Higher-Spin Symmetry Breaking and Einstein-Regime Limits

Weakly broken higher-spin symmetry gives small anomalous dimensions to formerly conserved currents and small AdS masses to the dual gauge fields. That deformation does not approach Einstein gravity: an infinite tower with $m_sL\ll1$ remains dynamical. An Einstein-like local bulk instead requires a parametrically large gap above spin two, controlled interactions, and the decoupling of every additional light string or Kaluza–Klein sector.

Required background. Vasiliev Higher-Spin Equations, Interactions, and Locality Obstructions supplies the unbroken bulk tower; Weakly Broken Higher-Spin Symmetry supplies the boundary nonconservation equations.

Helpful background. Higher-Spin Gaps and Einstein-Regime Obstructions states the low-spin criterion; Tensionless String Limits and Higher-Spin Enhancement gives the opposite, gap-closing limit.

From nonconservation to a bulk mass

An exactly conserved spin-$s$ current in a $d$-dimensional CFT has $\Delta_s=s+d-2$. Weak breaking takes

$$\partial\!\cdot J_s =\frac{1}{\sqrt N}\mathcal K_{s-1}, \qquad \Delta_s=s+d-2+\gamma_s, \qquad \gamma_s=O(N^{-1}).$$

The norm of the descendant $\partial\!\cdot J_s$ is proportional to $\gamma_s$, so its two-point function determines the leading anomalous dimension from the nonconservation operator. In AdS$_{d+1}$ the symmetric-spin mass–dimension relation is

$$m_s^2L^2=(\Delta_s+s-2)(\Delta_s-s-d+2) =\gamma_s(2s+d-4+\gamma_s).$$

Thus $m_s^2L^2\simeq(2s+d-4)\gamma_s$. The current and bulk field combine with the operator represented by $\mathcal K_{s-1}$ in an AdS Higgs mechanism. This is a quantitative translation of symmetry breaking, not a claim that the field has decoupled.

First application: compare the induced masses with a gap

For the critical $O(N)$ or Chern–Simons vector models, use a leading nonconservation equation to obtain $\gamma_s\sim c_s/N$. Then

$$m_sL\sim N^{-1/2}$$

for every fixed spin with nonzero $c_s$. At large $N$ the tower becomes lighter, whereas a low-spin EFT would require

$$\Delta_{s>2}\ge\Delta_{\mathrm{gap}}\gg1 \quad\Longleftrightarrow\quad m_{s>2}L\gg1.$$

The two regimes are parametrically opposite. Slightly broken higher-spin Ward identities can determine planar correlators with remarkable power Maldacena and Zhiboedov 2013, §§ 2–5, but they do not supply a trajectory to a sparse CFT.

Parity phases and double-trace boundary conditions change interactions and the scalar sector without lifting the entire tower to the cutoff. Increasing $1/N$ until $γ_s=O(1)$ removes perturbative control before producing a parametrically large gap. A string embedding could in principle reorganize the tower as tension grows, but then all additional string and Kaluza–Klein states and the order of the coupling limits must be followed.

Causality and truncation

Finite higher-derivative corrections from an isolated high-spin exchange can violate high-energy causality unless new higher-spin states enter at the same scale. In weakly coupled gravity this argument ties appreciable higher-curvature interactions to an infinite higher-spin tower Camanho et al. 2016, §§ 3–5. Removing all but finitely many light Vasiliev fields therefore loses the mechanism that can restore consistency. It is not a legitimate low-energy truncation merely because one computes only low-spin external legs.

An Einstein regime also requires $G_N/L^{d-1}\ll1$, curvature small in string units $\alpha'/L^2\ll1$, and a Kaluza–Klein gap above $L^{-1}$. Vector-model $1/N$ controls the first condition but supplies neither of the latter two.

Adversarial control: keep infinitely many light spins

Assume $\gamma_s=c_s/N$ for all fixed $s$ and integrate out every $s>2$ field. The putative Wilson coefficients contain inverse powers of $m_s^2\sim1/(NL^2)$ and do not form a suppressed local derivative expansion. Test a Regge or time-delay observable: sensitivity to arbitrarily high spins remains. A claim of an Einstein limit fails unless a separate deformation makes the whole tower parametrically heavy and preserves consistency during the transition.

The evidence ceiling is a controlled map from weak current nonconservation to small bulk masses and a clear obstruction to a finite low-spin truncation. No universal deformation from a vector-model/Vasiliev point to pure Einstein gravity is known. A genuine route must specify the full spectrum, interactions, $N$, $g_s$, $\alpha'$, curvature, Kaluza–Klein hierarchy, and nonuniform limits; it hands off to a concrete top-down construction rather than to symmetry breaking alone.

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

• Camanho, X. O., Edelstein, J. D., Maldacena, J., and Zhiboedov, A. (2016). “Causality Constraints on Corrections to the Graviton Three-Point Coupling.” Journal of High Energy Physics 2016(2), 020. 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.