Large N Factorization and Fock Space

Bulk effective field theory needs a Hilbert space before it needs an action. The Hilbert space of a weakly coupled bulk theory is approximately a Fock space: one-particle states, two-particle states, three-particle states, and so on, with weak interactions mixing them.

Where is this Fock space in the CFT?

The answer is large-$N$ factorization. In the strict $N\to\infty$ limit, suitably normalized single-trace operators behave like generalized free fields. Their products create multiparticle states. At finite but large $N$, connected correlators are small but nonzero, and the Fock space becomes weakly interacting.

The slogan is:

$$\text{single trace} \longleftrightarrow \text{one particle}, \qquad \text{multi trace} \longleftrightarrow \text{multiparticle state}.$$

This page explains the statement in Hilbert-space language.

In radial quantization, operators create states. At large $N$, single-trace primaries create one-particle bulk states, while normal-ordered products create multiparticle states. Connected correlators suppressed by powers of $1/N$ become weak bulk interactions.

Why this matters

A classical-looking bulk is not only about equations of motion. It is about the organization of quantum states.

In a free scalar field theory in AdS, a quantum state can contain any number of particles:

$$|0\rangle, \qquad |\phi\rangle, \qquad |\phi\phi\rangle, \qquad |\phi\phi\phi\rangle, \qquad \ldots$$

The same structure must be visible in the CFT. Since a CFT Hilbert space is generated by local operators acting on the vacuum, the bulk Fock-space question becomes:

$$\text{Which CFT operators create one-particle and multiparticle bulk states?}$$

The answer is controlled by the large-$N$ expansion.

Normalizing single-trace operators

In a matrix large-$N$ gauge theory, a typical gauge-invariant single-trace operator has the schematic form

$$\mathcal T(x)=\operatorname{Tr}\left(\Phi_1(x)\Phi_2(x)\cdots\Phi_k(x)\right).$$

Its two-point function often scales like a positive power of $N$. For holographic purposes it is convenient to define a normalized operator $\mathcal O$ with two-point function of order one:

$$\langle \mathcal O(x)\mathcal O(0)\rangle \sim N^0.$$

For adjoint matrix theories, this usually means rescaling a trace operator by a suitable power of $N$. With this normalization, connected correlators of single-trace operators satisfy

$$\boxed{ \langle \mathcal O_1\mathcal O_2\cdots\mathcal O_n\rangle_c \sim N^{2-n}. }$$

Thus

$$\langle \mathcal O\mathcal O\rangle_c\sim N^0, \qquad \langle \mathcal O\mathcal O\mathcal O\rangle_c\sim \frac1N, \qquad \langle \mathcal O\mathcal O\mathcal O\mathcal O\rangle_c\sim \frac1{N^2}.$$

This is the same scaling as a weakly coupled bulk theory with interaction strength

$$\kappa \sim \sqrt{G_N/L^{d-1}} \sim \frac1N.$$

Factorization

Large-$N$ factorization says that products of separated single-trace operators factorize at leading order:

$$\langle \mathcal O_1\mathcal O_2\rangle = \langle \mathcal O_1\rangle\langle \mathcal O_2\rangle +O(N^0)$$

for unnormalized operators, or more usefully, for normalized operators with vanishing one-point functions,

$$\langle \mathcal O_1\mathcal O_2\mathcal O_3\mathcal O_4\rangle = \langle \mathcal O_1\mathcal O_2\rangle\langle \mathcal O_3\mathcal O_4\rangle + \langle \mathcal O_1\mathcal O_3\rangle\langle \mathcal O_2\mathcal O_4\rangle + \langle \mathcal O_1\mathcal O_4\rangle\langle \mathcal O_2\mathcal O_3\rangle +O\!\left(\frac1{N^2}\right).$$

This is the Wick-like behavior of a generalized free field.

The word “generalized” matters. A generalized free field has the correlator factorization of a free field, but its two-point function can have an arbitrary scaling dimension $\Delta$ allowed by conformal symmetry:

$$\langle \mathcal O(x)\mathcal O(0)\rangle = \frac{C_{\mathcal O}}{|x|^{2\Delta}}.$$

It is not necessarily a free local field in the boundary spacetime. Rather, it is the boundary shadow of a free bulk field in AdS.

