}=1$ SQCD with gauge group $SU(N_c)$ and $6d~mathcal N = (1,0)$ SQCD with gauge group $Sp(N_c)$ at the respective UV fixed points.
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Magnetic quivers and Hasse diagrams for Higgs branches of rank $r$ 4d $mathcal{N}=2$ SCFTs arising from $mathbb{Z}_{ell}$ $mathcal{S}$-fold constructions are discussed. The magnetic quivers are derived using three different methods: 1) Using clues like dimension, global symmetry, and the folding parameter $ell$ to guess the magnetic quiver. 2) From 6d $mathcal{N}=(1,0)$ SCFTs as UV completions of 5d marginal theories, and specific FI deformations on their magnetic quiver, which is further folded by $mathbb{Z}_{ell}$. 3) From T-duality of Type IIA brane systems of 6d $mathcal{N}=(1,0)$ SCFTs and explicit mass deformation of the resulting brane web followed by $mathbb{Z}_{ell}$ folding. A choice of the ungauging scheme, either on a long node or on a short node, yields two different moduli spaces related by an orbifold action, thus suggesting a larger set of SCFTs in four dimensions than previously expected.
We study the compactification of the 6d ${cal N}=(2,0)$ SCFT on the product of a Riemann surface with flux and a circle. On the one hand, this can be understood by first reducing on the Riemann surface, giving rise to 4d ${cal N}=1$ and ${cal N}=2$ class ${cal S}$ theories, which we then reduce on $S^1$ to get 3d ${cal N}=2$ and ${cal N}=4$ class ${cal S}$ theories. On the other hand, we may first compactify on $S^1$ to get the 5d ${cal N}=2$ Yang-Mills theory. By studying its reduction on a Riemann surface, we obtain a mirror dual description of 3d class ${cal S}$ theories, generalizing the star-shaped quiver theories of Benini, Tachikawa, and Xie. We comment on some global properties of the gauge group in these reductions, and test the dualities by computing various supersymmetric partition functions.
For any gauge theory, there may be a subgroup of the gauge group which acts trivially on the matter content. While many physical observables are not sensitive to this fact, the identification of the precise gauge group becomes crucial when the magnetic spectrum of the theory is considered. This question is addressed in the context of Coulomb branches for $3$d $mathcal{N}=4$ quiver gauge theories, which are moduli spaces of dressed monopole operators. Since monopole operators are characterized by their magnetic charge, the identification of the gauge group is imperative for the determination of the magnetic lattice. It is well-known that the gauge group of unframed unitary quivers is the product of all unitary nodes in the quiver modded out by the diagonal $mathrm{U}(1)$ acting trivially on the matter representation. This reasoning generalises to the notion that a choice of gauge group associated to a quiver is given by the product of the individual nodes quotiented by any subgroup that acts trivially on the matter content. For unframed (unitary-) orthosymplectic quivers composed of $mathrm{SO}(textrm{even})$, $mathrm{USp}$, and possibly $mathrm{U}$ gauge nodes, the maximal subgroup acting trivially is a diagonal $mathbb{Z}_2$. For unframed unitary quivers with a single $mathrm{SU}(N)$ node it is $mathbb{Z}_N$. We use this notion to compute the Coulomb branch Hilbert series of many unitary-orthosymplectic quivers. Examples include nilpotent orbit closures of the exceptional E-type algebras and magnetic quivers that arise from brane physics. This includes Higgs branches of theories with 8 supercharges in dimensions $4$, $5$, and $6$. A crucial ingredient in the calculation of exact refined Hilbert series is the alternative construction of unframed magnetic quivers from resolved Slodowy slices, whose Hilbert series can be derived from Hall-Littlewood polynomials.
We compute the prepotential for gauge theories descending from ${cal N}=4$ SYM via quiver projections and mass deformations. This accounts for gauge theories with product gauge groups and bifundamental matter. The case of massive orientifold gauge theories with gauge group SO/Sp is also described. In the case with no gravitational corrections the results are shown to be in agreement with Seiberg-Witten analysis and previous results in the literature.
We study half-BPS surface operators in four dimensional N=2 SU(N) gauge theories, and analyze their low-energy effective action on the four dimensional Coulomb branch using equivariant localization. We also study surface operators as coupled 2d/4d quiver gauge theories with an SU(N) flavour symmetry. In this description, the same surface operator can be described by different quivers that are related to each other by two dimensional Seiberg duality. We argue that these dual quivers correspond, on the localization side, to distinct integration contours that can be determined by the Fayet-Iliopoulos parameters of the two dimensional gauge nodes. We verify the proposal by mapping the solutions of the twisted chiral ring equations of the 2d/4d quivers onto individual residues of the localization integrand.
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