Higher-Spin Correlator Tests and Ward Identities

Correlators test a higher-spin dictionary only when operator normalization, tensor structures, scalar quantization, parity, contact terms, and perturbative order are fixed. A two-point match sets the bulk coupling; three- and four-point data can then be predictions. Agreement in a coefficient removable by a pseudo-local field redefinition or local counterterm is not independent evidence.

Required background. Higher-Spin and Vector-Model Dualities fixes the operator map; Ward Identities, Weyl Anomalies, and Contact Terms fixes renormalized conservation laws.

Helpful background. Large-N Crossing, Double-Trace Data, and Contact Ambiguities treats four-point nonuniqueness; Vasiliev Higher-Spin Equations, Interactions, and Locality Obstructions explains the field-basis problem.

Normalization before interaction data

For a spin-$s$ current encoded with a null polarization $z$, conformal symmetry fixes

$$\langle J_s(x_1,z_1)J_s(x_2,z_2)\rangle =C_s\frac{H_{12}^{s}}{(x_{12}^2)^{\Delta_s}}, \qquad H_{12}=z_1\!\cdot z_2-2\frac{(z_1\!\cdot x_{12})(z_2\!\cdot x_{12})}{x_{12}^2}.$$

For the free vector model $C_s\propto N$. Rescaling $J_s$ rescales every three-point coefficient, so a bulk cubic coupling is meaningful only after the bulk kinetic term and $C_s$ are fixed. Current conservation imposes differential constraints at separated points and contact terms at coincident insertions,

$$\partial\!\cdot\langle J_s(x,z)\,\mathcal O_1(x_1)\cdots\rangle =\sum_i\delta^{(3)}(x-x_i)\,\langle\delta_s\mathcal O_i\cdots\rangle.$$

The right-hand side fixes charge normalization. It cannot be discarded when comparing integrated Ward identities.

First application: scalar–current–current data

Take a scalar singlet $J_0$ and two identical conserved currents $J_s$. Conformal invariance and conservation reduce the separated-point result to a finite basis,

$$\langle J_0J_sJ_s\rangle =C^{\mathrm{even}}_{0ss}\,\mathcal T^{\mathrm{even}}_{0ss} +C^{\mathrm{odd}}_{0ss}\,\mathcal T^{\mathrm{odd}}_{0ss},$$

where the parity-odd structure is absent in parity-invariant type-A theory. A bulk vertex of the schematic form

$$S_3\supset g_{0ss}\int_{\mathrm{AdS}_4}\!\sqrt g\, \varphi_0\,\varphi_s\,\nabla^{s}\varphi_s+\text{gauge completion}$$

produces a Witten integral. After dividing external legs by the square roots of their two-point residues,

$$\widehat C_{0ss}^{\mathrm{bulk}} =g_{0ss}\frac{\mathcal K(\Delta_0,s)}{\sqrt{Z_0Z_s^2}},$$

with the kinematic factor $\mathcal K$ fixed by the AdS integral. The vector model gives the same normalized quantity from Wick contractions in the free theory. Fixing one overall bulk coupling at $s=2$ leaves the spin dependence as a nontrivial test; the Vasiliev result reproduces the free-boson family Giombi and Yin 2010, §§ 4–6.

Changing the scalar from $\Delta=1$ to $\Delta=2$ requires the alternate-quantization transform, not a relabeling of the external line. In parity-violating Chern–Simons matter, $C^{\mathrm{odd}}_{0ss}$ fixes the bulk phase while the even structures provide further tests. At finite $N$, nonconservation terms and anomalous dimensions must enter the Ward identity at the same order as bulk loops.

Four-point tests and their limits

A connected singlet four-point function is $O(1/N)$ and decomposes into higher-spin exchanges plus contact solutions. Crossing and higher-spin Ward identities strongly constrain it, but a contact Witten diagram can shift polynomial or coincident data without changing selected exchange poles. Conversely, an infinite pseudo-local field redefinition may reshuffle exchange and contact representations. The invariant comparison is the full renormalized boundary correlator at separated points, including all channels and the declared quantization.

Tree matching probes classical bulk interactions. One-loop comparison requires the spectrum, ghosts, measure, boundary conditions, and counterterms; a single determinant is not a complete quantum test. There is still no higher-spin gap or controlled finite derivative truncation.

Adversarial control: move a contact term

Add an allowed local boundary counterterm, then repeat the tensor projection. Separated-point $C_{0ss}$ must be unchanged while coincident contact terms shift. Next apply a pseudo-local bulk improvement that changes the displayed $g_{0ss}$; if the complete separated-point correlator remains fixed, the bare coefficient was basis dependent. If a claimed match disappears under either admissible operation, it was not independent evidence.

The evidence ceiling is strong tree-level agreement for many normalized two- and three-point functions, together with higher-spin Ward-identity control and selected loop checks. It is not a proof of a unique bulk vertex basis, quartic locality, or nonperturbative duality. Generic Witten-diagram renormalization and CFT crossing retain their separate domains.

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

• Giombi, S., and Yin, X. (2010). “Higher Spin Gauge Theory and Holography: The Three-Point Functions.” Journal of High Energy Physics 2010(9), 115. 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.