Single-Trace and Multi-Trace Operators

Goal

The previous page introduced generalized free fields: the universal leading behavior of normalized single-particle operators in a large-$N$ CFT. This page explains where those operators come from in matrix large-$N$ theories and how their products become the multi-particle Hilbert space of AdS.

The slogan is

$$\boxed{ \text{trace number in the CFT} \quad\longleftrightarrow\quad \text{particle number in AdS} \qquad (N=\infty). }$$

This slogan is powerful, but it must be handled carefully. “Single trace” is a microscopic label in gauge theories. “Single particle” is the more invariant holographic idea. At large $N$, the two notions usually coincide in familiar matrix theories such as $\mathcal N=4$ super-Yang-Mills, but the exact statement is about the organization of the CFT Hilbert space and OPE data.

By the end of this page, the following dictionary should feel natural:

$$\mathcal O_A \quad\longleftrightarrow\quad \text{one-particle bulk state},$$ $$[\mathcal O_A\mathcal O_B]_{n,\ell} \quad\longleftrightarrow\quad \text{two-particle bulk state},$$ $$[\mathcal O_{A_1}\cdots\mathcal O_{A_p}] \quad\longleftrightarrow\quad \text{$p$-particle bulk state}.$$

Finite-$N$ corrections are equally important. They make particles interact, generate anomalous dimensions, and mix operators with the same quantum numbers.

The basic matrix-theory picture

In a large-$N$ gauge theory, the elementary fields are matrices in color space. For example, an adjoint scalar of an $SU(N)$ gauge theory can be written as

$$\Phi(x)=\Phi^a(x)T^a,$$

or, in explicit color indices, as

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

Gauge-invariant local operators are built by contracting color indices. The simplest contractions are traces:

$$\operatorname{Tr}\Phi^2, \qquad \operatorname{Tr}\Phi^3, \qquad \operatorname{Tr}\left(F_{\mu\nu}F^{\mu\nu}\right), \qquad \operatorname{Tr}\left(\Phi D_\mu\Phi D_\nu\Phi\right).$$

A single-trace operator has one color trace. A double-trace operator is a product of two traces at the same spacetime point, after renormalization and projection onto a conformal primary. A multi-trace operator is a product of several traces.

Schematic examples are

$$\mathcal O_A(x) \sim \operatorname{Tr}\left(\Phi^{L_A}\right)(x),$$ $$\mathcal O_A(x)\mathcal O_B(x) \sim \operatorname{Tr}\left(\Phi^{L_A}\right)(x) \operatorname{Tr}\left(\Phi^{L_B}\right)(x),$$

and

$$\mathcal O_A(x)\mathcal O_B(x)\mathcal O_C(x) \sim \operatorname{Tr}\left(\Phi^{L_A}\right) \operatorname{Tr}\left(\Phi^{L_B}\right) \operatorname{Tr}\left(\Phi^{L_C}\right)(x).$$

The word “schematic” matters. Operators with derivatives, spin, flavor indices, $R$-symmetry indices, and fermions require projections onto irreducible Lorentz and internal-symmetry representations. In a CFT, one ultimately wants conformal primaries, not merely arbitrary products of fields.

Normalization convention

There are several conventions in the literature. In this course, unless stated otherwise, a single-trace primary $\mathcal O_A$ is normalized so that

$$\langle \mathcal O_A(x)\mathcal O_B(0)\rangle = \frac{\delta_{AB}}{x^{2\Delta_A}}.$$

With this normalization, connected correlators of single-trace primaries scale as

$$\boxed{ \langle \mathcal O_{A_1}\cdots\mathcal O_{A_k}\rangle_{\rm conn} \sim N^{2-k}. }$$

Equivalently, since holographic matrix theories have $C_T\sim N^2$,

$$\langle \mathcal O_{A_1}\cdots\mathcal O_{A_k}\rangle_{\rm conn} \sim C_T^{1-k/2}.$$

Thus

$$\langle \mathcal O_A\mathcal O_B\rangle\sim N^0, \qquad \langle \mathcal O_A\mathcal O_B\mathcal O_C\rangle\sim \frac{1}{N},$$

and

$$\langle \mathcal O_A\mathcal O_B\mathcal O_C\mathcal O_D\rangle_{\rm conn} \sim \frac{1}{N^2}.$$

This is the normalization in which single-trace operators behave as generalized free fields at $N=\infty$.

