Spinning Fields, Forms, and Mixed-Symmetry Operators

A spinning AdS field maps not just to a dimension but to a complete boundary conformal representation: Lorentz Young symmetry, trace conditions, conservation or partial conservation, parity data, and gauge equivalences. For symmetric tensors and differential forms, compact mass–dimension formulas are useful. For mixed symmetry, the quadratic Casimir and gauge complex must be matched explicitly; counting raw tensor components gives the wrong operator content.

Required background. Bulk fields and boundary operators supplies the scalar indicial method. Spin and tensor representations supplies boundary Lorentz representations. Helpful background. Spinning correlators and tensor structures explains how polarization and conservation constraints appear in correlators.

First application: a massive vector and the current limit

Consider a Proca field on Lorentzian AdS$_{d+1}$,

$$S=-\frac{1}{4g_A^2}\int\sqrt{\lvert g\rvert} \left(F_{MN}F^{MN}-2m^2A_MA^M\right).$$

For $m\neq0$, the Proca equation implies the constraint $\nabla_MA^M=0$; in the massless limit, one may instead impose radial gauge and retain the Gauss constraint. Decompose the boundary components into transverse and longitudinal parts. A transverse mode $A_i\sim z^\delta a_i(x)$ obeys the indicial equation

$$\delta(\delta-d+2)=m^2L^2.$$

Writing the two roots as $\delta=d-\Delta-1$ and $\delta=\Delta-1$ gives

$$m^2L^2=(\Delta-1)(\Delta-d+1).$$

The slower coefficient is the vector source in standard quantization; the faster coefficient is the response. In the massless limit, the physical root is $\Delta=d-1$. Gauge invariance removes the longitudinal source redundancy, while the radial Gauss constraint becomes

$$\partial_i\langle J^i\rangle=0$$

in flat boundary space without an anomaly. Thus current conservation is not obtained by counting $d$ unconstrained boundary components; it is the shortening condition paired with bulk gauge redundancy.

Using a null auxiliary polarization $Z^i$ with $Z^2=0$ packages a symmetric traceless operator as $\mathcal O(x,Z)$. The same polarization convention must be used in bulk-to-boundary propagators and boundary correlators. Rescaling $Z$ or changing the projector normalization changes tensor coefficients, not the representation itself.

Symmetric tensors and differential forms

For a totally symmetric traceless spin-$s\geq1$ field in the conventional Fierz–Pauli mass parameter,

$$m^2L^2=(\Delta+s-2)(\Delta-s-d+2).$$

This convention and its curvature shifts are derived in Metsaev 2004, §§2–3. At $m^2=0$, the unitary root $\Delta=s+d-2$ is conserved. For $s=2$, bulk diffeomorphism invariance maps to stress-tensor conservation and $\Delta=d$; the corresponding shortening and unitarity restrictions are reviewed by Minwalla 1998, §§2–3. Special negative masses can produce partially massless gauge symmetries in AdS representations, but their boundary interpretation and unitarity depend on dimension and real form; they are not generic conserved currents.

For a massive $p$-form with boundary $p$-form components,

$$m^2L^2=(\Delta-p)(\Delta+p-d).$$

A massless $p$-form gauge field selects $\Delta=d-p$ for the standard conserved $(p-1)$-form-symmetry current, subject to the chosen electric or magnetic boundary condition. Hodge dualization may exchange boundary conditions and global sectors. It is therefore not enough to quote an equal number of local degrees of freedom.

Mixed Young symmetry needs Casimir matching

Let the boundary primary transform in a Lorentz representation with Young diagram $\mathbf Y$. The AdS field carries the corresponding $SO(d,2)$ lowest-weight module $[\Delta,\mathbf Y]$. Its wave operator is related to the difference between the $SO(d,2)$ quadratic Casimir and the Lorentz Casimir of $\mathbf Y$. Trace constraints, divergences, algebraic Bianchi identities, and gauge-for-gauge transformations depend on the entire diagram.

Consequently, there is no safe universal operation “replace $s$ by the number of boxes.” One must specify:

• row and column symmetries and all traces;
• the bulk mass convention, including curvature shifts;
• gauge parameters and reducibility;
• admissible boundary falloffs and quantization;
• the boundary shortening condition, if any.

Spinning AdS propagators implement these projectors explicitly, as developed by Costa, Gonçalves, and Penedones 2014, §§2–3.

Adversarial check: polarization overcounting

Suppose one treats every component of $A_M$ as a physical polarization. The result includes a radial mode and a longitudinal boundary mode even when $m=0$. It then predicts a generic dimension-$(d-1)$ vector primary rather than a conserved current and fails both the Gauss constraint and the current Ward identity. Removing only $A_z$ is still insufficient: residual gauge transformations and the constraint eliminate the longitudinal physical degree of freedom.

The strongest claim supported by the mass formula alone is the candidate weight of an unconstrained representation. A valid holographic map requires the constraint equations, gauge quotient, normalizability, and boundary unitarity condition to agree.

Controlled limits and handoff

The formulas classify free linearized fields on AdS and their boundary representations. Partial masslessness, mixed boundary conditions, interactions, and higher-spin consistency require additional analysis; a mass formula never replaces the constraint algebra. Currents, Stress Tensor, and Bulk Gauge and Metric Fields applies the massless shortening limit to conserved sources and Ward identities.

Exercise

Apply the symmetric-spin formula to a massless spin-two field and identify the boundary shortening condition.

Solution

For $s=2$ and $m^2=0$, the factors are $\Delta(\Delta-d)=0$. The physical standard root is $\Delta=d$. Bulk diffeomorphism invariance and the radial constraints map to a symmetric traceless operator satisfying $\partial_iT^{ij}=0$, namely the stress tensor. The $\Delta=0$ algebraic root is not the standard unitary stress-tensor representation.

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

• Costa, Miguel S., Vasco Gonçalves, and João Penedones. “Spinning AdS Propagators.” Journal of High Energy Physics 2014, 064 (2014). arXiv. DOI.
• Metsaev, R. R. “Massive Totally Symmetric Fields in AdS(d).” Physics Letters B 590 (2004): 95–104. arXiv. DOI.
• Minwalla, Shiraz. “Restrictions Imposed by Superconformal Invariance on Quantum Field Theories.” Advances in Theoretical and Mathematical Physics 2 (1998): 781–846. arXiv. DOI.