operators in $\mathbb{M}^{3|2{\cal N}}$, for $1\leq\mathcal{N}\leq 6$. In
section 3.1, we reviewed massive UIRs of the three-dimensional super-Poincaré
algebra. In section 3.2, we presented the ${\cal N}$-extended generalisation
of the massive on-shell supermultiplets (3.16) which realise the massive UIR
$\mathbb{D}(m:\sigma,\frac{n}{2},{\cal N})$. These ${\cal N}$-extended
supermultiplets appeared for the first time in [1]. We also studied higher-
spin massless superfields (3.25) in $\mathbb{M}^{3|2}$, which we defined as
gauge superfields which do not propagate any physical degrees on the equations
of motion. This feature was shown explicitly for massless superfields (3.25)
at the level of superfields and component fields.

In section 3.3, we computed the superspin projection operators
$\mathbb{M}^{3|2{\cal N}}$, for $1\leq\mathcal{N}\leq 6$ [1]. These
superprojectors map any unconstrained superfield to the space of transverse
superfields (pure superspin states) (3.16a). If the superspin projection
operator is applied to a superfield which satisfies the first-order
differential equation (3.16b), then the resulting massive superfield (3.16)
furnishes the massive UIR $\mathbb{D}(m:\sigma,\frac{n}{2},{\cal N})$. If
instead the superfield satisfies the Klein-Gordon equation, then the projected
field realises the reducible representation $\mathbb{D}(m:-,\frac{n}{2},{\cal
N})\oplus\mathbb{D}(m:+,\frac{n}{2},{\cal N})$. In order to extract out the
UIR with definite superhelicity, one needs to make use of the superhelicity
projectors (3.49).

In section 3.4, we made use of the superspin projection operators to construct
the linearised rank-$n$ super-Cotton tensor, for $n\geq 1$, in terms of the
unconstrained gauge prepotential $H_{\alpha(n)}$. The general form of these
super-Cotton tensors, which were presented for generic ${\cal N}$ in [1], are
given by eq. (3.106). In particular, a new expression for the rank-$n$ super-
Cotton tensor in the case of $\mathcal{N}=2$ supersymmetry was computed. This
expression is much simpler than the one originally given in [122]. The
rank-$n$ super-Cotton tensors, for $3\leq{\cal N}\leq 6$, were derived for the
first time in [1]. The corresponding results are given by eqs. (3.120),
(3.128), (3.137) and (3.141), respectively.

The linearised super-Cotton tensor of ${\cal N}=4$ conformal supergravity
required special attention since it is a scalar superfield $W$. It was also
computed in [1] for the first time, and is given by eq. (3.130). Making use of
$W$ allowed us to construct the linearised action for ${\cal N}=4$ conformal
supergravity, which is given by eq. (3.132). In the $\mathcal{N}=3$ case, we
also constructed the gauge-invariant action (3.125) which describes the
dynamics of the superconformal gravitino multiplet. In the case of conformal
supergravity with $\mathcal{N}\geq 5$, the super-Cotton tensor has a different
tensorial form than that of $W_{\alpha(n)}$, see eq. (3.99). Consequently, it
cannot be directly obtained from our results.

In section 3.5 we reviewed the formulation of massless (half-)integer
superspin actions in $\mathbb{M}^{3|2{\cal N}}$, which were first given in
[123]. In section 3.6 we proposed ${\cal N}$-extended generalisations for new
topologically massive theories [1], see eq. (3.170). The computation of the
linearised higher-spin Cotton tensors lead to explicit realisations of these
models for $3\leq{\cal N}\leq 6$ for the first time [1].

## Appendix

##  Appendix 3.A Superconformal primary multiplets

The ${\cal N}$-extended superconformal symmetry in three dimensions was
studied in detail by Park [182]. In this appendix our presentation follows
[166, 183]. In ${\cal N}$-extended Minkowski superspace
${\mathbb{M}}^{3|2{\cal N}}$, superconformal transformations, $z^{A}\to
z^{A}+\delta z^{A}=z^{A}+\xi^{A}(z)$, are generated by superconformal Killing
vectors. By definition, a superconformal Killing vector

$\displaystyle\xi=\xi^{a}(z)\,\partial_{a}+\xi^{\alpha}_{I}(z)\,D_{\alpha}^{I}$
(3.173)

is a real vector field obeying the condition $[\xi,D_{\alpha}^{I}]\propto
D_{\beta}^{J}$. This condition implies

$\displaystyle[\xi,D_{\alpha}^{I}]=-(D^{I}_{\alpha}\xi^{\beta}_{J})D^{J}_{\beta}=\frac{1}{2}\omega_{\alpha}{}^{\beta}(z)D_{\beta}^{I}+\Lambda^{IJ}(z)D_{\alpha}^{J}-\frac{1}{2}\sigma(z)D^{I}_{\alpha}~{},$
(3.174)

where we have defined

$\displaystyle\omega_{\alpha\beta}$ $\displaystyle:=$
$\displaystyle-\frac{2}{{\cal
N}}D^{J}_{(\alpha}\xi_{\beta)}^{J}=-\frac{1}{2}\partial^{\gamma}{}_{(\alpha}\xi_{\beta)\gamma}~{},$
(3.175a) $\displaystyle\Lambda^{IJ}$ $\displaystyle:=$
$\displaystyle-2D_{\alpha}^{[I}\xi^{J]\alpha}~{},$ (3.175b)
$\displaystyle\sigma$ $\displaystyle:=$ $\displaystyle\frac{1}{{\cal
N}}D^{I}_{\alpha}\xi^{\alpha I}=\frac{1}{3}\partial_{a}\xi^{a}~{}.$ (3.175c)

Here the parameters $\omega_{\alpha\beta}=\omega_{\beta\alpha}$,
$\Lambda^{IJ}=-\Lambda^{JI}$ and $\sigma$ correspond to $z$-dependent Lorentz,
${\rm SO}({\cal N})$ and scale transformations. These transformation
parameters are related to each other as follows:

$\displaystyle D^{I}_{\alpha}\omega_{\beta\gamma}$ $\displaystyle=$
$\displaystyle 2\varepsilon_{\alpha(\beta}D^{I}_{\gamma)}\sigma~{},$ (3.176a)
$\displaystyle D^{I}_{\alpha}\Lambda^{JK}$ $\displaystyle=$
$\displaystyle-2\delta^{I[J}D_{\alpha}^{K]}\sigma~{}.$ (3.176b)

Let $\Phi_{\cal A}^{\cal I}(z)$ be a superfield that transforms in a
representation $T$ of the Lorentz group with respect to its index ${\cal A}$
and in a representation $D$ of the $R$-symmetry group ${\rm SO}({\cal N})$
with respect to the index ${\cal I}$. Such a superfield is called primary of
dimension $d$ if its superconformal transformation law is

$\displaystyle\delta\Phi_{\cal A}^{\cal I}=-\xi\Phi_{\cal A}^{\cal
I}-d\,\sigma\Phi_{\cal A}^{\cal
I}+\frac{1}{2}\omega^{\alpha\beta}(M_{\alpha\beta})_{\cal A}{}^{\cal
B}\Phi_{\cal B}^{\cal I}+\frac{1}{2}\Lambda^{IJ}(R^{IJ})^{\cal I}{}_{\cal
J}\Phi_{\cal A}^{\cal J}~{}.~{}~{}$ (3.177)

The ${\rm SO}({\cal N})$ generator $R^{IJ}$ acts on an ${\rm SO}({\cal
N})$-vector $V^{K}$ as

$\displaystyle R^{IJ}V^{K}=2\delta^{K[I}V^{J]}~{}.$ (3.178)

Making use of this transformation law allows one to determine the dimensions
in (3.105).

##  Chapter 4 Four-dimensional anti-de Sitter (super)space

It was demonstrated in chapters 2 and 3 that the (super)spin projection
operators possess many interesting applications in three- and four-dimensional
Minkowski (super)space. For example, they can be used to decompose an
arbitrary (super)field into its irreducible components. Moreover, the
(super)spin projection operators were also shown to be fundamental in the
formulation of (S)CHS theory. Thus it is natural to consider extensions of
these operators to curved backgrounds, with the simplest case being four-
dimensional (anti-)de Sitter space (A)dS4. However, when deriving the spin
projection operators in (A)dS4, one immediately encounters computational
difficulties due to the presence of non-vanishing curvature.

Despite these challenges, the spin projection operators were recently found in
(A)dS4 by Kuzenko and Ponds [90] in 2020. One of the important outcomes of
[90] was a new understanding of the so-called partially massless fields in
AdS4. Specifically, it was shown that all necessary information required to
generate partially massless fields is encoded within the poles of the spin
projection operators. This chapter is devoted to the study of spin projection
operators in four-dimensional anti-de Sitter space, and their supersymmetric
generalisations to ${\cal N}=1$ and ${\cal N}=2$ AdS superspace.

This chapter is based on the publication [3], and the work in progress [5]. It
is organised as follows. Section 4.1 is devoted to reviewing and expanding
upon the spin projection operators and their corresponding applications in
AdS4. In subsection 4.1.1, we provide a brief overview of the UIRs of the AdS4
algebra $\mathfrak{so}(3,2)$. Next, we introduce salient facts concerning
field theory in AdS4. In particular, we discuss how the UIRs of
$\mathfrak{so}(3,2)$ are realised on the space of totally symmetric fields.
This is given in subsection 4.1.2. In section 4.1.3, we study the spin
projection operators and their applications. Here, we also introduce an
alternative, yet equivalent, form of the spin projection operators of [90].
The novelty of these operators is that they are expressed solely in terms of
the Casimir operators of $\mathfrak{so}(3,2)$. These operators are then used
to recast the known CHS theory in a manifestly factorised form. This is
accomplished in subsection 4.1.4.

Section 4.2 is devoted to extending the results of section 4.1 to four-
dimensional ${\cal N}=1$ anti-de Sitter superspace AdS4|4. In section 4.2.1 we
review the UIRs of the ${\cal N}=1$ AdS4 superalgebra $\mathfrak{osp}(1|4)$.
Essential aspects of AdS4|4 are detailed in section 4.2.2, which will allow
for a discussion of on-shell supermultiplets in section 4.2.3. The component
structure of these on-shell supermultiplets are analysed in section 4.2.4. We
compute the superspin projection operators in AdS4|4 and their corresponding
applications in section 4.2.5. In section 4.2.6 we elucidate the relationship
between the superspin projection operators and SCHS theory. In section 4.2.7
we study several off-shell models in AdS4|4. Most notably, we present an off-
shell model for the massive gravitino (superspin-$1$) multiplet.

Section 4.3 is devoted to extending various results from sections 4.1 and 4.2
to ${\cal N}=2$ anti-de Sitter superspace AdS4|8. In section 4.3.1, we
introduce pertinent aspects concerning AdS4|8. In section 4.3.2, we review the
free SCHS action in AdS4|8, which was recently derived by Kuzenko and Raptakis
in [87]. The superspin projection operators are then extracted from this SCHS
theory in section 4.3.3. Finally, we propose a dictionary for on-shell
superfields in section 4.3.4 using the poles of the superspin projection
operators. A summary of the results obtained is given in section 4.4. This
chapter is also accompanied by two technical appendices, 4.A and 4.B. Appendix
4.A contains a list of identities that are indispensable for the derivation of
many of the results in section 4.2. Appendix 4.B is devoted to the derivation
of partially massless gauge transformations in AdS4|4.

###  4.1 Four-dimensional anti-de Sitter space

In this section we review and elaborate upon the spin projection operators in
AdS4 [90] and their corresponding applications. These operators are AdS4
generalisations of the spin projectors in $\mathbb{M}^{4}$, which were
detailed in section 2.1. Before introducing the spin projection operators, it
is necessary to introduce the irreducible representations of
$\mathfrak{so}(3,2)$.

####  4.1.1 Irreducible representations of the anti-de Sitter algebra

Anti-de Sitter space in $d$ dimensions AdSd can be identified as a hyperboloid
embedded in $(d+1)$-dimensional ambient space $\mathbb{R}^{d+1}$

$\eta^{AB}X_{A}X_{B}=-l^{2}~{}.$ (4.1)

Here the index of the ambient space coordinate $X_{A}$ take the values
$A=0,1,2,\cdots,d$ and $l$ is the radius of AdSd, with $l^{2}>0$. The indices
are raised and lowered by the ambient metric tensor
$\eta^{AB}=\eta_{AB}=\text{diag}(-1,1,\cdots,1,-1)$.

The Lie algebra $\mathfrak{so}(d-1,2)$ of the AdSd isometry group is spanned
by the antisymmetric generators $J_{AB}=-J_{BA}$, which obey the commutation
relations

$[J_{AB},J_{CD}]={\rm
i}\big{(}\eta_{AC}J_{BD}-\eta_{AD}J_{BC}+\eta_{BD}J_{AC}-\eta_{BC}J_{AD}\big{)}~{}.$
(4.2)

The normalisation factor of the commutation relation (4.2) is chosen to ensure
that in the limit where the AdSd radius tends to infinity
$l\rightarrow\infty$, the algebra (4.2) contracts to the Poincaré algebra
(2.5). This can be checked directly by splitting the generators of
$\mathfrak{so}(d-1,2)$ as follows

$J_{AB}\longrightarrow\big{(}J_{ab},J_{ad}\big{)}\equiv\big{(}J_{ab},lP_{a}\big{)}~{},$
(4.3)

where $a,b=0,1,\cdots,d-1$.111Note that the index denoted $d$ refers to the
dimension $d$ and is not to be confused with a free index. The generators
$J_{0d}$ and $J_{ab}$ in (4.3) can be identified as the AdSd analogues of the
energy and angular momentum operators respectively.

Let us now restrict our attention to four dimensions $d=4$. The quadratic
$C_{1}$ and quartic $C_{2}$ Casimir operators of $\mathfrak{so}(3,2)$ are
[185, 184]

$C_{1}=\frac{1}{2}J^{AB}J_{AB}~{},\qquad
C_{2}=-W^{A}W_{A}~{},\qquad[C_{1},J_{AB}]=[C_{2},J_{AB}]=0~{},$ (4.4)

where $W_{A}=\frac{1}{8}\varepsilon_{ABCDE}J^{BC}J^{DE}$.

The irreducible representations $\mathfrak{so}(3,2)$ have been extensively
studied in the pioneering works [186, 187, 188, 184, 189, 190, 185, 191, 192].
We are interested in the lowest-energy (or lowest-weight) irreducible
representations222For a modern discussion on the lowest-energy UIRs of
$\mathfrak{so}(3,2)$, see the works [193, 194, 102, 17]. of
$\mathfrak{so}(3,2)$ since the energy they carry is bounded from below. The
irreducible representations of $\mathfrak{so}(3,2)$ are specified by the
lowest value $E_{0}$ of the energy $E$333The energy $E$ is chosen to be
dimensionless; it can be restored by rescaling $E$ by $E\to l^{-1}\,E$. and
spin $s$, and are commonly denoted by $D(E_{0},s)$. The parameters $E_{0}$ and
$s$ are determined by the eigenvalues of the Casimir operators (4.4) [184,
185]

$\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle
E_{0}(E_{0}-3)+s(s+1)~{},$ (4.5a) $\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle s(s+1)(E_{0}-1)(E_{0}-2)~{}.$ (4.5b)

Unlike in Minkowski space, unitarity imposes a bound on the allowed values of
energy in AdS4. According to the theorems proved in [185, 191], the
irreducible representation $D(E_{0},s)$ is unitary444In any UIR of
$\mathfrak{so}(d-1,2)$, the generators $J_{AB}$ are Hermitian and the norm for
the lowest-energy states is positive definite, i.e. see [17]. if and only if
one of the following conditions hold:

$\text{ (i)}\quad s=0~{},~{}E_{0}\geq\frac{1}{2}~{},\qquad\text{(ii)}\quad
s=\frac{1}{2}~{},~{}E_{0}\geq 1~{},\qquad\text{(iii)}\quad s\geq
1~{},~{}E_{0}\geq s+1~{}.$ (4.6)

The representations $D\big{(}\frac{1}{2},0\big{)}={\rm Rac}$ and
$D\big{(}1,\frac{1}{2}\big{)}={\rm Di}$ are known as the Dirac singletons
[187].555It was found by Flato and Fronsdal [197, 196] that the singletons are
the square roots of massless particles in the sense that all two-singleton
states are massless. The representations $D(s+1,s)$ for $s>0$ and
$D(1,0)\oplus D(2,0)$ are called massless since they contract to the massless
discrete helicity representations of the Poincaré group [192]. These
representations prove to be restrictions of certain unitary representations of
the conformal algebra $\mathfrak{so}(4,2)$ to $\mathfrak{so}(3,2)$ [192, 195].
The remaining representations $D(E_{0},s)$ are usually referred to as the
massive AdS4 representations.

####  4.1.2 Irreducible field representations

The geometry of ${\rm AdS}_{4}$ can be described in terms of torsion-free
Lorentz-covariant derivatives of the form

$\displaystyle\nabla_{a}=e_{a}+\omega_{a}~{},\qquad
e_{a}=e_{a}{}^{m}\partial_{m}~{},$ (4.7)

where $e_{a}{}^{m}$ is the inverse vielbein and $\omega_{a}$ is the Lorentz
connection

$\displaystyle\omega_{a}=\frac{1}{2}\,\omega_{a}{}^{bc}M_{bc}=\omega_{a}{}^{\beta\gamma}M_{\beta\gamma}+\bar{\omega}_{a}{}^{{\dot{\beta}}{\dot{\gamma}}}\bar{M}_{{\dot{\beta}}{\dot{\gamma}}}~{}.$
(4.8)

The covariant derivative (4.7) satisfies the commutation relation666Note that
we use the convention $[\nabla_{a},\nabla_{b}]=\frac{1}{2}{\cal
R}_{ab}{}^{ce}M_{ce}$, where ${\cal R}_{abce}$ is the Riemann tensor which we
normalise as ${\cal R}_{abce}=-2\mu\bar{\mu}\eta_{a[c}\eta_{e]b}$.

