Hasse Diagram Inversion in 3D N=4 Theories

• Hasse diagram inversion is a combinatorial operation that reverses the partial order and swaps ADE labels (minimal nilpotent orbit closures with Kleinian singularities) in moduli space diagrams.
• The Higgs–Coulomb inversion conjecture posits that for certain 3D N=4 quiver gauge theories, the Coulomb branch diagram is the inverted form of the Higgs branch diagram, providing predictive duality insights.
• The full moduli-space Hasse diagram is constructed by gluing red (Coulomb) slices with blue (inverted Higgs) subdiagrams, capturing the complete structure including mixed branches.

Hasse diagram inversion is a combinatorial operation on Hasse diagrams of moduli spaces of $\mathrm{3d}$ $\mathcal{N}=4$ quiver gauge theories. In "Hasse Diagrams for $\mathbf{3d}$ $\mathbf{\mathcal{N}=4}$ Quiver Gauge Theories -- Inversion and the full Moduli Space" (Grimminger et al., 2020), the operation is introduced together with a conjecture that, for a class of theories, the Coulomb-branch and Higgs-branch Hasse diagrams are related by inversion. The same framework is used to introduce a Hasse diagram for the entire moduli space, including Coulomb, Higgs, and mixed branches. For theories whose Higgs and Coulomb branch Hasse diagrams are related by inversion, the full moduli-space diagram is obtained directly from this relation; for theories whose branch diagrams are not related by inversion, the full moduli-space Hasse diagram can nevertheless be produced using different methods, and examples are given for theories with enhanced Coulomb branches (Grimminger et al., 2020).

1. Definition of the inversion operation

A Hasse diagram $\mathfrak{H}$ is said to be invertible if every elementary slice labelling it is either a closure of a minimal nilpotent orbit of ADE, denoted $a_n,d_n,e_n$, or a Kleinian singularity of ADE, denoted $A_n,D_n,E_n$ (Grimminger et al., 2020). In this setting, inversion is defined as a map

$$\mathfrak{I}:\{\text{invertible Hasse diagrams}\}\;\longrightarrow\;\{\text{invertible Hasse diagrams}\}.$$

The map acts in two steps. First, it reverses the partial order, turning the diagram upside-down. Second, it exchanges each lowercase slice with its uppercase counterpart: $$a_n\longleftrightarrow A_n,\qquad d_n\longleftrightarrow D_n,\qquad e_n\longleftrightarrow E_n.$$

Concretely, if

$$H\;=\;\Bigl\{\,\bullet\xrightarrow{a_{\alpha_1}}\bullet\xrightarrow{a_{\alpha_2}}\dots\Bigr\},$$

then

$\mathcal{N}=4$0

The operation is therefore purely combinatorial. Its input is the stratified Hasse diagram together with ADE labels on elementary slices, and its output is another invertible Hasse diagram obtained by flipping the order and exchanging minimal nilpotent-orbit closures with Kleinian singularities. The paper summarizes this as: flip the diagram, and swap lowercase nilpotent orbits with uppercase Kleinian singularities (Grimminger et al., 2020).

2. Higgs–Coulomb inversion conjecture

For a class of $\mathcal{N}=4$1 $\mathcal{N}=4$2 quiver gauge theories, the conjecture is

$\mathcal{N}=4$3

and dually,

$\mathcal{N}=4$4

Equivalently,

$\mathcal{N}=4$5

The conjecture is formulated at the level of Hasse diagrams of symplectic stratifications rather than as a direct isomorphism statement between varieties. In the terminology used in the paper, invertibility means that every slice is a known ADE building block, so flipping and exchanging them yields the dual branch (Grimminger et al., 2020). This suggests that the conjecture organizes Higgs–Coulomb duality through the elementary singular slices appearing in the stratification.

The same summary identifies the proposed relation with the realization of the $\mathcal{N}=4$6 mirror or Higgs–Coulomb duality on the level of symplectic stratifications (Grimminger et al., 2020). A plausible implication is that inversion provides a compact encoding of branch duality whenever the corresponding Hasse diagrams are built entirely from ADE minimal nilpotent and Kleinian slices.

3. Canonical examples

The paper gives several explicit examples in which inversion can be checked directly (Grimminger et al., 2020).

For SQED with $\mathcal{N}=4$7 flavours, with quiver $\mathcal{N}=4$8, the Higgs branch has two leaves

$\mathcal{N}=4$9

with slice $\mathbf{3d}$0, while the Coulomb branch has two leaves

$\mathbf{3d}$1

with slice $\mathbf{3d}$2. The corresponding Hasse diagrams are

$\mathbf{3d}$3

and these are related by $\mathbf{3d}$4 together with inversion.

