Magnetic Lattices for Orthosymplectic Quivers

TL;DR

This work develops a precise framework for identifying the gauge group of quiver gauge theories with eight supercharges, showing that the diagonal trivially-acting subgroup H inside the kernel kerφ can be modded out to yield distinct Coulomb branches 𝒞_H(Q). The authors formulate 𝒞_H(Q) = 𝒞_{kerφ}(Q)/N_H and demonstrate that different choices of H lead to orbifold relations among the Coulomb branches, with a particular emphasis on a diagonal ℤ_2 in unframed orthosymplectic quivers. Central to the analysis is the monopole formula for the Coulomb-branch Hilbert series, which relies on the precise magnetic lattice; they show how the lattice structure for unitary-orthosymplectic quivers includes integer and half-integer sectors, controlled by Z_k diag quotients. They compute exact refined HS and highest-weight generating functions for several families, notably the minimal nilpotent E_n orbits and their ℤ_2 orbifolds, and connect these Coulomb branches to Higgs branches of higher-dimensional theories (4d N=2 class S, 5d N=1 at infinite coupling, 6d N=(1,0)). A key methodological contribution is the use of resolved Slodowy slices and Hall-Littlewood polynomials to derive HS/HWG data, providing independent cross-checks. The results reveal that a single quiver can realize multiple moduli spaces via discrete gauging and that different magnetic quivers can describe the same symplectic singularity, offering a structured path to explore orbifolds and dual descriptions in string-theory contexts and higher-dimensional SCFTs.

Abstract

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.

Figures (46)

• Figure 1: Maximal subgroups $H$ of the product group $G$ of different types of unframed simply laced quivers.
• Figure 2: Contributions to the conformal dimension $\Delta$ that appear in the monopole formula \ref{['mono']}. (\ref{['subtab:1']}) summarises the vector mulitplet contributions, (\ref{['subtab:2']}) collects the hypermultiplet contributions, and (\ref{['subtab:3']}) provides the combined parts for certain special representations. Note in particular that a bifundamental of $\mathrm{SO}(2r)_{m} \times \mathrm{U}(k)_{n}$ contributes exactly the same as a bifundamental of $\mathrm{SO}(2r)_{m} \times \mathrm{USp}(2k)_{n}$ and of $\mathrm{USp}(2r)_{m} \times \mathrm{U}(k)_{n}$, as required by the fact that $\mathrm{U}(k)$ should be seen as a subgroup of $\mathrm{USp}(2k)$.
• Figure 3: The different lattices used in this paper are collected in (\ref{['subtab:new1']}); while the notation is clarified in (\ref{['subtab:new2']}). The lattices $\mathbb{Z}^n_{\Sigma=0}$ and $\mathbb{Z}^{n}/\delta$ (of rank $n-1$) are dual, as are the lattices $\mathbb{Z}^n_{2|\Sigma}$ and $\mathbb{Z}^n_{\frac{1}{2}}$ (of rank $n$). Note that, as explained in the text, $\Lambda^{\mathfrak{g}}_w$ and $\Lambda^{\mathfrak{g}}_{cw}$ are not lattices for the non semisimple group $\mathrm{U}(n)$, whose center is not discrete, so they are not shown in the figure. In this case, the root lattice and the weight lattice have ranks differing by one.
• Figure 4: Colored nodes used in the paper.
• Figure 5: In all the diagrams, the red dots show the weight lattice, and the black circles show the dual lattice, which is the magnetic lattice involved in the monopole formula. The arrow denotes the action of the Weyl group. \ref{['subfig:a']}: The stars show the root lattice of $\mathrm{USp}(2)$. This notion does not extend to the full group $\mathrm{SO}(2) \times \mathrm{USp}(2)$ because of the Abelian factor. We do not show the roots on the other diagrams. \ref{['subfig:b']}: $\mathrm{USp}(2)$ is replaced by $\mathrm{SO}(3)$. \ref{['subfig:c']}: $\mathrm{SO}(2)$ is replaced by $\mathrm{SO}(2)/\mathbb{Z}_2 \simeq \mathrm{SO}(2)$, which rescales the weights. \ref{['subfig:d']}: Combinations of the two $\mathbb{Z}_2$ modifications of \ref{['subfig:b']} and \ref{['subfig:c']}. The weight lattice has index $4$ compared to \ref{['subfig:a']}. \ref{['subfig:e']}: Finally this is the weight and coweight lattices for the quiver group.