If one uses unnormalized traces, powers of $N$ move into the definition of the operator. That is harmless, but it often obscures the holographic dictionary. The invariant statement is the scaling of normalized correlators.

Where the counting comes from

The origin of the scaling is the double-line expansion. In a matrix theory, propagators carry two color lines. A Feynman graph can be drawn as a two-dimensional surface. The power of $N$ is determined by the topology of that surface.

A connected genus-$g$ contribution with $k$ normalized single-trace insertions scales as

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

The leading contribution is planar, $g=0$, so

$$\langle \mathcal O_{A_1}\cdots\mathcal O_{A_k}\rangle_{\rm conn, planar} \sim N^{2-k}.$$

This is the CFT side of the string perturbation expansion:

$$\frac{1}{N} \quad\leftrightarrow\quad \text{string coupling},$$

and, in a theory with a classical gravity limit,

$$\frac{1}{N^2} \quad\leftrightarrow\quad G_N.$$

More precisely, in AdS units,

$$G_N\sim \frac{R_{\rm AdS}^{d-1}}{C_T}.$$

So the same parameter that suppresses connected correlators suppresses bulk quantum gravity effects.

Single trace means one-particle, not necessarily supergravity

A single-trace operator creates one object in the bulk. That object may be a light supergravity particle, a Kaluza-Klein mode, a massive string excitation, or something heavier. The word “particle” here means one bulk quantum, not necessarily a low-energy graviton-like excitation.

For example, in $\mathcal N=4$ SYM, half-BPS single traces have the schematic form

$$\mathcal O_p^{I}(x) \sim C^I_{i_1\cdots i_p}\operatorname{Tr} \left(\Phi^{i_1}\cdots\Phi^{i_p}\right)(x),$$

where $C^I$ projects onto a symmetric traceless representation of $SO(6)_R$. These operators are dual to Kaluza-Klein modes on $S^5$.

Other single traces, such as operators with many derivatives or large spin-chain excitation number, can be dual to stringy states rather than supergravity fields. At strong ‘t Hooft coupling, many such stringy single-trace operators become heavy. A sparse low-dimension single-trace spectrum is one of the signals that the bulk effective theory is local and gravitational at low energies.

Thus the refined dictionary is:

CFT object	Bulk object
light single-trace primary	light one-particle field in AdS
heavy single-trace primary	massive stringy one-particle state
stress tensor $T_{\mu\nu}$	graviton
conserved current $J_\mu$	gauge field
double-trace primary	two-particle state
$p$-trace primary	$p$-particle state

The stress tensor is special because of Ward identities, but holographically it is still a single-particle operator: it creates the graviton.

At $N=\infty$, normalized single-trace primaries behave as one-particle creators, while multi-trace primaries behave as multi-particle states. Finite-$N$ effects weakly mix sectors with the same global quantum numbers and generate anomalous dimensions.

Multi-trace operators are not naive products

A product such as $\mathcal O_A(x)\mathcal O_B(x)$ is singular. To define a local operator, one must renormalize the coincident-point product. In a CFT, one also wants a conformal primary. The resulting double-trace primary is denoted

$$[\mathcal O_A\mathcal O_B]_{n,\ell}.$$

For scalar single-trace primaries, its schematic form is

$$[\mathcal O_A\mathcal O_B]_{n,\ell} \sim \mathcal O_A\,\partial_{\{\mu_1}\cdots\partial_{\mu_\ell\}} (\partial^2)^n\mathcal O_B + \cdots.$$

The braces denote symmetric traceless projection on the spin-$\ell$ indices. The ellipsis contains terms with derivatives distributed differently between the two operators, plus trace subtractions and descendant subtractions. These terms are fixed by the requirement that the result be a primary:

$$K_\mu [\mathcal O_A\mathcal O_B]_{n,\ell}(0)=0.$$

At leading large $N$, the dimension is additive:

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

For identical bosonic scalars, Bose symmetry allows only even $\ell$ in the OPE $\mathcal O_A\times\mathcal O_A$. If the two operators are distinct, both even and odd spins can appear, subject to the full set of global-symmetry selection rules.

Higher multi-trace primaries are built similarly:

$$[\mathcal O_{A_1}\cdots\mathcal O_{A_p}]_{\rm primary}.$$

At $N=\infty$, their leading dimensions are sums of one-particle dimensions plus nonnegative integer excitation energies. In radial quantization, this is just the statement that a free Fock space has additive energies.