Radial quantization: operators create states

In a CFT, radial quantization maps local operators to states on the cylinder:

$$\mathcal O(0)|0\rangle \quad\longleftrightarrow\quad |\mathcal O\rangle.$$

A primary operator of dimension $\Delta$ creates an energy eigenstate on $\mathbb R\times S^{d-1}$:

$$H_{\mathrm{cyl}}|\mathcal O\rangle = \Delta |\mathcal O\rangle.$$

In global AdS, a scalar particle of mass $m$ has normal-mode energies

$$\omega_{n,\ell} = \Delta+2n+\ell,$$

where

$$m^2L^2=\Delta(\Delta-d).$$

The state-operator map then matches beautifully:

$$\text{primary }\mathcal O \quad\longleftrightarrow\quad \text{lowest one-particle AdS mode},$$

and

$$\text{descendants }\partial_{\mu_1}\cdots\partial_{\mu_k}\mathcal O \quad\longleftrightarrow\quad \text{excited one-particle modes}.$$

Multi-trace operators as multiparticle states

A two-particle state is created by a double-trace operator. Schematically,

$$[\mathcal O_A\mathcal O_B]_{n,\ell} \sim \mathcal O_A\,\partial^{2n}\partial_{\{\mu_1}\cdots\partial_{\mu_\ell\}}\mathcal O_B - \text{traces and descendants}.$$

Here $n$ is a radial excitation number and $\ell$ is the relative angular momentum. At $N=\infty$, its dimension is

$$\Delta_{AB;n,\ell}^{(0)} = \Delta_A+ \Delta_B+2n+\ell.$$

This is precisely the energy of two noninteracting AdS particles:

$$E_{AB;n,\ell}L = \Delta_A+ \Delta_B+2n+\ell.$$

At finite large $N$, interactions shift the dimension:

$$\Delta_{AB;n,\ell} = \Delta_A+ \Delta_B+2n+\ell + \frac{1}{N^2}\gamma_{AB;n,\ell}^{(1)} +\cdots.$$

The anomalous dimension $\gamma_{AB;n,\ell}^{(1)}$ is the CFT version of a bulk binding energy or scattering phase shift.

Similarly, triple-trace operators create three-particle states, and so on:

$$:\mathcal O_A\mathcal O_B\mathcal O_C: \quad\longleftrightarrow\quad |A,B,C\rangle_{\mathrm{bulk}}.$$

Fock space at $N=\infty$

At strict $N=\infty$, the connected correlators beyond two points vanish:

$$\langle \mathcal O_1\cdots\mathcal O_n\rangle_c=0, \qquad n>2.$$

The CFT operator algebra generated by the low-dimension single-trace operators becomes generalized free. This is the boundary signature of a free bulk theory.

One can think of the CFT Hilbert space, restricted to the low-energy sector, as approximately

$$\mathcal H_{\mathrm{low}} \approx \mathcal F\left(\mathcal H_{\mathrm{1p}}\right),$$

where $\mathcal H_{\mathrm{1p}}$ is the one-particle space generated by light single-trace primaries and their descendants, and $\mathcal F$ denotes the Fock-space construction.

In words:

$$\text{large-}N\text{ generalized free sector} \quad\longleftrightarrow\quad \text{free bulk particles in AdS}.$$

Interactions at finite large $N$

At finite large $N$, connected correlators are small but nonzero. This is the boundary imprint of bulk interactions.

After canonical normalization, a bulk scalar action has the schematic form

$$S = \int d^{d+1}x\sqrt{-g}\, \left[ \frac12(\nabla\phi)^2+\frac12m^2\phi^2 +\frac{1}{N}g_3\phi^3 +\frac{1}{N^2}g_4\phi^4 +\cdots \right].$$

The cubic term gives

$$\langle \mathcal O\mathcal O\mathcal O\rangle_c \sim \frac1N.$$

The exchange of two cubic vertices or one quartic vertex gives

$$\langle \mathcal O\mathcal O\mathcal O\mathcal O\rangle_c \sim \frac1{N^2}.$$

In Hilbert-space language, interactions mix Fock sectors. For example, a one-particle state can mix weakly with two-particle states if the quantum numbers allow it:

$$|\mathcal O\rangle \longrightarrow |\mathcal O\rangle + \frac1N\sum_{A,B,n,\ell}c_{AB;n,\ell} |A,B;n,\ell\rangle + \cdots.$$

The Fock-space picture is therefore approximate, not exact.

The role of the large gap

Large-$N$ factorization builds a Fock space, but it does not by itself make that Fock space local in a weakly curved bulk. To get local bulk EFT, one also needs a gap to additional single-trace states.

Without a large gap, there may be infinitely many light fields, including higher-spin fields. The bulk can still exist, but it may be stringy or higher-spin at the AdS scale.

