Spinning Fields, Forms, and Mixed-Symmetry Operators

A spinning AdS field maps to more than a scaling dimension. Its boundary operator also has a definite rotation or Lorentz representation, trace conditions, possible null descendants, and parity or duality data. In the bulk, those features are enforced by differential constraints and, at special masses, by gauge equivalences. This page derives the vector dictionary, fixes the auxiliary-polarization convention, organizes general bosonic tensors by their Young diagram, and then tests that organization on differential forms and mixed symmetry. Throughout, $d\geq3$, the bulk is AdS$_{d+1}$, falloffs refer to coordinate components in the inherited Poincaré coordinates, and the displayed operator dimensions use the ordinary unitary branch unless stated otherwise.

Required background. Bulk fields and boundary operators supplies the scalar indicial method and the flat-limit mass convention. Spin and tensor representations supplies boundary rotation representations. Helpful background. Spinning correlators and tensor structures explains how polarization and conservation constraints appear in correlators.

The transverse Proca equation

Begin with a massive vector $A_M$ and $F_{MN}=\partial_MA_N-\partial_NA_M$,

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

Varying $A_N$ gives

$$\nabla_MF^{MN}+m_A^2A^N=0.$$

Taking one more divergence and using the antisymmetry of $F^{MN}$ yields

$$m_A^2\nabla_MA^M=0.$$

For $m_A^2\neq0$, this is the Proca constraint: it follows from the equation of motion and is not a gauge choice. At fixed boundary momentum $k_i$, decompose the boundary coordinate component into a transverse part, $k^iA_i^{\mathrm T}=0$, and a longitudinal part. The transverse sector decouples from $A_z$ and obeys

$$\left[ -z^2\partial_z^2+(d-3)z\partial_z +z^2\Box_\eta+m_A^2L^2 \right]A_i^{\mathrm T}=0.$$

For $A_i^{\mathrm T}=z^\delta a_i^{\mathrm T}+\cdots$, the fixed-$k$ boundary-derivative term is subleading as $z\to0$, so

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

This equation is for a lower-index coordinate component. A tangent-frame component contains an extra factor of $z/L$, which shifts its displayed exponent by one; mixing the two conventions is a common source of apparent disagreement. The coordinate-component roots may be written as

$$\delta_{\mathrm{src}}=d-\Delta-1, \qquad \delta_{\mathrm{resp}}=\Delta-1,$$

and hence

$$m_A^2L^2=(\Delta-1)(\Delta-d+1), \qquad \nu_A=\sqrt{\frac{(d-2)^2}{4}+m_A^2L^2}, \qquad \Delta_\pm=\frac d2\pm\nu_A.$$

The one-unit shift relative to a scalar is not an arbitrary convention: a lower boundary index itself carries one power under a dilation. For noninteger $\nu_A>0$, the transverse expansion has the form

$$A_i^{\mathrm T} =z^{d-\Delta-1}\left(a_i^{\mathrm T}+z^2a_{(2)i}^{\mathrm T}+\cdots\right) +z^{\Delta-1}\left(b_i^{\mathrm T}+\cdots\right).$$

When $\nu_A$ is a positive integer, the even-power recursion can generate a source-local $z^{\Delta-1}\log(z\mu)$ term. At $\nu_A=0$, the two roots coalesce and the independent pair instead begins as $z^{(d-2)/2}(a_{\log i}\log(z\mu)+b_i+\cdots)$. Thus, on the standard branch, $a_i$ is the source while the nonlocal part of the renormalized canonical momentum is the response; the coefficient $b_i$ alone can be shifted by local counterterms at resonance.

The transverse projection isolates the indicial roots but does not force a generic massive-vector source to be transverse. If the full boundary coefficients are $a_i$ and $b_i$, the Proca constraint fixes the leading radial data, away from the exceptional denominators, as

$$A_z =-z^{d-\Delta}\frac{\partial\cdot a}{\Delta-1} +z^\Delta\frac{\partial\cdot b}{\Delta-d+1}+\cdots.$$

The radial component is constrained data, not something one may gauge away in the massive theory. These equations and the constraint follow directly from the coordinate Proca system in l’Yi 1999, §3, pp. 5–10, especially eqs. (20)–(21), (25)–(30), and (35)–(38), Open PDF.

