Large N Gauge Theory

Large-$N$ gauge theory is the first place where AdS/CFT starts to look less like magic. A gauge theory with gauge group $SU(N)$ or $U(N)$ contains fields with color indices. When those fields are in the adjoint representation, they are $N\times N$ matrices. Feynman diagrams then carry not only spacetime and momentum information, but also color topology.

The large-$N$ limit turns this topology into an organizing principle. Instead of expanding only in a small coupling constant, we reorganize diagrams by the genus of a two-dimensional surface. Planar diagrams dominate, nonplanar handles are suppressed by powers of $1/N^2$, and the resulting expansion has the same form as a closed-string perturbation series.

This is the field-theory reason why holography does not merely relate a gauge theory to a classical spacetime. At large $N$, gauge theory begins to behave like a weakly coupled string theory. In favorable cases, such as strongly coupled $\mathcal N=4$ super-Yang–Mills theory, that string theory has a low-energy limit described by classical gravity in AdS.

Why this matters

AdS/CFT has two separate semiclassical limits:

$$\text{large } N \quad\Longrightarrow\quad \text{weak bulk quantum loops},$$

and

$$\text{large 't Hooft coupling } \lambda \quad\Longrightarrow\quad \text{weak stringy curvature corrections}.$$

This page explains the first arrow. The next pages will refine the operator dictionary, but the basic slogan is already visible:

$$\frac{1}{N} \quad\text{acts like a bulk interaction strength}.$$

More precisely, for canonically normalized single-trace operators,

$$\langle \mathcal O_1 \mathcal O_2 \rangle_c \sim N^0, \qquad \langle \mathcal O_1 \mathcal O_2 \mathcal O_3 \rangle_c \sim \frac{1}{N}, \qquad \langle \mathcal O_1\cdots \mathcal O_n\rangle_c \sim N^{2-n}.$$

This is exactly the pattern one expects from weakly interacting single-particle fields in a bulk theory: two-point functions normalize the particles, three-point functions measure cubic interactions, and higher connected amplitudes become increasingly suppressed.

The ‘t Hooft limit

Consider a gauge theory with gauge group $SU(N)$ or $U(N)$. The Yang–Mills coupling $g_{\mathrm{YM}}$ is not the best variable to keep fixed as $N$ becomes large. The useful coupling is the ‘t Hooft coupling

$$\lambda = g_{\mathrm{YM}}^2 N.$$

The ‘t Hooft large-$N$ limit is

$$N\to \infty, \qquad \lambda = g_{\mathrm{YM}}^2 N \;\text{fixed}.$$

Equivalently,

$$g_{\mathrm{YM}}^2 \sim \frac{\lambda}{N}.$$

This scaling is not cosmetic. It is what keeps the leading diagrams finite and nontrivial. If $g_{\mathrm{YM}}$ were held fixed instead, the number of color degrees of freedom would grow while the interaction strength stayed too large. If $g_{\mathrm{YM}}$ were sent to zero too quickly, the large-$N$ theory would become trivial.

The ‘t Hooft limit holds fixed the effective coupling felt by a typical color loop.

Adjoint fields are matrices

An adjoint field may be written as

$$X(x) = X^a(x) T^a,$$

or, more explicitly, as a matrix

$$X^i{}_{j}(x), \qquad i,j=1,\ldots,N.$$

The two color indices are the key. A fundamental field $q^i$ carries one color line. An adjoint field $X^i{}_{j}$ carries two color lines, one for the upper index and one for the lower index. This turns ordinary Feynman diagrams into ribbon graphs.

The adjoint propagator schematically carries color structure

$$\langle X^i{}_{j}(x) X^k{}_{l}(y) \rangle \propto \delta^i_l\delta^k_j \times \text{spacetime propagator}.$$

The important point is not the detailed spacetime factor. The important point is the pair of Kronecker deltas. They show how color flows through the diagram.

Large-$N$ perturbation theory is organized by the topology of double-line color graphs. A connected diagram with genus $g$ and $b$ boundaries scales as $N^{2-2g-b}$, up to a function of the ‘t Hooft coupling $\lambda=g_{\mathrm{YM}}^2N$.

