Hasse Diagrams for $\mathbf{3d}$ $\mathbf{\mathcal{N}=4}$ Quiver Gauge Theories -- Inversion and the full Moduli Space

TL;DR

This work develops a unified framework to study the moduli spaces of 3d $\mathcal{N}=4$ quiver gauge theories through the lens of Hasse diagrams, introducing the inversion operation $\mathcal{I}$ that, for a broad class of theories, relates the Coulomb and Higgs branch diagrams. It further extends the diagrammatic tool to the full moduli space, incorporating mixed branches, by combining Coulomb and Higgs information via transverse slices and a quiver-subtraction (Kraft–Procesi) perspective. The authors demonstrate the invertible case across several single-node quivers (unitary, special unitary, and symplectic) and provide explicit formulas for the full moduli-space diagrams; for non-invertible or “bad” theories they show how inversion, brane methods, and careful analysis of enhanced Coulomb branches yield consistent predictions, including detailed examples in O- and G2-type theories and a $C_N$ flavor theory. The study highlights the geometric–physical interplay between symplectic leaves, transverse spaces, and Higgsing dynamics, and outlines open questions about non-invertible cases, non-reduced Higgs branches, and extensions to higher dimensions and other dualities, with potential impact on understanding singularities in moduli spaces and on symplectic duality.

Abstract

We study Hasse diagrams of moduli spaces of $\mathrm{3d}$ $\mathcal{N}=4$ quiver gauge theories. The goal of this work is twofold: 1) We introduce the notion of inverting a Hasse diagram and conjecture that the Coulomb branch and Higgs branch Hasse diagrams of certain theories are related through this operation. 2) We introduce a Hasse diagram to map out the entire moduli space of the theory, including the Coulomb, Higgs and mixed branches. For theories whose Higgs and Coulomb branch Hasse diagrams are related by inversion it is straight forward to generate the Hasse diagram of the entire moduli space. We apply inversion of the Higgs branch Hasse diagram in order to obtain the Coulomb branch Hasse diagram for bad theories and obtain results consistent with the literature. For theories whose Higgs and Coulomb branch Hasse diagrams are not related by inversion it is nevertheless possible to produce the Hasse diagram of the full moduli space using different methods. We give examples for Hasse diagrams of the entire moduli space of theories with \emph{enhanced} Coulomb branches.

Figures (7)

• Figure 1: One has to make a distinction between the various geometric spaces dealt with in the realm 3d $\mathcal{N}=4$ gauge theories. Examples of symplectic singularities are given through both the hyper-Kähler quotient and the Coulomb branch construction. The space of symplectic singularites which can be constructed as a Coulomb branch and as a hyper-Kähler quotient, is the realm of the famous 3d mirror symmetry. However, there are Coulomb branches for which no hyper-Kähler quotient construction is known and vice versa. Furthermore there are hyper-Kähler quotients and Coulomb branches, which are not symplectic singularities. If the hyper-Kähler quotient is a union of symplectic singularities, the individual cones may be described as the Coulomb branches of a set of magnetic quivers. The hyper-Kähler quotient need not accurately describe the Higgs branch of the quantum moduli space of a theory, an example is given in Section \ref{['Bad']}.
• Figure 2: Affine ADE Dynkin quivers. a) $\hat{A}_n$ b) $\hat{D}_n$ c) $\hat{E}_6$ d) $\hat{E}_7$ e) $\hat{E}_8$. Their Coulomb branches are the minimal nilpotent orbit closure of the corresponding algebra, written $a_n$, $d_n$, and $e_n$ respectively. Their Higgs branches are the Kleinian singularities corresponding to the algebra, written $A_n$, $D_n$, and $E_n$ respectively. It should be clear from context, when a capital letter refers to the Kleinian singularity rather than a Dynkin diagram or an algebra. The Hasse diagrams for both the Coulomb and Higgs branches are given in \ref{['eq:aff_Hasse']}.
• Figure 3: Brane set up for $O(1)$ with 1 flavour. The green line represents an $O5^+$ plane. The moduli space of $O(1)$ with 1 flavour consists of only its Higgs branch. The Higgs branch consists of two leaves depicted in the brane constructions (a) and (b). In (c) the magnetic quiver is presented, its Coulomb branch is $\mathbb{C}^2/\mathbb{Z}_2$
• Figure 4: Brane configurations corresponding to the leaves in \ref{['eq:nminA3hasse']}. In Cabrera:2016vvv Hasse diagrams were computed moving along the top row. The commutativity of red and blue lines in the full Hasse diagram allows to construct the Hasse diagram of a single branch in this way. See the appendix \ref{['app:O']} for a discussion of how to get moduli spaces and Hasse diagrams from brane constructions
• Figure 5: Brane configurations corresponding to the leaves in \ref{['eq:nminA3dualhasse']}. See Appendix \ref{['app:O']} for more information.

Theorems & Definitions (3)

• Definition 1
• Definition 2
• Example 1