The massive constraint does not imply that the dual operator is conserved. Real vector powers require only $m_A^2L^2\geq-(d-2)^2/4$, whereas ordinary vector unitarity requires $\Delta\geq d-1$ and therefore $m_A^2\geq0$. For $m_A^2>0$, the unitary root has $\Delta>d-1$ and describes a long vector primary. The radial and longitudinal coefficients cooperate to satisfy $\nabla_MA^M=0$ without producing a null boundary divergence descendant.

Maxwell gauge symmetry and the current Ward identity

At $m_A^2=0$, the two candidate dimensions are

$$\Delta_+=d-1, \qquad \Delta_-=1.$$

For $d>2$, $\Delta_+=d-1$ saturates the vector unitarity bound and is the conserved-current module. The $\Delta_-=1$ solution is the source or shadow branch, not a second ordinary unitary vector primary. Non-Dirichlet boundary conditions can instead produce a boundary gauge field or exchange electric and magnetic sectors; that is a different statement from declaring the $\Delta=1$ vector itself to be an ordinary local primary Marolf and Ross 2006, §§3–4, especially eqs. (3.17)–(3.18), (3.31)–(3.36), and (4.1)–(4.8), Open PDF.

In radial gauge $A_z=0$, the generic massless expansion begins

$$A_i(z,x)=A_{(0)i}(x)+\cdots +z^{d-2}A_{(d-2)i}(x)+\cdots,$$

with logarithmic modifications in resonant dimensions. The $N=z$ Maxwell equation is the radial Gauss constraint. If $\Pi^i$ denotes the renormalized radial electric momentum, it gives

$$\partial_i\Pi^i=0, \qquad \Pi^i_{\mathrm{ren}}\longleftrightarrow\langle J^i\rangle,$$

and therefore, on flat space without an anomaly or charged insertions,

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

A gauge parameter that vanishes at the boundary is a redundancy of a fixed-source bulk problem. A boundary value $\lambda_{(0)}(x)$ instead changes the source by $A_{(0)i}\mapsto A_{(0)i}+\partial_i\lambda_{(0)}$; invariance of the generating functional under that background-gauge transformation is the boundary Ward identity. The gauge choice, residual source equivalence, and Gauss constraint are three related but distinct statements.

The polarization count exposes the same mechanism. In $D=d+1$ bulk dimensions,

$$N_{\mathrm{Proca}}=D-1=d, \qquad N_{\mathrm{Maxwell}}=D-2=d-1.$$

The massive differential constraint removes one of the $D$ raw components. In the massless theory, a constraint and a gauge quotient remove two. On the boundary, a vector still has $d$ algebraic components at a point; conservation is a null differential descendant and leaves $d-1$ transverse components only at generic nonzero momentum. Merely setting $A_z=0$ leaves $d$ bulk components and therefore still overcounts Maxwell by one.

A fixed auxiliary-polarization convention

For a symmetric traceless boundary operator, this page uses a commuting auxiliary vector $Z^i$ in the complexified boundary tangent space and no factorial:

$$\mathcal O(x,Z) =\mathcal O_{i_1\cdots i_s}(x)Z^{i_1}\cdots Z^{i_s}, \qquad Z^2=0, \qquad \mathcal O(x,\alpha Z)=\alpha^s\mathcal O(x,Z).$$

Restricting the polynomial to $Z^2=0$ quotients trace terms. In Euclidean signature a nonzero null $Z$ is necessarily complex: it is an algebraic bookkeeping device, not a physical polarization state. The restriction does not impose conservation. When the same operator is lifted to embedding space, the convention is

$$P^2=Z^2=P\cdot Z=0, \qquad \mathcal O(\lambda P,\alpha Z) =\lambda^{-\Delta}\alpha^s\mathcal O(P,Z), \qquad Z\sim Z+\beta P.$$

The last equivalence enforces embedding-space transversality; it is not a claim that a massive bulk field has gauge symmetry. Bulk propagators, boundary correlators, and projectors must use the same normalization before their tensor coefficients can be compared.

For an antisymmetric $p$-tensor it is convenient to use one Grassmann polarization,

$$\mathcal O(x,\Theta) =\frac{1}{p!}\mathcal O_{i_1\cdots i_p}(x) \Theta^{i_1}\cdots\Theta^{i_p},$$

where $\Theta^i\Theta^j=-\Theta^j\Theta^i$. A general Young diagram needs one commuting family per row or, equivalently, one anticommuting family per column, followed by the appropriate Young and trace projections. Costa and Hansen develop both encodings and their conversion in Costa and Hansen 2015, §§2.1–2.5, pp. 3–11, Open PDF.