Double-line notation

In ordinary Feynman diagrams, a propagator is drawn as a single line. For an adjoint field, it is more informative to draw the propagator as two oppositely oriented color lines:

$$X^i{}_{j} \quad\leadsto\quad \text{one line for } i,\;\text{one line for } j.$$

When diagrams are drawn this way, each closed color loop contributes a factor of $N$ because it represents a sum over a free color index:

$$\sum_{i=1}^{N} \delta^i_i = N.$$

Thus the power of $N$ associated with a diagram can be read combinatorially:

• each closed color face contributes $N$;
• each propagator contributes a factor proportional to $g_{\mathrm{YM}}^2 \sim \lambda/N$;
• each interaction vertex contributes compensating factors determined by the action;
• at fixed $\lambda$, the remaining $N$-dependence depends only on topology.

For a connected vacuum ribbon graph, the result is

$$\mathcal A \sim N^{F-E+V} f(\lambda),$$

where $F$ is the number of faces, $E$ is the number of edges, and $V$ is the number of vertices of the ribbon graph. But

$$F-E+V = \chi,$$

where $\chi$ is the Euler characteristic of the two-dimensional surface on which the ribbon graph can be drawn without crossings. For a closed oriented surface of genus $g$,

$$\chi = 2-2g.$$

Therefore

$$\mathcal A_g \sim N^{2-2g} f_g(\lambda).$$

Planar diagrams have $g=0$ and scale as $N^2$. Diagrams with one handle have $g=1$ and scale as $N^0$. Diagrams with two handles scale as $N^{-2}$, and so on.

The expansion of the connected vacuum functional takes the form

$$\log Z = \sum_{g=0}^{\infty} N^{2-2g} F_g(\lambda).$$

This is the large-$N$ topological expansion.

Why planar diagrams dominate

At large $N$ with fixed $\lambda$,

$$N^2 \gg N^0 \gg N^{-2} \gg \cdots.$$

So the leading contribution comes from genus-zero diagrams. These are called planar diagrams, because they can be drawn on the plane or sphere without crossing double-line color strands.

The name “planar” is topological, not geometric in spacetime. A planar diagram may involve complicated momentum integrals and many vertices. What matters is that its color ribbon graph can be placed on a sphere without handles.

The large-$N$ expansion is therefore not the same as weak-coupling perturbation theory. Even when $\lambda$ is not small, one may still hope that the theory has an expansion in powers of $1/N$:

$$\text{large }N\text{ expansion} \neq \text{small }\lambda\text{ expansion}.$$

This distinction is essential in AdS/CFT. Classical gravity is expected when $N$ is large and $\lambda$ is large. Perturbative gauge-theory Feynman diagrams are useful when $\lambda$ is small. The most interesting holographic regime is precisely where ordinary perturbation theory fails, but the $1/N$ expansion remains meaningful.

Boundaries and operator insertions

So far we discussed vacuum diagrams. Correlation functions of single-trace operators introduce marked boundaries on the ribbon surface.

A typical single-trace operator is

$$\operatorname{Tr}(X_1 X_2\cdots X_J) = X_1{}^{i_1}{}_{i_2} X_2{}^{i_2}{}_{i_3} \cdots X_J{}^{i_J}{}_{i_1}.$$

The trace ties color indices into a loop. In the ribbon graph, that loop behaves like a boundary component. For connected correlators of canonically normalized single-trace operators, the large-$N$ scaling is

$$\langle \mathcal O_1\cdots \mathcal O_n\rangle_c = \sum_{g=0}^{\infty} N^{2-2g-n} G_{g,n}(\lambda).$$

At leading genus,

$$\langle \mathcal O_1\cdots \mathcal O_n\rangle_c \sim N^{2-n}.$$

Thus

$$\langle \mathcal O_1\mathcal O_2\rangle_c \sim N^0, \qquad \langle \mathcal O_1\mathcal O_2\mathcal O_3\rangle_c \sim \frac{1}{N}, \qquad \langle \mathcal O_1\cdots\mathcal O_4\rangle_c \sim \frac{1}{N^2}.$$

This is the standard normalization used when one wants single-trace operators to create single-particle bulk states with order-one two-point functions.