Normal ordering and orthogonality

It is useful to think of multi-trace operators as normal-ordered products. For two distinct normalized scalar primaries, write schematically

$$:\mathcal O_A\mathcal O_B:(x) = \lim_{y\to x} \left[ \mathcal O_A(x)\mathcal O_B(y) - \text{singular subtractions} \right].$$

The subtractions remove identity pieces, lower-trace pieces, and descendant contamination. In an exact generalized-free sector, this is directly analogous to normal ordering in an ordinary free theory.

For example, suppose $\mathcal O_A$ and $\mathcal O_B$ are distinct generalized free scalar primaries with diagonal two-point functions. Then the leading two-point function of the simple scalar double trace is

$$\left\langle :\mathcal O_A\mathcal O_B:(x)\,:\mathcal O_A\mathcal O_B:(0)\right\rangle = \frac{1}{x^{2(\Delta_A+\Delta_B)}}.$$

If $A=B$, the Wick contraction gives an extra factor of $2$, so the unit-normalized scalar double trace is

$$\frac{1}{\sqrt{2}}:\mathcal O_A^2:.$$

This is exactly the same combinatorics as the normalization of identical two-particle states.

At strict $N=\infty$, operators with different particle number are orthogonal after proper normal ordering:

$$\langle \text{one-particle}\,|\,\text{two-particle}\rangle=0,$$ $$\langle \text{$p$-particle}\,|\,\text{$q$-particle}\rangle=0 \qquad (p\neq q).$$

At finite $N$, this orthogonality is corrected by powers of $1/N$.

The OPE knows about both single and multi-trace sectors

Consider the OPE of two normalized single-trace scalar primaries. Schematically,

$$\mathcal O_A\times\mathcal O_B \sim \delta_{AB}\mathbf 1 + \frac{1}{N}\sum_C \lambda_{ABC}\mathcal O_C + \sum_{n,\ell}\lambda^{(0)}_{AB;n,\ell} [\mathcal O_A\mathcal O_B]_{n,\ell} + \cdots.$$

The important point is the different $N$-scaling:

OPE contribution	Scaling in $\mathcal O_A\times\mathcal O_B$	Bulk meaning
identity	$O(1)$	disconnected propagation
double-trace tower	$O(1)$	two-particle states
single-trace exchange	$O(1/N)$	cubic bulk coupling
anomalous double-trace data in four-point functions	$O(1/N^2)$	tree-level interaction

This is why the generalized-free four-point function has a nontrivial conformal-block expansion even though its connected part vanishes. The leading $O(1)$ OPE data in the $\mathcal O_A\times\mathcal O_B$ channel are mostly multi-trace data. The genuinely interacting single-particle exchange data enter one order later.

For identical scalar $\mathcal O$, the leading large-$N$ OPE takes the form

$$\mathcal O\times\mathcal O \sim \mathbf 1 + \sum_{n=0}^{\infty}\sum_{\ell=0,2,4,\ldots} \lambda^{(0)}_{n,\ell} [\mathcal O\mathcal O]_{n,\ell} + O\left(\frac{1}{N}\right).$$

At finite $N$,

$$\Delta_{n,\ell} = 2\Delta+2n+\ell + \frac{1}{N^2}\gamma_{n,\ell}^{(1)} + \cdots$$

in matrix theories where connected four-point functions scale as $1/N^2$. The anomalous dimensions $\gamma_{n,\ell}^{(1)}$ are the CFT imprint of bulk interactions.

Operator mixing

At $N=\infty$, trace number is a good organizing principle. At finite $N$, it is not an exact quantum number. Operators with the same Lorentz spin, scaling dimension range, and global-symmetry quantum numbers can mix.

Suppose there is a single-trace primary $\mathcal S$ and a double-trace primary $\mathcal D$ with the same quantum numbers. Their dilatation operator may take the schematic form

$$\Gamma = \begin{pmatrix} \Delta_{\mathcal S}^{(0)}+\delta_{\mathcal S} & \mu/N \\ \mu/N & \Delta_{\mathcal D}^{(0)}+\delta_{\mathcal D} \end{pmatrix},$$

where $\delta_{\mathcal S}$ and $\delta_{\mathcal D}$ contain diagonal anomalous-dimension corrections. If the two leading dimensions are separated by an order-one gap,

$$\Delta_{\mathcal S}^{(0)}-\Delta_{\mathcal D}^{(0)}=O(1),$$

then the mixing angle is small:

$$\theta \sim \frac{\mu/N}{\Delta_{\mathcal S}^{(0)}-\Delta_{\mathcal D}^{(0)}}.$$

If the two levels are nearly degenerate, however, the mixing can be order one even though the off-diagonal matrix element is $1/N$-suppressed. This is ordinary degenerate perturbation theory, but in AdS/CFT it has a clear physical meaning: a one-particle state can mix strongly with a two-particle state when their energies are close.

