Large-N Gauge Theory

The organizing idea

The previous page explained why gravity suggests an unusual counting of states: black-hole entropy scales like an area in Planck units. The next ingredient is a field-theory mechanism that can produce a controlled expansion looking like quantum gravity. That mechanism is the large-$N$ expansion of matrix gauge theories.

The basic observation is simple but deep. In an $SU(N)$ or $U(N)$ gauge theory, fields in the adjoint representation are matrices. Their Feynman diagrams do not merely form ordinary graphs; after color indices are drawn explicitly, they become ribbon graphs. A ribbon graph has a topology. It can be drawn on a sphere, a torus, or a higher-genus surface. The power of $N$ carried by the diagram is determined by this topology.

In the large-$N$ limit, planar diagrams dominate. Nonplanar diagrams are suppressed by powers of $1/N^2$. This is exactly the pattern of a closed-string perturbation expansion, where each additional handle on the worldsheet costs a factor of $g_s^2$. Thus large-$N$ gauge theory supplies the first structural reason that a gauge theory might have a string description:

$$\text{large-}N\text{ gauge theory} \quad\Longrightarrow\quad \text{sum over two-dimensional surfaces}.$$

This statement is not yet AdS/CFT. It does not say what the target space of the string is, whether the string is weakly curved, or whether the bulk theory is Einstein gravity. But it explains why the gauge-theory parameter $N$ is naturally related to the quantum-gravity loop expansion.

Adjoint fields carry two color indices, so their propagators are double lines. A ribbon graph contributes a power $N^{\chi}$, where $\chi=F-E+V=2-2g-b$ is the Euler characteristic of the surface on which the graph is drawn. The expansion of the gauge-theory free energy, $\log Z=\sum_g N^{2-2g}F_g(\lambda)$, mirrors a closed-string genus expansion with $g_s\sim 1/N$ at fixed $\lambda$.

The ‘t Hooft limit

Consider a gauge theory with gauge group $SU(N)$ or $U(N)$ and fields in the adjoint representation. The Yang-Mills coupling is $g_{\mathrm{YM}}$, and the natural large-$N$ coupling is the ‘t Hooft coupling

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

The ‘t Hooft limit is

$$N\to\infty, \qquad \lambda\ \text{fixed}.$$

This is the correct limit because a typical color loop gives a factor of $N$, while a gauge interaction brings powers of $g_{\mathrm{YM}}$. Holding $g_{\mathrm{YM}}$ fixed would make diagrams blow up with arbitrary powers of $N$. Holding $\lambda=g_{\mathrm{YM}}^2N$ fixed balances the color combinatorics against the coupling constants.

A schematic adjoint gauge-theory action can be written as

$$S = \frac{N}{\lambda} \int d^d x\,\operatorname{tr}\!\left( \frac{1}{4}F_{\mu\nu}F^{\mu\nu} +\frac{1}{2}(D\Phi)^2 +\cdots \right),$$

where $\operatorname{tr}$ is the trace in the fundamental representation. The precise field content depends on the theory. For example, in $\mathcal N=4$ super-Yang-Mills there are gauge fields, six adjoint scalars, and four adjoint Weyl fermions. The large-$N$ counting, however, is much more general than supersymmetry.

A crucial warning: in the ‘t Hooft limit,

$$g_{\mathrm{YM}}^2=\frac{\lambda}{N}\to 0,$$

but the theory is not necessarily weakly coupled. The physical interaction strength controlling planar dynamics is $\lambda$, not $g_{\mathrm{YM}}^2$ alone. Planar diagrams with arbitrarily many vertices can survive at $N=\infty$. Large $N$ is a topological expansion, not automatically a perturbative expansion in $\lambda$.

This distinction is central in AdS/CFT. Classical supergravity usually requires both

$$N\gg 1 \qquad\text{and}\qquad \lambda\gg 1.$$

The first condition suppresses quantum gravity loops. The second suppresses stringy corrections in powers of $\alpha'/L^2$.

