Single-Trace, Multi-Trace, and Collective-Field Organization for Bulk Criteria

At leading large $N$, an isolated normalized single-trace primary can label an approximate one-particle state, while multi-trace primaries organize the corresponding multiparticle sector. This interpretation is conditional rather than absolute. The inner-product matrix must first be normalized, the dilatation operator must then be diagonalized, and near-degenerate levels can mix through an $N$-independent angle even though the off-diagonal coupling is only $\mathcal O(N^{-1})$.

Required background. Large-N Factorization and Classical Bulk Scaling supplies the connected-correlator hierarchy.

Helpful background. From the Local OPE to Conformal Data defines the CFT spectrum and OPE coefficients. Local Field Redefinitions and the Equivalence Theorem explains which bulk statements survive a change of variables.

Read the taxonomy first, then follow the Gram-matrix calculation into the near-degenerate regime. The worked basis change separates invariant data—the generalized spectrum and metric-completed projections—from candidate-basis data such as individual overlaps, off-diagonal entries, and fixed-gap mixing angles.

Single traces, double traces, and collective coordinates

In an adjoint matrix theory, a single-trace candidate is a renormalized gauge-invariant composite with one color trace, normalized so that its two-point function is order one. A leading-order double-trace candidate is the normalized $N=\infty$ conformal-primary projection of the product after descendant contamination and null directions have been removed. At finite $N$, lower-trace components are determined by the Gram and dilatation problem rather than removed in advance. Vector models use a different organization; there, singlet bilocals can be the efficient collective coordinates.

Large-N operator organizations and their conditional bulk interpretations

Candidate coordinate	Large-N construction	Conditional bulk reading	Main limitation
Normalized single-trace primary	One color trace, with order-one two-point norm	Approximate one-particle label	Requires spectral isolation and weak mixing
Normalized leading-order double-trace candidate	Product of two single traces, projected to a primary at N = ∞	Approximate two-particle label	Resonances can destroy trace-number labels
Singlet bilocal in a vector model	Sum over one vector index, kept as a function of two points	Collective coordinate for a higher-spin sector	It is not one local particle or a finite-field Einstein EFT

Let $\mathcal O_i$ and $\mathcal O_j$ be normalized scalar single-trace primaries, and write

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

for this normalized $N=\infty$ conformal-primary projection of their double-trace family. Here $n\geq0$ counts radial excitation and $\ell\geq0$ is orbital spin; only even $\ell$ occurs when the two constituents are identical bosonic scalars. Spinning constituents require additional tensor-family labels. Under radial quantization, write

$$\lvert s\rangle=\mathcal O_s(0)\lvert0\rangle, \qquad \lvert ij;n,\ell\rangle =[\mathcal O_i\mathcal O_j]_{n,\ell}(0)\lvert0\rangle.$$

In an orientable adjoint-matrix ’t Hooft expansion, at fixed $n$ and $\ell$ and away from resonant mixing,

$$\Delta_{ij,n,\ell} =\Delta_i^{(0)}+\Delta_j^{(0)}+2n+\ell +\frac{\gamma^{(1)}_{ij,n,\ell}}{N^2}+\mathcal O(N^{-4}).$$

The $N=\infty$ contribution is the energy of two free AdS particles in global units, while $\gamma^{(1)}$ records the leading interaction shift in this matrix normalization. The exact $N=\infty$ tower follows from generalized-free factorization Fitzpatrick and Kaplan 2013, § 2.2, eqs. (18) and (23), printed pp. 9–10, Open PDF. The displayed $N^{-2}$ correction is matrix-theory-specific; Heemskerk et al. 2009, §§ 3.2–3.3, eqs. (3.3) and (3.9), printed pp. 10–12, Open PDF exhibit it in a more restrictive model with a low scalar, a $\mathbb Z_2$ symmetry, and a large gap. Other large-$N$ expansions can use different powers.

