Large $N$ limits of supersymmetric quantum field theories: A pedagogical overview

TL;DR

The work provides a structured, computation-oriented tour of large $N$ limits in SUSY QFTs across $d=3,4,5$, emphasizing localization to sphere partition functions and saddle-point analyses of resulting matrix models. It categorizes large-$N$ regimes into planar (’t Hooft), M-theory, and long quiver limits, and demonstrates concrete calculations for notable theories such as ABJM, 4d $\mathcal{N}=4$ SYM, and 3d/5d SQCD, including phase-transition phenomena. The methods unify the treatment across dimensions and quiver types, offering explicit density-based solutions and exact results (where available) that match holographic expectations in AdS/CFT. The findings highlight both universal features (saddle-point structure, density formulations) and model-dependent nuances (convergence conditions, scaling exponents $\chi$, and phase structures) with direct implications for holography and SUSY dynamics at strong coupling.

Abstract

The different large $N$ limits of supersymmetric quantum field theories in three, four, and five dimensions are reviewed. We distinguish between the planar limit of SQCD theories, the M-theory limit suited in three and five dimensions, and the long quiver limit. The method to solve exactly the sphere partition functions in each type of limit is spelled out in a pedagogical way. After a comprehensive general treatment of the saddle point approximation in the large $N$ limit, we present an extensive list of examples and detail the calculations. The scope of this overview is to provide an entry-level, computation-oriented understanding of the techniques featured in the field theory side of the AdS/CFT correspondence.

Figures (19)

• Figure 1.1: Organization of the contents. The central node, shaded, contains the core concepts. The warm up sections can be safely skipped by the reader familiar with basic concepts of supersymmetry and saddle point approximation. Solid arrows indicate the suggested reading pattern.
• Figure 2.1: The function $S_{\mathrm{eff}}^{\Gamma} (\phi)$ defined in \ref{['eq:GammaPot']}, shown as a function of $\phi$ for $N=10$ (black), $N=16$ (red) and $N=21$ (purple).
• Figure 2.2: The function $\frac{1}{N}S_{\mathrm{eff}}^{\mathrm{HS}}(\sigma_\ast)$ evaluated on the saddle point.
• Figure 3.1: Portion of a quiver representing a supersymmetric gauge theory. The two round nodes represent the gauge groups $U(N_1)$ and $U(N_2)$. The edge between the two round nodes represents a bifundamental hypermultiplet of $U(N_1)\times U(N_2)$, and the edge between the square node and the round node represents $\mathsf{F}$ additional hypermultiplets in the fundamental representation of $U(N_1)$.
• Figure 4.1: Sketch of the double-scaling limit. In this picture, the parameter space of 't Hooft couplings is the $(\lambda_1,\lambda_2)$-plane. The critical locus is shown in blue. In black, a curve $u \mapsto \lambda (u)$ in this space that intersects the critical locus at a single point $\lambda_{\mathrm{crit}}$. The double-scaling limit consists in sending $N \to \infty$ and $\lvert \lambda (u) - \lambda_{\mathrm{crit}}\rvert \to 0$ such that $N^{\gamma_{\mathrm{crit}}} [\lambda (u) - \lambda_{\mathrm{crit}}]$ stays finite, for a power $\gamma_{\mathrm{crit}} >0$ that depends only on the universality class of the model. The scaled variable $\lambda_{\text{double-scaled}} (u)$ becomes the control parameter of the problem in the double-scaling limit.