SU(N) gauge theories at large N

TL;DR

This article surveys the SU($N$) gauge theories in the large-$N$ limit, tracing the original planar expansion of ’t Hooft to modern non-perturbative lattice approaches and holographic dualities. It lays out core theoretical structures—planar dominance, factorization, master-field concepts, loop equations, and volume reduction—then reviews extensive lattice results across 4D, 3D, and 2D that test and refine large-$N$ predictions for confinement, spectra, topology, and finite-temperature behavior. The lattice results consistently show confinement persisting with mild $1/N$ corrections, string-like flux tubes, and a spectrum of stable mesons and glueballs with interactions suppressed as powers of $1/N$, while thermodynamics and topological properties exhibit characteristic large-$N scaling and phase structures. The review also connects these findings to holographic models and orbifold/orientation equivalences, highlighting how large-$N$ methods illuminate non-perturbative QCD-like dynamics and guiding future explorations of gauge theories with varied matter content and spacetime dimensionality.

Abstract

We review the theoretical developments and conceptual advances that stemmed from the generalization of QCD to the limit of a large number of color charges, originally proposed by 't Hooft. Then, after introducing the gauge-invariant non-perturbative formulation of non-Abelian gauge theories on a spacetime lattice, we present a selection of results from recent lattice studies of theories with a different number of colors, and the findings obtained from their extrapolation to the 't Hooft limit. We conclude with a brief discussion and a summary.

Figures (21)

• Figure 1: Different types of three-loop Feynman diagrams contributing to the gluon propagator; in the 't Hooft limit, only the diagram on the left panel yields a non-negligible contribution.
• Figure 2: Large-$N$ counting rules offer a simple interpretation for the Okubo-Zweig-Iizuka rule: the figure shows an example of Feynman diagrams relevant for the propagation of a meson: the diagram on the right, in which the process goes through an intermediate stage where only virtual gluons appear, is suppressed by a power of $1/N$ with respect to the one on the left.
• Figure 3: A simplified cartoon of the gauge/string correspondence: the figure shows a set of $N$ coincident D3-branes (D3, at the bottom of the figure), supporting excitations described by open strings ($o$), which describe gauge interactions. The D3-branes are heavy objects, so they curve the six extra dimensions of the spacetime in which they are defined. For the sake of clarity, the cartoon only displays a sketch of the extra dimensions through a given point $x$ on the branes. Starting from $x$, the metric in eq. (\ref{['brane_metric']}) shows that at small distance $r$ from the branes ($r \ll R$) the geometry of the ten-dimensional spacetime is approximately that of a five-dimensional anti-de Sitter spacetime ($AdS_5$), times a five-dimensional sphere ($S^5$), whose coordinates are here symbolically denoted as $\Omega_5$ (while for $R \ll r$ it tends to a ten-dimensional Minkowski spacetime). The gravitational excitations in this space are described by closed strings ($c$). The boundary ($b$) of the $AdS_5$ spacetime is obtained for $r \to \infty$, and is conformally equivalent to four-dimensional Minkowski spacetime. The holographic correspondence expressed by eq. (\ref{['AdS_CFT_mapping']}) states that the string partition function with a bulk field $\mathcal{J}$, which reduces to a field $J$ on the boundary---up to a scaling factor expressed by eq. (\ref{['boundary_condition']})---, is equivalent to the generating functional of the gauge theory defined in four-dimensional Minkowski spacetime, with a source $J$ coupled to a gauge-invariant operator $\mathcal{O}$, as in eq. (\ref{['source_term']}).
• Figure 4: A comparison of the results of simulations of the twisted EK model performed in ref. GonzalezArroyo:2012fx using the twist formulation discussed in ref. GonzalezArroyo:2010ss (green symbol), with those from lattice simulations performed in a large volume (in the conventional formulation of the theory, without exploiting the volume-independence properties), taken from ref. Allton:2008ty (red symbols), and their extrapolation as a function of the number of colors (blue line). The quantity which is plotted is the ratio (extrapolated to the continuum limit) between the $\Lambda_{\tiny{\hbox{QCD}}}$ parameter in the $\overline{\textrm{MS}}$ scheme, and the square root of the string tension, i.e., of the asymptotic force between static color sources at large separations.
• Figure 5: Lattice simulations of large-$N$ gauge theories with $n_f=1$ and $n_f=2$ flavors of dynamical adjoint fermions in a small volume indicate that center symmetry is preserved in a wide region of quark masses, whose width remains finite in the continuum limit. The figure, taken from ref. Bringoltz:2011by, is a sketch of the symmetry realizations in different regions of the space of the simulation parameters. In particular, the regions in which center symmetry is preserved in all the four directions are denoted by $Z_N^4$. The quantity on the vertical axis ($b$) is the inverse of the bare lattice 't Hooft coupling, so that (by virtue of asymptotic freedom) the continuum limit is obtained by taking $b \to \infty$---while the region corresponding to small values of $b$ is the strong-coupling region of the lattice theory. The quantity on the horizontal axis ($\kappa$) is a simulation parameter related to the fermion mass: in particular, the latter is infinite for $\kappa=0$ (corresponding to the pure Yang-Mills theory), while it decreases when $\kappa$ is increased towards the dashed region, reaching zero at the critical value $\kappa_c$. The figure, based on information obtained from simulations with $n_f=2$ dynamical quarks in the adjoint representation of the gauge group in a small volume, shows that, in the presence of dynamical adjoint fermions, the spontaneous breakdown of center symmetry is avoided, for a rather wide range of quark masses, in a region that extends to the continuum limit.