In the standard orientable single-trace ’t Hooft organization assumed here, normalized connected counting permits single–double candidate overlaps at order $1/N$. After the subtractions implicit in the leading-order double-trace projection,

$$\langle s\vert ij;n,\ell\rangle \sim \langle \mathcal O_s\mathcal O_i\mathcal O_j\rangle_{\mathrm c} =\mathcal O(N^{-1}),$$

whereas the leading double-trace norm contains a disconnected product of two two-point functions and is $\mathcal O(1)$. The $\mathcal O(N^{-1})$ Gram overlap follows directly from the connected three-point hierarchy. Assigning the same generic power to the candidate-basis dilatation matrix additionally assumes the usual single-trace large-$N$ organization of the Hamiltonian or dilatation generator; leading multi-trace deformations are outside this statement Heemskerk et al. 2009, § 2.4, footnote 9, printed pp. 7–8, Open PDF. A symmetry or selection rule can also remove the leading coupling. This discussion concerns a basis assembled from the $N=\infty$ trace sectors. Distinct exact conformal primaries with distinct dimensions already have zero two-point overlap.

The particle language is therefore most reliable when a finite set of low-dimension single traces generates an approximate Fock space below a declared gap. It is not implied by the word “trace” alone Fitzpatrick and Kaplan 2013, § 1, printed pp. 1–4, Open PDF.

Normalize before diagonalizing

Take candidate operators $A_A$ with the same exact global quantum numbers. They may be regulated composites or another convenient basis that has not yet been diagonalized. Under the state–operator correspondence of radial quantization, insertions at the origin create states and Hermitian conjugation is implemented by inversion. This supplies two Hermitian matrices,

$$G_{AB}=\langle A_A\vert A_B\rangle, \qquad D_{AB}=\langle A_A\vert\widehat D\vert A_B\rangle.$$

The Gram matrix is positive semidefinite in a unitary theory. Remove null directions, then choose a matrix $S$ on the positive subspace satisfying

$$S^\dagger G S=\mathbf 1; \qquad S=G^{-1/2}$$

as one convenient choice. This whitening step removes nonorthogonality but does not determine scaling dimensions. If the candidate space is closed under mixing through the target order, its eigenoperators solve

$$Dv_\alpha=\Delta_\alpha Gv_\alpha.$$

Equivalently, principal whitening asks us to diagonalize

$$H=G^{-1/2}DG^{-1/2}.$$

This is the standard reduction of a Hermitian generalized eigenproblem with positive-definite metric Anderson et al. 1999, Table 2.13. Whitening is not unique: $S=G^{-1/2}U$ also works for any unitary $U$. Consequently, the generalized eigenvalues are invariant, while individual off-diagonal entries and eigenvector components depend on the declared orthonormal reference convention.

If important operators have been omitted, the calculation is only a truncated variational problem. Mixing with the excluded space must be smaller than the claimed accuracy. Conflating an off-diagonal entry of $G$ with one of $H$ mistakes nonorthogonality for dynamical mixing. If the $A_A$ are already exact conformal primaries with distinct dimensions, conformal symmetry has performed the diagonalization: their two-point Gram matrix is diagonal.

A first-order calculation makes the distinction concrete. Let $\varepsilon=1/N$ and, for real $a$ and $b$, take

$$G=\mathbf 1+\varepsilon a\sigma_x, \qquad D= \begin{pmatrix} \Delta_s&\varepsilon b\\ \varepsilon b&\Delta_d \end{pmatrix} +\mathcal O(\varepsilon^2).$$

Then

$$G^{-1/2}=\mathbf 1-\frac{\varepsilon a}{2}\sigma_x +\mathcal O(\varepsilon^2), \qquad H_{12}=\varepsilon\mu+\mathcal O(\varepsilon^2),$$

with

$$\mu=b-\frac a2(\Delta_s+\Delta_d).$$

Thus neither the Gram overlap $a$ nor the unwhitened entry $b$ is the post-normalization coupling. Even $\mu$ is an off-diagonal entry in the principal-whitening reference convention, not a basis-invariant observable.