For affine ADE quivers $\mathbf{3d}$5, the Coulomb branch is the minimal orbit closure $\mathbf{3d}$6, while the Higgs branch is the Kleinian singularity $\mathbf{3d}$7. Their Hasse diagrams are

$\mathbf{3d}$8

A more structured example is the three-leaf quiver

$\mathbf{3d}$9

Its Coulomb-branch Hasse diagram, read bottom up, is

$\mathbf{\mathcal{N}=4}$0

while its Higgs-branch Hasse diagram, read bottom up, is

$\mathbf{\mathcal{N}=4}$1

In this case one checks that

$\mathbf{\mathcal{N}=4}$2

These examples exhibit the basic mechanism of inversion in increasing complexity: a single elementary slice in SQED, the affine ADE correspondence, and a multi-step stratification in the three-leaf example.

4. Bad theories and Coulomb-branch prediction

The inversion proposal is also applied to bad theories (Grimminger et al., 2020). The explicit example given is $\mathbf{\mathcal{N}=4}$3 with $\mathbf{\mathcal{N}=4}$4 flavours. Its classical Higgs branch splits into two $\mathbf{\mathcal{N}=4}$5 cones, with Hasse diagram

$\mathbf{\mathcal{N}=4}$6

Inverting this diagram, while forgetting orientation, yields two distinct minimal slices $\mathbf{\mathcal{N}=4}$7. This predicts a Coulomb branch with two bases, in agreement with the abelianisation analysis of Assel et al. cited in the paper (Grimminger et al., 2020). The full Hasse diagram of the quantum moduli space then has two Coulomb cones and two Higgs cones, intersecting along the predicted $\mathbf{\mathcal{N}=4}$8 slice.

This use of inversion is significant because the paper states that the Higgs-branch Hasse diagram can be inverted in order to obtain the Coulomb-branch Hasse diagram for bad theories, with results consistent with the literature (Grimminger et al., 2020). The claim is therefore not only structural but predictive: the combinatorial transformation yields Coulomb-branch singularity data that match known analyses.

A common misconception would be to treat inversion as a universal theorem for all $\mathbf{\mathcal{N}=4}$9 $\mathfrak{H}$0 quiver gauge theories. The paper does not make that claim. It states a conjecture for a class of theories, and it separately notes that there are theories whose Higgs and Coulomb branch Hasse diagrams are not related by inversion, although the full moduli-space Hasse diagram can still be produced by different methods (Grimminger et al., 2020).

5. Construction of the full moduli-space Hasse diagram

When $\mathfrak{H}$1 and $\mathfrak{H}$2 are related by inversion, the combined Hasse diagram $\mathfrak{H}$3 of the entire moduli space is constructed by a gluing procedure (Grimminger et al., 2020). The prescription is:

• Draw $\mathfrak{H}$4 in red: these are the Coulomb directions.
• Attach inverted upward subdiagrams in blue: at each node $\mathfrak{H}$5 of $\mathfrak{H}$6, attach the subtree $\mathfrak{H}$7.
• Impose the dual construction from the Higgs side: at each node attach red copies of inverted Higgs-branch subtrees, ensuring that red and blue slices commute.

For the three-leaf example, the resulting full moduli-space Hasse diagram has three top nodes, identified as the closures of Coulomb, mixed, and Higgs branches (Grimminger et al., 2020). In this sense the full diagram records not only the pure branches but also the mixed-branch structure and the way these strata are connected by elementary slices.

The paper presents this construction as straightforward precisely when the Coulomb and Higgs branch diagrams are related by inversion. A plausible implication is that inversion supplies not only a branch-to-branch correspondence but also a systematic local rule for assembling the global poset of the full moduli space.

6. Physical interpretation and scope

The physical interpretation given in the paper is phrased in terms of brane constructions and transitions between leaves of the Hasse diagram (Grimminger et al., 2020). Leaves correspond to phases in brane constructions, each leaf carrying unbroken gauge plus matter. Elementary slices are the new massless moduli needed to move between leaves; these are identified with Kraft–Procesi transitions. Invertibility means that every slice is a known ADE building block, so flipping and exchanging them yields the dual branch.

Within this interpretation, Hasse-diagram inversion is not merely a graph operation but a rule for reorganizing the singularity content of the moduli space. The paper’s summary attributes three consequences to this framework: predictions for Coulomb-branch singularities of bad theories, construction of the full moduli-space Hasse diagrams including mixed branches by gluing inverted subdiagrams, and consistent checks against brane realization and known mathematical results on affine Grassmannians and hyper-Kähler quotients (Grimminger et al., 2020).

The scope of the proposal is correspondingly delimited. For theories whose Higgs and Coulomb branch Hasse diagrams are related by inversion, the full moduli-space Hasse diagram follows directly. For theories whose branch diagrams are not related by inversion, the full moduli space can nevertheless be described by different methods, and the paper gives examples with enhanced Coulomb branches (Grimminger et al., 2020). This suggests that inversion is a distinguished mechanism inside a broader program of describing the entire moduli space of $\mathfrak{H}$8 $\mathfrak{H}$9 quiver gauge theories via Hasse diagrams.