Matrix fields and double-line notation

An adjoint field is a matrix,

$$\Phi=\Phi^a T^a, \qquad \Phi^i{}_{j},$$

with one fundamental index and one anti-fundamental index. In a $U(N)$ theory, the color structure of the free propagator is schematically

$$\left\langle \Phi^i{}_{j}\,\Phi^k{}_{l} \right\rangle \propto \delta^i{}_{l}\delta^k{}_{j}.$$

This is why one draws the propagator as two oppositely oriented color lines. The first Kronecker delta carries one index along the propagator; the second carries the other index back. Vertices join these strands cyclically because the interactions appear inside traces.

For $SU(N)$ rather than $U(N)$, the adjoint field is traceless. The propagator contains the projector

$$\delta^i{}_{l}\delta^k{}_{j} - \frac{1}{N}\delta^i{}_{j}\delta^k{}_{l}.$$

The second term subtracts the singlet $U(1)$ component. It is suppressed in many leading large-$N$ questions, but it matters for exact finite-$N$ identities, trace relations, and sometimes for global subtleties. In most AdS/CFT calculations, the distinction between $SU(N)$ and $U(N)$ is invisible at leading order in $1/N$.

The important point is that every closed color-index loop gives

$$\delta^i{}_{i}=N.$$

Therefore the power of $N$ in a diagram is found by counting closed color loops, propagators, and vertices.

Topological counting

Take a connected vacuum ribbon graph made only of adjoint fields. Let

• $V$ be the number of interaction vertices,
• $E$ be the number of propagators,
• $F$ be the number of closed color-index loops, also called faces.

With the normalization of the action above, a schematic counting gives

$$\mathcal A_\Gamma \sim N^F \left(\frac{\lambda}{N}\right)^E \left(\frac{N}{\lambda}\right)^V.$$

Thus

$$\mathcal A_\Gamma \sim N^{F-E+V}\lambda^{E-V}.$$

The power of $\lambda$ depends on the details of the vertices and on field normalizations. The power of $N$ is topological. The quantity

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

is the Euler characteristic of the two-dimensional surface obtained by thickening the graph into a ribbon graph. For a connected orientable closed surface of genus $g$,

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

Therefore a vacuum graph of genus $g$ contributes as

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

The leading diagrams are planar diagrams, $g=0$, which scale like $N^2$. Diagrams that require one handle, $g=1$, scale like $N^0$. Two handles give $N^{-2}$, and so on.

Equivalently, the vacuum free energy has the expansion

$$\log Z = N^2 F_0(\lambda)+F_1(\lambda)+\frac{1}{N^2}F_2(\lambda)+\cdots.$$

This is the first major bridge to string theory. A closed-string perturbation expansion has the schematic form

$$\log Z_{\mathrm{string}} = \frac{1}{g_s^2}\mathcal F_0+ \mathcal F_1+ g_s^2\mathcal F_2+ \cdots.$$

Comparing the two expansions suggests

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

at fixed $\lambda$. In the canonical AdS$_5$/CFT$_4$ example, the more precise relation includes the ‘t Hooft coupling because $g_s\sim g_{\mathrm{YM}}^2\sim \lambda/N$, up to convention-dependent numerical factors. For genus counting at fixed $\lambda$, however, the important scaling is $g_s\propto N^{-1}$.

Boundaries: operator insertions and fundamental matter

A ribbon graph may also have boundaries. There are two common sources of boundaries.

First, a single-trace operator insertion such as

$$\operatorname{tr}(\Phi^k)(x)$$

creates a marked cyclic boundary for color lines. Second, if the theory contains fields in the fundamental representation, a closed fundamental loop forms a boundary rather than a face made from adjoint color flow.

For a connected orientable surface with genus $g$ and $b$ boundaries,

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

So the corresponding color factor behaves like

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

This formula explains two standard large-$N$ facts.

First, each additional trace insertion reduces the naive power of $N$ by one before one chooses the normalization of the external operators. This is the origin of large-$N$ scaling rules for single-trace correlators.

Second, with $N_f$ fundamental flavors held fixed as $N\to\infty$, fundamental loops are suppressed relative to adjoint loops. A diagram with $b$ fundamental boundaries carries a factor roughly

$$N^{2-2g-b}N_f^b.$$

For fixed $N_f$, each boundary costs a factor of $N_f/N$. This is the field-theory origin of the probe-flavor approximation in holography: flavor branes with $N_f\ll N$ do not backreact at leading order on the geometry. In the Veneziano limit,

$$N\to\infty, \qquad \frac{N_f}{N}\ \text{fixed},$$

fundamental matter is not quenched, and the dual flavor sector should backreact.