$\displaystyle\big{[}\nabla_{a},\nabla_{b}\big{]}=-\mu\bar{\mu}M_{ab}\quad\Longleftrightarrow\quad\big{[}\nabla_{\alpha{\dot{\alpha}}},\nabla_{\beta{\dot{\beta}}}\big{]}=-2\mu\bar{\mu}\big{(}\varepsilon_{\alpha\beta}\bar{M}_{{\dot{\alpha}}{\dot{\beta}}}+\varepsilon_{{\dot{\alpha}}{\dot{\beta}}}M_{\alpha\beta}\big{)}~{},$
(4.9)

where the parameter $\mu\bar{\mu}>0$ is related to the AdS4 scalar curvature
$\mathcal{R}$ via $\mathcal{R}=-12\mu\bar{\mu}$, and to the AdS4 radius $l$
via $l^{-2}=\mu\bar{\mu}$.777The complex parameter $\mu$ appears explicitly
only in the algebra of AdS4|4 covariant derivatives (4.75). Note that changing
the sign of the curvature $\mathcal{R}$ yields de-Sitter space, while
Minkowski space is obtained in the limit $\mathcal{R}\rightarrow 0$.

We will make extensive use of the Casimir operators (4.4) of
$\mathfrak{so}(3,2)$ in the subsequent sections, which take the following form
in the field representation

$\displaystyle\mathcal{Q}=$
$\displaystyle~{}\Box-\mu\bar{\mu}\big{(}M^{2}+\bar{M}^{2}\big{)}~{},\ $
$\displaystyle\big{[}\mathcal{Q},\nabla_{\alpha{\dot{\alpha}}}\big{]}$
$\displaystyle=0~{},$ (4.10a) $\displaystyle\mathcal{W}^{2}=$
$\displaystyle-\frac{1}{2}\big{(}\mathcal{Q}+2\mu\bar{\mu}\big{)}\big{(}M^{2}+\bar{M}^{2}\big{)}+\nabla^{\alpha{\dot{\alpha}}}\nabla^{\beta{\dot{\beta}}}M_{\alpha\beta}\bar{M}_{{\dot{\alpha}}{\dot{\beta}}}$
$\displaystyle-\frac{1}{4}\mu\bar{\mu}\big{(}M^{2}M^{2}+\bar{M}^{2}\bar{M}^{2}+6M^{2}\bar{M}^{2}\big{)}~{},$
$\displaystyle\big{[}\mathcal{W}^{2},\nabla_{\alpha{\dot{\alpha}}}\big{]}$
$\displaystyle=0~{}.$ (4.10b)

Here
$\Box:=-\frac{1}{2}\nabla^{\alpha{\dot{\alpha}}}\nabla_{\alpha{\dot{\alpha}}}$
is the d’Alembert operator in AdS4. Note that we have introduced the notation
$\mathcal{Q}\equiv l^{-2}C_{1}$ and $\mathcal{W}^{2}\equiv l^{-2}C_{2}$ for
the Casimir operators (4.4) of $\mathfrak{so}(3,2)$ in the field
representation, with dimensions being restored.

Given two positive integers $m$ and $n$, a tensor field
$\phi_{\alpha(m){\dot{\alpha}}(n)}$ of Lorentz type
$(\frac{m}{2},\frac{n}{2})$ in AdS4 is said to be on-shell if it satisfies the
conditions

$\displaystyle\nabla^{\beta{\dot{\beta}}}\phi_{\beta\alpha(m-1){\dot{\beta}}{\dot{\alpha}}(n-1)}$
$\displaystyle=0~{},$ (4.11a)
$\displaystyle\big{(}\mathcal{Q}-\rho^{2}\big{)}\phi_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=0~{}.$ (4.11b)

Here the real parameter $\rho$, which carries mass dimension one, is called
the pseudo-mass.

It must be noted that there is an ambiguity in the way mass is defined in
AdS4. There exist additional terms proportional to the scalar curvature
$\mu\bar{\mu}$ in AdS4, which have no direct counterpart in the flat-space
limit $\mu\bar{\mu}\rightarrow 0$. Since these contributions also have mass
dimension two, it follows that the mass-shell equation (4.11b) can be
redefined modulo $\mu\bar{\mu}$ contributions without any effect on the
corresponding mass-shell equation in the flat space limit. In other words,
there exists an infinite family of on-shell fields in AdS4 which reduce to the
same on-shell field in $\mathbb{M}^{4}$.

To address this ambiguity, let us introduce the notion of physical mass
$\rho_{\text{phys}}$ whose meaning is derived from gauge symmetry. In
particular, the definition of physical mass is related to pseudo-mass via the
relation

$\displaystyle\rho^{2}_{\text{phys}}=\rho^{2}-\tau_{(1,m,n)}\mu\bar{\mu}~{},\qquad\tau_{(1,m,n)}=\frac{1}{2}(m+n+2)(m+n-2)~{},$
(4.12)

where $\tau_{(1,m,n)}$ is a partially massless value (4.27). As we will see,
this mass convention ensures that an on-shell field (4.11) with physical mass
$\rho^{2}_{\text{phys}}=0$, which we will call a massless field, is compatible
with the maximal gauge symmetry (4.22). Hence in the flat-space limit, a
massless field in AdS4 reduces to a massless gauge field (2.33) in
$\mathbb{M}^{4}$.

Contact with the representation theory can be established by comparing the
eigenvalue (4.11b) of the quadratic Casimir operator $\mathcal{Q}$ associated
with the on-shell field (4.11) with the eigenvalue (4.5a) of the Casimir
operator $l^{-2}C_{1}$ in the lowest-energy irreducible representation. Doing
this yields the following relationship888See [193, 194] for pedagogical
derivations of the result (4.13).  between the pseudo-mass $\rho$ and minimal
energy $E_{0}$

$\displaystyle\rho^{2}=\big{[}E_{0}(E_{0}-3)+s(s+1)\big{]}\mu\bar{\mu}~{},$
(4.13)

where $s=\frac{1}{2}(m+n)$. It follows from (4.12) that the minimal energy is
related to the physical mass through

$\displaystyle\rho_{\text{phys}}^{2}$
$\displaystyle=\big{[}E_{0}(E_{0}-3)-(s+1)(s-2)\big{]}\mu\bar{\mu}~{}.$ (4.14)

The action of the Casimir operator $\mathcal{W}^{2}$ (4.10b) on an
unconstrained field yields

$\displaystyle\mathcal{W}^{2}\phi_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=$ $\displaystyle
s(s+1)\big{(}\mathcal{Q}-(s-1)(s+2)\mu\bar{\mu}\big{)}\phi_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle+mn\nabla_{\alpha{\dot{\alpha}}}\nabla^{\beta{\dot{\beta}}}\phi_{\beta\alpha(m-1){\dot{\beta}}{\dot{\alpha}}(n-1)}~{}.$

If the field satisfies the on-shell conditions (4.11), then (4.1.2) reduces to

$\mathcal{W}^{2}\phi_{\alpha(m){\dot{\alpha}}(n)}=s(s+1)\big{(}\rho^{2}-(s-1)(s+2)\mu\bar{\mu}\big{)}\phi_{\alpha(m){\dot{\alpha}}(n)}~{}.$
(4.16)

Expressing (4.16) in terms of the minimal energy via the relation (4.13) gives

$\mathcal{W}^{2}\phi_{\alpha(m){\dot{\alpha}}(n)}=s(s+1)(E_{0}-2)(E_{0}-1)\mu\bar{\mu}\phi_{\alpha(m){\dot{\alpha}}(n)}~{}.$
(4.17)

Note that the eigenvalue (4.17) of the Casimir operator $\mathcal{W}^{2}$ on
the space of on-shell fields coincides identically with the eigenvalue (4.5b)
of the quartic Casimir operator $l^{-2}C_{2}$ in the lowest-energy irreducible
representation. It follows from (4.11b) and (4.17) that an on-shell field
(4.11) furnishes the irreducible representation $D(E_{0},s)$ of
$\mathfrak{so}(3,2)$. Thus, we say that an on-shell field carries spin
$s=\frac{1}{2}(m+n)$ and pseudo-mass $\rho$. From a field-theoretic
perspective, is natural to use the pseudo-mass $\rho$ as a representation
label in place of $E_{0}$. However, we choose to stay with the label $E_{0}$
in this chapter.

It follows from (4.14) that there are two distinct values of $E_{0}$ leading
to the same value of $\rho_{\text{phys}}^{2}$,

$\displaystyle\big{(}E_{0}\big{)}_{\pm}=\frac{3}{2}\pm\frac{1}{2}\sqrt{4\frac{\rho^{2}_{\text{phys}}}{\mu\bar{\mu}}+(m+n-1)^{2}}~{}.$
(4.18)

Hence the minimal energy carried by an on-shell field (4.92) is not unique.
However, we have yet to take unitarity into account. For $m+n>1$, the
unitarity bound $E_{0}\geq\frac{1}{2}(m+n+2)$ (cf. (4.6)) can be expressed in
terms of pseudo-mass and physical mass as follows

$\displaystyle\rho^{2}\geq\tau_{(1,m,n)}\mu\bar{\mu}\quad\implies\quad\rho^{2}_{\text{phys}}\geq
0~{}.$ (4.19)

In accordance with (4.19), the solution $(E_{0})_{-}$ always violates the
unitarity bound for $m+n>1$. Thus when referring to a unitary representation
with $s>\frac{1}{2}$ and pseudo-mass $\rho^{2}$, we are implicitly referring
to the representation corresponding to $(E_{0})_{+}$999Using the properties of
gauge symmetry and unitarity to uniquely determine the minimal energy $E_{0}$
for massless fields in AdS4 was first completed in [190, 29]. Such studies
were later generalised to AdSd in the case of bosonic [198] and fermionic
[199] fields.

$\displaystyle
E_{0}=\frac{3}{2}+\frac{1}{2}\sqrt{4\frac{\rho^{2}_{\text{phys}}}{\mu\bar{\mu}}+(m+n-1)^{2}}~{},\qquad
m+n>1~{}.$ (4.20)

For $s=0,\frac{1}{2}$ (or, equivalently, $m+n=0,1$), the unitary bound is
$E_{0}\geq s+\frac{1}{2}$, and the solution $(E_{0})_{-}$ in (4.18) does not
violate the unitarity bound for certain values of $\rho^{2}_{\text{phys}}$.
For $s=0$ the values of $\rho^{2}_{\text{phys}}$ leading to
$(E_{0})_{-}\geq\frac{1}{2}$ are restricted by the condition
$-\frac{1}{4}\mu\bar{\mu}\leq\rho^{2}_{\text{phys}}\leq\frac{3}{4}\mu\bar{\mu}$,
which is known as the Breitenlohner-Freedman bound [200].

Below we review the different species of on-shell fields in AdS4, which are
characterised by the pseudo-mass they carry.

#####  4.1.2.1 Massless fields

Given positive integers $m$ and $n$, an on-shell field (4.92) is said to be
massless if it carries the pseudo-mass

$\displaystyle\rho^{2}=\tau_{(1,m,n)}\mu\bar{\mu}~{},\qquad\tau_{(1,m,n)}=\frac{1}{2}(m+n-2)(m+n+2)~{}.$
(4.21)

The system of equations (4.11) with $\rho^{2}$ (4.21) is compatible with the
gauge symmetry

$\displaystyle\delta_{\zeta}\phi_{\alpha(m){\dot{\alpha}}(n)}=\nabla_{\alpha{\dot{\alpha}}}\zeta_{\alpha(m-1){\dot{\alpha}}(n-1)}~{},$
(4.22)

where the gauge parameter $\zeta_{\alpha(m-1){\dot{\alpha}}(n-1)}$ is complex
unconstrained. The maximal gauge symmetry (4.22) is the AdS4 analogue of the
gauge symmetry (2.33c) associated with massless fields in $\mathbb{M}^{4}$.
Note that the relationship (4.12) between physical mass and pseudo-mass was
chosen to ensure that massless fields in AdS4 carry vanishing physical mass
$\rho^{2}_{\text{phys}}=0$. It is easy to see that this physical mass
saturates the bound (4.19) and hence defines a unitary representation of
$\mathfrak{so}(3,2)$. It follows from (4.13) and (4.21) that a massless field
has minimal energy $E_{0}=s+1$.

Analogous to the discussion on massless fields in $\mathbb{M}^{4}$ (see
section 2.1.3.2), it follows that massless gauge fields in AdS4 do not realise
irreducible representations of $\mathfrak{so}(3,2)$ due to the presence of
gauge symmetry. These redundant gauge degrees of freedom are automatically
eliminated if one works with the gauge-invariant field strengths (4.56a),

$\displaystyle\mathfrak{W}_{\alpha(m+n)}(\phi)$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\nabla_{\alpha_{n}}{}^{{\dot{\beta}}_{n}}\phi_{\alpha_{n+1}\ldots\alpha_{m+n}){\dot{\beta}}(n)}~{},$
(4.23a) $\displaystyle\bar{\mathfrak{W}}_{{\dot{\alpha}}(m+n)}({\phi})$
$\displaystyle=$
$\displaystyle\nabla_{({\dot{\alpha}}_{1}}{}^{\beta_{1}}\cdots\nabla_{{\dot{\alpha}}_{m}}{}^{\beta_{m}}{\phi}_{\beta(m){\dot{\alpha}}_{m+1}\ldots{\dot{\alpha}}_{m+n})}~{},$
(4.23b)

instead of the gauge field $\phi_{\alpha(m){\dot{\alpha}}(n)}$. Making use of
the constraint (4.11a) satisfied by $\phi_{\alpha(m){\dot{\alpha}}(n)}$, it
may be shown that the field strengths (4.23) are constrained by

$\displaystyle\nabla_{{\dot{\beta}}}{}^{\beta}\mathfrak{W}_{\beta\alpha(m+n-1)}(\phi)=0\qquad$
$\displaystyle\Longrightarrow$
$\displaystyle\qquad\big{(}\mathcal{Q}-\tau_{(1,m,n)}\mu\bar{\mu}\big{)}\mathfrak{W}_{\alpha(m+n)}(\phi)=0~{},$
(4.24a)
$\displaystyle\nabla_{\beta}{}^{\dot{\beta}}\bar{\mathfrak{W}}_{{\dot{\beta}}{\dot{\alpha}}(m+n-1)}(\phi)=0\qquad$
$\displaystyle\Longrightarrow$
$\displaystyle\qquad\big{(}\mathcal{Q}-\tau_{(1,m,n)}\mu\bar{\mu}\big{)}\bar{\mathfrak{W}}_{{\dot{\alpha}}(m+n)}(\phi)=0~{}.$
(4.24b)

The latter constraints are consistent with (4.11b) only if the pseudo-masses
obey (4.21).

The actions describing massless spin-$s$ and spin-$(s+\frac{1}{2})$ particles
in AdS4 were computed in the seminal works [29] and [30],
respectively.101010See [110] for the corresponding actions in two-component
spinor notation. On the equations of motion, it can be shown that the only
surviving field strengths for the massless spin-$s$ theory are (4.24) for the
cases $m=n=s$. While for the massless spin-$(s+\frac{1}{2})$ model, the non-
vanishing field strengths are (4.24a) with $n=m-1=s$ and (4.24b) with
$m=n-1=s$. As was shown in [110], these field strengths satisfy the
constraints on the equations of motion

$\displaystyle\nabla_{{\dot{\beta}}}{}^{\beta}\mathfrak{W}_{\beta\alpha(2s-1)}(\phi)$
$\displaystyle=0~{},$
$\displaystyle\nabla_{\beta}{}^{\dot{\beta}}\bar{\mathfrak{W}}_{{\dot{\beta}}{\dot{\alpha}}(2s-1)}(\phi)$
$\displaystyle=0~{},$ (4.25a)
$\displaystyle\nabla_{{\dot{\beta}}}{}^{\beta}\mathfrak{W}_{\beta\alpha(2s)}(\phi)$
$\displaystyle=0~{},$
$\displaystyle\nabla_{\beta}{}^{\dot{\beta}}\bar{\mathfrak{W}}_{{\dot{\beta}}\alpha(2s)}(\phi)$
$\displaystyle=0~{},$ (4.25b)

which are exactly the on-shell conditions (4.24).