When the trace label fails

Choose a Gram-orthonormal reference basis that approaches the single- and double-trace sectors as $N\to\infty$. At the first nontrivial order, its two-state block is

$$H= \begin{pmatrix} \Delta_s & \varepsilon\mu\\ \varepsilon\mu^* & \Delta_d \end{pmatrix} +\mathcal O(\varepsilon^2).$$

After removing the phase of $\mu$, the displayed $2\times2$ truncation has eigenvalues and a rotation angle relative to that reference basis,

$$\Delta_\pm =\frac{\Delta_s+\Delta_d}{2} \pm \sqrt{\left(\frac{\delta\Delta}{2}\right)^2+\varepsilon^2\lvert\mu\rvert^2}, \qquad \tan 2\theta =\frac{2\varepsilon\lvert\mu\rvert}{\delta\Delta},$$

where $\delta\Delta=\Delta_s-\Delta_d$. The eigenvalues are invariant; $\mu$ and $\theta$ are reference-basis quantities. Their useful content is the scale comparison:

Mixing regimes in the single–double-trace two-state block

Gap regime	Spectral result in the displayed block	Status of the large-N trace label
δΔ → δ0 ≠ 0	Mixing shifts eigenvalues by O(N−2)	A continuously chosen trace basis needs only an O(N−1) rotation, but that angle is convention dependent
δΔ = c/N, c ≠ 0	The level splitting is O(N−1)	If μ approaches a nonzero limit, the rotation approaches an N-independent angle; if a selection rule sets that limit to zero, no mixing follows at this order
δΔ = o(N−1) or exact degeneracy	The whole degenerate block contributes at leading order	Diagonalize the complete block before assigning particle labels

For a fixed gap, omitted diagonal terms of order $N^{-2}$ are needed for a complete eigenvalue at the same accuracy as the mixing-induced shift. When $\delta\Delta=c/N$ and $\mu\to\mu_0\neq0$, the rotation approaches an $N$-independent angle $\theta_0$ with $\tan2\theta_0=2\lvert\mu_0\rvert/c$, while

$$\Delta_+-\Delta_- =\frac1N\sqrt{c^2+4\lvert\mu_0\rvert^2} +\mathcal O(N^{-2}).$$

The angle’s numerical size still depends on $\lvert\mu_0/c\rvert$; “unsuppressed” need not mean close to $45^\circ$. This absence of $1/N$ suppression is robust under perturbatively near-identity basis changes that preserve the $N=\infty$ trace sectors. The next calculation isolates the basis dependence and explains the robustness claim.

Basis dependence and an invariant stress test

Let $K=(\mathcal O_s,\mathcal O_d)$ be a row of candidate operators, where $\mathcal O_d$ is the projected double trace, and make the trace-changing basis transformation

$$K_R=KR, \qquad R= \begin{pmatrix} 1&0\\ \varepsilon\alpha&1 \end{pmatrix}.$$

The first transformed operator is $\mathcal O_s+\varepsilon\alpha\mathcal O_d$, so this is nonlinear when expressed in single-trace coordinates.

Transforming the two-state pair

To isolate the algebra, temporarily treat

$$G=\mathbf1, \qquad D= \begin{pmatrix} \Delta_s&\varepsilon\mu\\ \varepsilon\mu^*&\Delta_d \end{pmatrix}$$

as the exact two-state model. The transformed matrices are

$$G_R=R^\dagger G R = \begin{pmatrix} 1+\varepsilon^2\lvert\alpha\rvert^2&\varepsilon\alpha^*\\ \varepsilon\alpha&1 \end{pmatrix},$$ $$D_R=R^\dagger D R = \begin{pmatrix} \Delta_s+\varepsilon^2(\mu\alpha+\alpha^*\mu^*+\lvert\alpha\rvert^2\Delta_d) &\varepsilon(\mu+\alpha^*\Delta_d)\\ \varepsilon(\mu^*+\alpha\Delta_d)&\Delta_d \end{pmatrix}.$$