Large-$N$ factorization

The topological expansion implies a second key property: gauge-invariant observables become classical variables at large $N$.

Let $\mathfrak o_I$ denote a normalized single-trace observable whose expectation value remains order one at large $N$. Schematically one may think of

$$\mathfrak o_I(x) \sim \frac{1}{N}\operatorname{tr}\!\left(\Phi^k(x)+\cdots\right),$$

although the exact normalization depends on the operator. Couple it to a source by writing

$$Z[J] = \left\langle \exp\!\left( N^2\int d^d x\,J^I(x)\mathfrak o_I(x) \right) \right\rangle.$$

The large-$N$ expansion says

$$\log Z[J] = N^2 w_0[J]+w_1[J]+O(N^{-2}).$$

Connected correlators of the $\mathfrak o_I$ are obtained by differentiating $\log Z$ and dividing by the source-coupling factors. Therefore

$$\left\langle \mathfrak o_1\cdots \mathfrak o_n \right\rangle_{\mathrm{conn}} \sim N^{2-2n}.$$

In particular,

$$\left\langle \mathfrak o_1\mathfrak o_2 \right\rangle = \left\langle \mathfrak o_1\right\rangle \left\langle \mathfrak o_2\right\rangle +O(N^{-2}).$$

This is large-$N$ factorization. It is analogous to the statement that fluctuations around a classical saddle are small. The role of $\hbar$ is played by $1/N^2$.

A second normalization is often more convenient in AdS/CFT. Define a particle-normalized operator

$$\mathcal O_I=N\mathfrak o_I,$$

so that its connected two-point function is order one when the one-point function vanishes:

$$\left\langle \mathcal O_I(x)\mathcal O_J(0) \right\rangle_{\mathrm{conn}} \sim N^0.$$

Then

$$\left\langle \mathcal O_1\cdots\mathcal O_n \right\rangle_{\mathrm{conn}} \sim N^{2-n}.$$

This is the normalization behind the common statement that cubic bulk couplings are order $1/N$ and quartic bulk couplings are order $1/N^2$.

Both normalizations are useful. The first makes factorization look like classical probability. The second makes the CFT operators look like canonically normalized single-particle fields. Confusing these two normalizations is one of the most common sources of wrong powers of $N$.

Single-trace operators as single-particle states

In matrix large-$N$ theories, the basic gauge-invariant local operators are traces:

$$\operatorname{tr}(F_{\mu\nu}F^{\mu\nu}), \qquad \operatorname{tr}(\Phi^I\Phi^J), \qquad \operatorname{tr}(\bar\psi\Gamma D\psi), \qquad \ldots$$

A single trace is the gauge-theory analog of one closed string. A product of traces is the analog of a multi-string or multi-particle state. In the CFT operator language, this means

$$\text{single-trace primary} \quad\longleftrightarrow\quad \text{single-particle bulk field},$$

while

$$\text{multi-trace operator} \quad\longleftrightarrow\quad \text{multi-particle bulk state}.$$

At $N=\infty$, multi-trace operators behave almost like composites of independent single-trace operators. For example, in a large-$N$ CFT, a double-trace primary schematically of the form

$$[\mathcal O_i\mathcal O_j]_{n,\ell}$$

has dimension

$$\Delta_{ij,n,\ell} = \Delta_i+ \Delta_j+2n+\ell+O(N^{-2}),$$

where $n$ counts radial excitations and $\ell$ is spin. The $O(N^{-2})$ correction is the anomalous dimension produced by bulk interactions. In Witten-diagram language, it appears from tree-level exchange and contact diagrams. In bulk language, it is a binding-energy correction between two particles in AdS.

This is a remarkably concrete statement. Large $N$ does not merely say that some diagrams are small. It says that the Hilbert space organizes itself approximately like a Fock space:

$$\text{single particles} \oplus \text{two-particle states} \oplus \text{three-particle states} \oplus \cdots,$$

with interactions suppressed by powers of $1/N$.