A warning about normalization is useful here. Some authors use unnormalized operators such as $\operatorname{Tr}X^J$, while others insert powers of $N$ or $\sqrt N$ so that two-point functions are order one. The physics is not changed by this convention, but the explicit powers of $N$ in correlators move around. Whenever comparing formulas, first check how the operators and sources are normalized.

Large-$N$ factorization

The most important consequence of the previous scaling is factorization. Suppose $A$ and $B$ are gauge-invariant single-trace or multi-trace operators normalized so that their expectation values are order one. Then connected correlations are suppressed:

$$\langle A B\rangle = \langle A\rangle\langle B\rangle +O\!\left(\frac{1}{N^2}\right),$$

for bosonic gauge-invariant observables of the usual single-trace type.

This resembles a classical limit. In ordinary quantum mechanics, connected fluctuations are suppressed when $\hbar\to 0$. In large-$N$ gauge theory, connected fluctuations of suitable gauge-invariant observables are suppressed when $1/N\to 0$.

This is one reason the bulk dual can become classical. The boundary theory remains a quantum field theory, but its gauge-invariant collective variables behave more and more classically as $N$ grows.

The next page will examine this point more carefully because it is the bridge from large-$N$ field theory to bulk Fock space.

The string-theory interpretation

The genus expansion of large-$N$ gauge theory looks like the perturbative expansion of a closed string theory. A closed-string partition function has the schematic form

$$Z_{\mathrm{string}} = \sum_{g=0}^{\infty} g_s^{2g-2} Z_g,$$

where $g_s$ is the string coupling and $g$ is the genus of the string worldsheet.

The gauge-theory expansion has the form

$$\log Z_{\mathrm{gauge}} = \sum_{g=0}^{\infty} N^{2-2g}F_g(\lambda).$$

Comparing the powers gives the rough identification

$$g_s \sim \frac{1}{N}.$$

In the canonical AdS$_5$/CFT$_4$ example, the more precise parametric relation is

$$g_s \sim \frac{\lambda}{N},$$

up to conventional numerical factors. At fixed large $\lambda$, large $N$ still means weak string coupling.

This does not mean every large-$N$ gauge theory has a simple classical gravity dual. The large-$N$ expansion suggests a string description, but classical Einstein gravity requires more. Roughly, the string theory must have a regime in which massive string modes are much heavier than the AdS curvature scale. In the boundary language, that corresponds to a large gap in the spectrum of higher-spin single-trace operators.

So the hierarchy is:

$$\text{large }N \quad\Rightarrow\quad \text{weakly coupled string expansion},$$

but

$$\text{large }N + \text{large gap/strong coupling} \quad\Rightarrow\quad \text{local classical gravity}.$$

Why $N^2$ degrees of freedom?

An adjoint matrix has approximately $N^2$ components. This simple fact has many holographic consequences.

For example, the free energy density of a deconfined large-$N$ adjoint plasma typically scales as

$$F \sim N^2.$$

The central charge or stress-tensor two-point coefficient of a holographic CFT also scales as

$$C_T \sim N^2.$$

On the gravity side, the coefficient controlling classical gravitational dynamics is the inverse Newton constant in AdS units:

$$\frac{L^{d-1}}{G_N} \sim N^2.$$

This relation is one of the most important normalization facts in the entire course. It says that the bulk classical action is large:

$$S_{\mathrm{bulk}} \sim \frac{L^{d-1}}{G_N} \sim N^2.$$

A large action suppresses quantum fluctuations around a saddle point. Thus the same $N^2$ that counts adjoint degrees of freedom in the gauge theory becomes the semiclassical parameter of the bulk gravitational path integral.