Invariant spectrum and movable fixed-gap mixing

The generalized characteristic polynomial transforms as

$$\det(D_R-\lambda G_R) =\lvert\det R\rvert^2\det(D-\lambda G),$$

so its roots are exactly unchanged. Principal whitening of the transformed pair instead gives

$$\bigl(G_R^{-1/2}D_RG_R^{-1/2}\bigr)_{12} =\varepsilon\left[ \mu-\frac{\alpha^*}{2}(\Delta_s-\Delta_d) \right] +\mathcal O(\varepsilon^2).$$

For a fixed nonzero gap, choosing $\alpha^*=2\mu/(\Delta_s-\Delta_d)$ removes the leading principal-whitening off-diagonal entry. Hence the fixed-gap $\mathcal O(N^{-1})$ angle cannot be called physical. If instead $\Delta_s-\Delta_d=c\varepsilon$ with $c\neq0$, any $\alpha=\mathcal O(1)$ changes that entry only at $\mathcal O(\varepsilon^2)$. Canceling $\varepsilon\mu_0$ for $\mu_0\neq0$ would require $\alpha^*=2\mu_0/(c\varepsilon)$, hence $R_{21}=\varepsilon\alpha=\mathcal O(N^0)$; the transformation would not approach the identity and would not preserve the declared $N=\infty$ trace sectors. If $c=0$, this shear does not change the leading off-diagonal entry at all. That is why the absence of $1/N$ suppression is robust even though a particular angle is not invariant.

Metric-completed overlaps

Complete contractions provide a second invariant. For an external-overlap vector $f_A=\langle A_A\vert X\rangle$,

$$f_R=R^\dagger f, \qquad f_R^\dagger G_R^{-1}f_R=f^\dagger G^{-1}f.$$

The candidate overlaps themselves change, but their metric-completed contraction over the same operator subspace does not. Because $\mathcal O_s+\varepsilon\alpha\mathcal O_d$ is generally not a primary eigenoperator, its three-point overlaps are not new conformal-primary OPE coefficients. After rediagonalization, nondegenerate normalized-primary dimensions and OPE data are unchanged up to phase conventions. In an exactly degenerate eigenspace, individual components can rotate, so invariant contraction matrices or sums over the degenerate subspace are the appropriate data.

Boundary bases and matched bulk variables

A boundary operator-basis change and a local bulk field redefinition are related dictionary choices, but they are not the same operation. As explained in Local Field Redefinitions and the Equivalence Theorem, the bulk change of variables must transform the action, functional measure or Jacobian, boundary conditions, source map, boundary terms, and counterterms consistently. Then the renormalized generating functional expressed in matched physical sources is unchanged. Separated-point correlators agree, while local contact terms may depend on counterterm and source conventions Criado and Pérez-Victoria 2019, § 2, printed pp. 5–8, Open PDF.

An explicit $AdS_5\times S^5$ / $\mathcal N=4$ SYM protected-operator example shows how bulk field redefinitions, boundary terms, and single–double-trace operator admixtures fit together Arutyunov and Frolov 2000, §§ 2–3 and conclusion, printed pp. 5, 7–8, and 13–14, Open PDF. It is a useful application, not a universal theorem for every CFT. In general, exchange and contact contributions may be redistributed, while an off-shell cubic coefficient by itself remains basis information rather than a separately invariant observable.