The best low-energy bulk picture requires both:

$$\begin{array}{rcl} \text{large }N &\Longrightarrow& \text{weak interactions between Fock quanta},\\[1mm] \text{large gap} &\Longrightarrow& \text{local derivative expansion}. \end{array}$$

A useful combined statement is:

$$\boxed{ \text{large }N + \Delta_{\mathrm{gap}}\gg 1 \quad\Longrightarrow\quad \text{weakly interacting local bulk Fock space below the cutoff}. }$$

Multi-trace mixing and the need to diagonalize

The phrase “double-trace operator” hides a technical point. Operators with the same quantum numbers can mix. For example, a single-trace operator $\mathcal O_C$ and a double-trace operator $[\mathcal O_A\mathcal O_B]_{n,\ell}$ may share the same spin and approximate dimension.

The true energy eigenstates are obtained by diagonalizing the dilatation operator:

$$D|\Psi_a\rangle = \Delta_a|\Psi_a\rangle.$$

At large $N$, mixing is often perturbatively small. But near degeneracies can enhance it. Bulk language calls the same effect particle mixing, decay, or resonance physics.

Thus, the dictionary is sharpest after one specifies a basis of properly normalized, approximately diagonal operators.

Finite $N$: where Fock space breaks

A Fock space has infinitely many independent multiparticle states. A finite-$N$ holographic CFT does not. At sufficiently high energies, finite-$N$ constraints become important.

Several effects signal the breakdown of the naive Fock picture:

• trace relations: for finite-size matrices, traces are not all independent;
• stringy exclusion effects: not every large quantum number state has a naive independent bulk-particle interpretation;
• black-hole formation: sufficiently energetic states are better described as black holes than as a dilute gas of particles;
• entropy bounds: the number of independent bulk states in a region is constrained by gravitational entropy;
• nonperturbative effects: corrections can scale like $e^{-N^2}$, invisible in ordinary perturbation theory.

The Fock-space picture is therefore a low-energy and perturbative description:

$$E L \ll N^2 \quad\text{and}\quad E L \ll \text{black-hole threshold in the chosen regime},$$

with more refined thresholds depending on dimension, charges, angular momenta, and the operator sector.

Gravitational dressing and locality

There is another subtlety. In a theory with gravity, a strictly local gauge-invariant bulk operator does not exist in the same way as a local operator in a nongravitational QFT. A bulk field must be gravitationally dressed to the boundary or to some reference structure.

In the large-$N$ limit, this dressing is weak, and one can use approximate local bulk fields. But at finite $N$, exact gauge-invariant bulk observables are relational and nonlocal.

The CFT sees this through the fact that a bulk local operator is represented by a complicated, generally nonlocal boundary operator. In perturbation theory, this can be constructed order by order. Nonperturbatively, the notion of a local bulk operator is limited by gravitational constraints.

A simple dictionary of states

Here is the state dictionary in its most useful first-pass form.

CFT object	Bulk interpretation
vacuum $	0\rangle$ on the cylinder
light single-trace primary $\mathcal O$	one-particle state
descendants of $\mathcal O$	excited one-particle AdS modes
double-trace primary $[\mathcal O_A\mathcal O_B]_{n,\ell}$	two-particle state with radial excitation $n$ and angular momentum $\ell$
multi-trace operator	multiparticle bulk state
connected three-point function $\sim 1/N$	cubic bulk interaction
double-trace anomalous dimension $\sim 1/N^2$	binding energy/scattering data
high-energy dense spectrum	black-hole/stringy regime

This table is a map, not a substitute for diagonalizing the CFT.

Dictionary checkpoint

The central translation of this page is:

$$\boxed{ \begin{array}{ccl} \mathcal O\text{ single trace} &\longleftrightarrow& \text{one-particle bulk state},\\[1mm] :\mathcal O_A\mathcal O_B: &\longleftrightarrow& \text{two-particle bulk state},\\[1mm] \langle \mathcal O_1\cdots\mathcal O_n\rangle_c\sim N^{2-n} &\longleftrightarrow& \text{weak bulk interactions},\\[1mm] \gamma_{AB;n,\ell}/N^2 &\longleftrightarrow& \text{bulk binding energy/scattering data}. \end{array}}$$

This is the Hilbert-space backbone of perturbative AdS quantum gravity.

Common confusions

“Multi-trace means composite, so it cannot be fundamental.”

Correct, but irrelevant. A two-particle state is also composite. Multi-trace operators are not fundamental bulk fields; they create multiparticle states built from the one-particle fields dual to single-trace operators.

“A product of two primaries is automatically a primary.”

No. A product of two primaries contains descendants and traces. The double-trace primary $[\mathcal O_A\mathcal O_B]_{n,\ell}$ is a particular linear combination chosen to transform as a conformal primary.