This is how resonances and multi-particle thresholds appear in CFT language.

Single-particle versus single-trace in an abstract CFT

In a general CFT, one may not have a Lagrangian or color traces. For holography, the more invariant concept is a single-particle primary.

A practical large-$N$ definition is this: a primary $\mathcal O_A$ is single-particle if it belongs to a chosen basis of normalized primaries whose connected correlators obey

$$\langle \mathcal O_{A_1}\cdots\mathcal O_{A_k}\rangle_{\rm conn} \sim C_T^{1-k/2},$$

and if it cannot be written, at leading order, as a normal-ordered product of lower-particle primaries.

This definition is recursive. Once the single-particle primaries are identified, multi-particle primaries are built from their products and primary projections.

For matrix gauge theories, this abstract definition agrees with the usual single-trace/multi-trace language for low-dimension operators at large $N$. For vector models, the terminology changes: the natural single-particle operators are often bilinears rather than traces, and the bulk dual, when it exists, is usually a higher-spin theory rather than ordinary Einstein gravity. The large-$N$ logic of factorization and multi-particle composites is still present.

Finite-$N$ trace relations

At finite $N$, traces are not all independent. This is a basic algebraic fact: finite-size matrices obey polynomial identities. For one matrix, the Cayley-Hamilton theorem implies that sufficiently long traces can be expressed in terms of lower traces. For several noncommuting matrices, the relations are more complicated, but finite-$N$ constraints still exist.

In the large-$N$ limit, these relations disappear for operators of fixed dimension. That is why it is consistent to discuss an infinite tower of multi-trace states at $N=\infty$.

But if the operator dimension grows with $N$, finite-$N$ constraints become physically important. In AdS/CFT, this is related to stringy exclusion effects, giant gravitons, and the fact that a finite-$N$ CFT has fewer independent states than the naive infinite-$N$ Fock space.

The safe rule for this course is

$$\text{fixed operator dimension first, then }N\to\infty.$$

In that regime, single-trace and multi-trace language is reliable.

Multi-trace deformations versus multi-trace operators

A multi-trace operator is a local operator in the CFT spectrum. A multi-trace deformation is a change of the action by such an operator. For example,

$$S_{\rm CFT} \longrightarrow S_{\rm CFT} + f\int d^d x\,\mathcal O^2(x).$$

These are related but not identical concepts. The operator $\mathcal O^2$ belongs to the spectrum whether or not we deform by it. Adding it to the action changes the theory.

In AdS/CFT, double-trace deformations are especially important because they modify boundary conditions for the dual bulk field. We will return to this idea later. On this page, the main point is spectral: multi-trace operators are the CFT names of multi-particle states.

The large-$N$ Hilbert space as a Fock space

Radial quantization gives the cleanest picture. A primary $\mathcal O_A$ creates a state

$$|A\rangle=\mathcal O_A(0)|0\rangle$$

with cylinder energy

$$E_A=\Delta_A.$$

A double-trace primary creates a two-particle state:

$$[\mathcal O_A\mathcal O_B]_{n,\ell}(0)|0\rangle \quad\longleftrightarrow\quad |A,B;n,\ell\rangle.$$

At $N=\infty$,

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

At finite $N$,

$$E_{AB;n,\ell} = E_{AB;n,\ell}^{(0)} + \frac{1}{N^2}\gamma_{AB;n,\ell}^{(1)} + \cdots.$$

This is precisely the structure of perturbation theory for weakly interacting particles in global AdS. The CFT dilatation operator is the global-AdS Hamiltonian.

Thus the large-$N$ Hilbert space is approximately

$$\mathcal H_{\rm CFT} \simeq \mathcal F\left(\mathcal H_{\rm single\ particle}\right), \qquad N=\infty,$$

where $\mathcal F$ denotes a Fock-space construction. The approximation breaks down when finite-$N$ constraints, strong mixing, black-hole states, or very high energies become important.