The top block controls the unitary threshold

After Euclidean continuation, denote the conformal module by $\mathcal D(\Delta;\mathbf Y)$ and write its boundary tensor representation as a canonical $SO(d)$ Young diagram

$$\mathbf Y=(s_1,s_2,\ldots), \qquad s_1=\cdots=s_r>s_{r+1},$$

so $r$ is the height of its top rectangular block. Define

$$\Delta_0=s_1+d-r-1.$$

For the ordinary bosonic tensor modules considered here, the flat-limit or Stueckelberg mass parameter $m_{\mathbf Y}^2$ is normalized to vanish at the unitary AdS massless point and obeys

$$m_{\mathbf Y}^2L^2 =\Delta(\Delta-d)-\Delta_0(\Delta_0-d) =(\Delta-\Delta_0)(\Delta+\Delta_0-d).$$

The ordinary CFT unitarity condition is

$$\Delta\geq\Delta_0.$$

At $\Delta=\Delta_0$, a Young-projected level-one divergence becomes null. The corresponding unitary AdS field is gauge invariant, and its gauge parameter has the Young shape obtained by removing one box from the last row of the top block. This mass formula is a representation parameter, not an instruction to read a bare coefficient from an arbitrary component Lagrangian. Metsaev derives the lowest-energy relation in Metsaev 2004, §5, pp. 8–9, especially eqs. (5.64), (5.66), (5.69), and (5.74), Open PDF; his $d$ is the bulk dimension $d+1$ used here, and his $E_0$ is $\Delta$. The mixed-tensor bound and projected conservation condition are developed in Costa and Hansen 2015, §4, pp. 29–31, especially eqs. (4.1), (4.3), and (4.9)–(4.10), Open PDF.

Several familiar formulas are now special cases:

• Symmetric traceless integer spin $s\geq1$. Here $\mathbf Y=(s)$, $r=1$, and $\Delta_0=s+d-2$, giving

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

  The root $\Delta=s+d-2$ is the conserved unitary module; the other root $2-s$ is the source or shadow weight. The detailed derivation and admissibility tests belong to the preceding page.

• Vector. Setting $s=1$ recovers $(m_AL)^2=(\Delta-1)(\Delta-d+1)$ and the massless current threshold $\Delta_0=d-1$.

• Antisymmetric $p$-tensor in the canonical range. Here $\mathbf Y=(1^p)$, $r=p$, and $\Delta_0=d-p$, giving $(m_pL)^2=(\Delta-p)(\Delta+p-d)$.

For a symmetric field, a transverse-traceless wave equation may instead use a curvature-shifted Laplacian parameter $\kappa_s^2$. With the Euclidean-continuation convention $D^2\varphi_{\mathrm{TT}}=\kappa_s^2\varphi_{\mathrm{TT}}$,

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

Thus $\kappa_1^2L^2=m_A^2L^2-d$ and $\kappa_2^2L^2=m_{\mathrm{FP}}^2L^2-2$. Maxwell and Fronsdal fields have $m_s^2=0$ even though their covariant tensor Laplacians contain curvature shifts. Symmetric-traceless propagator conventions are compared explicitly in Costa, Gonçalves, and Penedones 2014, §3, especially eqs. (22)–(27) and (34)–(37), Open PDF.

Differential forms and higher-form currents

For $1\leq p\leq d$, let $B_{M_1\cdots M_p}$ be a massive $p$-form with $H_{p+1}=\mathrm dB_p$ and action

$$S_p=-\frac{1}{2g_p^2}\int\mathrm d^{d+1}x\sqrt{\lvert g\rvert} \left( \frac{H_{M_0\cdots M_p}H^{M_0\cdots M_p}}{(p+1)!} -\frac{m_p^2B_{M_1\cdots M_p}B^{M_1\cdots M_p}}{p!} \right).$$

This defines the Maxwell-like mass parameter $m_p^2$, normalized so that $m_p^2=0$ restores $B_p\mapsto B_p+\mathrm d\Lambda_{p-1}$. Its equation of motion is

$$\nabla_{M_0}H^{M_0M_1\cdots M_p} +m_p^2B^{M_1\cdots M_p}=0.$$

For $m_p^2\neq0$, it implies the co-closure constraint $\nabla_{M_1}B^{M_1\cdots M_p}=0$. A boundary-transverse coordinate component satisfies

$$\left[ -z^2\partial_z^2+(d-1-2p)z\partial_z +z^2\Box_\eta+m_p^2L^2 \right]B^{\mathrm T}_{i_1\cdots i_p}=0.$$

