Linear Quivers at Large-$N$

TL;DR

This work develops a dimension-spanning framework for long linear quivers with eight supercharges, solved at large $N$ via localisation on $\mathbb{S}^d$. A universal Poisson equation for the eigenvalue density governs the long-quiver limit across $d\ge3$, with a rank-function Fourier data $R_k$ encoding quiver detail. In $d=4$ they treat finite and infinite coupling, computing anomaly coefficients, sphere partition functions, extremal correlators, Wilson loops, and a holographic comparison through a simplified Gaiotto–Maldacena background; defects of various dimensionalities are analyzed and mirror-type relations between quivers are established. The interpolating function $\tilde{F}_d$ connects the $d=3$ and $d=5$ results and recovers the Weyl anomaly $a$ at $d=4$, demonstrating universal large-$N$ behavior and supporting a holographic dual description for long quivers across dimensions. Two explicit examples illustrate the framework, confirming the analytical structure and highlighting the potential for string-theory embeddings and further holographic comparisons.

Abstract

Quiver theories constitute an important class of supersymmetric gauge theories with well-defined holographic duals. Motivated by holographic duality, we use localisation on $S^d$ to study long linear quivers at large-N. The large-N solution shows a remarkable degree of universality across dimensions, including $d = 4$ where quivers are genuinely superconformal. In that case we upgrade the solution of long quivers to quivers of any length.

Figures (8)

• Figure 1: Linear quiver of length $P-1$. Circular nodes indicate gauge groups, square nodes indicate flavour symmetries.
• Figure 2: Two sample plots of the rank function $\mathcal{R} (\eta)$. Left: first example, also known as $+_{N,P}$ theory. Right: second example, also known as $T_P$ theory if $N=1$.
• Figure 3: Left: balanced linear quivers that satisfy \ref{['eq:assumptionMir']}. Right: the corresponding quivers characterised by $\mathcal{R}^{\vee} (\eta^{\vee})$. From top to bottom: $(P,K)=(4,16),(4,8),(4,8),(6,12)$. In the right quiver of the first line the dots omit 6 gauge nodes of rank 8; in the right quiver of the fourth line the dots omit 2 gauge nodes of rank 12. In $d=3$ the second and third pairs of quivers are mirror pairs.
• Figure 4: Two quivers that satisfy \ref{['eq:assumptionMir']} are related by partial Higgsing. Left: The graph of the rank function $\mathcal{R}(\eta)$ is triangular, hence $\mathcal{R}^{\vee} (\eta^{\vee})$ describes the genuine mirror in $d=3$. Right: The rank functions $\mathcal{R} (\eta)$ and $\mathcal{R}^{\vee} (\eta^{\vee})$ are not related by duality.
• Figure 5: Linear quiver of length $P-1$. Circular nodes indicate gauge groups, square nodes indicate flavour symmetries. When the quiver is balanced, it describes a four-dimensional $\mathcal{N}=2$ SCFT.

Theorems & Definitions (33)

• Definition 2.1
• proof : Digression
• Definition 2.2
• Definition 2.3
• Definition 2.5
• proof : Digression
• proof
• Lemma 2.7
• Definition 2.8
• Definition 2.9