Magnetic Quiver Framework

• Magnetic Quiver Framework is a combinatorial and physical method that translates brane system data into quiver diagrams to study Higgs branch moduli spaces.
• It systematically captures phase transitions, discrete gauging, and symmetry enhancements, enabling a clear stratification of moduli spaces.
• The approach supports duality checks using Hilbert series computations and operations like polymerisation and quotient quiver subtraction for practical applications.

The Magnetic Quiver Framework provides a rigorous combinatorial and physical machinery for encoding, analyzing, and stratifying the Higgs branch moduli spaces of gauge theories with extended supersymmetry, particularly those arising as worldvolume theories on stacks of M5 branes probing ADE-type singularities or other high-dimensional brane setups. It establishes an explicit equivalence between geometric objects (such as closures of nilpotent orbits and symplectic singularities) and the Coulomb branches of associated 3d $\mathcal{N}=4$ quiver gauge theories, derived directly from brane system data and phase transitions inherent to the parent theory.

1. Conceptual Foundations and Construction of Magnetic Quivers

Magnetic quivers are auxiliary 3d $\mathcal{N}=4$ quiver gauge theories constructed so that their Coulomb branches precisely reproduce the Higgs branches of higher-dimensional gauge theories at both finite and infinite gauge coupling. The essential formula relating these spaces is

$$\mathcal{H}_{\text{Higgs}}^{(D)} \cong \mathcal{C}_{\text{Coulomb}}^{(3d)}(\mathcal{Q}_{\text{mag}})$$

where $\mathcal{Q}_{\text{mag}}$ is the magnetic quiver, typically read off from a Type IIA or Type IIB brane configuration engineered to reflect all relevant gauge couplings, flavor symmetries, and discrete data (Cabrera et al., 2019, Bourget et al., 2020).

The magnetic quiver is constructed by systematically mapping physical objects (D-branes, NS5-branes, orientifold planes) and their arrangements, including suspended stacks and brane intersections, into gauge nodes and matter representations in the quiver. Notably, the rules for node assignment and bifundamental matter are sensitive to both the underlying brane projection (unitary, orthogonal, symplectic) and the discrete brane content (for instance, negative branes in D-type setups (Hanany et al., 2022)).

The framework generalizes prior approaches by encoding all possible Higgs branch phases—including strong coupling transitions, discrete gaugings, and exceptional global symmetry enhancements—at the level of the quiver diagram and its associated moduli space geometry.

2. Phase Structure, Transitions, and Hasse Diagrams

A central merit of the Magnetic Quiver Framework is its ability to describe the stratification of Higgs branch moduli space as the parent theory’s parameters (notably, gauge couplings) are tuned. Phase transitions are of several types:

• Finite vs. Infinite Gauge Coupling: At generic points on the tensor branch, the Higgs branch is described by conventional hyper-Kähler quotients. As NS5-brane separations are tuned to zero (infinite coupling), new massless degrees of freedom (tensionless strings) appear, yielding an enlarged Higgs branch. The magnetic quiver explicitly encodes these transitions via node augmentation (Cabrera et al., 2019).
• Discrete Gauging: Coincident NS5-brane pairs can lead to nontrivial discrete gauging (permuting pairs), with the moduli space dimension preserved but sectors identified under discrete symmetries (typically $S_n$) (Cabrera et al., 2019, Hanany et al., 2023).
• Exceptional Instanton Transitions: Small $E_8$ instanton transitions yield moduli spaces realizable as nilpotent orbit closures, most transparently seen in the magnetic quiver growing an $E_8$ bouquet node at infinite coupling (Cabrera et al., 2019, Hanany et al., 2022).

The entire phase structure is organized into a Hasse diagram, with strata labeled by the types and number of transitions, and organized by partitions encoding discrete gaugings (e.g., $\{n_i\}$ labeling conjugacy classes) (Cabrera et al., 2019, Bourget et al., 2020).

3. Extension to Orthogonal and Symplectic Gauge Groups

The framework is extended to systems with O6 and O5 orientifold planes, which induce projections onto orthosymplectic gauge groups in the parent theory (Cabrera et al., 2019, Bourget et al., 2020). This necessitates conjectured rules to associate magnetic gauge nodes (types $b_n, c_n, d_n$) depending on orientifold charge and brane content:

• Magnetic orientifolds convert electric O6 planes into magnetic counterparts according to explicit tables, yielding symplectic or orthogonal nodes with requisite matter content (antisymmetric, bifundamental, etc.).
• Linking number conservation is modified in the presence of orientifolds (half-brane transitions change O6 type while maintaining charge conservation) (Cabrera et al., 2019).

The resulting magnetic quivers often alternate between orthogonal and symplectic nodes and may require discrete quotients (such as $\mathbb{Z}_2$) to properly account for gauge group centers acting trivially on matter (Bourget et al., 2020).