The indicial ansatz $B^{\mathrm T}_{i_1\cdots i_p}\sim z^\delta b_{i_1\cdots i_p}$ gives

$$\delta(\delta-d+2p)=m_p^2L^2,$$

with coordinate-component powers

$$\delta_{\mathrm{src}}=d-\Delta-p, \qquad \delta_{\mathrm{resp}}=\Delta-p,$$

and therefore

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

The full massive equation, consistency condition, and boundary-component radial equation appear in l’Yi 1999, §2, pp. 3–4, eqs. (5)–(17), and §3, p. 8, eqs. (50)–(52), Open PDF.

For $1\leq p<d/2$ and the ordinary electric or Dirichlet boundary condition, $m_p^2=0$ gives

$$B_{i_1\cdots i_p} =B_{(0)i_1\cdots i_p}+\cdots +z^{d-2p}B_{(d-2p)i_1\cdots i_p}+\cdots,$$

and the unitary operator has

$$\Delta=d-p, \qquad \partial^{i_1}J_{i_1\cdots i_p}=0.$$

This conserved rank-$p$ current generates a $(p-1)$-form global symmetry; $B_{(0)}$ is its background $p$-form gauge field. The terminology and the roles of charged defects, currents, and background fields are fixed in Gaiotto, Kapustin, Seiberg, and Willett 2015, §3, pp. 11–12, Open PDF.

The domain restriction matters. At $p=d/2$, the two powers coalesce and a logarithmic solution replaces the naive pair, and the source can run Hofman and Iqbal 2018, §II.A, especially eqs. (2.8)–(2.17), Open PDF. Self-duality can split the middle-form representation, while electric-versus-magnetic boundary conditions select different boundary sectors. For $p>d/2$, first Hodge-dualize to the canonical $SO(d)$ Young label. Boundary conditions can exchange electric and magnetic sectors, and global flux sectors require more than the local mass formula.

Partial conservation below the unitary bound

For symmetric spin $s$, define a depth label $t=0,1,\ldots,s-1$ so that $t=0$ is the fully massless Fronsdal field. The discrete partially massless values are

$$m_{s,t}^2L^2=-t(2s+d-t-4), \qquad \Delta_{s,t}=s+d-t-2.$$

The bulk gauge transformation has $t+1$ derivatives at leading order and a rank-$(s-t-1)$ parameter. On the boundary, its analogue is a projected $(t+1)$-fold conservation condition. For $t>0$,

$$\Delta_{s,t}=(s+d-2)-t$$

lies below the ordinary symmetric-tensor unitarity bound. A partially massless AdS representation may therefore be algebraically consistent while its boundary module is nonunitary in an ordinary reflection-positive CFT. The spectrum and depth dictionary are tabulated in Gwak, Kim, and Rey 2016, appendix B, p. 60, eqs. (185)–(187) and table 4, Open PDF.

For example, $s=2$, $d=4$, and $t=1$ give

$$m_{2,1}^2L^2=-3, \qquad \Delta_{2,1}=3.$$

The scalar-parameter bulk gauge symmetry maps to a double-divergence shortening condition, but $\Delta=3$ lies below the spin-two bound $\Delta\geq4$. Partial conservation must not be mistaken for an ordinary unitary stress tensor.

Mixed Young symmetry and the AdS gauge complex

The quadratic conformal Casimir separates the dimension from the tensor label:

$$C_2[\Delta,\mathbf Y] =\Delta(\Delta-d)+C_2^{SO(d)}(\mathbf Y).$$

This eigenvalue is representation invariant. Converting it into a local wave operator requires a choice of tensor realization and curvature terms, which is why replacing $s$ by the number of boxes cannot produce a universal component equation.

The top-block rule also has a genuine dynamical consequence. A flat-space mixed-symmetry gauge field generally has one gauge parameter for each removable corner of its Young diagram. A unitary irreducible AdS massless field retains the gauge symmetry associated with the top block. In the flat limit, that AdS module generally decomposes into several massless Poincaré fields rather than one. Brink, Metsaev, and Vasiliev derive the general mismatch in Brink, Metsaev, and Vasiliev 2000, §§2–3, pp. 3–13, especially eqs. (30)–(31) and (35)–(42), Open PDF. Their explicit three-cell-hook analysis appears in §4, pp. 14–19, especially eqs. (44)–(49), (70)–(75), and (77)–(81), Open PDF.

