Supersymmetric Linear Quivers

Updated 20 August 2025

Supersymmetric linear quivers are graphical frameworks that represent gauge groups and matter fields as nodes and edges, crucial for modeling diverse supersymmetric theories.
Matrix model localization techniques reveal key eigenvalue cancellation conditions that are central to matching supergravity duals and understanding large-N dynamics.
The integration of representation theory, brane constructions, and geometric methods provides deep insights into BPS spectra, moduli spaces, and operator dualities in these systems.

A supersymmetric linear quiver is a gauge-theoretic structure in which gauge groups and their matter content are graphically encoded as nodes and edges arranged in a line. Each node represents a gauge group factor (typically unitary, orthogonal, or symplectic), and edges correspond to matter fields, most commonly bifundamental or fundamental hypermultiplets. Supersymmetric linear quivers provide a flexible encoding of a wide range of gauge theories that manifest different dimensions and supersymmetry classes, and admit a rich spectrum of dualities, geometric interpretations, and applications in both field theory and string theory contexts.

1. Matrix Model Solutions and the Constraints of Supersymmetry

The partition function of supersymmetric quiver gauge theories—especially in 3d with N = 3 (or higher) supersymmetry—can be computed exactly via localization to a matrix model. For 3d $\mathcal{N} = 3$ Chern-Simons-matter theories, the Kapustin–Willett–Yaakov (KWY) matrix model realizes the $S^3$ partition function as an eigenvalue integral: $Z = \int \prod_{a, l} d\mu_{a, l} \, \prod_{a} L_V(G_a, k_a, \mu_a) \prod_I L_M(\ldots)$ where $L_V$ gives the vector multiplet (CS and gauge) contributions and $L_M$ encodes matter contributions, e.g., for bifundamental hypermultiplets. For $U(N)$ gauge nodes,

$L_V(U(N),k,\mu) = e^{i\pi k \sum_m \mu_m^2} \prod_{l < m} [2 \sinh \pi(\mu_l - \mu_m)]^2.$

In the presence of bifundamental matter, saddle-point techniques determine the large $N$ eigenvalue distributions. A crucial requirement for emergent supergravity duals with $N^{3/2}$ free energy scaling is the cancellation of long-range eigenvalue interactions. Imposing this condition constrains the allowed quiver topologies: nodes with effective ranks $n_a$ (i.e., $N_a = n_aN$ ) must satisfy

$2 n_a = \sum_{b : (a,b) \in E} n_b,$

or, rewritten, $\sum_b \widehat{A}_{ab} n_b = 0$ with $\widehat{A}$ an affine Cartan matrix. This selection principle restricts admissible linear quivers to those associated with affine ADE Dynkin diagrams or (upon folding) their non-simply laced cousins (Gulotta et al., 2012).

2. Quiver Algebra, Representation Theory, and BPS Spectrum

In four-dimensional $\mathcal{N} = 2$ setups, the spectrum of BPS states is encoded in the representation theory of quivers with superpotentials. Each node is associated with a one-dimensional lattice vector $e_v$ in the electromagnetic charge lattice $\Gamma$ , and arrows are determined by the Dirac pairing $B_{uv} = \langle e_u, e_v\rangle_{\text{Dirac}}$ . Representations $X$ with dimension vector $\mathbf{d} = \sum_v (\dim X_v) e_v$ satisfying F-term constraints from the superpotential $W$ (i.e., $J(Q,W) = \mathbb{C}Q/(\partial W)$ ) enumerate classical BPS configurations. The stability of BPS objects is determined by central charge data: $Z(X) = -\frac{1}{g_{\text{YM}}^2} \sum_\alpha C_\alpha m_\alpha(X) + O(1), \quad m_\alpha(X) = \langle \dim X, \mathfrak{q}_\alpha \rangle_{\text{Dirac}}.$ Half-hypermultiplet representations and their intricate coupling constraints can be handled recursively via Higgs decoupling chains and Dirac quantization, yielding extended quiver structures encoding the desired spectrum (e.g., SU(6) $\oplus$ ½(20), SO(12) $\oplus$ ½(32), E $_7$ $\oplus$ ½(56)) (Cecotti, 2012).

3. Dualities, Folding/Unfolding, and Classification

Supersymmetric linear quivers are central to the web of Seiberg-like dualities in 3d $\mathcal{N}=2$ , $\mathcal{N}=3$ , and $4d$ $\mathcal{N}=1,2$ gauge theories. In 3d Chern-Simons-matter theories, generalized Seiberg duality is realized as a Weyl reflection on the underlying affine Dynkin diagram,

$k_b \rightarrow k_b - (\alpha_a, \alpha_b) k_a,$

with analogous transformations for the eigenvalue distributions. Furthermore, the “folding/unfolding” technique systematically maps unitary, orthogonal, and symplectic group quivers into one another by employing $\mathbb{Z}_2$ outer automorphisms (at both matrix model and quiver diagram level), tracking eigenvalue duplication and sign reversals. This machinery produces non-simply laced quivers, captures transitions between simply laced and folded types, and is essential for matching partition functions and free energies of dual theories. Folding is physically realized in the brane construction context as the addition or removal of orientifold/orbifold planes (O3/O5) acting on the D3-brane configuration, corresponding to discrete symmetries on the quiver (Gulotta et al., 2012).