A useful toy model of the counting

To see the counting without the complications of gauge fixing and vector indices, consider a matrix field $X^i{}_{j}$ with an action schematically normalized as

$$S[X] = \frac{N}{\lambda} \int d^d x\; \operatorname{Tr} \left[ \frac12 (\partial X)^2 + \frac12 X^2 + \frac{1}{3}X^3 + \cdots \right].$$

This normalization makes the $N$-counting transparent. The propagator carries a factor

$$\frac{\lambda}{N},$$

while an interaction vertex carries a factor

$$\frac{N}{\lambda}.$$

A graph with $E$ propagators, $V$ vertices, and $F$ closed color faces gives

$$N^F \left(\frac{\lambda}{N}\right)^E \left(\frac{N}{\lambda}\right)^V = N^{F-E+V}\lambda^{E-V}.$$

At fixed $\lambda$, the power of $N$ is

$$N^{F-E+V}=N^\chi.$$

The detailed power of $\lambda$ depends on the graph and the interaction vertices. The power of $N$ knows only the topology.

This is the core of the ‘t Hooft expansion.

Relation to confinement and mesons

Large-$N$ ideas were originally developed as a way to understand strong interactions. In QCD-like theories with fundamental quarks, quark loops add boundaries to the ribbon surface. A diagram with genus $g$ and $b$ quark boundaries scales like

$$\mathcal A_{g,b} \sim N^{2-2g-b}.$$

Thus each additional quark loop costs a factor of $1/N$. In the strict large-$N$ limit, mesons become stable narrow resonances, and interactions among color-singlet hadrons are suppressed.

This historical connection is useful but should not be confused with the AdS/CFT examples studied later. The canonical AdS$_5$/CFT$_4$ theory is not QCD. It is conformal, supersymmetric, and contains only adjoint fields in its simplest form. The common lesson is the topological organization of gauge dynamics.

What large $N$ does and does not do

Large $N$ does something profound: it creates a controlled expansion parameter even in strongly coupled gauge theories. But it does not automatically solve the theory.

At fixed $\lambda$, the planar contribution $F_0(\lambda)$ may still be a complicated function. If $\lambda$ is small, ordinary perturbation theory can approximate it. If $\lambda$ is large, perturbation theory in $\lambda$ fails. AdS/CFT becomes powerful precisely because it can sometimes compute $F_0(\lambda)$ at large $\lambda$ using classical gravity.

So there are two expansions:

$$\text{topological expansion:} \qquad \frac{1}{N},$$

and

$$\text{coupling expansion:} \qquad \lambda\;\text{or}\;\frac{1}{\lambda}.$$

In the classical gravity regime of AdS$_5$/CFT$_4$,

$$N\gg 1, \qquad \lambda\gg 1.$$

The first condition suppresses bulk loops. The second suppresses string-scale corrections.

Dictionary checkpoint

The large-$N$ dictionary is:

Gauge-theory statement	Bulk interpretation
adjoint fields are matrices	color flow becomes ribbon topology
$\lambda=g_{\mathrm{YM}}^2N$ fixed	smooth ‘t Hooft large-$N$ limit
planar diagrams dominate	genus-zero string worldsheets dominate
each handle costs $1/N^2$	closed-string loops are suppressed
$1/N$ suppresses connected interactions	bulk fields interact weakly
$C_T\sim N^2$	$L^{d-1}/G_N\sim N^2$
large $N$ factorization	classical bulk saddle behavior

The single most important sentence is this:

$$\boxed{\text{Large }N\text{ is the boundary origin of the bulk semiclassical expansion.}}$$

Common confusions

“Large $N$ means the gauge theory is weakly coupled.”

No. The ‘t Hooft coupling $\lambda=g_{\mathrm{YM}}^2N$ may be small or large. Large $N$ controls the topology of color diagrams. It does not by itself make the planar theory easy.

“Planar diagrams are just diagrams that look planar on the page.”

No. Planarity refers to the double-line color graph. A diagram is planar if its ribbon graph can be drawn on a sphere without crossing color lines.

“The string coupling is exactly $1/N$.”

Parametrically, large $N$ suppresses string loops. In the canonical AdS$_5$/CFT$_4$ normalization, $g_s$ is proportional to $\lambda/N$, up to numerical conventions. What is robust is that the closed-string genus expansion is controlled by powers of $1/N^2$ at fixed appropriate coupling.

