Free and Critical Vector Models at the Higher-Spin Dictionary Interface

The free and critical $O(N)$ models share the same leading large-$N$ tower of even-spin conserved currents, yet their scalar singlets, generating functionals, and $1/N$ interactions differ. They supply the canonical alternate and standard boundary conditions for the $m^2L^2=-2$ scalar in type-A higher-spin theory. Equality of the leading spin list is therefore only the beginning of the dictionary.

Required background. Vector-Model Bilocals as Higher-Spin Bulk Data gives the singlet organization; Ultraviolet and Infrared Fixed Points: Criteria and Evidence fixes what it means to compare the two fixed points.

Helpful background. Weakly Broken Higher-Spin Symmetry treats current nonconservation; Completeness and the Operator Basis keeps the singlet spectrum distinct from a complete CFT Hilbert space.

Two fixed points and two scalar operators

The free theory has

$$S_{\mathrm{free}}=\frac12\int d^3x\,(\partial\phi^i)^2, \qquad \Delta_\phi=\frac12.$$

Its lowest singlet scalar $J_0=\phi^i\phi^i/\sqrt N$ has $\Delta=1$, and its even-spin bilinears $J_s$ have $\Delta_s=s+1$. The quartic deformation $(\phi^i\phi^i)^2$ drives the theory to the critical Wilson–Fisher fixed point. Introducing an auxiliary field $\sigma$ gives

$$S=\int d^3x\left[\frac12(\partial\phi^i)^2 +\frac{1}{2\sqrt N}\sigma\phi^i\phi^i-\frac{1}{4g}\sigma^2\right].$$

At the critical point the scalar primary represented by $\sigma$ has $\Delta_\sigma=2+O(1/N)$. The equation of motion removes $\phi^i\phi^i$ as an independent primary. The even currents remain conserved at $N=\infty$, but acquire anomalous dimensions and weak nonconservation at finite $N$.

The AdS scalar satisfies

$$m^2L^2=\Delta(\Delta-3)=-2,$$

so both $\Delta_-=1$ and $\Delta_+=2$ are allowed. The free singlet sector maps to $\Delta_-$ boundary conditions and the critical singlet sector to $\Delta_+$. This is the concrete content of the free/critical boundary-condition relation proposed by Klebanov and Polyakov Klebanov and Polyakov 2002, pp. 214–218.

First application: dimensions and the current tower

At leading $N=\infty$,

$$\begin{array}{c|cc} &\text{free }O(N)&\text{critical }O(N)\\ \hline \text{singlet scalar}&\Delta=1&\Delta=2\\ \text{even current }J_s&\Delta=s+1&\Delta=s+1 \end{array}$$

while connected correlators of canonically normalized singlets scale as $N^{1-n/2}$. Thus both theories have a classical higher-spin bulk at leading large $N$, but different scalar boundary conditions and different cubic data involving the scalar. The critical theory is not a free boundary CFT merely because its currents are conserved at $N=\infty$.

At finite $N$, a schematic nonconservation equation reads

$$\partial\!\cdot J_s=\frac{1}{\sqrt N}\sum_{s_1,s_2}c_{s;s_1s_2}\,[J_{s_1}J_{s_2}]+O(N^{-1}),$$

which implies $\gamma_s=O(1/N)$. Slightly broken higher-spin symmetry then fixes broad families of planar correlators but does not restore exact current conservation Maldacena and Zhiboedov 2013, §§ 2–5. In the bulk, $\gamma_s$ corresponds to a small higher-spin mass in AdS units, with interactions still involving infinitely many spins.

Generating functionals are related, not identical

At large $N$, coupling a source to $J_0$ and introducing $\sigma$ relates the two scalar generating functionals by a Legendre transform, up to local counterterms. Consequently the scalar two-point kernel is inverted in its nonlocal part. Higher-point functions inherit contact terms and exchange contributions that must be transformed consistently. The relation does not identify the microscopic operator algebras or nonsinglet sectors.

Neither model supplies an $\alpha'/L^2$ expansion: the infinite current tower remains at dimensions of order one. $N\gg1$ suppresses bulk loops through $G_N/L^2\sim1/N$, but it does not justify a low-spin truncation, a Kaluza–Klein decoupling, or a nonperturbative completion.

Adversarial control: retain the first 1/N correction

Compute or import a nonzero $\gamma_s$ and the corresponding divergence two-point function. Any claim of exact higher-spin symmetry in the finite-$N$ critical theory then fails. Next compare a scalar three-point coefficient before and after the Legendre transform: it changes even though the leading spin spectrum does not. A spectrum-only claim cannot distinguish the fixed points.

The evidence ceiling is a precise leading and subleading large-$N$ relation between two boundary CFT singlet sectors and two scalar quantizations, supported by dimensions, Ward identities, and correlators. It is not a proof that one bulk functional integral exists nonperturbatively for both choices, nor that either admits an Einstein limit. The detailed RG interpolation hands off to the double-trace page.

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

• Klebanov, I. R., and Polyakov, A. M. (2002). “AdS Dual of the Critical O(N) Vector Model.” Physics Letters B 550, 213–219. DOI.
• Maldacena, J., and Zhiboedov, A. (2013). “Constraining Conformal Field Theories with a Slightly Broken Higher Spin Symmetry.” Classical and Quantum Gravity 30, 104003. DOI.