A Tale of Two Orbits: Non-Simply Laced Mirror

Abstract: A three-dimensional $\mathcal{N}=4$ gauge theory is constructed whose Higgs branch realizes the affine closure of the cotangent bundle of the minimal nilpotent orbit of $\mathfrak{sl}n$. This space is a symplectic singularity recently identified by Fu and Liu as a $\mathrm{U}(1)$ hyperkähler quotient of the closure of the minimal nilpotent orbit of $\mathfrak{so}{2n+2}$. The theory arises by gauging an $\mathrm{SO}(2)\cong\mathrm{U}(1)$ subgroup of the flavour symmetry of $\mathrm{SU}(2)$ SQCD with $n+1$ flavours. The Hilbert series is computed and the stratification is determined. A non-simply laced magnetic quiver is proposed whose Coulomb branch reproduces the same singularity. Evidence is thereby provided for a mirror pair involving a non-simply laced quiver, further tested through quiver subtraction and Hasse diagram inversion. A related $\mathbb{Z}_2$ quotient of the magnetic lattice is also analysed, and the exceptional behaviour in the case $n=2$, where $A_1 \cong C_1$, is explained. This construction provides a concrete example in which the Higgs-branch structure associated with a non-simply laced magnetic quiver can be inferred and validated through its mirror dual.

• The paper introduces a gauge-theoretic construction that realizes the affine closure of the cotangent bundle of the minimal nilpotent orbit in a 3D N=4 theory.
• It computes Hilbert series for Higgs and Coulomb branches, confirming mirror symmetry through a non-simply laced magnetic quiver formulation.
• The work employs Hasse diagram inversion and discrete gauging techniques to systematically map the symplectic singularities and stratification of moduli spaces.

Summary of "A Tale of Two Orbits: Non-Simply Laced Mirror" (2605.14695)

Introduction and Context

The paper addresses the construction and study of moduli spaces in three-dimensional $\mathcal{N}=4$ supersymmetric gauge theories, with emphasis on Higgs and Coulomb branches as hyperkähler varieties and symplectic singularities. The systematic use of quiver gauge theories enables the realization of nilpotent orbit closures and their stratifications. Mirror symmetry and symplectic duality provide dual geometric perspectives, exchanging the Higgs and Coulomb branches, particularly relevant for the understanding of symplectic singularities associated with non-simply laced quivers—a theme that remains less developed compared to the simply laced case due to the lack of direct Lagrangian constructions.

Recent work by Fu and Liu demonstrated that the affine closure of the cotangent bundle of the minimal nilpotent orbit of $\mathfrak{sl}_n$ can be interpreted as a $\mathrm{U}(1)$ hyperkähler quotient of the closure of the minimal nilpotent orbit of $\mathfrak{so}_{2n+2}$. The present work realizes this correspondence gauge-theoretically: a 3d $\mathcal{N}=4$ theory $\mathcal{T}$ is constructed whose Higgs branch matches the aforementioned symplectic singularity, and its mirror is proposed to be a non-simply laced magnetic quiver $\mathcal{T}^\vee$.

Construction of the Gauge Theory and Higgs Branch Analysis

The theory $\mathcal{T}$ consists of an $\mathrm{SU}(2)$ gauge node coupled to an $\mathrm{SO}(2) \cong \mathrm{U}(1)$ gauge node, with $\mathfrak{sl}_n$0 fundamental flavours and an additional $\mathfrak{sl}_n$1 flavour node. The continuous flavour symmetry is $\mathfrak{sl}_n$2. This setup implements a $\mathfrak{sl}_n$3 gauging of the flavour symmetry of $\mathfrak{sl}_n$4 SQCD with $\mathfrak{sl}_n$5 flavours.

The Higgs branch $\mathfrak{sl}_n$6 is shown to realize the affine closure of the cotangent bundle of the minimal nilpotent orbit of $\mathfrak{sl}_n$7:

$\mathfrak{sl}_n$8

where the quotient is realized as a $\mathfrak{sl}_n$9 hyperkähler quotient of the closure of the minimal nilpotent orbit of $\mathrm{U}(1)$0. This identification is central to the paper.

The Hilbert series for the Higgs branch is computed explicitly for $\mathrm{U}(1)$1, establishing the operator counting:

• For $\mathrm{U}(1)$2:

$\mathrm{U}(1)$3

• For $\mathrm{U}(1)$4:

$\mathrm{U}(1)$5

The stratification of the Higgs branch is analyzed using the Higgs mechanism, displaying a hierarchy of theories obtained via vacuum expectation values assigned to hypermultiplets. The resulting stratification and Hasse diagrams of the Higgs branch correspond to the symplectic leaf structure of the underlying singularity.