The OPE as a particle-number expansion

The operator product expansion can be reorganized by particle number. For two single-particle operators,

$$\mathcal O_A\times\mathcal O_B = \text{identity} + \text{two-particle primaries} + \frac{1}{N}\text{one-particle primaries} + \cdots.$$

This ordering may look surprising: why do two-particle operators appear at order one, while single-particle exchange is suppressed? The answer is that two generalized-free particles are already present in the disconnected theory. A single-particle exchange between two external particles requires an interaction vertex, and interaction vertices are $1/N$-suppressed.

For a four-point function of single-trace scalars, the expansion is

$$\mathcal G(u,v) = \mathcal G_{\rm MFT}(u,v) + \frac{1}{N^2}\mathcal G^{(1)}(u,v) + \cdots.$$

The leading term contains the identity and double-trace blocks. The correction $\mathcal G^{(1)}$ contains single-trace exchange and corrections to double-trace data.

This is the conceptual bridge to Witten diagrams:

CFT expansion	AdS perturbation theory
identity block	disconnected propagation
leading double-trace blocks	free two-particle states
single-trace exchange at $1/N^2$ in four-point functions	tree-level exchange Witten diagram
anomalous dimensions of double traces	binding energies and phase shifts
$1/N^4$ corrections	loop Witten diagrams

The precise power of $1/N$ depends on whether one discusses OPE coefficients or full connected correlators. For normalized single traces, cubic OPE coefficients scale as $1/N$, while their contribution to a four-point function scales as $1/N^2$.

AdS/CFT checkpoint

The core holographic dictionary is

$$\boxed{ \mathcal O_A \leftrightarrow \phi_A }$$

for a single-particle bulk field or string state, and

$$\boxed{ [\mathcal O_A\mathcal O_B]_{n,\ell} \leftrightarrow \text{two-particle state of }\phi_A\text{ and }\phi_B. }$$

The large-$N$ expansion controls how sharply this particle-number interpretation holds. At $N=\infty$, trace number is conserved and the Hilbert space is Fock-like. At finite $N$, trace number can change, reflecting bulk interactions:

$$\text{one particle} \leftrightarrow \text{two particles}$$

through processes whose amplitudes are suppressed by powers of $1/N$.

This is why large-$N$ CFT is the right language for quantum gravity in AdS. It gives a nonperturbative operator algebra, but at large $N$ that algebra reorganizes itself into the perturbative Hilbert space of particles, strings, and interactions.

Summary

Single-trace operators are the basic gauge-invariant operators of matrix large-$N$ theories. With order-one two-point normalization, their connected correlators scale as

$$\langle \mathcal O_{A_1}\cdots \mathcal O_{A_k}\rangle_{\rm conn} \sim N^{2-k} \sim C_T^{1-k/2}.$$

At $N=\infty$, they create one-particle states. Multi-trace primaries create multi-particle states. For scalar double traces,

$$[ \mathcal O_A\mathcal O_B ]_{n,\ell} \quad\text{has}\quad \Delta_{AB;n,\ell}^{(0)} = \Delta_A+\Delta_B+2n+\ell.$$

Finite-$N$ effects mix operators with the same quantum numbers and shift multi-trace dimensions:

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

These anomalous dimensions are the CFT form of bulk binding energies and scattering data. The whole point of large-$N$ holography is that a complicated CFT operator algebra becomes, to leading order, a Fock-space problem in AdS.

Exercises

Exercise 1. Connected scaling and factorization

Assume normalized single-trace operators obey

$$\langle \mathcal O_1\cdots\mathcal O_k\rangle_{\rm conn} \sim N^{2-k}.$$

Show that the four-point function factorizes at leading order, while its connected part is suppressed by $1/N^2$.

Solution

For $k=2$,

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

For $k=4$,

$$\langle \mathcal O_1\mathcal O_2\mathcal O_3\mathcal O_4\rangle_{\rm conn} \sim N^{-2}.$$

The full four-point function decomposes into connected and disconnected pieces:

$$\begin{aligned} \langle 1234\rangle &= \langle 12\rangle\langle 34\rangle + \langle 13\rangle\langle 24\rangle + \langle 14\rangle\langle 23\rangle + \langle 1234\rangle_{\rm conn}. \end{aligned}$$

The disconnected products are order $N^0$, while the connected term is order $N^{-2}$. Thus

$$\langle 1234\rangle = \langle 12\rangle\langle 34\rangle + \langle 13\rangle\langle 24\rangle + \langle 14\rangle\langle 23\rangle + O(N^{-2}).$$

This is large-$N$ factorization.

