Vasiliev Higher-Spin Equations, Interactions, and Locality Obstructions

Vasiliev systems package infinitely many higher-spin fields into master fields on spacetime augmented by noncommuting auxiliary spinors. Their unfolded equations are gauge-consistent and reproduce free Fronsdal fields around AdS, but extracting spacetime interaction vertices requires eliminating auxiliary directions and choosing field variables. The resulting infinite derivative tails make ordinary EFT locality a substantive unresolved condition rather than an automatic consequence of formal consistency.

Required background. Higher-Spin and Vector-Model Dualities fixes the proposed spectrum and large-$N$ map.

Helpful background. Weakly Broken Higher-Spin Symmetry supplies the boundary constraints; Local Field Redefinitions and the Equivalence Theorem explains which redefinitions preserve observables; Loop Witten Diagrams and Bulk EFT Renormalization gives the contrast with a derivative-truncated bulk EFT.

Master fields and the AdS vacuum

In four dimensions one introduces a spacetime one-form $W(x;Y,Z)$, auxiliary one-forms $S_A(x;Y,Z)$, and a zero-form $B(x;Y,Z)$. Functions of oscillator variables multiply with a Moyal-type star product; schematically,

$$(f\star g)(Y)=f(Y)\exp\!\left[i\overleftarrow\partial_A C^{AB}\overrightarrow\partial_B\right]g(Y).$$

The equations impose a flat master connection together with covariant constancy and a deformed oscillator constraint,

$$dW+W\star W=0, \qquad dB+W\star B-B\star\pi(W)=0, \qquad S\star S=i\,dZ^A dZ_A+\mathcal V(B).$$

The automorphism $\pi$ and interaction function $\mathcal V$ encode model and parity data. Gauge transformations act as $\delta W=d\epsilon+[W,\epsilon]_\star$ with corresponding twisted-adjoint action on $B$. These compact equations are not a local spacetime Lagrangian.

Choose an AdS vacuum $W_0$ satisfying $dW_0+W_0\star W_0=0$, $B_0=0$. Expanding $W=W_0+w$ and $B=C+\cdots$, the oscillator components of $w$ yield one gauge potential for every allowed spin, while $C$ contains gauge-invariant generalized Weyl tensors and the scalar. Eliminating auxiliary components recovers the Fronsdal equations. Vasiliev’s construction establishes nonlinear unfolded consistency Vasiliev 1990, pp. 378–382; it does not by itself select an admissible functional space, boundary condition, quantum measure, or nonperturbative completion.

First application: the first nonlinear source

Write a perturbative expansion in the canonically normalized bulk coupling $g_{\mathrm{HS}}\sim N^{-1/2}$,

$$W=W_0+g_{\mathrm{HS}}w^{(1)}+g_{\mathrm{HS}}^2w^{(2)}+\cdots, \qquad B=g_{\mathrm{HS}}C^{(1)}+g_{\mathrm{HS}}^2C^{(2)}+\cdots.$$

At first order the equations are linear and give the scalar plus Fronsdal tower. At second order, terms such as $w^{(1)}\star C^{(1)}$ source $C^{(2)}$. Solving the $Z$ dependence and projecting onto spacetime fields generates a current source for spin $s$,

$$\mathcal F_s[\varphi_s] =g_{\mathrm{HS}}\sum_{n=0}^{\infty}a_{s,n}L^{2n}\nabla^{2n}J_s(C,C),$$

where $\mathcal F_s$ is the Fronsdal operator. The infinite series is pseudo-local: AdS supplies powers of $L$, but there is no heavy scale $M\gg L^{-1}$ making successive derivatives uniformly small. A local cubic vertex can sometimes be isolated after improvements, yet the allowed tail depends on the functional class of field redefinitions.

Boundary three-point functions give a better invariant than any one coefficient $a_{s,n}$. Holographic reconstruction from vector-model correlators fixes cubic couplings modulo on-shell trivial and contact terms Sleight and Taronna 2017, §§ 3–5. At quartic order and beyond, exchanges, contact terms, and pseudo-local improvements mix, so a local derivative expansion has not been established.

Locality criteria

An admissible field redefinition must preserve boundary conditions, asymptotic charges, and separated-point correlators, and its derivative tail must lie in a specified summability class. Calling every formal series removable would also remove physical vertices; forbidding every infinite series would reject the unfolded variables themselves. A useful locality claim must therefore state the field basis, convergence or asymptotic rule, and invariant observable.

No $\alpha'/L^2\ll1$ hierarchy exists here: the higher-spin tower is massless. The curvature is of order $L^{-2}$ and can be weak in Planck units when $N\gg1$, but Planck suppression does not create a higher-spin gap. Any Kaluza–Klein or string truncation requires a separate embedding.

Adversarial control: a pseudo-local redefinition

Apply

$$\varphi_s\mapsto\varphi_s+g_{\mathrm{HS}}\sum_{n\ge0}b_{s,n}L^{2n}\nabla^{2n}J_s$$

and recompute both the apparent primary cubic coefficient and the renormalized separated-point three-point function. If the coefficient changes while the correlator does not, that coefficient is basis dependent and cannot diagnose locality. If the correlator changes, the redefinition was not admissible under the stated boundary conditions. This test forces the functional-class assumption into the open.

The evidence ceiling is nonlinear classical unfolded consistency plus many successful boundary correlator matches. A conventional local action, a controlled low-spin truncation, loop completeness, and a unique nonperturbative theory are not established. Those claims hand off respectively to locality analysis, boundary correlator tests, and definition criteria.

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

• Sleight, C., and Taronna, M. (2017). “Higher-Spin Interactions from Conformal Field Theory: The Complete Cubic Couplings.” Journal of High Energy Physics 2017(2), 095. DOI.
• Vasiliev, M. A. (1990). “Consistent Equation for Interacting Gauge Fields of All Spins in 3+1 Dimensions.” Physics Letters B 243, 378–382. DOI.