Mirror Theory and Non-Simply Laced Quivers

The paper proposes, based on stratification matching and Hilbert series agreement, a mirror theory $\mathrm{U}(1)$6 represented by a non-simply laced quiver with $\mathrm{U}(1)$7 nodes and double edges corresponding to non-simply laced connections. The mirror proposal is supported by:

• Hilbert series computation on the Coulomb branch, matching the Higgs branch series of $\mathrm{U}(1)$8
• Hasse diagram inversion, consistent with symplectic duality, matching the stratification patterns

The monopole formula is applied to compute the Coulomb branch Hilbert series of $\mathrm{U}(1)$9, incorporating non-simply laced contributions through asymmetric terms in the conformal dimension. For $\mathfrak{so}_{2n+2}$0, the Hilbert series coincides with that of $\mathfrak{so}_{2n+2}$1, demonstrating equivalence of branches.

This construction is noteworthy for providing a concrete example in which the Higgs branch geometry associated with a non-simply laced magnetic quiver can be inferred and verified through its mirror dual, facilitating analysis of such cases despite the absence of a direct Lagrangian description.

Symplectic Duality and Inversion Procedure

Building on recent advancements in symplectic duality, the paper reconstructs the Higgs branch of $\mathfrak{so}_{2n+2}$2 from its Coulomb branch via inversion of the Hasse diagram, employing the concept that the geometry of the moduli space is encoded in the data of transverse slices associated with quiver gauge theories.

Importantly, the inversion carefully addresses subtleties arising from trivial slices and FI parameter contributions, demonstrating that even theories with trivial geometry may affect the global structure of the moduli space after inversion. The procedure is rigorously established for both Higgs and Coulomb branches, confirming duality and branch equivalence.

Moreover, the results show that the mirror pair $\mathfrak{so}_{2n+2}$3 exhibits a strong numerical correspondence for Hilbert series and Hasse diagrams, reinforcing conjectural mirror symmetry for these theories.

Discrete Quotients and Alternative Constructions

The paper explores discrete quotients related to different ungauging choices. If a $\mathfrak{so}_{2n+2}$4 quotient is applied to the magnetic lattice, the resulting Coulomb branch is a double cover, and the new quiver's Coulomb branch corresponds to a slice in the affine Grassmannian, with the singularity being a double cover of the slice associated with $\mathfrak{so}_{2n+2}$5 in $\mathfrak{so}_{2n+2}$6.

Hasse diagrams of these quotient theories are analyzed, clarifying that discrete gauging of magnetic symmetries modifies only the Coulomb branch, not the Higgs branch. Distinctions between the descriptions of leaves and slices for quotient theories are shown to persist even when their Coulomb branches are equal, highlighting the necessity of careful gauge-theoretic labeling for inversion.

Special Case: $\mathfrak{so}_{2n+2}$7

The $\mathfrak{so}_{2n+2}$8 case diverges from the general pattern, as $\mathfrak{so}_{2n+2}$9 and $\mathcal{N}=4$0 coincide, altering the identification of the singularity. The theory $\mathcal{N}=4$1 is self-mirror, with identical Higgs and Coulomb branches, corresponding to the quotient $\mathcal{N}=4$2. The minimal nilpotent orbit closure in this scenario aligns with the $\mathcal{N}=4$3 singularity within the $\mathcal{N}=4$4 family. The inversion procedure for isolated singularities becomes tautological, emphasizing the need for comprehensive understanding of Higgs branches for general non-simply laced theories.

Implications and Future Directions

The construction and analysis provided offer definitive evidence for mirror symmetry between gauge-theoretic and geometric realizations of non-simply laced symplectic singularities. The explicit stratification matching and Hilbert series correspondence establish a template for studying cases beyond the simply laced paradigm, contributing to the broader program of extending quiver-based constructions to non-simply laced moduli spaces.

Practically, this facilitates the indirect analysis of moduli spaces absent direct Lagrangian descriptions via mirror duality and symplectic inversion. Theoretically, it suggests that symplectic duality and Hasse diagram inversion can provide a systematic, slice-by-slice reconstruction of inaccessible Higgs branches.

Further developments may include:

• Generalizing the inversion procedure for more intricate non-simply laced quivers
• Extending mirror symmetry checks to other families of symplectic singularities
• Developing a first-principles mathematical description of Higgs branches for general non-simply laced quivers
• Exploring mass and FI parameter deformations within the duality framework
• Investigating brane or folding constructions for a more comprehensive understanding of physical realizations

Conclusion

This work establishes, via explicit gauge-theoretic and geometric constructions, the realization and duality of non-simply laced symplectic singularities in 3d $\mathcal{N}=4$5 theories. Through Higgs branch stratification, Hilbert series computations, and symplectic duality, the paper demonstrates equivalence between the Higgs branch of an $\mathcal{N}=4$6-$\mathcal{N}=4$7 theory and the Coulomb branch of its non-simply laced mirror quiver. The results provide a foundation for analyzing moduli spaces in settings where direct Lagrangian approaches are unavailable and suggest robust methods for extending these ideas to broader classes of gauge theories and symplectic singularities.