#####  4.1.2.2 Partially massless fields

Given two positive integers $m$ and $n$, the tensor field
$\phi^{(t)}_{\alpha(m){\dot{\alpha}}(n)}$ is said to be partially massless
with depth-$t$ if it satisfies the on-shell conditions (4.11) such that its
pseudo-mass takes the special value [94, 92, 93, 90]

$\displaystyle\rho^{2}=\tau_{(t,m,n)}\mu\bar{\mu}~{},\qquad 1\leq
t\leq\text{min}(m,n)~{},$ (4.26)

where the constants $\tau_{(t,m,n)}$ are the partially massless values defined
by

$\displaystyle\tau_{(t,m,n)}:=\frac{1}{2}\Big{(}(m+n-t+3)(m+n-t-1)+(t-1)(t+1)\Big{)}~{}.$
(4.27)

The attributing feature of partially massless fields is that, for a fixed
depth $t$, the system of equations (4.11) with $\rho^{2}$ (4.26) admit a
depth-$t$ gauge symmetry of the form

$\displaystyle\delta_{\zeta}\phi^{(t)}_{\alpha(m){\dot{\alpha}}(n)}=\nabla_{(\alpha_{1}({\dot{\alpha}}_{1}}\cdots\nabla_{\alpha_{t}{\dot{\alpha}}_{t}}\zeta^{(t)}_{\alpha_{t+1}\dots\alpha_{m}){\dot{\alpha}}_{t+1}\dots\dot{\alpha}_{n})}~{}.$
(4.28)

This is true as long as the gauge parameter
$\zeta^{(t)}_{\alpha(m-t){\dot{\alpha}}(n-t)}$ is also on-shell with the same
pseudo-mass,

$\displaystyle\big{(}\mathcal{Q}-\tau_{(t,m,n)}\mu\bar{\mu}\big{)}\zeta^{(t)}_{\alpha(m-t){\dot{\alpha}}(n-t)}$
$\displaystyle=0~{},$ (4.29a)
$\displaystyle\nabla^{\beta{\dot{\beta}}}\zeta^{(t)}_{\beta\alpha(m-t-1){\dot{\beta}}{\dot{\alpha}}(n-t-1)}$
$\displaystyle=0~{}.$ (4.29b)

Note that strictly massless fields (4.21) carry depth $t=1$.

On account of the inequality

$\displaystyle\tau_{(1,m,n)}>\tau_{(t,m,n)}~{},\qquad 2\leq
t\leq\text{min}(m,n)~{},$ (4.30)

it follows from (4.19) that true partially massless representations are non-
unitary. In particular, this means that there are two minimal energy values,

$\displaystyle(E_{0})_{\pm}=\frac{3}{2}\pm\frac{1}{2}(m+n-2t+1)~{},$ (4.31)

which are equally valid since they both violate the unitarity bound. In this
chapter, whenever this ambiguity arises, we always implicitly choose the
positive branch, $E_{0}\equiv(E_{0})_{+}$. To distinguish the partially
massless fields, we denote true partially massless fields as those with depth
$2\leq t\leq\text{min}(m,n)$ and are thus non-unitary. To distinguish the true
partially-massless representations with depth $t$ and Lorentz type
$(\frac{m}{2},\frac{n}{2})$, we will employ the notation

$\displaystyle P\big{(}t,m,n\big{)}~{},\qquad 2\leq t\leq\text{min}(m,n)~{}.$
(4.32)

Such a representation carries minimal energy $E_{0}=\frac{1}{2}(m+n)-t+2$.

It is common in the higher-spin literature to define the partially massless
on-shell conditions in terms of the AdS4 d’Alembertian $\Box$, rather than the
Casimir operator $\mathcal{Q}$. Converting to these conventions, we find that
the conditions (4.11b) and (4.29a) take the respective forms

$\displaystyle\big{(}\Box-\big{[}(m-t)(n-t)-(t+2)\big{]}\mu\bar{\mu}\big{)}\phi^{(t)}_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=0~{},$ (4.33a)
$\displaystyle\big{(}\Box-(mn+t-2)\mu\bar{\mu}\big{)}\zeta^{(t)}_{\alpha(m-t){\dot{\alpha}}(n-t)}$
$\displaystyle=0~{}.$ (4.33b)

#####  4.1.2.3 Massive fields

An on-shell field (4.11) is said to be massive if its pseudo-mass is
constrained by

$\displaystyle\rho^{2}>\tau_{(1,m,n)}\mu\bar{\mu}\quad\implies\quad\rho_{\text{phys}}^{2}>0~{},$
(4.34)

but is otherwise arbitrary. This restriction ensures the unitarity of the
representation.

It must be noted that there also exists massive fields which carry negative
physical mass $\rho_{\text{phys}}^{2}<0$, given that $\rho_{\text{phys}}^{2}$
does not coincide with the partially massless modes
$\rho^{2}\neq\tau_{(t,m,n)}\mu\bar{\mu}$, where $2\leq t\leq\text{min}(m,n)$.
In accordance with (4.19), such a field will realise a non-unitary irreducible
representation of $\mathfrak{so}(3,2)$.

####  4.1.3 Spin projection operators

In this section we review, and expand upon, the spin projection operators in
AdS4. Let us denote by $V_{(m,n)}$ the space of tensor fields
$\phi_{\alpha(m){\dot{\alpha}}(n)}$ of Lorentz type
$(\frac{m}{2},\frac{n}{2})$ in AdS4. For integers $m,n\geq 2$, we define the
spin projection operator $\Pi^{\perp}_{(m,n)}$ by its action on $V_{(m,n)}$
via the rule

$\displaystyle\Pi^{\perp}_{(m,n)}:V_{(m,n)}$ $\displaystyle\longrightarrow$
$\displaystyle V_{(m,n)}~{},$ (4.35a) $\displaystyle\phi_{\alpha(n)}$
$\displaystyle\longmapsto$
$\displaystyle\Pi^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}=:\Pi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}(\phi)~{},$
(4.35b)

For fixed $m$ and $n$, the operator $\Pi^{\perp}_{(m,n)}$ is defined by the
following properties:

  1. 1.

Idempotence: The operator $\Pi^{\perp}_{(m,n)}$ squares to itself on
$V_{(m,n)}$,

$\Pi^{\perp}_{(m,n)}\Pi^{\perp}_{(m,n)}=\Pi^{\perp}_{(m,n)}~{}.$ (4.36a)

  2. 2.

Transversality: The operator $\Pi^{\perp}_{(m,n)}$ maps
$\phi_{\alpha(m){\dot{\alpha}}(n)}$ to a transverse field,

$\nabla^{\beta{\dot{\beta}}}\Pi^{\perp}_{\beta\alpha(m-1){\dot{\beta}}{\dot{\alpha}}(n-1)}(\phi)=0~{}.$
(4.36b)

The spin projection operator $\Pi^{\perp}_{(m,n)}$ maps a field
$\phi_{\alpha(m){\dot{\alpha}}(n)}$ on $V_{(m,n)}$, where
$\phi_{\alpha(m){\dot{\alpha}}(n)}$ is on the mass-shell (4.11b), to an on-
shell field (4.11)

$\displaystyle(\mathcal{Q}-\rho^{2})\Pi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}(\phi)$
$\displaystyle=$ $\displaystyle 0~{},$ (4.37a)
$\displaystyle\nabla^{\beta{\dot{\beta}}}\Pi^{\perp}_{\beta\alpha(m-1){\dot{\beta}}{\dot{\alpha}}(n-1)}(\phi)$
$\displaystyle=$ $\displaystyle 0~{}.$ (4.37b)

Thus the spin projection operator $\Pi^{\perp}_{(m,n)}$ singles out the
component of such a $\phi_{\alpha(m){\dot{\alpha}}(n)}$ which furnishes the
irreducible representation $D(E_{0},s)$ of $\mathfrak{so}(3,2)$.

The spin projection operators take the following form on
$V_{(m,n)}$111111Switching the curvature $\mu\bar{\mu}$ to $-\mu\bar{\mu}$ in
(4.38) yields the projection operators on de-Sitter space [90].

$\displaystyle\Pi^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=$
$\displaystyle\Big{[}\prod_{t=1}^{n}\big{(}\mathcal{Q}-\tau_{(t,m,n)}\mu\bar{\mu}\big{)}\Big{]}^{-1}\mathbb{P}_{\alpha(m){\dot{\alpha}}(n)}(\phi)~{},$
(4.38a)
$\displaystyle\widehat{\Pi}^{\perp}_{(m,n)}\phi_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=$
$\displaystyle\Big{[}\prod_{t=1}^{m}\big{(}\mathcal{Q}-\tau_{(t,m,n)}\mu\bar{\mu}\big{)}\Big{]}^{-1}\widehat{\mathbb{P}}_{\alpha(m){\dot{\alpha}}(n)}(\phi)~{},$
(4.38b)

where the projected fields $\mathbb{P}_{\alpha(m){\dot{\alpha}}(n)}(\phi)$ and
$\widehat{\mathbb{P}}_{\alpha(m){\dot{\alpha}}(n)}(\phi)$ are

$\displaystyle\mathbb{P}_{\alpha(m){\dot{\alpha}}(n)}(\phi)$ $\displaystyle=$
$\displaystyle\nabla_{({\dot{\alpha}}_{1}}{}^{\beta_{1}}\cdots\nabla_{{\dot{\alpha}}_{n})}{}^{\beta_{n}}\nabla_{(\beta_{1}}{}^{{\dot{\beta}}_{1}}\cdots\nabla_{\beta_{n}}{}^{{\dot{\beta}}_{n}}\phi_{\alpha_{1}\ldots\alpha_{m}){\dot{\beta}}(n)}~{},$
(4.39a) $\displaystyle\widehat{\mathbb{P}}_{\alpha(m){\dot{\alpha}}(n)}(\phi)$
$\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\nabla_{\alpha_{m})}{}^{{\dot{\beta}}_{m}}\nabla_{({\dot{\beta}}_{1}}{}^{\beta_{1}}\cdots\nabla_{{\dot{\beta}}_{m}}{}^{\beta_{m}}\phi_{\beta(m){\dot{\alpha}}_{1}\ldots{\dot{\alpha}}_{n})}~{}.$
(4.39b)

It was shown in [90] that the projectors (4.38a) and (4.38b) are equivalent on
$V_{(m,n)}$

$\Pi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}(\phi)=\widehat{\Pi}^{\perp}_{\alpha(m){\dot{\alpha}}(n)}(\phi)~{}.$
(4.40)

In accordance with this, we are free to work with either the spin projection
operator $\Pi^{\perp}_{(m,n)}$ (4.38a) or $\widehat{\Pi}^{\perp}_{(m,n)}$
(4.38b) on ${V}_{(m,n)}$. From this point onwards, we will only consider the
spin projection operator $\Pi_{(m,n)}^{\perp}$ on $V_{(m,n)}$. Furthermore, it
can be shown that $\Pi_{(m,n)}^{\perp}$ acts like the unit operator on the
space of transverse fields

$\nabla^{\beta{\dot{\beta}}}\phi^{\perp}_{\beta\alpha(m-1){\dot{\beta}}{\dot{\alpha}}(n-1)}=0\qquad\Longrightarrow\qquad\Pi_{(m,n)}^{\perp}\phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}=\phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}~{}.$
(4.41)

The projectors (4.38) were recently derived by Kuzenko and Ponds in [90]. They
demonstrated that the parameter $\tau_{(t,m,n)}$, which determines the poles
of the spin projection operators, coincides with the partially massless value
(4.27).121212The partially massless values (4.27) appearing in the denominator
of the projectors (4.38) differ to those given in [90] by a factor of a half.
This is due to the fact that this thesis uses a different normalisation factor
for the AdS4 curvature. This observation provided a new understanding of
partially massless fields. Thus, the spin projection operators provide an
alternative method for computing partially massless values. This result will
be essential in the formulation of partially massless superfields in AdS4|4,
since the partially massless values are unknown.

We would like to point out that $\tau_{(t,m,n)}$ appears in the poles of the
projector (4.38) with $1\leq t\leq\text{max}(m,n)$. In accordance with the
pseudo-mass (4.26) of a partially massless field, it is apparent that the
poles of the projector coincide with partially massless modes for $1\leq
t\leq\text{min}(m,n)$ and additional (non-unitary) massive modes for
$\text{min}(m,n)<t\leq\text{max}(m,n)$. Depending on the integers $m$ and $n$,
one can always choose to work with a particular projector such that its poles
only coincide with the partially massless values (4.27). Specifically, given
$m>n$, it proves natural to work with the projector $\Pi^{\perp}_{(m,n)}$, as
the bounds of the product in (4.38a) includes all values of $t$ within the
range $1\leq t\leq\text{min}(m,n)$. Hence, the poles of the projector only
contains partially massless modes. Using a similar argument, if $n>m$, it is
natural to work with $\widehat{\Pi}^{\perp}_{(m,n)}$ (4.38b).

The spin projection operators (4.38) can be recast solely in terms of the
Casimir operators (4.10a) and (4.10b) of $\mathfrak{so}(3,2)$ as follows

$\displaystyle\tilde{\Pi}^{\perp}_{(m,n)}$ $\displaystyle=$
$\displaystyle\frac{m!}{n!(m+n)!}\prod_{j=0}^{n-1}\frac{\big{(}\mathcal{W}^{2}-(s-j)(s-j-1)[{\cal
Q}-(s-j-2)(s-j+1)\mu\bar{\mu}]\big{)}}{\big{(}{\cal
Q}-\tau_{(j+1,m,n)}\mu\bar{\mu}\big{)}}~{},\hskip 42.67912pt$ (4.42a)
$\displaystyle=$
$\displaystyle\frac{n!}{m!(m+n)!}\prod_{j=0}^{m-1}\frac{\big{(}\mathcal{W}^{2}-(s-j)(s-j-1)[{\cal
Q}-(s-j-2)(s-j+1)\mu\bar{\mu}]\big{)}}{\big{(}{\cal
Q}-\tau_{(j+1,m,n)}\mu\bar{\mu}\big{)}}~{}.$ (4.42b)

The spin projection operator $\tilde{\Pi}^{\perp}_{(m,n)}$ is a novel
realisation of the projectors (4.38) given in [90]. Alike the projection
operators $\Pi_{(m,n)}^{\perp}$, all information concerning partially massless
fields is encoded within the poles of the projectors
$\tilde{\Pi}^{\perp}_{(m,n)}$. Furthermore, it can be shown that the zeros of
$\tilde{\Pi}^{\perp}_{(m,n)}$ actually coincide with the eigenvalues generated
by ${\cal W}^{2}$ when acting on the space of transverse fields
$\phi^{\perp}_{\alpha(m-j){\dot{\alpha}}(n-j)}$, with $1\leq j\leq n$ and
$m\geq n$. It can be shown that this also holds in the case $m\leq n$ with
$1\leq j\leq m$.

One can show that the projection operators (4.42) act as the identity operator
on the space of transverse fields $\phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}$

$\nabla^{\beta{\dot{\beta}}}\phi^{\perp}_{\beta\alpha(m-1){\dot{\beta}}{\dot{\alpha}}(n-1)}=0\qquad\Longrightarrow\qquad\tilde{\Pi}^{\perp}_{(m,n)}\phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}=\phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}~{}.$
(4.43)

The projectors $\tilde{\Pi}^{\perp}_{(m,n)}$ exhibit some non-trivial
properties when acting on fields that belong to spaces other than $V_{(m,n)}$.
Specifically, the projectors $\Pi^{\perp}_{(m,n)}$ and
$\tilde{\Pi}^{\perp}_{(m,n)}$ are no longer equivalent on the space
$V_{(j,k)}$, where $2\leq j\leq m-1$ and $2\leq k\leq n-1$. Additionally,
these operators are no longer idempotent and transverse on $V_{(j,k)}$. It can
be shown that the $\tilde{\Pi}^{\perp}_{(m,n)}$ annihilates any field
belonging to $V_{(j,k)}$

$\tilde{\Pi}_{(m,n)}\phi_{\alpha(j){\dot{\alpha}}(k)}=0~{},$ (4.44)

where $1\leq j\leq m-1$ and $1\leq k\leq n-1$.

#####  4.1.3.1 Longitudinal projectors and lower-spin extractors

The orthogonal complement of $\Pi^{\perp}_{(m,n)}$ is defined on $V_{(m,n)}$
by the rule

$\Pi^{\parallel}_{(m,n)}=\mathds{1}-\Pi^{\perp}_{(m,n)}~{},$ (4.45)

which, by construction, satisfies the properties

$\Pi_{(m,n)}^{\parallel}\Pi_{(m,n)}^{\parallel}=\Pi_{(m,n)}^{\parallel}~{},$
(4.46a)
$\qquad\Pi_{(m,n)}^{\parallel}\Pi_{(m,n)}^{\perp}=\Pi_{(m,n)}^{\perp}\Pi_{(m,n)}^{\parallel}=0~{}.$
(4.46b)

It can be shown that $\Pi_{(m,n)}^{\parallel}$ projects onto the longitudinal
part of $\phi_{\alpha(m){\dot{\alpha}}(n)}$

$\Pi_{(m,n)}^{\parallel}\phi_{\alpha(m){\dot{\alpha}}(n)}=\nabla_{\alpha{\dot{\alpha}}}\phi_{\alpha(m-1){\dot{\alpha}}(n-1)}~{},$
(4.47)

where $\phi_{\alpha(m-1){\dot{\alpha}}(n-1)}$ is an unconstrained field. Due
to properties (4.46a) and (4.47), the operator $\Pi_{(m,n)}^{\parallel}$ is
called the longitudinal projector.