In $4d$ $\mathcal{N}=1$ and $\mathcal{N}=2$ linear quivers, the Bailey lemma acts at the level of the elliptic hypergeometric superconformal index and induces a duality web by recursive "quivering" of base $s$ -confining theories. The corresponding index relations and operator mappings, including the Seiberg duality exchange $n \leftrightarrow m$ , systematically enumerate all dual quiver descriptions for given $(N_c, N_f)$ data (Brünner et al., 2016).

4. Cohomological, Geometric, and Operator Aspects

On the quantum mechanical level, the moduli space $\mathcal{M}$ of supersymmetric linear quivers (as in quiver quantum mechanics) is a complete intersection subject to D- and F-term constraints. Its topology controls refined BPS state counts via Poincaré polynomials,

$Q(\mathcal{M};y) = (-y)^{-d}\sum_{p=0}^{2d}b_p(\mathcal{M})(-y)^p,$

with the full cohomology recoverable from the Coulomb branch fixed point formula (multi-center localization) plus pure-Higgs (middle cohomology) "single-centered" contributions. This decomposition is mirrored in multi-centered black hole physics, where scaling solutions and wall-crossing phenomena correspond to structure in the quiver moduli space’s Betti numbers (Manschot et al., 2012).

Chiral operator counting, fusion coefficients, and the entire structure of the free and interacting chiral rings are efficiently formulated using permutation-based "split-node quivers," symmetric group representations (via Young diagrams, Littlewood-Richardson coefficients), and relations to topological field theory (S $_n$ TFT) on thickened quiver surfaces. At large $N$ , elegant infinite-product formulas capture the scaling of operator spectra (Pasukonis et al., 2013).

5. Brane Constructions, Geometry, and Moduli Spaces

The realization of supersymmetric linear quivers in string theory utilizes Hanany-Witten brane setups, with D3-branes suspended between NS5 and D5 branes in Type IIB. This engineering establishes a geometric interpretation of linear quivers, with brane moves corresponding to gauging and decoupling of global symmetries, and the moduli space identified as the Higgs (or Coulomb) branch of the corresponding gauge theory. The symmetry enhancement, folding, and the emergence of non-simply laced quivers are mapped to the inclusion of orientifold and orbifold planes in this setup, and the moduli spaces often have a geometric interpretation as symmetric products of hyperkähler cones or more general spaces with tri-Sasaki-Einstein metrics (Gulotta et al., 2012, Dey et al., 2014).

Mirror symmetry in 3d $\mathcal{N}=4$ linear quivers exchanges Higgs and Coulomb branch data, maps FI parameters to masses, and is manifest in dual brane realizations as S-duality. Gauging of flavor symmetries allows derivation of more intricate quiver types (D, E, star-shaped), all with computable partition functions and Hilbert series; dualities are verified by matching S $^3$ partition functions and Hilbert series (Dey et al., 2014).

6. Operator Dualities and Monopole Mappings

Chiral ring data (including monopoles, baryons, baryon-monopoles) and precise mapping of gauge-invariant operators across dualities in $\mathcal{N}=2$ quivers are controlled by nodewise rules: undressed monopoles extend or shorten across dualizations, while baryon and baryon-monopole structures appear in quivers with orthogonal or unitary nodes. Supersymmetric indices and R-charge assignments provide stringent non-perturbative checks of these operator identifications, crucial for establishing dualities, especially in the context of deconfinement, adjoint matter, and the interplay between continuous and discrete global symmetries (Benvenuti et al., 2020).

7. Lattice Realizations and Nonperturbative Aspects

Discrete lattice formulations of supersymmetric linear quivers are achievable via topologically twisted formulations and geometric discretization, preserving a nilpotent scalar supersymmetry. Lattice fields are assigned to links, faces, and higher cells according to their spin and representation properties, and quiver structure is captured by associating adjoint variables to each lattice and bifundamental fields to links connecting distinct gauge group lattices. These lattice models enable controlled nonperturbative investigation of dualities, strong coupling, and supersymmetric gauge theory dynamics, opening paths to numerical simulation and exploration of the quantum moduli spaces (Joseph, 2013).

Supersymmetric linear quivers, through their algebraic, geometric, and physical richness, serve as a critical laboratory for exploring the interplay between gauge dynamics, dualities, string theory, and nonperturbative quantum field theory phenomena. Their analysis invokes matrix models, representation theory, combinatorics, geometric engineering, and localization, yielding precision tools for the paper of moduli spaces, BPS spectra, duality webs, and the geometric correspondence with brane configurations and their associated gravity duals.