4. Magnetic Quivers with Boundary Conditions and Negative Branes

For brane configurations with boundary conditions specified by partitions (in A-type: $SU(k)$; in D-type: $SO(2k)$), the magnetic quiver is assembled from blocks associated to such partitions (via $T_\rho$ theories) glued to central ADHM-like components. In D-type cases, the necessary inclusion of negative brane numbers (for certain partition boundary conditions) gives rise to nodes with negative imbalance ("bad nodes") which, after analytic continuation or reinterpretation, yield consistent anomaly coefficients and Higgs branch dimensions (Hanany et al., 2022). The construction is schematized as: $$\begin{aligned} Q_{\rho_L, \rho_R}^{\text{A-type}} &= \left( T_{\rho_L}[U(k)] \times Y^A_{n,k} \times T_{\rho_R}[U(k)] \right) \underset{U(k)}{///}, \ Q_{\rho_L, \rho_R}^{\text{D-type}} &= \left( T_{\rho_L}[SO(2k)] \times Y^D_{n,2k} \times T_{\rho_R}[SO(2k)] \right) \underset{SO(2k)/\mathbb{Z}_2}{///} \end{aligned}$$ with discrete quotients reflecting orientifold identifications (Hanany et al., 2022).

5. Discrete Quotation, Polymerisation, and Quotient Quiver Subtraction

Recent developments introduce sophisticated diagrammatic operations on magnetic quivers, critically broadening the scope of the framework:

• Discrete Quotients (Orbifold Operations): Operations whereby permutation symmetry ($S_n$) and cyclic symmetry ($\mathbb{Z}_q^{n-1}$) are gauged to produce moduli spaces as orbifolds of simpler complete-graph (bouquet) quivers. This centralizes the monopole formula with Molien sums over discrete symmetries (Hanany et al., 2023).
• Polymerisation Techniques: Chain and cyclic polymerisation procedures gauge diagonal SU/U$(k)$ subgroups of Coulomb branch global symmetries, implemented as hyper-Kähler quotients. These methods enable systematic construction of Kronheimer-Nakajima and ADHM-type quivers and produce instanton moduli spaces for SU$(N)$ instantons on $\mathbb{C}^2$ and A-type singularities (Hanany et al., 17 Jun 2024).
• Orthosymplectic Quotient Quiver Subtraction: New operations whereby a chosen orthosymplectic quotient quiver (representing e.g., SU(2), SU(3), SO(7) subgroups) is subtracted from a target magnetic quiver, effecting the gauging of the subgroup at the level of the Coulomb branch. This generalizes Kraft–Procesi transitions and systematically produces moduli spaces as symplectic quotients (Bennett et al., 23 Sep 2024, Bennett et al., 25 Mar 2025).

These techniques provide rigorous methods for generating new quivers associated to closures of nilpotent orbits, affine Grassmannian slices, and moduli spaces in class S constructions, including framed and unframed orthosymplectic cases.

6. Mathematical Formulations, Hilbert Series, and Computational Tools

The identification of the Higgs branch with the Coulomb branch of a magnetic quiver is demonstrated via explicit computation of Hilbert series using the monopole formula. The Hilbert series

$$HS_{\mathcal{C}}(t) = \sum_{\mathfrak{m}} t^{\Delta(\mathfrak{m})} P_G(t, \mathfrak{m})$$

counts dressed monopole operators (labeled by magnetic charge $\mathfrak{m}$), with $P_G$ encoding residual gauge symmetry. For non-simply laced quivers or those with multiplicity, contributions are adjusted for edge weights, and discrete group quotients are implemented via Molien averaging.

For orthosymplectic quivers, the choice of gauge group (e.g., quotienting by $\mathbb{Z}_2^{\text{diag}}$) alters the magnetic lattice and, consequently, the spectrum of operators and moduli space geometry (Bourget et al., 2020). These considerations are essential in matching moduli spaces described by various magnetic quiver presentations.

Advanced operations such as polymerisation and quotient quiver subtraction are validated by comparing Hilbert series to expected closures of nilpotent orbits or instanton moduli spaces, with plethystic logarithms revealing generators and relations associated to the global symmetry.

7. Duality, Mirror Symmetry, and Interchangeability of Quiver Constructions

A consequential result established by the framework is the IR duality between apparently distinct magnetic quiver constructions for the same moduli space, notably between unitary and orthosymplectic quivers derived from different brane setups (O5/O7 planes, framed/unframed quivers) (Nawata et al., 2021, Bennett et al., 23 Sep 2024, Bennett et al., 25 Mar 2025). Matching of sphere partition functions, supersymmetric indices, and spectra of line defects confirms that multiple quiver presentations are physically equivalent in the IR and equally valid for computing the vacuum structure and operator content.

These dualities imply a substantial redundancy in quiver descriptions, allowing researchers to select representations best suited to computational or conceptual clarity.

• Key original frameworks and extensions: (Cabrera et al., 2019, Bourget et al., 2020, Hanany et al., 2022, Hanany et al., 2023, Hanany et al., 17 Jun 2024).
• Discrete symmetry actions and quiver quotienting: (Hanany et al., 2023).
• Polymerisation and construction of instanton moduli spaces: (Hanany et al., 17 Jun 2024).
• Orthosymplectic quotient quiver subtraction and generalizations: (Bennett et al., 23 Sep 2024, Bennett et al., 25 Mar 2025).
• Dual magnetic quivers and IR equivalence: (Nawata et al., 2021).
• Mathematical symmetries and isomorphisms of quiver schemes: (Terada et al., 16 Apr 2024).

The Magnetic Quiver Framework thus synthesizes brane system data, combinatorial quiver techniques, and advanced computational tools (Hilbert series, monopole formula, discrete quotients) to yield a systematic, algorithmic method for analyzing the nonperturbative geometry of vacuum moduli spaces in supersymmetric gauge theories, with applications to Higgs branch physics, instanton moduli spaces, and the geometric representation of dualities and symmetry enhancements in string and M-theory contexts.