For the hook $\mathbf Y=(2,1)$, the top block has $r=1$, so

$$\Delta_0=d, \qquad m_{(2,1)}^2L^2=\Delta(\Delta-d).$$

At $\Delta=d$, the null projected divergence and the unitary AdS gauge parameter have shape $(1,1)$, obtained by removing one box from the first row. The other removable corner belongs to the richer flat-space gauge system; treating both corners as independent AdS gauge symmetries would describe the wrong module.

Reducibility must also be retained. A massless $p$-form already has the chain

$$\Lambda_{p-1}\sim\Lambda_{p-1}+\mathrm d\Lambda_{p-2}, \qquad \Lambda_{p-2}\sim\Lambda_{p-2}+\mathrm d\Lambda_{p-3}, \qquad\ldots,$$

and general mixed diagrams can carry analogous gauge-for-gauge maps. Row and column symmetries, traces, algebraic Bianchi identities, gauge parameters, reducibility, falloffs, and boundary shortening are one package; dropping any one of them can change the representation.

In even boundary dimension, a diagram of maximal allowed height may further split into self-dual and anti-self-dual representations. Parity data and the real form must then be specified in addition to the Young shape. These low-dimensional identifications are why the canonical $SO(d)$ label must be chosen before applying the top-block formula.

A compact spin-sensitive dictionary

The table summarizes the free linearized tests. On a narrow screen, scroll horizontally without shrinking the text.

Spin-sensitive bulk-to-boundary dictionary

Bulk field or regime	Candidate-dimension relation	Constraint or gauge quotient	Boundary representation	Failure if omitted
Massive vector	mA2L2 = (Δ−1)(Δ−d+1)	On-shell Proca divergence constraint; no gauge quotient	Long vector with Δ > d−1 on the ordinary unitary branch	A raw-component count includes a spurious polarization.
Maxwell vector	mA2 = 0 and Δ = d−1	Gauge quotient plus radial Gauss constraint	Conserved current; source defined modulo a background gauge transformation	Removing only Az leaves one longitudinal mode too many.
Symmetric traceless spin s	ms2L2 = (Δ+s−2)(Δ−s−d+2)	Massive transverse-traceless constraints; Fronsdal gauge quotient at zero mass	Long module above Δ = s+d−2; conserved module at equality	A curvature-shifted Laplacian eigenvalue can be mistaken for the physical mass.
Canonical p-form	mp2L2 = (Δ−p)(Δ+p−d)	Massive co-closure; gauge and gauge-for-gauge chain at zero mass	For p < d/2, a conserved rank-p current at Δ = d−p	The form degree, symmetry degree, or electric/magnetic sector can be misidentified.
Mixed Young diagram Y	mY2L2 = (Δ−Δ0)(Δ+Δ0−d)	Young projectors, traces, top-block gauge map, and any reducibility	A primary of dimension Δ in Y, with Δ ≥ Δ0 for the ordinary unitary module	Counting boxes loses the actual representation and its null descendants.

Common pitfalls

Treating every constraint as a gauge choice. Proca co-closure is an on-shell condition for a massive field. Maxwell gauge equivalence exists only at the gauge point, and its Gauss equation remains an independent constraint.

Calling every real indicial root a CFT operator. Reality supplies candidate weights. Boundary conditions, normalizability, the correct tensor unitarity bound, and the gauge complex decide whether an ordinary unitary operator exists.

Using the number of Young boxes as “spin.” The symmetric diagram $(2)$ and the antisymmetric diagram $(1,1)$ both have two boxes, but they have different component counts, unitarity thresholds, mass formulas, and gauge parameters.

Ignoring Hodge and chirality exceptions. A noncanonical column can duplicate a lower-rank representation, while a maximal-height diagram in even $d$ can split. Dualize or resolve chirality before applying a formula.

Exercises

Starting from the Proca equation, recover the transverse radial equation and both coordinate-component falloffs. What changes at $m_A^2=0$?