Central charge, Newton’s constant, and classical gravity

A matrix CFT has order $N^2$ degrees of freedom. This shows up in thermodynamics, anomalies, and stress-tensor correlation functions. For example, the stress-tensor two-point coefficient behaves as

$$C_T\sim N^2$$

in adjoint large-$N$ gauge theories.

On the gravity side, the coefficient multiplying the classical bulk action is

$$\frac{1}{G_{d+1}}.$$

For an AdS$_{d+1}$ dual with radius $L$, the dimensionless measure of the gravitational coupling is

$$\frac{L^{d-1}}{G_{d+1}}.$$

The holographic dictionary therefore identifies the large central charge with the inverse Newton coupling:

$$C_T \sim \frac{L^{d-1}}{G_{d+1}} \sim N^2.$$

This is why the large-$N$ limit is the classical-gravity limit. Bulk loops are suppressed by powers of

$$G_{d+1}/L^{d-1}\sim \frac{1}{N^2}.$$

But this is only the loop expansion. To obtain a weakly curved Einstein-like bulk, one also needs a large gap to higher-spin and stringy states. In the canonical D3-brane example, that gap is controlled by the ‘t Hooft coupling $\lambda$:

$$\frac{L^2}{\alpha'}\sim \sqrt{\lambda}.$$

So the rough hierarchy is

$$\begin{array}{ccl} N\gg 1 &\Longleftrightarrow& \text{classical bulk, suppressed quantum loops},\\ \lambda\gg 1 &\Longleftrightarrow& \text{weakly curved bulk, suppressed stringy corrections}. \end{array}$$

Large $N$ alone gives something string-like. Large $N$ plus strong coupling and a sparse light spectrum give something close to classical Einstein gravity.

A small worked example: why four-point functions start at $1/N^2$

Use particle-normalized single-trace operators $\mathcal O_i$ with

$$\langle\mathcal O_i\mathcal O_j\rangle_{\mathrm{conn}} \sim N^0.$$

The connected three-point function scales as

$$\langle\mathcal O_1\mathcal O_2\mathcal O_3\rangle_{\mathrm{conn}} \sim \frac{1}{N}.$$

Therefore the corresponding cubic bulk interaction has coupling of order $1/N$. A connected four-point function receives contributions from two basic kinds of tree-level bulk diagrams:

1. an exchange diagram with two cubic vertices,
2. a contact diagram with one quartic vertex.

The exchange diagram scales as

$$\left(\frac{1}{N}\right)^2 = \frac{1}{N^2}.$$

The quartic coupling must also scale as $1/N^2$ if the bulk action has the form

$$S_{\mathrm{bulk}} \sim N^2\int d^{d+1}x\,\mathcal L(\phi),$$

with canonically normalized fluctuations obtained by expanding fields with a factor of $1/N$. Hence

$$\langle \mathcal O_1\mathcal O_2\mathcal O_3\mathcal O_4 \rangle_{\mathrm{conn}} \sim \frac{1}{N^2}.$$

This is the CFT origin of tree-level Witten diagrams. A disconnected four-point function is order $N^0$, being a product of two two-point functions, while the connected part is suppressed by $1/N^2$. The connected part is where bulk interactions live.

What large $N$ buys you in AdS/CFT

The following dictionary is only schematic, but it is one of the most useful summaries in the subject.

Large-$N$ gauge-theory feature	Bulk interpretation
$\log Z\sim N^2$	Classical bulk action scales like $1/G_N$
Planar dominance	Classical string worldsheet or tree-level closed-string physics
Genus-$g$ suppression $N^{2-2g}$	Closed-string loop expansion with $g_s\sim 1/N$
Large-$N$ factorization	Classical bulk saddle; suppressed quantum fluctuations
Single-trace primaries	Single-particle bulk fields or single-string states
Multi-trace operators	Multi-particle bulk states
Connected $n$-point functions $\sim N^{2-n}$ in particle normalization	Bulk $n$-point interactions suppressed by powers of $1/N$
$C_T\sim N^2$	$L^{d-1}/G_{d+1}\sim N^2$
Fundamental loops suppressed for fixed $N_f$	Probe flavor branes with negligible backreaction

The table also hints at a limitation. A large-$N$ expansion is necessary for a weakly coupled bulk, but it is not sufficient for a simple geometric bulk. Vector large-$N$ theories, for example, can have duals involving higher-spin gauge fields rather than ordinary Einstein gravity. Matrix large-$N$ theories are naturally string-like, but whether the string has a local weakly curved target-space description is a further dynamical question.