Exercise 2. Unit normalization of an identical double trace

Let $\mathcal O$ be a generalized free scalar with

$$\langle \mathcal O(x)\mathcal O(0)\rangle=\frac{1}{x^{2\Delta}}.$$

Show that the scalar double-trace operator

$$\mathcal D(x)=\frac{1}{\sqrt{2}}:\mathcal O^2:(x)$$

has unit-normalized two-point function at leading order:

$$\langle \mathcal D(x)\mathcal D(0)\rangle = \frac{1}{x^{4\Delta}}.$$

Solution

Using Wick factorization and normal ordering, contractions inside the same normal-ordered product are removed. The only leading contractions pair the two fields at $x$ with the two fields at $0$:

$$\left\langle :\mathcal O^2:(x):\mathcal O^2:(0)\right\rangle = 2\left(\frac{1}{x^{2\Delta}}\right)^2 = \frac{2}{x^{4\Delta}}.$$

Therefore

$$\langle \mathcal D(x)\mathcal D(0)\rangle = \frac{1}{2}\frac{2}{x^{4\Delta}} = \frac{1}{x^{4\Delta}}.$$

The factor $1/\sqrt{2}$ is the same Bose-symmetry normalization as for two identical particles.

Exercise 3. Additive dimensions from the cylinder

Let $\mathcal O_A$ and $\mathcal O_B$ be scalar primaries of dimensions $\Delta_A$ and $\Delta_B$. Use radial quantization to explain why

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

Solution

Radial quantization maps a primary operator to a state on $S^{d-1}$:

$$\mathcal O_A(0)|0\rangle=|A\rangle, \qquad E_A=\Delta_A.$$

At $N=\infty$, two-particle energies add. A two-particle state built from $A$ and $B$ has base energy

$$\Delta_A+\Delta_B.$$

Orbital angular momentum $\ell$ adds $\ell$ units of energy, and the radial excitation number $n$ adds $2n$ units. Therefore

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

By the state-operator correspondence, cylinder energy equals scaling dimension, so

$$\Delta_{AB;n,\ell}^{(0)}=E_{AB;n,\ell}^{(0)}.$$

Exercise 4. Small mixing angle

Consider two operators with the same quantum numbers and dilatation matrix

$$\Gamma = \begin{pmatrix} \Delta_S & \mu/N \\ \mu/N & \Delta_D \end{pmatrix},$$

where $\Delta_S-\Delta_D=O(1)$. Compute the leading mixing angle.

Solution

For a real symmetric $2\times2$ matrix, the mixing angle satisfies

$$\tan 2\theta = \frac{2\mu/N}{\Delta_S-\Delta_D}.$$

If the off-diagonal term is small compared to the level splitting, then

$$\theta \simeq \frac{\mu/N}{\Delta_S-\Delta_D}.$$

Thus the mixing is suppressed by $1/N$ when the two levels are not nearly degenerate. If $\Delta_S-\Delta_D$ becomes of order $1/N$, this perturbative estimate fails and the mixing can become order one.

Exercise 5. Single-trace exchange in a four-point function

Suppose the OPE coefficient of three normalized single-trace operators scales as

$$\lambda_{ABC}\sim \frac{1}{N}.$$

Explain why exchange of a single-trace primary $\mathcal O_C$ in a four-point function of normalized single traces contributes at order $1/N^2$.

Solution

A conformal-block contribution from exchanging $\mathcal O_C$ in the $\mathcal O_A\mathcal O_B$ and $\mathcal O_D\mathcal O_E$ channels is proportional to a product of two OPE coefficients:

$$\lambda_{ABC}\lambda_{DEC}\,G_{\Delta_C,\ell_C}(u,v).$$

Each cubic single-trace OPE coefficient scales as $1/N$, so

$$\lambda_{ABC}\lambda_{DEC} \sim \frac{1}{N^2}.$$

Thus single-trace exchange appears at order $1/N^2$ in the connected four-point function. This matches the bulk statement that a tree-level exchange Witten diagram is suppressed by one power of the gravitational coupling $G_N\sim 1/N^2$.