Solution

For a boundary-transverse mode, $A_z$ and the longitudinal boundary component decouple. Using $\sqrt{\lvert g\rvert}=L^{d+1}z^{-(d+1)}$, $g^{zz}=-z^2/L^2$, and $g^{ij}=z^2\eta^{ij}/L^2$ in $\nabla_MF^{Mi}+m_A^2A^i=0$ gives

$$\left[-z^2\partial_z^2+(d-3)z\partial_z +z^2\Box_\eta+m_A^2L^2\right]A_i^{\mathrm T}=0.$$

At fixed boundary momentum, substituting $A_i^{\mathrm T}\sim z^\delta$ gives $\delta(\delta-d+2)=m_A^2L^2$. If the operator dimension is $\Delta$, the roots are $d-\Delta-1$ and $\Delta-1$, so $(m_AL)^2=(\Delta-1)(\Delta-d+1)$. At zero mass, the powers are $0$ and $d-2$, while the Proca constraint is replaced by gauge equivalence plus the Maxwell Gauss constraint. The unitary operator root is $\Delta=d-1$.

In AdS$_7$/CFT$_6$, take a massless two-form with the ordinary electric boundary condition. Find both algebraic dimensions, select the unitary current, and identify the degree of the associated global symmetry.

Solution

Here $d=6$ and $p=2$, so

$$0=(\Delta-2)(\Delta-4).$$

The roots are $2$ and $4$. Because $p<d/2$, the antisymmetric-tensor unitarity threshold is $\Delta_0=d-p=4$. The ordinary electric branch therefore gives a conserved rank-two current $J_{ij}$ of dimension $4$. Its background field is a two-form, so it generates a one-form global symmetry. The $\Delta=2$ root is the source or shadow weight, not an ordinary unitary rank-two primary.

In a five-dimensional boundary CFT, compare the two-box diagrams $(2)$ and $(1,1)$. Count their algebraic components and find their unitary thresholds. Why does box counting fail?

Solution

A symmetric traceless rank-two tensor has

$$\frac{5(5+1)}{2}-1=14$$

components. An antisymmetric two-form has

$$\binom52=10$$

components. For $(2)$, $s_1=2$ and $r=1$, so $\Delta_0=2+5-1-1=5$. For $(1,1)$, $s_1=1$ and $r=2$, so $\Delta_0=1+5-2-1=3$. Equal box number therefore fixes neither the $SO(5)$ representation nor its shortening threshold; the row and column symmetries are essential data.

Controlled handoffs

Currents, Stress Tensor, and Bulk Gauge and Metric Fields fixes renormalized response normalizations and develops the current and stress-tensor Ward identities. Spinning and Tensor Witten Diagrams develops bulk-to-boundary propagators, tensor structures, and interaction diagrams. Higher-Form Currents, Charges, Backgrounds, and Ward Identities develops global sectors and charged defects. Higher-Spin Algebras, Gauge Fields, and Interactions treats nonlinear consistency beyond the free representation dictionary.

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

• Brink, Lars, R. R. Metsaev, and Mikhail A. Vasiliev. “How Massless Are Massless Fields in AdS$_d$?” Nuclear Physics B 586 (2000): 183–205. DOI. Open PDF.
• Costa, Miguel S., Vasco Gonçalves, and João Penedones. “Spinning AdS Propagators.” Journal of High Energy Physics 2014, 064 (2014). DOI. Open PDF.
• Costa, Miguel S., and Tobias Hansen. “Conformal Correlators of Mixed-Symmetry Tensors.” Journal of High Energy Physics 2015, 151 (2015). DOI. Open PDF.
• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 2015, 172 (2015). DOI. Open PDF.
• Gwak, Seungho, Jaewon Kim, and Soo-Jong Rey. “Massless and Massive Higher Spins from Anti-de Sitter Space Waveguide.” Journal of High Energy Physics 2016, 024 (2016). DOI. Open PDF.
• Hofman, Diego M., and Nabil Iqbal. “Generalized Global Symmetries and Holography.” SciPost Physics 4, 005 (2018). DOI. Open PDF.
• l’Yi, W. S. “Coordinate-Space Holographic Projection of Fields and an Application to Massive Vector Fields.” arXiv:hep-th/9808051, revised 1999. Open PDF.
• l’Yi, W. S. “Correlators of Currents Corresponding to the Massive $p$-Form Fields in AdS/CFT Correspondence.” Physics Letters B 448 (1999): 218–226. DOI. Open PDF.
• Marolf, Donald, and Simon F. Ross. “Boundary Conditions and Dualities: Vector Fields in AdS/CFT.” Journal of High Energy Physics 2006, 085 (2006). DOI. Open PDF.
• Metsaev, R. R. “Massive Totally Symmetric Fields in AdS(d).” Physics Letters B 590 (2004): 95–104. DOI. Open PDF.