Weakly Broken Higher-Spin Symmetry

A symmetric-traceless spin-$s$ primary at the unitarity bound is conserved; moving slightly above the bound turns its divergence from a null state into a small-norm operator. This recombination relates the current’s anomalous dimension to its nonconservation equation and converts approximate higher-spin charges into quantitative constraints on correlators. Exact and weakly broken symmetry have sharply different conclusions and must be kept separate.

Required background. Unitarity bounds and null states supplies shortening and recombination. Large-N CFT data supplies factorization and normalization. Helpful background. Subleading, double-scaling, and nonuniform limits explains when fixed-spin $1/N$ counting ceases to be uniform.

Conservation and recombination

Let $J_{\mu_1\cdots\mu_s}$ be a Hermitian symmetric-traceless primary in a unitary CFT with $d>2$ and $s\ge1$. Normalize its two-point function by

$$\langle J_s(x,z)J_s(0,z')\rangle =C_{J_s}\frac{\bigl(z\cdot I(x)\cdot z'\bigr)^s}{(x^2)^{\Delta_s}}, \qquad z^2=z'^2=0.$$

The unitarity bound is $\Delta_s\ge d-2+s$. At equality,

$$\partial^{\mu_1}J_{\mu_1\cdots\mu_s}=0$$

as an operator equation away from contact terms. For a nearly conserved current write

$$\Delta_s=d-2+s+\gamma_s, \qquad \partial\cdot J_s=g_s K_{s-1},$$

where $K_{s-1}$ is a spin-$(s-1)$ operator with the descendant dimension $d+s-1+\gamma_s$. Choose its two-point normalization $C_K$ in the same polarization convention. Differentiating the exact $J_s$ two-point function at both points gives

$$\langle\partial\cdot J_s(x,z)\,\partial\cdot J_s(0,z')\rangle =A_{s,d}\,\gamma_s C_{J_s} \frac{\bigl(z\cdot I(x)\cdot z'\bigr)^{s-1}}{(x^2)^{d+s-1}} +O(\gamma_s^2),$$

where $A_{s,d}>0$ is defined by this displayed two-point and divergence convention. Applying the nonconservation equation to the same correlator therefore yields the normalization-independent leading relation

$$\gamma_s=\frac{g_s^2 C_K}{A_{s,d}C_{J_s}}+O(g_s^4).$$

This formula is the simplest extraction of a higher-spin anomalous dimension. It also makes positivity visible: in a unitary theory with Hermitian $K$, the leading $\gamma_s$ is nonnegative. A different normalization of $K$ changes $g_s$ and $C_K$ inversely, leaving $g_s^2C_K$ fixed.

For unit-normalized operators in a vector large-$N$ theory, a double-trace $K$ commonly gives $g_s=O(N^{-1/2})$ and hence $\gamma_s=O(N^{-1})$. In the unnormalized convention where single-trace two-point functions are $O(N)$, the coefficient of a double-trace divergence can instead appear as $O(1/N)$. Quoting the coefficient without the operator norms obscures this equivalence.

The stress tensor is special: exact translations require $g_2=0$ and $\gamma_2=0$. Currents for exact internal symmetries likewise remain conserved. The weak breaking applies to the remaining tower, not to every spin indiscriminately.

Pseudocharge identities

For an exactly conserved current, integrating its flux over a sphere defines a charge whose Ward identity is a sum of local symmetry actions. With weak breaking, remove small balls around insertions and integrate the divergence over the remaining region. With outward orientations chosen consistently,

$$\sum_i \left\langle \mathcal O_1\cdots[Q_s,\mathcal O_i]\cdots\mathcal O_n \right\rangle =g_s\int d^dx\, \langle K_{s-1}(x)\mathcal O_1\cdots\mathcal O_n\rangle +\text{contact terms}.$$

This is not an ordinary conservation law. The right-hand side is physical and, at large $N$, often factorizes into lower-point functions. Applied order by order, it restricts allowed three-point structures and relates their coefficients. Contact terms and coincident-point subtractions must be fixed in the same scheme on both sides; they cannot be dropped merely because the separated-point divergence is small.

In a specific three-dimensional large-$N$ setting with a unique stress tensor, one scalar single-trace primary, an approximately conserved even-spin tower, and no additional single-trace operator allowed in the divergence, the constraints reduce current three-point functions to one-parameter families. In the quasi-fermion convention of Maldacena and Zhiboedov,

$$\langle J_{s_1}J_{s_2}J_{s_3}\rangle =\widetilde N\left[ \frac{\widetilde\lambda^2}{1+\widetilde\lambda^2} \langle JJJ\rangle_{\mathrm{bos}} +\frac{1}{1+\widetilde\lambda^2} \langle JJJ\rangle_{\mathrm{fer}} +\frac{\widetilde\lambda}{1+\widetilde\lambda^2} \langle JJJ\rangle_{\mathrm{odd}} \right].$$

Here the even structures are normalized to one real free boson or one Majorana fermion, the odd structure has the paper’s parity convention, and $\widetilde N$ fixes the current two-point normalization. The quasi-boson parameterization interchanges the free-boson and free-fermion endpoints. The derivation and its restrictive spectral assumptions are given in Maldacena and Zhiboedov 2012, §§2–5 and Appendix C. This formula is not a classification of arbitrary large-$N$ CFTs with a nearly conserved current.