“Factorization means the theory is free.”

At $N=\infty$, the low-energy single-trace sector behaves like a generalized free theory. But the full finite-$N$ theory is not free. The $1/N$ corrections encode bulk interactions.

“Every large-$N$ CFT has an Einstein Fock space.”

No. Large $N$ gives weak interactions, but the particle content may include infinitely many light higher-spin or stringy fields. A local Einstein-like Fock space also requires a sparse low-energy spectrum and a large gap.

“The Fock space is exact at finite $N$.”

No. It is an approximation valid in a perturbative low-energy regime. Finite-$N$ trace identities, black-hole states, and gravitational constraints all limit the naive Fock construction.

Exercises

Exercise 1

Let $\mathcal O$ be a normalized single-trace operator with

$$\langle \mathcal O(x)\mathcal O(0)\rangle\sim N^0.$$

Assume large-$N$ counting

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

What are the scalings of the connected three- and four-point functions? What bulk interactions do they correspond to?

Solution

For $n=3$,

$$\langle \mathcal O\mathcal O\mathcal O\rangle_c \sim N^{-1}.$$

This is the scaling of a cubic bulk interaction among canonically normalized fields.

For $n=4$,

$$\langle \mathcal O\mathcal O\mathcal O\mathcal O\rangle_c \sim N^{-2}.$$

This is the scaling of tree-level four-point bulk processes: either a quartic contact diagram or an exchange diagram built from two cubic vertices. Bulk loops are further suppressed by additional powers of $1/N^2$.

Exercise 2

At $N=\infty$, two scalar single-trace primaries $\mathcal O_A$ and $\mathcal O_B$ have dimensions $\Delta_A$ and $\Delta_B$. What is the leading dimension of the double-trace primary $[\mathcal O_A\mathcal O_B]_{n,\ell}$?

Solution

At $N=\infty$, the two bulk particles do not interact. The corresponding double-trace primary has dimension

$$\Delta_{AB;n,\ell}^{(0)} = \Delta_A+\Delta_B+2n+\ell.$$

Here $n$ is the radial excitation number and $\ell$ is the relative spin/angular momentum. At finite large $N$, interactions produce anomalous dimensions:

$$\Delta_{AB;n,\ell} = \Delta_A+\Delta_B+2n+\ell + \frac{1}{N^2}\gamma_{AB;n,\ell}^{(1)}+\cdots.$$

Exercise 3

Explain why a large gap is needed in addition to large-$N$ factorization if one wants a local Einstein-like bulk EFT.

Solution

Large-$N$ factorization makes connected correlators small and therefore gives weak interactions. But it does not guarantee that only a few light fields exist. A theory may have infinitely many light single-trace operators, including higher-spin operators, even at large $N$.

A local Einstein-like EFT requires a finite or controlled number of light fields and a derivative expansion. This requires a gap

$$\Delta_{\mathrm{gap}}\gg 1$$

to additional single-trace states, especially higher-spin or stringy states. The large gap suppresses higher-derivative corrections by powers of $1/\Delta_{\mathrm{gap}}$ and allows low-energy physics to be local on length scales much smaller than $L$ but larger than the cutoff scale.

Exercise 4

Suppose a double-trace operator has dimension

$$\Delta_{AB;n,\ell} = \Delta_A+ \Delta_B+2n+\ell+\frac{1}{N^2}\gamma_{AB;n,\ell}^{(1)}+ \cdots.$$

What is the bulk interpretation of $\gamma_{AB;n,\ell}^{(1)}$?

Solution

In global AdS, dimensions are energies in units of $1/L$. The leading term is the energy of two noninteracting particles. The correction

$$\frac{1}{N^2}\gamma_{AB;n,\ell}^{(1)}$$

is therefore the leading interaction energy. Depending on the kinematic regime, it can be interpreted as a binding energy, a perturbative energy shift, or scattering data for the two bulk particles.

Further reading

• G. ‘t Hooft, A Planar Diagram Theory for Strong Interactions.
• E. Witten, Baryons and Branes in Anti de Sitter Space, for finite-$N$ and stringy-exclusion-type lessons in holography.
• A. L. Fitzpatrick, E. Katz, D. Poland, and D. Simmons-Duffin, Effective Conformal Theory and the Flat-Space Limit of AdS.
• A. L. Fitzpatrick and J. Kaplan, Scattering States in AdS/CFT.
• I. Heemskerk, J. Penedones, J. Polchinski, and J. Sully, Holography from Conformal Field Theory.
• A. Hamilton, D. Kabat, G. Lifschytz, and D. A. Lowe, Local Bulk Operators in AdS/CFT.