“Every large-$N$ gauge theory has an Einstein gravity dual.”

No. Large $N$ suggests a string-like expansion, not necessarily a weakly curved gravitational description. A simple Einstein gravity dual requires additional dynamical conditions, such as strong coupling and a sparse spectrum of low-dimension single-trace higher-spin operators.

“$SU(N)$ and $U(N)$ make no difference.”

At leading large $N$, the distinction is often subleading for adjoint-sector observables. But it can matter for global issues, decoupled $U(1)$ factors, line operators, and precise dictionary questions. This course will usually ignore the distinction until it matters.

Exercises

Exercise 1: Euler counting

A connected vacuum ribbon graph has $V$ vertices, $E$ propagators, and $F$ closed color faces. Assume the action is normalized so that propagators contribute $\lambda/N$ and vertices contribute $N/\lambda$. Show that the graph scales as

$$N^{F-E+V}$$

at fixed $\lambda$, and identify this exponent with the Euler characteristic.

Solution

The color faces give a factor $N^F$. The propagators give $(\lambda/N)^E$. The vertices give $(N/\lambda)^V$. Multiplying these factors gives

$$N^F \left(\frac{\lambda}{N}\right)^E \left(\frac{N}{\lambda}\right)^V = N^{F-E+V}\lambda^{E-V}.$$

At fixed $\lambda$, the power of $N$ is $F-E+V$. For a ribbon graph drawn on a closed oriented surface, this is the Euler characteristic

$$\chi = F-E+V.$$

For genus $g$,

$$\chi = 2-2g,$$

so the graph scales as $N^{2-2g}$ times a function of $\lambda$.

Exercise 2: Cost of a handle

Compare a planar connected vacuum diagram with a genus-one connected vacuum diagram. By what power of $N$ is the genus-one diagram suppressed?

Solution

A planar connected vacuum diagram has $g=0$, so it scales as

$$N^{2-2g}=N^2.$$

A genus-one connected vacuum diagram has $g=1$, so it scales as

$$N^{2-2g}=N^0.$$

The ratio is

$$\frac{N^0}{N^2}=\frac{1}{N^2}.$$

Thus one handle costs $1/N^2$.

Exercise 3: Three-point interactions

Assume canonically normalized single-trace operators obey

$$\langle \mathcal O_1\cdots \mathcal O_n\rangle_c \sim N^{2-n}$$

at leading planar order. What is the scaling of a connected three-point function? What does this suggest for the corresponding cubic bulk coupling?

Solution

For $n=3$,

$$\langle \mathcal O_1\mathcal O_2\mathcal O_3\rangle_c \sim N^{-1}.$$

If the operators create canonically normalized single-particle bulk states, then the connected three-point function measures a cubic interaction among those bulk fields. The scaling suggests that the cubic bulk coupling is of order $1/N$.

Exercise 4: Why large $N$ is not enough for classical gravity

Explain why the existence of a $1/N$ expansion does not by itself imply a weakly curved Einstein gravity dual.

Solution

The $1/N$ expansion organizes the theory by topology and suggests a weakly coupled string expansion. However, a string theory can still be highly stringy: many massive string modes may have masses comparable to the curvature scale, and higher-derivative corrections may be important. A weakly curved Einstein gravity dual requires not only weak string coupling, controlled by large $N$, but also suppression of string-scale corrections. In the canonical example this is achieved at large ‘t Hooft coupling $\lambda$, where $L^2/\alpha'$ is large.

Further reading

• G. ‘t Hooft, A Planar Diagram Theory for Strong Interactions. The original source of the topological large-$N$ expansion.
• G. ‘t Hooft, Large N. A concise retrospective lecture on large-$N$ ideas.
• E. Witten, Baryons in the 1/N Expansion. A classic analysis of how hadronic physics simplifies at large $N$.
• O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz, Large N Field Theories, String Theory and Gravity. The standard broad review connecting large-$N$ field theory to AdS/CFT.