Collective variables and higher-spin sectors

Vector models and collective-field formulations may instead make bilocals natural. For $N$ real vector components, define the expectation-normalized bilocal and its order-one fluctuation by

$$\Psi(x,y) =\frac1N\sum_{a=1}^{N}\phi^a(x)\phi^a(y), \qquad B_{\mathrm{fluc}}(x,y) =\sqrt N\bigl(\Psi(x,y)-\langle\Psi(x,y)\rangle\bigr).$$

For $n\geq2$, at separated or regulated bilocal endpoints, every connected Wick contraction of $n$ bilocals in the free theory carries one closed vector-index sum and $n$ explicit factors of $N^{-1}$. Thus

$$\kappa_n(\Psi)\sim N^{-n}N=N^{1-n}, \qquad \kappa_n(B_{\mathrm{fluc}}) \sim N^{n/2}N^{1-n}=N^{1-n/2}.$$

The $n=1$ cumulant of $B_{\mathrm{fluc}}$ vanishes by centering. The same generic leading tree-level power counting follows around the conventional interacting vector-model saddle: the bilocal propagator is order one, while a $k$-fluctuation vertex scales as $N^{1-k/2}$ Das and Jevicki 2003, eqs. (1.5), (2.26), and (2.35)–(2.36), printed pp. 4 and 10–12, Open PDF. This differs from the matrix hierarchy $N^{2-n}$.

In the real free $O(N)$ model, the center and relative coordinates of the bilocal decompose into the even-spin singlet tower Das and Jevicki 2003, eqs. (1.6)–(1.9), printed pp. 4–5, Open PDF. A canonical map to higher-spin variables is developed in Jevicki, Jin, and Ye 2011, § 4, eqs. (118)–(127), printed pp. 11–14, Open PDF. At finite $N$, the bilocal variables are overcomplete and obey constraints; treating every bilocal component as independent overcounts the Hilbert space Das and Jevicki 2003, § 1, printed p. 5, Open PDF. Calling the bilocal “one field” therefore names a collective coordinate, not one local bulk particle or a finite-field Einstein EFT.

Scaled limits, evidence ceiling, and handoff

For each operator, state its normalization, primary and descendant status, trace organization, large-$N$ scaling, mixing partners, and spectral isolation. Then:

1. Compute the Gram matrix in the regulated candidate basis.
2. Remove null states and whiten the positive subspace.
3. Transform and diagonalize the dilatation matrix.
4. Compare every level splitting with its off-diagonal mixing element.
5. Test a near-identity trace-changing basis transformation and retain only invariant spectral or complete-correlator statements.
6. Treat boundary operator-basis changes and matched bulk field redefinitions as separate transformations.

The single-particle interpretation is taken at fixed operator length and away from splittings that vanish as fast as the mixing matrix element. Holding $\delta\Delta$ fixed before $N\to\infty$ yields a small reference-basis rotation; when $\mu\to\mu_0\neq0$, scaling $\delta\Delta\sim N^{-1}$ yields an $N$-independent rotation.

This organization licenses an approximate particle-number basis, not a unique off-shell field basis or a proof of bulk locality. The next article, From Genus Counting to a Holographic String Regime, asks when the matrix expansion has the topology and parameter control needed for a string interpretation.

Common pitfalls

Diagonalizing the Gram matrix and stopping. Gram whitening removes nonorthogonality. Scaling dimensions come from the generalized eigenproblem; off-diagonal entries and mixing angles still depend on the chosen orthonormal reference basis.

Calling every nonlinear admixture a primary. A boundary-basis change can be useful, and a separately matched bulk field redefinition may reorganize the same calculation, but the two operations are not identical. A generic boundary admixture is not a dilatation eigenoperator until the candidate sector has been rediagonalized.

Using trace number as an exact particle number. Trace organization is an asymptotic bookkeeping device. Near degeneracy, finite-$N$ identities, collective variables, and strong mixing can all defeat a literal particle label.