Exact symmetry is a different statement

In a unitary three-dimensional CFT satisfying assumptions including a unique stress tensor, one exactly conserved higher-spin current with $s>2$ generates an infinite tower and fixes stress-tensor and conserved-current correlators to free-boson or free-fermion forms Maldacena and Zhiboedov 2013, Abstract and §§1–6. The theorem uses exact nullness and exact charge identities. It cannot be applied at finite $g_s$, however small.

Weak breaking instead allows interacting families. Its conclusions are perturbative in the nonconservation and conditional on the spectrum closing under the pseudocharge identities. An omitted single-trace operator of the correct spin and twist can enter $K_{s-1}$, change the $N$ scaling, and enlarge the allowed three-point data.

The structured comparison below states the scope of the higher-spin control used here.

Target	Dimension or sector	Parameter	Observable	Computed order	Error estimate	Independent check	Evidence basis	Known failure
Exact higher-spin tower	unitary $d=3$ CFT under the theorem’s spectrum and stress-tensor hypotheses	exact conservation	$\partial\cdot J_s=0$ and $\gamma_s=0$	exact within the hypotheses	no breaking remainder; theorem scope remains conditional	all higher-spin charge identities	exact null descendants and Ward identities	inferring a Lagrangian or global equivalence beyond the constrained correlators
Stress tensor inside a weakly broken family	any finite $N$ with exact translations	zero breaking for $s=2$	$\partial^\mu T_{\mu\nu}=0$ and $\gamma_2=0$	exact up to distributional contacts	contact-term convention only	translation Ward identity	exact spacetime symmetry	including $T$ among weakly broken spins
Vector-model current with $s>2$	fixed spin at large $N$	$g_s=O(N^{-1/2})$ in unit normalization	$\gamma_s=g_s^2C_K/(A_{s,d}C_{J_s})+\cdots$	leading nontrivial order, usually $1/N$	higher breaking orders and operator mixing	descendant two-point norm	multiplet recombination plus factorization	uniformity when $s$ scales with $N$
Quasi-fermion three-point data	$d=3$, unique stress tensor, one leading scalar, declared single-trace spectrum	$1/N$ breaking and $\widetilde\lambda$	even-boson, even-fermion, and parity-odd coefficients	leading large $N$	subleading factorization and omitted-spectrum effects	pseudocharge closure on unused correlators	integrated nonconservation identities	applying the one-parameter result to extra single-trace matter
Quasi-boson three-point data	$d=3$ with a leading scalar near dimension one and the analogous spectrum	$1/N$ breaking and family parameter	constrained even and odd structures	leading large $N$	scalar-sector contact terms and higher orders	parity convention and scalar pseudocharge identities	integrated nonconservation identities	treating quasi-boson and quasi-fermion parameter conventions as identical
Nonunitary continuation	declared nonunitary operator space	small divergence, without positivity	descendant bilinear form and possibly complex $\gamma_s$	order stated by the continuation	norm-sign and contour dependence	direct two-point matrix calculation	conformal covariance without reflection positivity	using unitary norm positivity or exact-symmetry conclusions

Failure tests and handoff

Norm test. Differentiate the normalized two-point function and compare it with the proposed $K$ two-point function. The power of $x$, spin projector, sign, and $N$ scaling must agree.

Mixing test. List every spin-$(s-1)$ operator with the descendant dimension at the working order. If several exist, the nonconservation equation is vector-valued and the norm relation uses their full two-point matrix.

Closure test. Apply the integrated identity to a correlator not used to determine the coefficients. An unmatched structure signals missing operators, contact terms, or an inconsistent parity assignment.

Order test. Keep the divergence, anomalous dimensions, and factorized correlators to compatible orders. Squaring an $O(N^{-1/2})$ nonconservation coefficient while retaining only leading norms gives $O(1/N)$, not an all-orders result.

A bounded calculation can be used for checking the descendant-norm relation and truncation scaling once implemented. No numerical or symbolic result here is attributed to an execution of that calculation.

Exercises

Assume unit-normalized $J_s$ and $K_{s-1}$ with $g_s=aN^{-1/2}+O(N^{-3/2})$. Determine the leading scaling and coefficient of $\gamma_s$.

Solution

$C_{J_s}=C_K=1$ in this convention, so $\gamma_s=a^2/(A_{s,d}N)+O(N^{-2})$, allowing also subleading corrections to the two-point normalizations and nonconservation equation.

Why can an exactly conserved stress tensor coexist with weakly broken $J_s$ for $s>2$?

Solution

Stress-tensor conservation follows from exact translation invariance, whereas the higher-spin transformations are additional symmetries that interactions may break. Their multiplets can recombine without breaking the translation Ward identity.

References

• Maldacena, J., and Zhiboedov, A. (2012), “Constraining conformal field theories with a slightly broken higher spin symmetry,” Classical and Quantum Gravity 30, 104003. doi:10.1088/0264-9381/30/10/104003. Open PDF
• Maldacena, J., and Zhiboedov, A. (2013), “Constraining conformal field theories with a higher spin symmetry,” Journal of Physics A: Mathematical and Theoretical 46, 214011. doi:10.1088/1751-8113/46/21/214011. Open PDF