Let
$\phi^{\parallel}_{(m,n)}=\nabla_{\alpha{\dot{\alpha}}}\phi_{\alpha(m-1){\dot{\alpha}}(n-1)}$
be some longitudinal field. Since the spin projection operator
$\tilde{\Pi}^{\perp}_{(m,n)}$ (4.42) annihilates all fields of
rank-$(m-1,n-1)$, it follows that $\tilde{\Pi}^{\perp}_{(m,n)}$ annihilates
any rank-$(m,n)$ longitudinal field

$\tilde{\Pi}^{\perp}_{(m,n)}\phi^{\parallel}_{\alpha(m){\dot{\alpha}}(n)}=0~{}.$
(4.48)

Making use of this result, it can be shown that $\Pi_{(m,n)}^{\parallel}$ acts
like the unit operator on the space of longitudinal fields

$\tilde{\Pi}^{\parallel}_{(m,n)}\phi^{\parallel}_{\alpha(m){\dot{\alpha}}(n)}=0~{}.$
(4.49)

Since the projectors $\Pi^{\perp}_{(m,n)}$ and $\Pi_{(m,n)}^{\parallel}$
resolve the identity operator (4.45), it follows that any unconstrained field
$\phi_{\alpha(m){\dot{\alpha}}(n)}$ on $V_{(m,n)}$ can be decomposed in the
following manner

$\phi_{\alpha(m){\dot{\alpha}}(n)}=\phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}+\nabla_{\alpha{\dot{\alpha}}}\phi_{\alpha(m-1){\dot{\alpha}}(n-1)}~{},$
(4.50)

where $\phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}$ is transverse and
$\phi_{\alpha(m-1){\dot{\alpha}}(n-1)}$ is unconstrained and thus reducible.
Due to properties (4.43) and (4.48), it follows that the spin projection
operator $\Pi^{\perp}_{(m,n)}$ selects out the pure spin $s=\frac{1}{2}(m+n)$
component from the decomposition (4.50). We can continue this prescription
iteratively to obtain the following decomposition

$\phi_{\alpha(m){\dot{\alpha}}(n)}=\sum_{j=0}^{n-1}(\nabla_{\alpha{\dot{\alpha}}})^{j}\phi^{\perp}_{\alpha(m-j){\dot{\alpha}}(n-j)}+(\nabla_{\alpha{\dot{\alpha}}})^{n}\phi_{\alpha(m-n)}~{},$
(4.51)

where we have assumed, without loss of generality, that $m\geq n$. The set of
fields
$\big{\\{}\phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)},\phi^{\perp}_{\alpha(m-1){\dot{\alpha}}(n-1)},\cdots,\phi^{\perp}_{\alpha(m-n+1){\dot{\alpha}}}\big{\\}}$
are transverse and $\phi_{\alpha(m-n)}$ does not satisfy any differential
constraints, thus irreducible.

We can introduce operators which extract the bosonic
$\phi^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k)}$ and fermionic
$\phi^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k-1)}$ fields from the
decomposition (4.51). In particular, the spin-$(s-k)$ and
spin-$(s-k-\frac{1}{2})$ components can be extracted via

$\displaystyle\phi_{\alpha(s){\dot{\alpha}}(s)}$ $\displaystyle\mapsto$
$\displaystyle\phi^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k)}=\big{(}S^{\perp}_{(s-k)}\phi\big{)}_{\alpha(s-k){\dot{\alpha}}(s-k)}\equiv
S^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k)}(\phi)~{},$ (4.52a)
$\displaystyle\phi_{\alpha(s){\dot{\alpha}}(s-1)}$ $\displaystyle\mapsto$
$\displaystyle\phi^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k-1)}=\big{(}S^{\perp}_{(s-k-\frac{1}{2})}\phi\big{)}_{\alpha(s-k){\dot{\alpha}}(s-k-1)}\equiv
S^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k-1)}(\phi)~{},\hskip 17.07182pt$
(4.52b)

where we have defined the operators

$\displaystyle S^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k)}(\phi)$
$\displaystyle=$ $\displaystyle
a_{k}\Big{[}\prod_{j=1}^{k}\big{(}\mathcal{Q}-\tau_{(j,s-k+j,s-k+j)}\mu\bar{\mu}\big{)}\Big{]}^{-1}$
(4.53a)
$\displaystyle\times\big{(}\nabla^{\beta{\dot{\beta}}}\big{)}^{k}\tilde{\Pi}^{\perp}_{(s-k,s-k)}\phi_{\beta(k)\alpha(s-k){\dot{\beta}}(k){\dot{\alpha}}(s-k)}~{},$
$\displaystyle S^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k-1)}(\phi)$
$\displaystyle=$ $\displaystyle
b_{k}\Big{[}\prod_{j=1}^{k}\big{(}\mathcal{Q}-\tau_{(j,s-k+j,s-k+j-1)}\mu\bar{\mu}\big{)}\Big{]}^{-1}$
(4.53b)
$\displaystyle\times\big{(}\nabla^{\beta{\dot{\beta}}}\big{)}^{k}\tilde{\Pi}^{\perp}_{(s-k,s-k-1)}\phi_{\beta(k)\alpha(s-k){\dot{\beta}}(k){\dot{\alpha}}(s-k-1)}~{}.$
The constants $a_{k}$ and $b_{k}$ take the form
$a_{k}=(-1)^{k}\binom{s}{k}^{2}\binom{2s-k+1}{k}^{-1}~{},\quad
b_{k}=(-1)^{k}\binom{s}{k}\binom{s-1}{k}\binom{2s-k}{k}^{-1}~{}.$ (4.53c)

It is clear that the $S^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k)}(\phi)$ and
$S^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k-1)}(\phi)$ are transverse

$\displaystyle\nabla^{\beta{\dot{\beta}}}S^{\perp}_{\beta\alpha(s-k-1){\dot{\beta}}{\dot{\alpha}}(s-k-1)}(\phi)=0~{},\qquad\nabla^{\beta{\dot{\beta}}}S^{\perp}_{\beta\alpha(s-k-1){\dot{\beta}}{\dot{\alpha}}(s-k-2)}(\phi)=0~{}.$
(4.54)

The operators $S^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k)}(\phi)$ and
$S^{\perp}_{\alpha(s-k){\dot{\alpha}}(s-k-1)}(\phi)$ are called the
spin-$(s-k)$ and spin-$(s-k-\frac{1}{2})$ transverse extractors,131313Note
that the extractors (4.53) are not projectors because they are dimensionful.
respectively. These extractors were derived for the first time in this thesis.
In the flat-space limit, the operators (4.53) reduce to (2.87).

####  4.1.4 Conformal higher-spin theory

For integers $m,n\geq 1$, the CHS action in AdS4 is described in terms of a
totally symmetric complex spinor field $h_{\alpha(m){\dot{\alpha}}(n)}$, which
is defined modulo gauge transformations of the form141414For a modern
treatment on CHS theory in a conformally flat background, see [86].

$\delta_{\zeta}h_{\alpha(m){\dot{\alpha}}(n)}=\nabla_{\alpha{\dot{\alpha}}}\zeta_{\alpha(m-1){\dot{\alpha}}(n-1)}~{},$
(4.55)

where the gauge parameter $\zeta_{\alpha(m-1){\dot{\alpha}}(n-1)}$ is complex
unconstrained.

Associated with $h_{\alpha(m){\dot{\alpha}}(n)}$ and its complex conjugate
$\bar{h}_{\alpha(n){\dot{\alpha}}(m)}$ are the linearised higher-spin Weyl
tensors

$\displaystyle\mathfrak{W}_{\alpha(m+n)}(h)$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\nabla_{\alpha_{n}}{}^{{\dot{\beta}}_{n}}h_{\alpha_{n+1}\ldots\alpha_{m+n}){\dot{\beta}}(n)}~{},$
(4.56a) $\displaystyle\mathfrak{W}_{\alpha(m+n)}(\bar{h})$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\nabla_{\alpha_{m}}{}^{{\dot{\beta}}_{m}}\bar{h}_{\alpha_{m+1}\ldots\alpha_{m+n}){\dot{\beta}}(m)}~{},$
(4.56b)

which are invariant under the gauge transformations (4.55)

$\delta_{\zeta}\mathfrak{W}_{\alpha(m+n)}(h)=0~{},\qquad\delta_{\zeta}\mathfrak{W}_{\alpha(m+n)}(\bar{h})=0~{}.$
(4.57)

The Weyl tensors (4.56) are the fundamental building blocks of the gauge-
invariant CHS action, which is given by

$S_{\text{CHS}}^{(m,n)}[h,\bar{h}]={\rm i}^{m+n}\int{\rm
d}^{4}x~{}e~{}{\mathfrak{W}}^{\alpha(m+n)}(h){\mathfrak{W}}_{\alpha(m+n)}(\bar{h})~{}+~{}\text{c.c.}$
(4.58)

Integrating by parts, the action (4.58) can be written in the alternative form

$\displaystyle S_{\text{CHS}}^{(m,n)}[h,\bar{h}]$ $\displaystyle=$
$\displaystyle{\rm i}^{m+n}\int{\rm
d}^{4}x~{}e~{}\bar{h}^{\alpha(n){\dot{\alpha}}(m)}{\mathfrak{B}}_{\alpha(n){\dot{\alpha}}(m)}(h)+{\mathrm{c.c.}}~{},$
(4.59a) $\displaystyle=$ $\displaystyle{\rm i}^{m+n}\int{\rm
d}^{4}x~{}e~{}\bar{h}^{\alpha(n){\dot{\alpha}}(m)}\widehat{\mathfrak{B}}_{\alpha(n){\dot{\alpha}}(m)}({h})+{\mathrm{c.c.}}~{},$
(4.59b)

in which the linearised higher-spin Bach tensors appear

$\displaystyle{\mathfrak{B}}_{\alpha(n){\dot{\alpha}}(m)}(h)$ $\displaystyle=$
$\displaystyle\nabla_{({\dot{\alpha}}_{1}}{}^{\beta_{1}}\cdots\nabla_{{\dot{\alpha}}_{m})}{}^{\beta_{m}}{\mathfrak{W}}_{\alpha(n)\beta(m)}(h)~{},$
(4.60a)
$\displaystyle\widehat{\mathfrak{B}}_{\alpha(n){\dot{\alpha}}(m)}({h})$
$\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\nabla_{\alpha_{n})}{}^{{\dot{\beta}}_{n}}\overline{\mathfrak{W}}_{{\dot{\alpha}}(m){\dot{\beta}}(n)}({h})~{}.$
(4.60b)

The linearised higher-spin Bach tensors (4.60) are descendants of the Weyl
tensors (4.56), which in addition to being gauge-invariant

$\delta_{\zeta}{\mathfrak{B}}_{\alpha(n){\dot{\alpha}}(m)}(h)=0~{},\qquad\delta_{\zeta}\widehat{\mathfrak{B}}_{\alpha(n){\dot{\alpha}}(m)}(\bar{h})=0~{},$
(4.61)

they are also transverse

$\nabla^{\beta{\dot{\beta}}}{\mathfrak{B}}_{\beta\alpha(n-1){\dot{\beta}}{\dot{\alpha}}(m-1)}(h)=0~{},\qquad\nabla^{\beta{\dot{\beta}}}\widehat{\mathfrak{B}}_{\beta\alpha(n-1){\dot{\beta}}{\dot{\alpha}}(m-1)}(\bar{h})=0~{}.$
(4.62)

One can show that the linearised higher-spin Bach-tensors (4.60) are
equivalent [90]

${\mathfrak{B}}_{\alpha(n){\dot{\alpha}}(m)}(h)=\widehat{\mathfrak{B}}_{\alpha(n){\dot{\alpha}}(m)}({h})~{}.$
(4.63)

Since the higher-spin Bach tensors (4.60) are transverse by definition, it is
natural to recast them in terms of the unique transverse spin projection
operators (4.38) [90]

$\displaystyle\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(h)$ $\displaystyle=$
$\displaystyle\prod_{j=1}^{n}\big{(}\mathcal{Q}-\tau_{(j,m,n)}\mu\bar{\mu}\big{)}\big{(}\nabla_{\dot{\alpha}}{}^{\beta}\big{)}^{m-n}\Pi_{(m,n)}^{\perp}h_{\alpha(n)\beta(m-n){\dot{\alpha}}(n)}~{},~{}~{}m\geq
n~{}.$ (4.64a) $\displaystyle{\mathfrak{B}}_{\alpha(n){\dot{\alpha}}(m)}(h)$
$\displaystyle=$
$\displaystyle\prod_{j=1}^{m}\big{(}\mathcal{Q}-\tau_{(j,m,n)}\mu\bar{\mu}\big{)}\big{(}\nabla_{\alpha}{}^{\dot{\beta}}\big{)}^{n-m}\Pi_{(m,n)}^{\perp}h_{\alpha(m){\dot{\beta}}(n-m){\dot{\alpha}}(m)}~{},~{}~{}n\geq
m~{}.\hskip 28.45274pt$ (4.64b)

The Weyl tensors (4.56) can also be realised in terms of the spin projectors
(4.38),

$\displaystyle\mathfrak{W}_{\alpha(m+n)}(h)$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\nabla_{\alpha_{n}}{}^{{\dot{\beta}}_{n}}\Pi^{\perp}_{(m,n)}h_{\alpha_{n+1}\ldots\alpha_{m+n}){\dot{\beta}}(n)}~{},$
(4.65a) $\displaystyle\mathfrak{W}_{\alpha(m+n)}(\bar{h})$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\nabla_{\alpha_{m}}{}^{{\dot{\beta}}_{m}}\Pi^{\perp}_{(n,m)}\bar{h}_{\alpha_{m+1}\ldots\alpha_{m+n}){\dot{\beta}}(m)}~{}.$
(4.65b)

It follows from (4.64) that the CHS action (4.59) can be recast in terms of
the spin projection operators [90]

$\displaystyle S^{(m,n)}_{\text{CHS}}[h,\bar{h}]={\rm i}^{m+n}$
$\displaystyle\int{\rm
d}^{4}x~{}e~{}\bar{h}^{\alpha(n){\dot{\alpha}}(m)}\prod_{j=1}^{n}\big{(}\mathcal{Q}-\tau_{(j,m,n)}\mu\bar{\mu}\big{)}\big{(}\nabla_{\dot{\alpha}}{}^{\beta}\big{)}^{m-n}$
$\displaystyle\times\Pi_{(m,n)}^{\perp}h_{\alpha(n)\beta(m-n){\dot{\alpha}}(n)}+{\mathrm{c.c.}}~{},\qquad
m\geq n~{},$ (4.66a) $\displaystyle S^{(m,n)}_{\text{CHS}}[h,\bar{h}]={\rm
i}^{m+n}$ $\displaystyle\int{\rm
d}^{4}x~{}e~{}\bar{h}^{\alpha(n){\dot{\alpha}}(m)}\prod_{j=1}^{m}\big{(}\mathcal{Q}-\tau_{(j,m,n)}\mu\bar{\mu}\big{)}\big{(}\nabla_{\alpha}{}^{\dot{\beta}}\big{)}^{n-m}$
$\displaystyle\times\Pi_{(m,n)}^{\perp}h_{\alpha(m){\dot{\beta}}(n-m){\dot{\alpha}}(m)}+{\mathrm{c.c.}}~{},\qquad
n\geq m~{}.$ (4.66b)

Note that all the results given above can be expressed in terms of the
projector $\tilde{\Pi}^{\perp}_{(m,n)}$ (4.42), due to the fact that
$\tilde{\Pi}^{\perp}_{(m,n)}$ and ${\Pi}^{\perp}_{(m,n)}$ are equivalent on
$V_{(m,n)}$. This will give rise to novel realisations of the results given in
[90].

In particular, one of the benefits of writing the CHS action (4.58) in terms
of $\tilde{\Pi}^{\perp}_{(m,n)}$ is that they can easily be factorised into
second order differential equations. In the cases $m=n=s$, we find that the
CHS action (4.66) takes the following form,

$S^{(s,s)}_{\text{CHS}}[h]=2(-1)^{s}\int{\rm
d}^{4}x~{}e~{}h^{\alpha(s){\dot{\alpha}}(s)}\prod_{j=1}^{s}\big{(}\mathcal{Q}-\tau_{(j,s,s)}\mu\bar{\mu}\big{)}\tilde{\Pi}^{\perp}_{(s,s)}h_{\alpha(s){\dot{\alpha}}(s)}~{},$
(4.67)

where we have imposed the reality condition
$h_{\alpha(s){\dot{\alpha}}(s)}=\bar{h}_{\alpha(s){\dot{\alpha}}(s)}$.
Explicitly evaluating $\tilde{\Pi}^{\perp}_{(s,s)}$ in (4.67) yields

$\displaystyle S^{(s,s)}_{\text{CHS}}[h]=\frac{2(-1)^{s}}{(2s)!}$
$\displaystyle\int{\rm
d}^{4}x~{}e~{}h^{\alpha(s){\dot{\alpha}}(s)}\prod_{j=0}^{s-1}\big{(}\mathcal{W}^{2}-(s-j)(s-j-1)$
$\displaystyle\times[{\cal
Q}-(s-j-2)(s-j+1)\mu\bar{\mu}]\big{)}h_{\alpha(s){\dot{\alpha}}(s)}~{}.$
(4.68)

Thus the CHS action factorises wholly into second-order differential operators
without having to enter the transverse gauge, which was necessary in the
analysis of [90]. However, for arbitrary $m$ and $n$, it is easy to see that
the CHS actions (4.66), whether expressed in terms of ${\Pi}^{\perp}_{(m,n)}$
or $\tilde{\Pi}^{\perp}_{(m,n)}$, do not completely factorise into products of
second-order operators.

The factorisation of the conformal operators (4.60) was observed long ago in
[91, 201, 202] for the lower-spin values $s=\frac{3}{2}$ and $s=2$. The
factorisation of the higher-spin conformal operators was conjectured in [77,
203] (see also [204]), and later proved by several groups [205, 206, 207, 90].

###  4.2 Four-dimensional ${\cal N}=1$ anti-de Sitter superspace

This section is devoted to the novel formulation of superspin projection
operators in AdS4|4. These operators can be considered supersymmetric
generalisations of the AdS4 spin projection operators of section 4.1.
Additionally, they can also be viewed as AdS4 extensions of the superspin
projection operators in $\mathbb{M}^{4|4}$, which were reviewed in section
2.2. Before studying these superprojectors and their applications, it is first
necessary review the irreducible representations of the ${\cal N}=1$ AdS4
superalgebra $\mathfrak{osp}(1|4)$.