Exercises

Whiten first, then mix

Let $\varepsilon=1/N$ and consider two real candidate states with

$$G= \begin{pmatrix} 1&\varepsilon a\\ \varepsilon a&1 \end{pmatrix}, \qquad D= \begin{pmatrix} \Delta_s&\varepsilon b\\ \varepsilon b&\Delta_d \end{pmatrix} +\mathcal O(\varepsilon^2).$$

Find $G^{-1/2}$ and the off-diagonal entry of $H=G^{-1/2}DG^{-1/2}$ through first order in $\varepsilon$. Assuming $\Delta_s-\Delta_d\to\delta_0\neq0$, which combination controls the rotation in this reference convention, and which part of the answer is invariant?

Solution — first-order whitening

Writing $G=\mathbf1+\varepsilon a\sigma_x$ gives

$$G^{-1/2}=\mathbf1-\frac{\varepsilon a}{2}\sigma_x+\mathcal O(\varepsilon^2).$$

Therefore

$$H= \begin{pmatrix} \Delta_s&\varepsilon\mu\\ \varepsilon\mu&\Delta_d \end{pmatrix} +\mathcal O(\varepsilon^2), \qquad \mu=b-\frac a2(\Delta_s+\Delta_d).$$

For the nonzero limiting gap stated in the question,

$$\tan2\theta=\frac{2\varepsilon\mu}{\Delta_s-\Delta_d}+\mathcal O(\varepsilon^2).$$

Thus neither the Gram entry $a$ nor the unwhitened dilatation entry $b$ alone controls the reference-basis rotation. The generalized eigenvalues are invariant, but $\mu$ and the corresponding fixed-gap angle change under an allowed change of orthonormal reference basis.

Scale the gap against the mixing

Take the whitened two-state block with $\mu\neq0$ fixed and

$$\Delta_s-\Delta_d=cN^{-p}, \qquad c\neq0,\qquad p\geq0.$$

Classify the reference-basis rotation and leading level splitting for $p<1$, $p=1$, and $p>1$. At which value does the rotation cease to vanish as $N\to\infty$?

Solution — three gap regimes

The ratio of the off-diagonal entry to the gap is

$$\frac{\lvert\mu\rvert/N}{\lvert c\rvert N^{-p}} =\frac{\lvert\mu\rvert}{\lvert c\rvert}N^{p-1}.$$

For $p<1$, this ratio vanishes and the reference-basis angle scales as $N^{p-1}$. Dropping the unspecified $\mathcal O(N^{-2})$ matrix remainder, the leading two-state truncation gives

$$\Delta_+-\Delta_- =\sqrt{c^2N^{-2p}+4\lvert\mu\rvert^2N^{-2}} =\lvert c\rvert N^{-p} \left[ 1+\frac{2\lvert\mu\rvert^2}{c^2}N^{2p-2} +\mathcal O(N^{4p-4}) \right].$$

The continuously labeled levels receive mixing-induced shifts

$$\delta\Delta_s =\frac{\lvert\mu\rvert^2}{c}N^{p-2} +o(N^{p-2}), \qquad \delta\Delta_d =-\frac{\lvert\mu\rvert^2}{c}N^{p-2} +o(N^{p-2}).$$

Each shift has magnitude $\lvert\mu\rvert^2\lvert c\rvert^{-1}N^{p-2}$. At $p=0$, unknown diagonal $\mathcal O(N^{-2})$ terms contribute at the same order, so this is only the mixing-induced part of the full correction.

For $p=1$, the angle has a finite limit,

$$\tan2\theta=\frac{2\lvert\mu\rvert}{c},$$

and the leading splitting is

$$\Delta_+-\Delta_- =\frac1N\sqrt{c^2+4\lvert\mu\rvert^2} +\mathcal O(N^{-2}).$$

For $p>1$, the original gap is smaller than the mixing entry. The reference-basis eigenvectors approach equal-magnitude mixtures and the leading splitting is $2\lvert\mu\rvert/N$. The rotation ceases to vanish at the threshold $p=1$.