Common mistakes

Mistake 1: Large $N$ means weak coupling.

No. The ‘t Hooft coupling $\lambda$ can be small, finite, or large. Large $N$ organizes diagrams by topology. It does not by itself make the planar theory easy.

Mistake 2: Planar means classical gravity.

Planar means genus zero in the gauge-theory/string expansion. Classical Einstein gravity also requires the string scale to be small compared with the AdS radius and requires the light spectrum to be sparse enough that only a few low-spin fields remain relevant.

Mistake 3: Every single-trace operator is light in the bulk.

Single-trace means single-string or single-particle, but the corresponding bulk state may be very heavy. In a strongly coupled holographic CFT, most stringy single-trace operators have dimensions that grow with a positive power of $\lambda$.

Mistake 4: The normalization of operators is harmless.

It is not. The same physical correlator can be described using classical-variable normalization, particle normalization, or stress-tensor normalization. Always say which one is being used before assigning powers of $N$.

Mistake 5: $SU(N)$ and $U(N)$ are always identical.

They agree for many leading large-$N$ observables, but finite-$N$ trace relations, decoupled $U(1)$ sectors, global forms of the gauge group, and line operators can distinguish them.

Exercises

Exercise 1: Euler characteristic from double-line counting

Consider a connected vacuum ribbon graph with $V$ vertices, $E$ propagators, and $F$ closed index loops. Use the schematic rules

$$\text{face}:N, \qquad \text{propagator}:\frac{\lambda}{N}, \qquad \text{vertex}:\frac{N}{\lambda}$$

to show that the amplitude scales as $N^{\chi}$ times a function of $\lambda$, where $\chi=F-E+V$.

Solution

Multiplying the factors gives

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

The exponent of $N$ is

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

For a connected orientable closed ribbon graph drawn on a genus-$g$ surface,

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

so

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

The precise power of $\lambda$ depends on the graph and on the interaction vertices, but the power of $N$ is topological.

Exercise 2: Handles and the string coupling

Suppose the connected vacuum free energy of a matrix gauge theory has the expansion

$$\log Z = N^2F_0(\lambda)+F_1(\lambda)+N^{-2}F_2(\lambda)+\cdots.$$

Compare this with a closed-string expansion and identify the scaling of $g_s$ with $N$.

Solution

A closed-string perturbation expansion has the form

$$\log Z_{\mathrm{string}} = \sum_{g=0}^{\infty}g_s^{2g-2}\mathcal F_g.$$

The sphere term scales as $g_s^{-2}$, the torus term as $g_s^0$, and the genus-two term as $g_s^2$. Matching this with

$$N^2, \qquad N^0, \qquad N^{-2},$$

we find

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

at fixed ‘t Hooft coupling. In the D3-brane duality, convention-dependent constants and powers of $\lambda$ refine this relation, but the genus-counting statement is $g_s\propto N^{-1}$ at fixed $\lambda$.

Exercise 3: Factorization from the generating functional

Let $\mathfrak o$ be a normalized single-trace observable with source coupling

$$Z[J] = \left\langle \exp\!\left(N^2\int J\mathfrak o\right) \right\rangle,$$

and suppose

$$\log Z[J]=N^2w_0[J]+O(N^0).$$

Show that

$$\langle \mathfrak o(x)\mathfrak o(y)\rangle_{\mathrm{conn}} \sim N^{-2}.$$

Solution

Because the source appears as $N^2J\mathfrak o$, differentiating once gives

$$\langle\mathfrak o(x)\rangle = \frac{1}{N^2}\frac{\delta\log Z}{\delta J(x)}.$$

Differentiating twice gives

$$\langle\mathfrak o(x)\mathfrak o(y)\rangle_{\mathrm{conn}} = \frac{1}{N^4} \frac{\delta^2\log Z}{\delta J(x)\delta J(y)}.$$