####  4.2.1 Irreducible representations of ${\cal N}=1$ AdS4 superalgebra

The unitary irreducible representations of $\mathfrak{osp}(1|4)$ were first
studied in [208] (see also [193, 194] for a detailed review). They are
conveniently described by their decomposition into irreducible representations
of $\mathfrak{so}(3,2)$ in analogy with the case of the super-Poincaré algebra
in $\mathbb{M}^{4|4}$ (cf. (2.111)). Thus from a field theoretic standpoint,
each term in the decomposition can be identified with a particle (or field)
carrying definite energy and spin. Unlike in $\mathbb{M}^{4}$, particles in
AdS4 belonging to the same supermultiplet do not carry the same energy.

The irreducible representations of $\mathfrak{osp}(1|4)$ are labelled by the
quantum numbers $E_{0}$ and superspin $s$, which we encode in the notation
$\mathfrak{S}(E_{0},s)$. There exist four types of unitary supermultiplets in
AdS4.151515Our notation for the AdS4 supermultiplets follows the one
introduced in [209]. Fronsdal [210] used a different notation for these
representations, $D^{S}(E_{0},s)$.  The first are known as massive
representations and have the following decomposition[208]

$\displaystyle\mathfrak{S}\big{(}{E}_{0},s\big{)}:=D\big{(}E_{0}+\frac{1}{2},s-\frac{1}{2}\big{)}\oplus$
$\displaystyle D\big{(}E_{0},s\big{)}\oplus D\big{(}E_{0}+1,s\big{)}\oplus
D\big{(}E_{0}+\frac{1}{2},s+\frac{1}{2}\big{)}~{},$ (4.69a)
$\displaystyle\quad s>0\,,\quad E_{0}>s+1\,,$ (4.69b)

where the last inequality is the unitarity bound. This decomposition implies
that a massive supermultiplet describes four particles in AdS4.

When $s=0$ the first term in the right-hand side of (4.69a) is absent in the
decomposition and the unitarity bound is also modified. The corresponding
representations are referred to as the Wess-Zumino representations:

$\displaystyle\mathfrak{S}\big{(}{E}_{0},0\big{)}:=D\big{(}E_{0},0\big{)}\oplus
D\big{(}E_{0}+1,0\big{)}\oplus
D\big{(}E_{0}+\frac{1}{2},\frac{1}{2}\big{)}\,,\qquad E_{0}>\frac{1}{2}\,.$
(4.70)

When the unitarity bound is saturated, $E_{0}=s+1$ in eq. (4.69) or
$E_{0}=\frac{1}{2}$ in eq. (4.70), the supermultiplets get shortened. This
yields the other two types of representations.

For $s>0$ the decomposition (4.69) contains fields of spin $s\geq 1$ and the
shortening can be attributed to the appearance of gauge invariance at the
field theoretic level. The corresponding representations are called massless
and are of the form

$\displaystyle\mathfrak{S}\big{(}s+1,s\big{)}:=D\big{(}s+1,s\big{)}\oplus
D\big{(}s+\frac{3}{2},s+\frac{1}{2}\big{)}\,,\qquad s>0\,.$ (4.71)

This implies that a massless supermultiplet in AdS4 consists of two physical
component fields.

Finally, the fourth type of representation occurs for $s=0,\
E_{0}=\frac{1}{2}$ and is called the Dirac supermultiplet [197, 211] (or
super-singleton)

$\displaystyle\mathfrak{S}\big{(}\frac{1}{2},0\big{)}=D\big{(}\frac{1}{2},0\big{)}\oplus
D\big{(}1,\frac{1}{2}\big{)}\,.$ (4.72)

It unifies the bosonic and fermionic singletons.

####  4.2.2 ${\cal N}=1$ AdS superspace

Historically, ${\cal N}=1$ AdS superspace AdS4|4 was originally introduced
[213, 214, 212] as the coset space ${\rm
AdS^{4|4}}:={{\mathsf{OSp}}(1|4)}/{{\mathsf{SO}}(3,1)}$. The same superspace
is equivalently realised as a unique maximally symmetric solution of the two
off-shell formulations for ${\cal N}=1$ AdS supergravity: (i) the well-known
minimal theory (see, e.g., [41, 105] for reviews); and (ii) the more recently
discovered non-minimal theory [215]. Here we prefer to use the second
definition, since it allows us to obtain all information about the geometry of
${\rm AdS}^{4|4}$ from the well-known supergravity results.

As usual, we denote by $z^{M}=(x^{m},\theta^{\mu},\bar{\theta}_{\dot{\mu}})$
the local coordinates of ${\rm AdS}^{4|4}$. The geometry of ${\rm AdS}^{4|4}$
is described in terms of covariant derivatives of the form

$\displaystyle{\cal D}_{A}=({\cal D}_{a},{\cal D}_{\alpha},\bar{\cal
D}^{\dot{\alpha}})=E_{A}+\Omega_{A}~{},\qquad
E_{A}=E_{A}{}^{M}\partial_{M}~{},$ (4.73)

where $E_{A}{}^{M}$ is the inverse superspace vielbein, and

$\displaystyle\Omega_{A}=\frac{1}{2}\,\Omega_{A}{}^{bc}M_{bc}=\Omega_{A}{}^{\beta\gamma}M_{\beta\gamma}+\bar{\Omega}_{A}{}^{{\dot{\beta}}{\dot{\gamma}}}\bar{M}_{{\dot{\beta}}{\dot{\gamma}}}~{},$
(4.74)

is the Lorentz connection. The covariant derivatives ${\cal D}_{A}$ obey the
following graded commutation relations:

$\displaystyle\\{{\cal D}_{\alpha},{\bar{\cal D}}_{{\dot{\alpha}}}\\}$
$\displaystyle=$ $\displaystyle-2{\rm i}{\cal D}_{\alpha{\dot{\alpha}}}~{},$
(4.75a) $\displaystyle\\{{\cal D}_{\alpha},{\cal D}_{\beta}\\}$
$\displaystyle=$
$\displaystyle-4\bar{\mu}M_{\alpha\beta}~{},\qquad\\{{\bar{\cal
D}}_{\dot{\alpha}},{\bar{\cal
D}}_{\dot{\beta}}\\}=4\mu\bar{M}_{{\dot{\alpha}}{\dot{\beta}}}~{},$ (4.75b)
$\displaystyle\,[{\cal D}_{\alpha},{\cal D}_{\beta{\dot{\beta}}}]$
$\displaystyle=$ $\displaystyle{\rm
i}\bar{\mu}\varepsilon_{\alpha\beta}{\bar{\cal
D}}_{{\dot{\beta}}},\qquad\,\hskip 6.25963pt[{\bar{\cal
D}}_{\dot{\alpha}},{\cal D}_{\beta{\dot{\beta}}}]=-{\rm
i}\mu\varepsilon_{{\dot{\alpha}}{\dot{\beta}}}{\cal D}_{\beta},$ (4.75c)
$\displaystyle\,[{\cal D}_{\alpha{\dot{\alpha}}},{\cal
D}_{\beta{\dot{\beta}}}]$ $\displaystyle=$
$\displaystyle-2\bar{\mu}\mu(\varepsilon_{\alpha\beta}\bar{M}_{{\dot{\alpha}}{\dot{\beta}}}+\varepsilon_{{\dot{\alpha}}{\dot{\beta}}}M_{\alpha\beta})~{}.$
(4.75d)

Here $\mu\neq 0$ is a complex parameter, which we recall is related to the
scalar curvature ${\cal R}$ of AdS4 by the rule ${\cal R}=-12\mu\bar{\mu}$.

In what follows, we make extensive use of the quadratic Casimir of
$\mathfrak{osp}(1|4)$, whose realisation on superfields takes the form [54]

$\mathbb{Q}:=\Box+\frac{1}{4}\big{(}\mu\mathcal{D}^{2}+\bar{\mu}\bar{\mathcal{D}}^{2}\big{)}-\mu\bar{\mu}\big{(}M^{\alpha\beta}M_{\alpha\beta}+\bar{M}^{{\dot{\alpha}}{\dot{\beta}}}\bar{M}_{{\dot{\alpha}}{\dot{\beta}}}\big{)}~{},\quad\big{[}\mathbb{Q},\mathcal{D}_{A}\big{]}=0~{}.$
(4.76)

Here $\Box=-\frac{1}{2}{\cal D}^{\alpha{\dot{\alpha}}}{\cal
D}_{\alpha{\dot{\alpha}}}$ is the d’Alembert operator in AdS4|4.

Let us introduce several types of constrained superfields on AdS4|4 which will
be crucial to our subsequent analysis. For integers $m,n\geq 1$, a totally
symmetric complex tensor superfield $\Gamma_{\alpha(m){\dot{\alpha}}(n)}$ is
said to be transverse linear if it satisfies the constraint

$\displaystyle{\bar{\cal
D}}^{\dot{\beta}}{\Gamma}_{\alpha(m){\dot{\beta}}{\dot{\alpha}}(n-1)}$
$\displaystyle=$ $\displaystyle 0\quad\Longrightarrow\quad({\bar{\cal
D}}^{2}-2(n+2)\mu){\Gamma}_{\alpha(m){\dot{\alpha}}(n)}=0~{}.$ (4.77)

On the other hand, the superfield $\Gamma_{\alpha(m){\dot{\alpha}}(n)}$ is
said to be transverse anti-linear if it obeys the constraint

$\displaystyle{\cal D}^{\beta}{\Gamma}_{\beta\alpha(m-1){\dot{\alpha}}(n)}$
$\displaystyle=$ $\displaystyle 0\quad\Longrightarrow\quad({\cal
D}^{2}-2(m+2)\bar{\mu}){\Gamma}_{\alpha(m){\dot{\alpha}}(n)}=0~{}.$ (4.78)

Given integers $m,n\geq 0$, a complex superfield
$G_{\alpha(m){\dot{\alpha}}(n)}$ is said to be longitudinal linear if it
satisfies the constraint

$\displaystyle{\bar{\cal
D}}_{({\dot{\alpha}}_{1}}{G}_{\alpha(m){\dot{\alpha}}_{2}\ldots{\dot{\alpha}}_{n+1})}$
$\displaystyle=$ $\displaystyle 0\quad\Longrightarrow\quad({\bar{\cal
D}}^{2}+2n\mu){G}_{\alpha(m){\dot{\alpha}}(n)}=0~{}.$ (4.79)

Similarly, the complex superfield $G_{\alpha(m){\dot{\alpha}}(n)}$ is said to
be longitudinal anti-linear if it obeys

$\displaystyle{\cal
D}_{(\alpha_{1}}{G}_{\alpha_{2}\dots\alpha_{m+1}){\dot{\alpha}}(n)}$
$\displaystyle=$ $\displaystyle 0\quad\Longrightarrow\quad({\cal
D}^{2}+2m\bar{\mu}){G}_{\alpha(m){\dot{\alpha}}(n)}=0~{}.$ (4.80)

The transverse (anti-)linear and longitudinal (anti-)linear superfields were
first introduced in AdS4|4 by Ivanov and Sorin in [212]. For $n=0$, the first
constraint in (4.77) is not defined, while the latter condition

$({\bar{\cal D}}^{2}-4\mu){\Gamma}_{\alpha(m){\dot{\alpha}}(n)}=0~{},$ (4.81)

defines a linear superfield. Note that the definition for an anti-linear
superfield follows from (4.81) via complex conjugation.

For $n=0$, the constraint (4.79) defines a chiral superfield,

${\bar{\cal
D}}_{{\dot{\alpha}}}G_{\alpha(m)}=0\quad\Longrightarrow\quad{\bar{\cal
D}}^{2}G_{\alpha(m)}=0~{},$ (4.82a) whilst for $m=0$, the condition (4.80)
defines an anti-chiral superfield ${\cal
D}_{\alpha}{G}_{{\dot{\alpha}}(n)}=0\quad\Longrightarrow\quad{\cal
D}^{2}{G}_{{\dot{\alpha}}(n)}=0~{}.$ (4.82b)

Unlike in $\mathbb{M}^{4|4}$, chiral and anti-chiral superfields only exist in
AdS4|4 for the cases $m\geq 0,~{}n=0$ and $n\geq 0,~{}m=0$, respectively. It
is easy to see that the other cases are inconsistent, as a consequence of the
algebra (4.75).

Given an integer $n\geq 1$, a complex superfield
${\mathfrak{V}}_{\alpha(m){\dot{\alpha}}(n)}$ can be uniquely expressed as the
sum of a transverse linear and a longitudinal linear superfield

$\displaystyle\mathfrak{V}_{\alpha(m){\dot{\alpha}}(n)}=$ $\displaystyle-$
$\displaystyle\frac{1}{2\mu(n+2)}{\bar{\cal D}}^{\dot{\beta}}{\bar{\cal
D}}_{({\dot{\beta}}}\mathfrak{V}_{\alpha(m){\dot{\alpha}}_{1}\dots\dot{\alpha}_{n})}-\frac{1}{2\mu(n+1)}{\bar{\cal
D}}_{({\dot{\alpha}}_{1}}{\bar{\cal
D}}^{|{\dot{\beta}}|}\mathfrak{V}_{\alpha(m){\dot{\alpha}}_{2}\dots\dot{\alpha}_{n}){\dot{\beta}}}~{}.~{}~{}~{}$
(4.83)

Note that we have introduced the notation where an index placed between
vertical bars, for example $|{\dot{\beta}}|$, implies that it is not subject
to symmetrisation. Choosing $\mathfrak{V}_{\alpha(m){\dot{\alpha}}(n)}$ to be
transverse linear ($\Gamma_{\alpha(m){\dot{\alpha}}(n)}$) or longitudinal
linear ($G_{\alpha(m){\dot{\alpha}}(n)}$), the above relation gives

$\displaystyle\Gamma_{\alpha(m){\dot{\alpha}}(n)}$ $\displaystyle=$
$\displaystyle{\bar{\cal
D}}^{\dot{\beta}}\xi_{\alpha(m){\dot{\beta}}{\dot{\alpha}}(n)}~{},$ (4.84a)
$\displaystyle G_{\alpha(m){\dot{\alpha}}(n)}$ $\displaystyle=$
$\displaystyle{\bar{\cal
D}}_{({\dot{\alpha}}_{1}}\zeta_{\alpha(m){\dot{\alpha}}_{2}...{\dot{\alpha}}_{n})}~{},$
(4.84b)

for some complex prepotentials $\xi_{\alpha(m){\dot{\alpha}}(n+1)}$ and
$\zeta_{\alpha(m){\dot{\alpha}}(n-1)}$. These relations provide general
solutions to the constraints (4.77) and (4.79) respectively.

Ivanov and Sorin also introduced the projectors $\mathcal{P}^{\perp}_{(n)}$
and $\mathcal{P}^{\parallel}_{(n)}$ in [212] which map the space of
unconstrained superfields $\mathfrak{V}_{\alpha(m){\dot{\alpha}}(n)}$ to the
space of transverse linear (4.77) and longitudinal linear superfields (4.79),
respectively. These operators take the form

$\displaystyle\mathcal{P}^{\perp}_{(n)}$ $\displaystyle=$
$\displaystyle\frac{1}{4(n+1)\mu}({\bar{\cal
D}}^{2}+2n\mu)~{},\qquad\qquad\quad{\bar{\cal
D}}^{\dot{\beta}}\mathcal{P}^{\perp}_{(n)}\mathfrak{V}_{\alpha(m){\dot{\beta}}{\dot{\alpha}}(n-1)}=0~{},$
(4.85a) $\displaystyle\mathcal{P}^{\parallel}_{(n)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4(n+1)\mu}({\bar{\cal
D}}^{2}-2(n+2)\mu)~{},\qquad{\bar{\cal
D}}_{({\dot{\alpha}}_{1}}\mathcal{P}^{\parallel}_{(n)}\mathfrak{V}_{\alpha(m){\dot{\alpha}}_{2}...{\dot{\alpha}}_{n+1})}=0~{},$
(4.85b)

and satisfy the projector properties

$\mathcal{P}^{\perp}_{(n)}\mathcal{P}^{\parallel}_{(n)}=0~{},\qquad\mathcal{P}^{\parallel}_{(n)}\mathcal{P}^{\perp}_{(n)}=0~{},\qquad\mathcal{P}^{\perp}_{(n)}+\mathcal{P}^{\parallel}_{(n)}=\mathds{1}~{}.$
(4.86)

It follows that that any superfield
$\mathfrak{V}_{\alpha(m){\dot{\alpha}}(n)}$ can be uniquely represented as a
sum of transverse linear and longitudinal linear superfields,

$\displaystyle\mathfrak{V}_{\alpha(m){\dot{\alpha}}(n)}=\Gamma_{\alpha(m){\dot{\alpha}}(n)}+G_{\alpha(m){\dot{\alpha}}(n)}~{},$
(4.87)

which is consistent with (4.83). Note that the projectors (4.85) are non-
analytic in $\mu$.