Challenge an “observable” off-diagonal entry

A collaborator calls the principal-whitening entry $H_{12}=\varepsilon\mu$ an observable. For fixed nonzero $\mu$ and a fixed nonzero gap $\delta\Delta=\Delta_s-\Delta_d$, use the near-identity transformation in the text to choose $\alpha$ so that the transformed principal-whitening entry vanishes through order $\varepsilon$. Then explain why the generalized spectrum is unchanged and why the same $\alpha=\mathcal O(1)$ cannot remove the leading mixing when $\delta\Delta=c\varepsilon$ with $c\neq0$. What happens for $c=0$?

Solution — movable entry, invariant spectrum

The transformed entry is

$$\bigl(G_R^{-1/2}D_RG_R^{-1/2}\bigr)_{12} =\varepsilon\left(\mu-\frac{\alpha^*}{2}\delta\Delta\right) +\mathcal O(\varepsilon^2).$$

At fixed $\delta\Delta\neq0$, choose

$$\alpha^*=\frac{2\mu}{\delta\Delta}.$$

This removes the order-$\varepsilon$ entry while keeping $R-\mathbf1=\mathcal O(\varepsilon)$. Nevertheless,

$$\det(D_R-\lambda G_R) =\lvert\det R\rvert^2\det(D-\lambda G),$$

so the generalized eigenvalues are unchanged. If $\delta\Delta=c\varepsilon$ with $c\neq0$, any $\alpha=\mathcal O(1)$ changes the bracket only by $\mathcal O(\varepsilon)$ and therefore changes the full off-diagonal entry only at $\mathcal O(\varepsilon^2)$. Canceling $\varepsilon\mu$ would require $\alpha=\mathcal O(\varepsilon^{-1})$, which is not a near-identity transformation and does not preserve the $N=\infty$ trace sectors. If $c=0$, the shear term proportional to $\delta\Delta$ vanishes and cannot change the leading off-diagonal entry at all. The individual entry is movable at fixed gap; the spectrum and the $N$-independent near-degenerate rotation for nonzero leading $\mu$ are robust.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Anderson, E.; Bai, Z.; Bischof, C.; Blackford, S.; Demmel, J.; Dongarra, J.; Du Croz, J.; Greenbaum, A.; Hammarling, S.; McKenney, A.; and Sorensen, D. LAPACK Users’ Guide. 3rd ed. Philadelphia: Society for Industrial and Applied Mathematics, 1999. doi:10.1137/1.9780898719604.
• Arutyunov, G., and S. Frolov. “On the Correspondence between Gravity Fields and CFT Operators.” Journal of High Energy Physics 04, 017 (2000). doi:10.1088/1126-6708/2000/04/017. Open PDF.
• Criado, J. C., and M. Pérez-Victoria. “Field Redefinitions in Effective Theories at Higher Orders.” Journal of High Energy Physics 03, 038 (2019). doi:10.1007/JHEP03(2019)038. Open PDF.
• Das, Sumit R., and Antal Jevicki. “Large-N Collective Fields and Holography.” Physical Review D 68, 044011 (2003). doi:10.1103/PhysRevD.68.044011. Open PDF.
• Fitzpatrick, A. Liam, and Jared Kaplan. “AdS Field Theory from Conformal Field Theory.” Journal of High Energy Physics 02, 054 (2013). doi:10.1007/JHEP02(2013)054. Open PDF.
• Heemskerk, Idse; Penedones, João; Polchinski, Joseph; and Sully, James. “Holography from Conformal Field Theory.” Journal of High Energy Physics 10, 079 (2009). doi:10.1088/1126-6708/2009/10/079. Open PDF.
• Jevicki, Antal; Jin, Kewang; and Ye, Qibin. “Collective Dipole Model of AdS/CFT and Higher Spin Gravity.” Journal of Physics A: Mathematical and Theoretical 44, 465402 (2011). doi:10.1088/1751-8113/44/46/465402. Open PDF.