Since $\log Z=N^2w_0+O(N^0)$, the second derivative of $\log Z$ is generically order $N^2$. Therefore

$$\langle\mathfrak o(x)\mathfrak o(y)\rangle_{\mathrm{conn}} \sim \frac{N^2}{N^4} = N^{-2}.$$

This gives large-$N$ factorization:

$$\langle\mathfrak o(x)\mathfrak o(y)\rangle = \langle\mathfrak o(x)\rangle\langle\mathfrak o(y)\rangle +O(N^{-2}).$$

Exercise 4: Changing operator normalization

Assume classical-variable normalization gives

$$\langle \mathfrak o_1\cdots\mathfrak o_n\rangle_{\mathrm{conn}} \sim N^{2-2n}.$$

Define particle-normalized operators $\mathcal O_i=N\mathfrak o_i$. Show that

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

Solution

Substituting $\mathcal O_i=N\mathfrak o_i$ gives

$$\langle \mathcal O_1\cdots\mathcal O_n\rangle_{\mathrm{conn}} = N^n \langle \mathfrak o_1\cdots\mathfrak o_n\rangle_{\mathrm{conn}}.$$

Using the assumed scaling,

$$N^n\cdot N^{2-2n}=N^{2-n}.$$

Thus particle-normalized two-point functions are order $N^0$, three-point functions are order $1/N$, and connected four-point functions are order $1/N^2$.

Exercise 5: Fundamental matter and probe flavor

Consider an adjoint gauge theory with $N_f$ fundamental flavors. A connected diagram with genus $g$ and $b$ closed fundamental loops scales as

$$N^{2-2g-b}N_f^b.$$

Explain why fundamental loops are suppressed when $N_f$ is fixed, and why they are not suppressed in the Veneziano limit.

Solution

Relative to a purely adjoint planar contribution, which scales as $N^2$, a diagram with $b$ fundamental loops and genus zero scales as

$$N^{2-b}N_f^b = N^2\left(\frac{N_f}{N}\right)^b.$$

If $N_f$ is held fixed as $N\to\infty$, then each fundamental boundary is suppressed by $N_f/N$. This is the quenched or probe-flavor regime.

In the Veneziano limit, $N_f/N$ is held fixed. Then

$$\left(\frac{N_f}{N}\right)^b$$

is order one, so diagrams with fundamental loops are not parametrically suppressed. In holography this corresponds to flavor branes whose stress tensor can backreact on the geometry.

Exercise 6: Matrix large $N$ versus vector large $N$

A theory with $N$ vector fields often has a large-$N$ expansion, but its number of elementary degrees of freedom scales like $N$, not $N^2$. Explain why this suggests a different kind of bulk dual from a matrix large-$N$ gauge theory.

Solution

In a matrix gauge theory, adjoint fields have two color indices, so the number of components scales like $N^2$. The perturbative expansion is organized by ribbon graphs, and the power of $N$ is related to the genus of a two-dimensional surface. This is why matrix large-$N$ theories are naturally connected to closed-string expansions.

In a vector model, fields have one index, so the number of components scales like $N$. The diagrams are not organized in the same way by closed orientable string worldsheets. The singlet sector can still have a large-$N$ expansion and can still admit a bulk interpretation, but the dual is often higher-spin-like rather than Einstein-like. This is one reason large $N$ alone is not enough to guarantee a conventional semiclassical gravity dual.

Further reading

The original source of the topological expansion is G. ‘t Hooft, A Planar Diagram Theory for Strong Interactions. For a concise review by ‘t Hooft, see Large N, arXiv:hep-th/0204069. For the role of large $N$ in AdS/CFT, see Aharony, Gubser, Maldacena, Ooguri, and Oz, Large N Field Theories, String Theory and Gravity, arXiv:hep-th/9905111. David Tong’s gauge-theory lecture notes also give a clear pedagogical account of double-line notation and large-$N$ counting.

The next page explains how these ribbon-graph surfaces become physically string-like in confining gauge theories and why AdS/CFT requires a more subtle notion of string than the QCD flux tube.