Vector-Model Bilocals as Higher-Spin Bulk Data

The singlet bilocal $\Psi(x,y)=N^{-1}\phi^i(x)\phi^i(y)$ reorganizes an $O(N)$ vector model into variables whose harmonic components carry the entire single-trace higher-spin tower. The change of variables can be exact in a regulated singlet integral, including a nontrivial Jacobian and finite-$N$ rank constraints. Interpreting its fluctuations as local fields in one higher dimension is an additional, saddle-dependent reconstruction—not a consequence of the variable change alone.

Required background. Vector Models, Auxiliary Fields, and Large-N Saddles supplies the original path integral; Single-Trace, Multi-Trace, and Collective-Field Organization for Bulk Criteria fixes large-$N$ counting.

Helpful background. Large-N CFT Data and Vector Models supplies the spin decomposition; CFT-to-Bulk Reconstruction: Uniqueness and Ambiguities explains why collective data need not choose unique local bulk coordinates.

The collective change of variables

For a regulated Euclidean model

$$S[\phi]=\frac12\int d^dx\,\bigl[(\partial\phi^i)^2+m^2\phi^i\phi^i\bigr] +\frac{\lambda}{4N}\int d^dx\,(\phi^i\phi^i)^2,$$

define the symmetric kernel $\Psi(x,y)$. On a lattice of $K$ sites it is a $K\times K$ Gram matrix of $N$ vectors. Changing variables from $\phi$ to the singlet orbit and $\Psi$ produces a Jacobian whose leading term is $\log J\propto N\operatorname{Tr}\log\Psi$. The regulated collective action therefore has the schematic exact form

$$S_{\mathrm{col}}[\Psi] =N\,S_0[\Psi]-\frac N2\operatorname{Tr}\log\Psi+S_{\mathrm{measure}}[\Psi],$$

where $S_0$ contains the kinetic and interaction kernels and subleading measure terms depend on regulator and orbit volume. Collective-field methods derive this Jacobian rather than discarding it Jevicki and Sakita 1980, §§ 2–3.

At $N=\infty$, stationarity gives the gap equation. In the free critical massless theory, translation invariance yields $\Psi_0(p)=1/p^2$ up to normalization. Set

$$\Psi=\Psi_0+N^{-1/2}\eta.$$

Expanding $\operatorname{Tr}\log(\Psi_0+N^{-1/2}\eta)$ gives

$$S_{\mathrm{col}}=N S[\Psi_0] +\frac14\operatorname{Tr}(\Psi_0^{-1}\eta\Psi_0^{-1}\eta) +N^{-1/2}S_3[\eta]+\cdots.$$

Thus collective propagators are products of two vector propagators and $n$-point collective vertices scale as $N^{1-n/2}$, the same counting expected of tree and loop higher-spin interactions.

First application: decompose the bilocal fluctuation

Introduce center and relative coordinates $X=(x+y)/2$, $r=x-y$. Expanding $\eta(X,r)$ in traceless harmonics of the direction $\hat r$ gives coefficients with spins $s=0,2,4,\ldots$ for the real $O(N)$ singlet. Near $r=0$ their leading local moments are precisely

$$\eta(X,r)\sim\sum_{s\ \mathrm{even}}r^{\mu_1}\cdots r^{\mu_s}J_{\mu_1\cdots\mu_s}(X)+\text{descendants}.$$

A canonical transformation can map bilocal phase-space variables to higher-spin fields in AdS and reproduce their quadratic dynamics Das and Jevicki 2003, §§ III–V. What is exact is the singlet reorganization and its $1/N$ expansion. The choice of emergent radial coordinate, gauge, and local field basis is not unique; different transforms can encode the same boundary bilocal.

Finite-N and sector limits

On the finite lattice, $\Psi$ is positive semidefinite and

$$\operatorname{rank}\Psi\le N.$$

When $K>N$, its matrix entries cannot be independent. Treating every bilocal mode as an unconstrained oscillator overcounts finite-$N$ states. In the continuum the analogous relations require a regulator before they are meaningful. Moreover, $\Psi$ is blind to nonsinglet operators carrying an $O(N)$ index. Gauging or projecting the global symmetry and specifying topological sectors are separate steps.

There is no derivative gap that turns the bilocal into a finite local EFT: all current spins remain light. Large $N$ controls collective loops, not $\alpha'$ corrections, Kaluza–Klein modes, or a string coupling unless an independent top-down model supplies those identifications.

Adversarial control: exceed the Gram rank

Regulate at $K>N$, propose a small fluctuation of $\Psi$ with rank $N+1$, and test whether it can be written as $N^{-1}\phi\phi^{T}$. It cannot. The unconstrained saddle expansion has left the exact finite-$N$ image. Next insert a nonsinglet operator $\phi^i$; no functional of $\Psi$ reconstructs its index. These failures locate the singlet, large-$N$ domain rather than refuting the collective description inside it.

The evidence ceiling is an exact regulated change of singlet variables and a powerful large-$N$ collective perturbation theory. It does not establish a unique local spacetime, Vasiliev’s nonlinear functional class, or a complete finite-$N$ bulk Hilbert space. Those questions hand off to reconstruction and interaction analyses.

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

• Das, S. R., and Jevicki, A. (2003). “Large-N Collective Fields and Holography.” Physical Review D 68, 044011. DOI.
• Jevicki, A., and Sakita, B. (1980). “The Quantum Collective Field Method and Its Application to the Planar Limit.” Nuclear Physics B 165, 511–527. DOI.