Finally, let us introduce the following operators:

$\displaystyle\mathcal{P}_{(0)}$
$\displaystyle:=-\frac{1}{8\mathbb{Q}}\mathcal{D}^{\beta}\big{(}\bar{\mathcal{D}}^{2}-4\mu\big{)}\mathcal{D}_{\beta}~{},$
(4.88a) $\displaystyle\mathcal{P}_{(+)}$
$\displaystyle:=\frac{1}{16\mathbb{Q}}\big{(}\bar{\mathcal{D}}^{2}-4\mu\big{)}\mathcal{D}^{2}~{},$
(4.88b) $\displaystyle\mathcal{P}_{(-)}$
$\displaystyle:=\frac{1}{16\mathbb{Q}}\big{(}\mathcal{D}^{2}-4\bar{\mu}\big{)}\bar{\mathcal{D}}^{2}~{}.$
(4.88c)

By making use of the identities in appendix 4.A, one may show that when
restricted to the space of scalar superfields, the operators (4.88) satisfy
the projector properties

$\displaystyle\mathds{1}=\sum_{i}\mathcal{P}_{(i)}~{},\qquad\mathcal{P}_{(i)}\mathcal{P}_{(j)}=\delta_{ij}\mathcal{P}_{(j)}~{},$
(4.89)

for $i=0,+,-$. Here $\mathcal{P}_{(+)}$ and $\mathcal{P}_{(-)}$ are the chiral
and anti-chiral projectors, respectively, while $\mathcal{P}_{(0)}$ projects
onto the space of LAL superfields. Using these projectors, it is always
possible to decompose an unconstrained complex scalar superfield
$\mathfrak{V}$ as

$\displaystyle\mathfrak{V}=\mathfrak{L}+\sigma+\bar{\rho}~{},$ (4.90)

where the complex superfield $\mathfrak{L}$ is simultaneously linear and anti-
linear, $\sigma$ is chiral and $\bar{\rho}$ is anti-chiral

$\displaystyle\big{(}\mathcal{D}^{2}-4\bar{\mu}\big{)}\mathfrak{L}=\big{(}\bar{\mathcal{D}}^{2}-4\mu\big{)}\mathfrak{L}=0~{},\qquad\bar{\mathcal{D}}_{{\dot{\alpha}}}\sigma=0~{},\qquad\mathcal{D}_{\alpha}\bar{\rho}=0~{}.$
(4.91)

If $\mathfrak{V}$ is real, then $\mathfrak{L}$ is also real and
$\bar{\rho}=\bar{\sigma}$. In the flat superspace limit $\mu\rightarrow 0$,
the projectors reduce to those constructed by Salam and Strathdee [56].

####  4.2.3 Irreducible superfield representations

Given integers $m,n\geq 1$, a tensor superfield
$\Phi_{\alpha(m){\dot{\alpha}}(n)}(z)\equiv\Phi_{\alpha(m){\dot{\alpha}}(n)}$
on AdS4|4 is said to be on-shell if it satisfies the conditions

$\displaystyle\mathcal{D}^{\beta}\Phi_{\beta\alpha(m-1){\dot{\alpha}}(n)}=0~{},\qquad\bar{\mathcal{D}}^{{\dot{\beta}}}\Phi_{\alpha(m){\dot{\beta}}{\dot{\alpha}}(n-1)}=0~{},$
(4.92a)
$\displaystyle\big{(}\mathbb{Q}-M^{2}\big{)}\Phi_{\alpha(m){\dot{\alpha}}(n)}=0~{}.$
(4.92b)

Hence, an on-shell superfield is simultaneously transverse linear (4.77) and
transverse anti-linear (4.78) (TLAL). It follows from the TLAL condition
(4.92a) that an on-shell superfield is also transverse

${\cal
D}^{\beta{\dot{\beta}}}\Phi_{\beta\alpha(m-1){\dot{\beta}}{\dot{\alpha}}(n-1)}=0~{}.$
(4.93)

By going to components, we will see in section 4.2.4 that an on-shell
superfield $\Phi_{\alpha(m){\dot{\alpha}}(n)}$ (4.92) furnishes the
irreducible representation
$\mathfrak{S}\big{(}E_{0}(M),\frac{1}{2}(m+n+1)\big{)}$ of
$\mathfrak{osp}(1|4)$, where

$\displaystyle
E_{0}(M)=1+\frac{1}{2}\sqrt{4\frac{M^{2}}{\mu\bar{\mu}}-(m+n)(m+n+4)+1}~{}.$
(4.94)

In accordance with this, an on-shell supermultiplet (4.92) is said to carry
superspin $s=\frac{1}{2}(m+n+1)$ and pseudo-mass $M$.

In the case when $m>n=0$ and $n>m=0$, the condition (4.92a) needs to be
changed to

$\displaystyle\mathcal{D}^{\beta}\Phi_{\beta\alpha(m-1)}$
$\displaystyle=0~{},\qquad\big{(}\bar{\mathcal{D}}^{2}-4\mu\big{)}\Phi_{\alpha(m)}=0~{},$
(4.95a)
$\displaystyle\bar{\mathcal{D}}^{{\dot{\beta}}}\bar{\Phi}_{{\dot{\beta}}{\dot{\alpha}}(n-1)}$
$\displaystyle=0~{},\qquad\big{(}\mathcal{D}^{2}-4\bar{\mu}\big{)}\bar{\Phi}_{{\dot{\alpha}}(n)}=0~{},$
(4.95b)

respectively, whilst the mass-shell condition (4.92b) remains the same. In the
scalar case $m=n=0$, the on-shell superfields are discussed in sections
4.2.7.1 and 4.2.7.2.

In analogy with the story in AdS4 (cf. (4.12)), we define the physical mass
$M_{\text{phys}}$ according to the rule

$\displaystyle
M^{2}_{\text{phys}}=M^{2}-\lambda_{(1,m,n)}\mu\bar{\mu}~{},\qquad\lambda_{(1,m,n)}:=\frac{1}{2}\big{(}m+n\big{)}\big{(}m+n+3\big{)}~{},$
(4.96)

where the constant $\lambda_{(1,m,n)}$ is a partially massless value. As
discussed in section 4.1.2, the pseudo-mass of an on-shell field in AdS4 is
subject to a unitarity bound (cf. (4.19)). Since the on-shell supermultiplet
$\Phi_{\alpha(m){\dot{\alpha}}(n)}$ contains a multitude of such fields, this
in turn induces a unitarity bound on the pseudo-mass of
$\Phi_{\alpha(m){\dot{\alpha}}(n)}$:

$M^{2}_{\text{phys}}\geq 0\quad\implies\quad
M^{2}\geq\lambda_{(1,m,n)}\mu\bar{\mu}~{}.$ (4.97)

Recall in section 2.2.2.2, for $m,n\geq 1$, there also existed other
constraint spaces on which tensor superfields realised irreducible
representations of the super-Poincaré algebra. Namely, the space of chiral
(2.162a) and anti-chiral (2.162b) transverse superfields. However for $m,n\geq
1$, there does not exist a direct analogue of such superfields in AdS4|4,
since (anti-)chiral superfields do not exist.

#####  4.2.3.1 Massless supermultiplets

An on-shell supermultiplet (4.92) is said to be massless if it carries the
pseudo-mass

$\displaystyle
M^{2}=\lambda_{(1,m,n)}\mu\bar{\mu}~{},\qquad\lambda_{(1,m,n)}:=\frac{1}{2}(m+n)(m+n+3)~{}.$
(4.98)

It can be shown that the system of equations (4.92), with pseudo-mass (4.98),
is compatible with the gauge symmetry

$\displaystyle\delta_{\zeta,\xi}\Phi_{\alpha(m){\dot{\alpha}}(n)}=\bar{\mathcal{D}}_{({\dot{\alpha}}_{1}}\zeta_{\alpha(m){\dot{\alpha}}_{2}\dots\dot{\alpha}_{n})}+\mathcal{D}_{(\alpha_{1}}\xi_{\alpha_{2}\dots\alpha_{m}){\dot{\alpha}}(n)}~{},$
(4.99a) where the complex gauge parameters
$\zeta_{\alpha(m){\dot{\alpha}}(n-1)}$ and
$\xi_{\alpha(m-1){\dot{\alpha}}(n)}$ are TLAL and satisfy the reality
conditions $\displaystyle{\cal
D}_{(\alpha_{1}}{}^{\dot{\beta}}\xi_{\alpha_{2}\dots\alpha_{m}){\dot{\beta}}{\dot{\alpha}}(n-1)}$
$\displaystyle=$ $\displaystyle\phantom{-}{\rm
i}(n+1)\mu\zeta_{\alpha(m){\dot{\alpha}}(n-1)}~{},$ (4.99b)
$\displaystyle{\cal
D}^{\beta}{}_{({\dot{\alpha}}_{1}}\zeta_{\beta\alpha(m-1){\dot{\alpha}}_{2}\dots\dot{\alpha}_{n})}$
$\displaystyle=$ $\displaystyle-{\rm
i}(m+1)\bar{\mu}\xi_{\alpha(m-1){\dot{\alpha}}(n)}~{}.$ (4.99c)

These on-shell conditions imply that $\zeta_{\alpha(m){\dot{\alpha}}(n-1)}$
and $\xi_{\alpha(m-1){\dot{\alpha}}(n)}$ satisfy the equations

$\displaystyle\big{(}\mathbb{Q}-\lambda_{(1,m,n)}\mu\bar{\mu}\big{)}\zeta_{\alpha(m){\dot{\alpha}}(n-1)}=0~{},\qquad\big{(}\mathbb{Q}-\lambda_{(1,m,n)}\mu\bar{\mu}\big{)}\xi_{\alpha(m-1){\dot{\alpha}}(n)}=0~{}.$
(4.100)

It follows from (4.100) and the gauge variation of (4.92b) that the gauge
parameters are only non-zero if $M$ obeys (4.98). The mass convention (4.96)
was chosen to ensure that a massless gauge superfield in AdS4|4 carries zero
physical mass $M^{2}_{\text{phys}}=0$. In accordance with the unitarity bound,
it follows that massless representations of $\mathfrak{osp}(1|4)$ are unitary.

In the case $m=n=s$, one can consistently impose the reality condition
$H_{\alpha(s){\dot{\alpha}}(s)}:=\Phi_{\alpha(s){\dot{\alpha}}(s)}=\bar{H}_{\alpha(s){\dot{\alpha}}(s)}$,
whereupon the gauge transformations (4.99a) take the form

$\displaystyle\delta_{\zeta}H_{\alpha(s){\dot{\alpha}}(s)}=\bar{\mathcal{D}}_{({\dot{\alpha}}_{1}}\zeta_{\alpha(s){\dot{\alpha}}_{2}\dots\dot{\alpha}_{s})}-\mathcal{D}_{(\alpha_{1}}\bar{\zeta}_{\alpha_{2}\dots\alpha_{s}){\dot{\alpha}}(s)}~{},$
(4.101a) whilst the reality conditions (4.99b) and (4.99c) for the gauge
parameters become $\displaystyle{\cal
D}^{\beta}{}_{({\dot{\alpha}}_{1}}\zeta_{\beta\alpha(s-1){\dot{\alpha}}_{2}\dots\dot{\alpha}_{s})}$
$\displaystyle=\phantom{-}{\rm
i}(s+1)\bar{\mu}\bar{\zeta}_{\alpha(s-1){\dot{\alpha}}(s)}~{},$ (4.101b)
$\displaystyle{\cal
D}_{(\alpha_{1}}{}^{\dot{\beta}}\bar{\zeta}_{\alpha_{2}\dots\alpha_{s}){\dot{\beta}}{\dot{\alpha}}(s-1)}$
$\displaystyle=-{\rm i}(s+1)\mu\zeta_{\alpha(s){\dot{\alpha}}(s-1)}~{}.$
(4.101c)

The presence of the gauge freedom (4.99a) means that the $\mathsf{OSp}(1|4)$
representation realised on the on-shell superfield (4.92) with $M^{2}$ given
by (4.98) is not irreducible. To obtain an irreducible representation, the
space of TLAL superfields (4.92) and (4.98) has to be factorised with respect
to the gauge modes. More specifically, two superfields
$\Phi_{\alpha(m){\dot{\alpha}}(n)}$ and
$\widetilde{\Phi}_{\alpha(m){\dot{\alpha}}(n)}$ are said to be equivalent if
they differ by a gauge transformation (4.99a). The genuine massless
representation is realised on the quotient space of the space of on-shell
superfields (4.92) and (4.98) with respect to this equivalence relation.

Alternatively, the gauge degrees of freedom are automatically eliminated if
one works with the gauge-invariant chiral field strength (4.170a),

$\displaystyle\mathfrak{W}_{\alpha(m+n+1)}(\Phi)$
$\displaystyle=-\frac{1}{4}\big{(}\bar{\mathcal{D}}^{2}-4\mu\big{)}\mathcal{D}_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\mathcal{D}_{\alpha_{n}}{}^{{\dot{\beta}}_{n}}\mathcal{D}_{\alpha_{n+1}}\Phi_{\alpha_{n+2}\dots\alpha_{m+n+1}){\dot{\beta}}(n)}~{},$
(4.102)

instead of the prepotential $\Phi_{\alpha(m){\dot{\alpha}}(n)}$. Assuming the
descendent field $\Phi_{\alpha(m){\dot{\alpha}}(n)}$ is on-shell, it follows
from the constraint (4.92a) that $\mathfrak{W}_{\alpha(m+n+1)}(\phi)$ is
constrained by

$\displaystyle
0=\mathcal{D}^{\beta}\mathfrak{W}_{\beta\alpha(m+n)}(\Phi)\quad\implies\quad
0=\big{(}\mathbb{Q}-\lambda_{(1,m,n)}\mu\bar{\mu}\big{)}\mathfrak{W}_{\alpha(m+n+1)}(\phi)~{},$
(4.103)

where the second relation follows from the chirality of the field strength.
Once again, the latter is consistent with (4.92b) only if $M$ satisfies
(4.98). The description in terms of $\mathfrak{W}_{\alpha(m+n+1)}(\Phi)$
provides the second way to formulate massless dynamics.

Let us return to the gauge-invariant chiral field strength (4.102) for
$n=2s+1$ and $n=2s$, which originate in the massless models for the
superspin-$(s+\frac{1}{2})$ and superspin-$s$ multiplets in AdS4|4 [110],
respectively. As was demonstrated in [110], on the mass shell they satisfy the
equations

$\displaystyle\mathcal{D}^{\beta}\mathfrak{W}_{\beta\alpha(2s)}(\Phi)$
$\displaystyle=$ $\displaystyle 0~{},$ (4.104a)
$\displaystyle\qquad\mathcal{D}^{\beta}\mathfrak{W}_{\beta\alpha(2s-1)}(\Phi)$
$\displaystyle=$ $\displaystyle 0~{},$ (4.104b)

which are exactly the on-shell constraints (4.103).

#####  4.2.3.2 Partially massless supermultiplets

The results presented in this section were obtained by Michael Ponds in [3].

An on-shell supermultiplet $\Phi^{(t)}_{\alpha(m){\dot{\alpha}}(n)}$ (4.92) is
said to be partially massless with super-depth $t$ if it carries the pseudo-
mass

$\displaystyle M^{2}=\lambda_{(t,m,n)}\mu\bar{\mu}~{},\qquad 1\leq
t\leq\text{min}(m+1,n+1)~{},$ (4.105)

where the constants $\lambda_{(t,m,n)}$ are the partially massless values
defined by

$\displaystyle\lambda_{(t,m,n)}$
$\displaystyle=\frac{1}{2}\Big{(}(m+n-t+1)(m+n-t+4)+t(t-1)\Big{)}~{}.$ (4.106)

We will demonstrate in section 4.2.4 that the surviving component fields of an
on-shell partially massless superfield with super-depth $t>1$ are themselves
partially massless with depths $t$ or $t-1$. The reason that we have chosen
the bound on $t$ to be $1\leq t\leq\text{min}(m+1,n+1)$ in the definition
(4.105) is because, as we will see, these are the only values for $t$ which
have a gauge symmetry.

From (4.105), we see that strictly massless superfields carry super-depth
$t=1$. Partially massless supermultiplets whose super-depth lies within the
range $2\leq t\leq\text{min}(m+1,n+1)$ have negative physical mass and
describe non-unitary representations of $\mathsf{OSp}(1|4)$. This may be seen
from the inequality

$\displaystyle\lambda_{(1,m,n)}>\lambda_{(t,m,n)}~{},\qquad 2\leq
t\leq\text{min}(m+1,n+1)~{}.$ (4.107)

To distinguish the latter we may sometimes refer to them as true partially
massless supermultiplets. In analogy with (4.32), we will denote the true
partially massless representation with super-depth $t$ and Lorentz type
$(\frac{m}{2},\frac{n}{2})$ by

$\displaystyle\mathfrak{P}(t,m,n)~{},\qquad 2\leq
t\leq\text{min}(m+1,n+1)~{}.$ (4.108)

According to (4.94), such a representation carries minimal energy
$E_{0}=\frac{1}{2}(m+n+1)-t+2$.

In the previous section it was observed that on-shell, (4.98) is the only
pseudo-mass value compatible with the massless gauge symmetry (4.99a). For
true partially massless supermultiplets the story is considerably more
complicated. This is because the correct gauge symmetry at the superspace
level is not yet known, much less the gauge invariant actions and field
strengths. However, by making use of the superprojectors which we will
introduce in section 4.2.5, it is possible to systematically derive the most
general gauge symmetry compatible with the on-shell conditions (4.92).161616To
emphasise, the partially massless values (4.105), which characterise partially
massless supermultiplets in AdS4|4, were extracted from the superspin
projection operators (4.149). Nothing was known about partially massless
superfields prior to the creation of these superprojectors.  In appendix 4.B
this procedure is carried out in detail for the real supermultiplet
$\Phi_{\alpha{\dot{\alpha}}}$. The results of this analysis, and that of the
supermultiplet $\Phi_{\alpha(2){\dot{\alpha}}(2)}$ obtained by analogy, are
summarised below

Super-depth | $t=1$ | $t=2$ |  $t=3$
---|---|---|---
$\delta\Phi_{\alpha{\dot{\alpha}}}$ | $\bar{\mathcal{D}}_{{\dot{\alpha}}}\zeta_{\alpha}-\mathcal{D}_{\alpha}\bar{\zeta}_{{\dot{\alpha}}}$ | $\mathcal{D}_{\alpha{\dot{\alpha}}}(\sigma+\bar{\sigma})$ | –
$\delta\Phi_{\alpha(2){\dot{\alpha}}(2)}$ | $\bar{\mathcal{D}}_{{\dot{\alpha}}}\zeta_{\alpha(2){\dot{\alpha}}}-\mathcal{D}_{\alpha}\bar{\zeta}_{\alpha{\dot{\alpha}}(2)}$ | $\mathcal{D}_{\alpha{\dot{\alpha}}}\big{(}\bar{\mathcal{D}}_{{\dot{\alpha}}}\xi_{\alpha}-\mathcal{D}_{\alpha}\bar{\xi}_{{\dot{\alpha}}}\big{)}$ | $\mathcal{D}_{\alpha{\dot{\alpha}}}\mathcal{D}_{\alpha{\dot{\alpha}}}(\eta+\bar{\eta})$
Table 4.2.1: Gauge symmetry for lower-rank PM supermultiplets.

In table 4.2.1, the most general gauge symmetries for the aforementioned
supermultiplets are given. In both cases it turns out that a gauge symmetry is
present only when the pseudo-mass takes the values specified in (4.105). The
gauge parameters are on-shell in the sense that they each possess the same
pseudo-mass as its parent gauge field and that they satisfy certain
irreducibility constraints. Specifically, in the super-depth $t=1$ case, the
gauge parameter $\zeta_{\alpha}$ in table 4.2.1 is simultaneously linear
(4.81) and transverse anti-linear (4.78) (LTAL), while
$\zeta_{\alpha(2){\dot{\alpha}}}$ is TLAL. They also obey the reality
conditions (4.101b) and (4.101c). The gauge parameter $\xi_{\alpha}$, which
corresponds to $\Phi_{\alpha(2){\dot{\alpha}}(2)}$ with super-depth $t=2$, is
LTAL and obeys the reality condition

$\displaystyle\mathcal{D}_{{\dot{\alpha}}}{}^{\beta}\xi_{\beta}=3\text{i}\bar{\mu}\bar{\xi}_{{\dot{\alpha}}}~{}.$
(4.109)

In the case of maximal super-depth, the gauge parameters $\sigma$ and $\eta$
are chiral and obey the reality conditions

$\displaystyle-\frac{1}{4}\big{(}\mathcal{D}^{2}-4\bar{\mu}\big{)}\sigma+2\bar{\mu}\bar{\sigma}$
$\displaystyle=$ $\displaystyle 0~{},$ (4.110a)
$\displaystyle-\frac{1}{4}\big{(}\mathcal{D}^{2}-4\bar{\mu}\big{)}\eta+3\bar{\mu}\bar{\eta}$
$\displaystyle=$ $\displaystyle 0~{}.$ (4.110b)

These conditions are the equations of motion which follow from various Wess-
Zumino models (c.f. (4.187)).

The method used to derive the above results (see appendix 4.B for the details)
is quite general in that it deduces all types of gauge symmetry an on-shell
supermultiplet can possess, and at which mass values they appear. However, for
higher-rank multiplets it quickly becomes infeasible due to the computational
expense. Nevertheless, from the results of the above analysis, we are now in a
position to make an ansatz for the gauge transformations of the half-integer
superspin-$(s+\frac{1}{2})$ multiplets
$\Phi^{(t)}_{\alpha(s){\dot{\alpha}}(s)}$.171717The gauge transformations for
partially massless superfields when $m\neq n$ have yet to be derived.  For
true partially-massless multiplets with super-depth $2\leq t\leq s+1$, they
are given by181818Interestingly, the higher-depth gauge transformations
(4.111a) are different to those proposed for the generalised superconformal
multiplets in [216]. The latter transformations possess the same functional
form as the second group of terms in (4.167).

$\displaystyle 2\leq t\leq s~{}$
$\displaystyle:~{}\qquad\delta_{\xi}\Phi^{(t)}_{\alpha(s){\dot{\alpha}}(s)}=\big{(}\mathcal{D}_{\alpha{\dot{\alpha}}}\big{)}^{t-1}\big{(}\bar{\mathcal{D}}_{{\dot{\alpha}}}\xi_{\alpha(s-t+1){\dot{\alpha}}(s-t)}-\mathcal{D}_{\alpha}\bar{\xi}_{\alpha(s-t){\dot{\alpha}}(s-t+1)}\big{)}~{},$
(4.111a) $\displaystyle t=s+1~{}$
$\displaystyle:~{}\qquad\delta_{\sigma}\Phi^{(s+1)}_{\alpha(s){\dot{\alpha}}(s)}=\big{(}\mathcal{D}_{\alpha{\dot{\alpha}}}\big{)}^{s}\big{(}\sigma+\bar{\sigma}\big{)}~{}.$
(4.111b)

The system of equations (4.92) and (4.105) is invariant under the
transformations (4.111a) only if the gauge parameters are TLAL and satisfy the
reality conditions

$\displaystyle{\cal
D}^{\beta}{}_{{\dot{\alpha}}}\xi_{\beta\alpha(s-t){\dot{\alpha}}(s-t)}$
$\displaystyle=\phantom{-}{\rm
i}(s+1)\bar{\mu}\bar{\xi}_{\alpha(s-t){\dot{\alpha}}(s-t+1)}~{},$ (4.112a)
$\displaystyle{\cal
D}_{\alpha}{}^{{\dot{\beta}}}\bar{\xi}_{\alpha(s-t){\dot{\alpha}}(s-t){\dot{\beta}}}$
$\displaystyle=-{\rm i}(s+1)\mu\xi_{\alpha(s-t+1){\dot{\alpha}}(s-t)}~{}.$
(4.112b)

The same is true for the gauge transformations (4.111b) given that $\sigma$ is
chiral, $\bar{\mathcal{D}}_{{\dot{\alpha}}}\sigma=0$, and that it obeys the
conditions

$\displaystyle-\frac{1}{4}\big{(}\mathcal{D}^{2}-4\bar{\mu}\big{)}\sigma+(s+1)\bar{\mu}\bar{\sigma}$
$\displaystyle=0~{},$ (4.113a)
$\displaystyle-\frac{1}{4}\big{(}\bar{\mathcal{D}}^{2}-4\mu\big{)}\bar{\sigma}+(s+1)\mu\sigma$
$\displaystyle=0~{}.$ (4.113b)

It may be shown that (4.112) and (4.113) imply that each gauge parameter
satisfies the same mass-shell equation as its parent gauge field, as required.
We note that upon substituting $t=1$ into (4.111a) and (4.112), one recovers
the massless gauge transformations (4.101a) with
$\xi_{\alpha(s){\dot{\alpha}}(s-1)}\equiv\zeta_{\alpha(s){\dot{\alpha}}(s-1)}$.

#####  4.2.3.3 Massive supermultiplets

An on-shell superfield (4.92) is said to be massive if it has a pseudo-mass
that satisfies

$\displaystyle M^{2}\geq\lambda_{(1,m,n)}\mu\bar{\mu}~{},$ (4.114)

but is otherwise arbitrary. By virtue of the unitarity bound (4.97), this
definition ensures that the physical mass is positive and that the
corresponding representation of $\mathsf{OSp}(1|4)$ is unitary. Note that we
can also introduce (non-unitary) massive fields in AdS4|4, which carry pseudo-
mass $M^{2}\leq\lambda_{(1,m,n)}\mu\bar{\mu}$ such that ,
$M^{2}\neq\lambda_{(t,m,n)}\mu\bar{\mu}$, for $2\leq
t\leq\text{max}(m+1,n+1)$. However, we will not be interested in these types
of massive supermultiplets.

Given integers $m,n\geq 1$, with $m+n\equiv 2s$, we denote by ${\cal
L}_{(m,n)}^{[M]}$ the space of on-shell superfields (4.92). The following
proposition holds: Provided the pseudo-mass satisfies

$\displaystyle M^{2}\neq\lambda_{(t,m,n)}~{},\qquad 1\leq
t\leq\text{max}(m+1,n+1)~{},$ (4.115)

the $\mathfrak{osp}(1|4)$ representations on the functional spaces ${\cal
L}_{(2s,0)}^{[M]},~{}{\cal L}_{(2s-1,1)}^{[M]},\dots,{\cal
L}_{(1,2s-1)}^{[M]},~{}{\cal L}_{(0,2s)}^{[M]}$ are equivalent to
$\mathfrak{S}\big{(}E_{0}(M),s+\frac{1}{2}\big{)}$, where $E_{0}(M)$ is given
by (4.94).

To prove the above claim, we consider an arbitrary superfield
$\Phi_{\alpha(m){\dot{\alpha}}(n)}\in{\cal L}_{(m,n)}^{[M]}$ and associate
with it the following descendants

$\displaystyle\Phi_{\alpha(m){\dot{\alpha}}(n)}~{}$ $\displaystyle\to$
$\displaystyle~{}\psi_{\alpha(m+1){\dot{\alpha}}(n-1)}:={\cal
D}_{\alpha_{m+1}}{}^{\dot{\beta}}\Phi_{\alpha(m){\dot{\beta}}{\dot{\alpha}}(n-1)}~{},$
(4.116a) $\displaystyle\Phi_{\alpha(m){\dot{\alpha}}(n)}~{}$
$\displaystyle\to$
$\displaystyle~{}\chi_{\alpha(m-1){\dot{\alpha}}(n+1)}:={\cal
D}^{\beta}{}_{{\dot{\alpha}}_{n+1}}\Phi_{\beta\alpha(m-1){\dot{\alpha}}(n)}~{}.$
(4.116b)

It is obvious that $\psi_{\alpha(m+1){\dot{\alpha}}(n-1)}$ is completely
symmetric in its undotted indices, while
$\chi_{\alpha(m-1){\dot{\alpha}}(n+1)}$ is completely symmetric in its dotted
indices. The descendant $\psi_{\alpha(m+1){\dot{\alpha}}(n-1)}$ proves to obey
the conditions (4.92) if $n\neq 1$. In the $n=1$ case,
$\psi_{\alpha(m+1){\dot{\alpha}}(n-1)}$ is an on-shell superfield of the type
(4.95a). The descendant $\chi_{\alpha(m-1){\dot{\alpha}}(n+1)}$ has analogous
properties. Therefore the relations (4.116a) and (4.116b) define linear
mappings from ${\cal L}_{(m,n)}^{[M]}$ to ${\cal L}_{(m+1,n-1)}^{[M]}$ and
${\cal L}_{(m-1,n+1)}^{[M]}$, respectively.

Making use of the identities (4.268) leads to the relations

$\displaystyle{\cal D}^{\beta}{}_{\dot{\gamma}}{\cal
D}_{\beta}{}^{\dot{\beta}}\Phi_{\alpha(m){\dot{\beta}}{\dot{\alpha}}(n-1)}$
$\displaystyle=$
$\displaystyle\big{(}M^{2}-\lambda_{(n+1,m,n)}\mu\bar{\mu}\big{)}\Phi_{\alpha(m){\dot{\gamma}}{\dot{\alpha}}(n-1)}~{},$
(4.117a) $\displaystyle{\cal D}_{\gamma}{}^{\dot{\beta}}{\cal
D}^{\beta}{}_{\dot{\beta}}\Phi_{\beta\alpha(m-1){\dot{\alpha}}(n)}$
$\displaystyle=$
$\displaystyle\big{(}M^{2}-\lambda_{(m+1,m,n)}\mu\bar{\mu}\big{)}\Phi_{\gamma\alpha(m-1){\dot{\alpha}}(n)}~{}.$
(4.117b)

In conjunction with the identities

$\displaystyle\lambda_{(n+1,m,n)}$ $\displaystyle=$
$\displaystyle\lambda_{(m+2,m+1,n-1)}~{},\qquad\lambda_{(m+1,m,n)}=\lambda_{(n+2,m-1,n+1)}~{},$
(4.118)

the relations (4.117) tell us that the linear transformations (4.116a) and
(4.116b) are one-to-one and onto, as long as
$M^{2}\neq\lambda_{(n+1,m,n)}\mu\bar{\mu}$ and
$M^{2}\neq\lambda_{(m+1,m,n)}\mu\bar{\mu}$.

Let us introduce the linear maps

$\displaystyle\Delta_{(m,n)}:{\cal L}_{(m,n)}^{[M]}\to{\cal
L}_{(m+1,n-1)}^{[M]}~{},\qquad\widetilde{\Delta}_{(m,n)}:{\cal
L}_{(m,n)}^{[M]}\to{\cal L}_{(m-1,n+1)}^{[M]}~{},$ (4.119)

defined by

$\displaystyle\Phi_{\alpha(m){\dot{\alpha}}(n)}~{}$ $\displaystyle\to$
$\displaystyle~{}\Phi_{\alpha(m+1){\dot{\alpha}}(n-1)}:=\big{(}\Delta_{(m,n)}\big{)}_{\alpha_{m+1}}{}^{\dot{\beta}}\Phi_{\alpha(m){\dot{\beta}}{\dot{\alpha}}(n-1)}~{},$
(4.120a) $\displaystyle\Phi_{\alpha(m){\dot{\alpha}}(n)}~{}$
$\displaystyle\to$
$\displaystyle~{}\Phi_{\alpha(m-1){\dot{\alpha}}(n+1)}:=\big{(}\widetilde{\Delta}_{(m,n)}\big{)}^{\beta}{}_{{\dot{\alpha}}_{n+1}}\Phi_{\beta\alpha(m-1){\dot{\beta}}{\dot{\alpha}}(n)}~{},$
(4.120b)

where we have introduced the dimensionless operators

$\displaystyle\big{(}\Delta_{(m,n)}\big{)}_{\alpha{\dot{\alpha}}}=\frac{\mathcal{D}_{\alpha{\dot{\alpha}}}}{\sqrt{\mathbb{Q}-\lambda_{(n+1,m,n)}\mu\bar{\mu}}}~{},\qquad\big{(}\widetilde{\Delta}_{(m,n)}\big{)}_{\alpha{\dot{\alpha}}}=\frac{\mathcal{D}_{\alpha{\dot{\alpha}}}}{\sqrt{\mathbb{Q}-\lambda_{(m+1,m,n)}\mu\bar{\mu}}}~{}.~{}~{}~{}$
(4.121)

The above relations are equivalent to

$\displaystyle\widetilde{\Delta}_{(m+1,n-1)}\Delta_{(m,n)}\Big{|}_{{\cal
L}_{(m,n)}^{[M]}}={\mathbbm{1}}~{},\qquad\Delta_{(m-1,n+1)}\widetilde{\Delta}_{(m,n)}\Big{|}_{{\cal
L}_{(m,n)}^{[M]}}={\mathbbm{1}}~{}.$ (4.122)

Therefore, the superfields

$\displaystyle\Phi_{\alpha(2s)},~{}\Phi_{\alpha(2s-1){\dot{\alpha}}},~{}\dots,~{}\Phi_{\alpha(m){\dot{\alpha}}(n)},~{}\dots,~{}\Phi_{\alpha{\dot{\alpha}}(2s-1)},~{}\Phi_{{\dot{\alpha}}(2s)}~{},$
(4.123)

are also on-shell (4.92) and hence furnish the representation
$\mathfrak{S}\big{(}E_{0}(M),s+\frac{1}{2}\big{)}$.

In proving the above assertion it was assumed that the pseudo-mass of
$\Phi_{\alpha(m){\dot{\alpha}}(n)}$ satisfies (4.115). This was to ensure that
when $\Phi_{\alpha(m){\dot{\alpha}}(n)}$ is on the mass-shell, the operators
(4.121) are well defined. When $\Phi_{\alpha(m){\dot{\alpha}}(n)}$ is
partially-massless with super-depth $t$, at some stage the maps between tensor
types become ill defined. Nevertheless, in this case it may be shown that the
corresponding representation may be equivalently realised on any of the
functional spaces
$\mathcal{L}^{[M]}_{(m+n-t+1,t-1)},\mathcal{L}^{[M]}_{(m+n-t,t)},\dots,\mathcal{L}^{[M]}_{(t,m+n-t)},\mathcal{L}^{[M]}_{(t-1,m+n-t+1)}$.

In general, $\Phi_{\alpha(m){\dot{\alpha}}(n)}$ and its conjugate
$\bar{\Phi}_{\alpha(n){\dot{\alpha}}(m)}$ describe two equivalent
representations of $\mathfrak{osp}(1|4)$. In order to obtain a single
representation, we need to impose a reality condition on
$\Phi_{\alpha(m){\dot{\alpha}}(n)}$. In the case when $m+n\equiv 2s$ is even,
$\Phi_{\alpha(m){\dot{\alpha}}(n)}$ can represented by
$\Phi_{\alpha(s){\dot{\alpha}}(s)}$ which we require to be real,

$\displaystyle\bar{\Phi}_{\alpha(s){\dot{\alpha}}(s)}=\Phi_{\alpha(s){\dot{\alpha}}(s)}~{}.$
(4.124)

If $m+n\equiv 2s-1$ is odd, $\Phi_{\alpha(m){\dot{\alpha}}(n)}$ can be
represented by $\Phi_{\alpha(s){\dot{\alpha}}(s-1)}$. In this case a reality
condition can be chosen in the form of a Dirac-type pair of equations

$\displaystyle\mathcal{D}_{({\dot{\alpha}}_{1}}{}^{\beta}\Phi_{\beta\alpha(s-1){\dot{\alpha}}_{2}\dots\dot{\alpha}_{s})}$
$\displaystyle={\rm e}^{{\rm
i}\gamma}M^{2}_{(s+1)}\bar{\Phi}_{\alpha(s-1){\dot{\alpha}}(s)}~{},$ (4.125a)
$\displaystyle\mathcal{D}_{(\alpha_{1}}{}^{{\dot{\beta}}}\bar{\Phi}_{\alpha_{2}\dots\alpha_{s}){\dot{\alpha}}(s-1){\dot{\beta}}}$
$\displaystyle={\rm e}^{-{\rm
i}\gamma}M^{2}_{(s+1)}\Phi_{\alpha(s){\dot{\alpha}}(s-1)}~{},$ (4.125b)

where $M^{2}_{(s+1)}:=M^{2}-\lambda_{(s+1,s,s-1)}\mu\bar{\mu}$ and $\gamma$ is
a constant real phase. The pair of equations (4.125) lead to the same mass-
shell equation (4.92).

####  4.2.4 Component analysis of on-shell supermultiplets

We now turn to the component analysis of the on-shell superfield
$\Phi_{\alpha(m){\dot{\alpha}}(n)}$ satisfying the conditions (4.92) with
arbitrary $M^{2}$. For this, we make use of the bar-projection of a tensor
superfield ${\mathfrak{V}}={\mathfrak{V}}(x,\theta,\bar{\theta})$ (with
suppressed indices) in AdS4|4, which is defined in the same way as its
counterpart in $\mathbb{M}^{4|4}$ (cf. (2.123)). The supermultiplet of fields
associated with $\mathfrak{V}$ is defined to consist of all independent fields
contained in $\big{\\{}{\mathfrak{V}}|,{\cal
D}_{\alpha}{\mathfrak{V}}|,\bar{\cal
D}_{{\dot{\alpha}}}{\mathfrak{V}}|,\dots\big{\\}}$.

The covariant derivative of AdS4, $\nabla_{a}$, is related to ${\cal D}_{a}$
according to the rule

$\displaystyle\nabla_{a}{\cal
V}:=(\mathcal{D}_{a}{\mathfrak{V}})|~{},\qquad{\cal V}:={\mathfrak{V}}|~{}.$
(4.126)

In practice, we assume a Wess-Zumino gauge condition to be imposed on the
geometric objects in (4.73) such that the background geometry is purely
bosonic,

$\displaystyle{\cal
D}_{a}|:=E_{a}{}^{M}|\partial_{M}+\frac{1}{2}\,\Omega_{a}{}^{bc}|M_{bc}=e_{a}{}^{m}\partial_{m}+\frac{1}{2}\,\omega_{a}{}^{bc}M_{bc}=\nabla_{a}~{}.$
(4.127)

The supersymmetry transformation of the component fields of $\mathfrak{V}$ is
computed according to the rule (4.277)

Below we make use of the non-supersymmetric AdS quadratic Casimir operator
(4.10a) which is related to the Casimir (4.76) via $\mathcal{Q}{\cal
V}=\Big{[}\Big{(}\mathbb{Q}-\frac{1}{4}\big{(}\mu\mathcal{D}^{2}+\bar{\mu}\bar{\mathcal{D}}^{2}\big{)}\Big{)}{\mathfrak{V}}\Big{]}\Big{|}$.

#####  4.2.4.1 Massive supermultiplets

In general there are four non-vanishing independent complex component fields,

$\displaystyle A_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle:=\Phi_{\alpha(m){\dot{\alpha}}(n)}|~{},$ (4.128a)
$\displaystyle B_{\alpha(m+1){\dot{\alpha}}(n)}$
$\displaystyle:=\mathcal{D}_{\alpha}\Phi_{\alpha(m){\dot{\alpha}}(n)}|~{},$
(4.128b) $\displaystyle C_{\alpha(m){\dot{\alpha}}(n+1)}$
$\displaystyle:=\bar{\mathcal{D}}_{{\dot{\alpha}}}\Phi_{\alpha(m){\dot{\alpha}}(n)}|~{},$
(4.128c) $\displaystyle E_{\alpha(m+1){\dot{\alpha}}(n+1)}$
$\displaystyle:=\bigg{(}\frac{1}{2}\big{[}\mathcal{D}_{\alpha},\bar{\mathcal{D}}_{{\dot{\alpha}}}\big{]}-\text{i}\frac{m-n}{m+n+2}\mathcal{D}_{\alpha{\dot{\alpha}}}\bigg{)}\Phi_{\alpha(m){\dot{\alpha}}(n)}|~{}.$
(4.128d)

They have each been defined in such a way that they are transverse

$\displaystyle 0$
$\displaystyle=\nabla^{\beta{\dot{\beta}}}A_{\beta\alpha(m-1){\dot{\beta}}{\dot{\alpha}}(n-1)}~{},$
(4.129a) $\displaystyle 0$
$\displaystyle=\nabla^{\beta{\dot{\beta}}}B_{\beta\alpha(m){\dot{\beta}}{\dot{\alpha}}(n-1)}~{},$
(4.129b) $\displaystyle 0$
$\displaystyle=\nabla^{\beta{\dot{\beta}}}C_{\beta\alpha(m-1){\dot{\beta}}{\dot{\alpha}}(n)}~{},$
(4.129c) $\displaystyle 0$
$\displaystyle=\nabla^{\beta{\dot{\beta}}}E_{\beta\alpha(m){\dot{\beta}}{\dot{\alpha}}(n)}~{},$
(4.129d)

which may be shown by making use of the on-shell conditions (4.92).
Furthermore, one may show that the bottom and top components satisfy the mass-
shell equations

$\displaystyle 0$
$\displaystyle=\Big{(}\mathcal{Q}-\big{[}M^{2}-\frac{1}{2}(m+n+4)\mu\bar{\mu}\big{]}\Big{)}A_{\alpha(m){\dot{\alpha}}(n)}~{},$
(4.130a) $\displaystyle 0$
$\displaystyle=\Big{(}\mathcal{Q}-\big{[}M^{2}+\frac{1}{2}(m+n)\mu\bar{\mu}\big{]}\Big{)}E_{\alpha(m+1){\dot{\alpha}}(n+1)}~{},$
(4.130b)

whilst the two middle component fields satisfy the differential equations

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\phantom{-}{\rm
i}\mu\nabla_{({\dot{\alpha}}_{1}}{}^{\beta}B_{\beta\alpha(m){\dot{\alpha}}_{2}\dots\dot{\alpha}_{n+1})}+\Big{(}\mathcal{Q}-\big{[}M^{2}-\frac{1}{2}(m-n+3)\mu\bar{\mu}\big{]}\Big{)}C_{\alpha(m){\dot{\alpha}}(n+1)}~{},\hskip
19.91684pt$ (4.131a) $\displaystyle 0$ $\displaystyle=$ $\displaystyle-{\rm
i}\bar{\mu}\nabla_{(\alpha_{1}}{}^{{\dot{\beta}}}C_{\alpha_{2}\dots\alpha_{m+1}){\dot{\alpha}}(n){\dot{\beta}}}+\Big{(}\mathcal{Q}-\big{[}M^{2}-\frac{1}{2}(n-m+3)\mu\bar{\mu}\big{]}\Big{)}B_{\alpha(m+1){\dot{\alpha}}(n)}~{}.~{}~{}~{}$
(4.131b)

The latter imply that only one of the two fields
$B_{\alpha(m+1){\dot{\alpha}}(n)}$ and $C_{\alpha(m){\dot{\alpha}}(n+1)}$ is
independent, and lead to the higher derivative mass-shell equations

$\displaystyle 0$
$\displaystyle=\Big{(}\mathcal{Q}-M^{2}_{+}\Big{)}\Big{(}\mathcal{Q}-M^{2}_{-}\Big{)}B_{\alpha(m+1){\dot{\alpha}}(n)}~{},$
(4.132a) $\displaystyle 0$
$\displaystyle=\Big{(}\mathcal{Q}-M^{2}_{+}\Big{)}\Big{(}\mathcal{Q}-M^{2}_{-}\Big{)}C_{\alpha(m){\dot{\alpha}}(n+1)}~{},$
(4.132b)

where we have denoted

$\displaystyle M^{2}_{\pm}$
$\displaystyle=M^{2}-\mu\bar{\mu}\pm\frac{\mu\bar{\mu}}{2}\sqrt{4\frac{M^{2}}{\mu\bar{\mu}}-(m+n)(m+n+4)+1}~{}.$
(4.133)

Using the relation (4.14), one may confirm that the above component results
are in agreement with the decomposition (4.69),

$\displaystyle\mathfrak{S}\big{(}{E}_{0},s\big{)}=D\big{(}E_{0}+\frac{1}{2},s-\frac{1}{2}\big{)}$
$\displaystyle\oplus D\big{(}E_{0},s\big{)}\oplus
D\big{(}E_{0}+1,s\big{)}\oplus
D\big{(}E_{0}+\frac{1}{2},s+\frac{1}{2}\big{)}~{},$ (4.134)

as dictated by representation theory. Here $s=\frac{1}{2}(m+n+1)$ and
$E_{0}\equiv E_{0}(M)$ is defined according to $\eqref{Energy}$.

#####  4.2.4.2 Partially massless supermultiplets

In this section we restrict our attention to on-shell true partially massless
supermultiplets with super-depth $2\leq t\leq\text{min}(m+1,n+1)$. The
massless case with $t=1$ will be considered separately in the next section.
Upon specifying the pseudo-mass to be given by (4.105), one can show that the
mass-shell equations (4.130) and (4.132) become

$\displaystyle 0$
$\displaystyle=\Big{(}\mathcal{Q}-\tau_{(t-1,m,n)}\mu\bar{\mu}\Big{)}A_{\alpha(m){\dot{\alpha}}(n)}~{},$
(4.135a) $\displaystyle 0$
$\displaystyle=\Big{(}\mathcal{Q}-\tau_{(t-1,m+1,n)}\mu\bar{\mu}\Big{)}\Big{(}\mathcal{Q}-\tau_{(t,m+1,n)}\mu\bar{\mu}\Big{)}B_{\alpha(m+1){\dot{\alpha}}(n)}~{},$
(4.135b) $\displaystyle 0$
$\displaystyle=\Big{(}\mathcal{Q}-\tau_{(t-1,m,n+1)}\mu\bar{\mu}\Big{)}\Big{(}\mathcal{Q}-\tau_{(t,m,n+1)}\mu\bar{\mu}\Big{)}C_{\alpha(m){\dot{\alpha}}(n+1)}~{},$
(4.135c) $\displaystyle 0$
$\displaystyle=\Big{(}\mathcal{Q}-\tau_{(t,m+1,n+1)}\mu\bar{\mu}\Big{)}E_{\alpha(m+1){\dot{\alpha}}(n+1)}~{}.$
(4.135d)

Here $\tau_{(t,m,n)}$ are the non-supersymmetric partially massless values
(4.12), which are related to the supersymmetric ones (4.106) through

$\displaystyle\lambda_{(t,m,n)}=\tau_{(t-1,m,n)}+\frac{1}{2}(m+n+4)~{}.$
(4.136)

In accordance with the discussion of on-shell partially massless fields given
in section 4.1.2.2, the following remarks hold:

  * •

The pair of equations (4.129a) and (4.135a) admits a depth $t_{A}=t-1$ gauge
symmetry

$\displaystyle\delta_{\zeta}A_{\alpha(m){\dot{\alpha}}(n)}=\nabla_{(\alpha_{1}({\dot{\alpha}}_{1}}\cdots\nabla_{\alpha_{t_{A}}{\dot{\alpha}}_{t_{A}}}\zeta_{\alpha_{t_{A}+1}\dots\alpha_{m}){\dot{\alpha}}_{t_{A}+1}\dots\dot{\alpha}_{n})}$
(4.137)

when $2\leq t\leq\text{min}(m,n)$.

  * •

The pair of equations (4.129d) and (4.135d) admits a depth $t_{E}=t$ gauge
symmetry

$\displaystyle\delta_{\xi}E_{\alpha(m+1){\dot{\alpha}}(n+1)}=\nabla_{(\alpha_{1}({\dot{\alpha}}_{1}}\cdots\nabla_{\alpha_{t_{E}}{\dot{\alpha}}_{t_{E}}}\xi_{\alpha_{t_{E}+1}\dots\alpha_{m+1}){\dot{\alpha}}_{t_{E}+1}\dots\dot{\alpha}_{n+1})}$
(4.138)

when $2\leq t\leq\text{min}(m+1,n+1)$.

  * •

Equation (4.129b) and the first branch of (4.135b) admit a depth $t_{B}=t-1$

$\displaystyle\delta_{\rho}B_{\alpha(m+1){\dot{\alpha}}(n)}=\nabla_{(\alpha_{1}({\dot{\alpha}}_{1}}\cdots\nabla_{\alpha_{t_{B}}{\dot{\alpha}}_{t_{B}}}\rho_{\alpha_{t_{B}+1}\dots\alpha_{m+1}){\dot{\alpha}}_{t_{B}+1}\dots\dot{\alpha}_{n})}$
(4.139)

gauge symmetry when $2\leq t\leq\text{min}(m+1,n+1)$.

  * •

Equation (4.129b) and the second branch of (4.135b) admit a depth
$t^{\prime}_{B}=t$

$\displaystyle\delta_{\rho}B_{\alpha(m+1){\dot{\alpha}}(n)}=\nabla_{(\alpha_{1}({\dot{\alpha}}_{1}}\cdots\nabla_{\alpha_{t^{\prime}_{B}}{\dot{\alpha}}_{t^{\prime}_{B}}}\rho_{\alpha_{t^{\prime}_{B}+1}\dots\alpha_{m+1}){\dot{\alpha}}_{t^{\prime}_{B}+1}\dots\dot{\alpha}_{n})}$
(4.140)

gauge symmetry when $2\leq t\leq\text{min}(m+1,n)$.

  * •

Comments similar to those above regarding $B_{\alpha(m+1){\dot{\alpha}}(n)}$,
apply to $C_{\alpha(m){\dot{\alpha}}(n+1)}$.

  * •

The above gauge symmetries hold only for gauge parameters which are transverse
and which satisfy the same mass-shell equation as the parent gauge field.

One can reverse the logic and ask the question: for what values of $M^{2}$
does there appear a gauge symmetry (of any depth) at the component level? The
answer is precisely those values which appear in the superprojectors (4.106)
(though, as above, special care must be taken when deducing the upper and
lower bounds on the range of possible depths).

From the above component analysis one can see that the true partially massless
representation $\mathfrak{P}\big{(}t,m,n\big{)}$ of $\mathfrak{osp}(1|4)$
decomposes into $\mathfrak{so}(3,2)$ subrepresentations as follows

$\mathfrak{P}\big{(}t,m,n\big{)}=P\big{(}t-1,m,n\big{)}\oplus
P\big{(}t-1,m+1,n\big{)}\oplus P\big{(}t,m+1,n\big{)}\oplus
P\big{(}t,m+1,n+1\big{)}~{}.$ (4.141)

For integer ($m=n+1=s$) and half-integer ($m=n=s$) superspin, this
decomposition is in agreement with the results of [217].

#####  4.2.4.3 Massless supermultiplets

To study the component structure of an on-shell massless supermultiplet of
superspin $s>0$, it is advantageous to work with the gauge-invariant field
strength $\mathfrak{W}_{\alpha(2s)}$ defined by (4.170a), instead of the
prepotential $\Phi_{\alpha(m){\dot{\alpha}}(n)}$, where $2s=m+n+1$. The
fundamental properties of $\mathfrak{W}_{\alpha(2s)}$,

$\displaystyle\bar{\cal
D}_{\dot{\beta}}\mathfrak{W}_{\alpha(2s)}=0~{},\qquad\mathcal{D}^{\beta}\mathfrak{W}_{\beta\alpha(2s-1)}=0\quad\Leftrightarrow\quad\big{(}{\cal
D}^{2}-4(s+1)\bar{\mu}\big{)}\mathfrak{W}_{\alpha(2s)}=0~{},$ (4.142)

imply the following equations:

$\displaystyle\mathcal{D}^{\beta{\dot{\beta}}}\mathfrak{W}_{\beta\alpha(2s-1)}$
$\displaystyle=$ $\displaystyle 0~{},$ (4.143a)
$\displaystyle\big{(}\mathbb{Q}-\lambda_{(1,2s-1,0)}\mu\bar{\mu}\big{)}\mathfrak{W}_{\alpha(2s)}$
$\displaystyle=$ $\displaystyle 0~{}.$ (4.143b)

The chiral field strength $\mathfrak{W}_{\alpha(2s)}$ has two independent
component fields

$\displaystyle C_{\alpha(2s)}$ $\displaystyle:=$
$\displaystyle\mathfrak{W}_{\alpha(2s)}|~{},$ (4.144a) $\displaystyle
C_{\alpha(2s+1)}$ $\displaystyle:=$ $\displaystyle{\cal
D}_{(\alpha_{1}}\mathfrak{W}_{\alpha_{2}\dots\alpha_{2s+1})}|~{}.$ (4.144b)

Their properties follow from the equations (4.142) and (4.143)

$\displaystyle\nabla^{\beta{\dot{\beta}}}C_{\beta\alpha(2s-1)}$
$\displaystyle=$ $\displaystyle
0\quad\implies\quad\big{(}\mathcal{Q}-\tau_{(1,2s,0)}\mu\bar{\mu}\big{)}C_{\alpha(2s)}=0~{},$
(4.145a) $\displaystyle\nabla^{\beta{\dot{\beta}}}C_{\beta\alpha(2s)}$
$\displaystyle=$ $\displaystyle
0\quad\implies\quad\big{(}\mathcal{Q}-\tau_{(1,2s+1,0)}\mu\bar{\mu}\big{)}C_{\alpha(2s+1)}=0~{}.$
(4.145b)

These relations tell us that the component field strengths $C_{\alpha(2s)}$
and $C_{\alpha(2s+1)}$ furnish the massless representation
$\mathfrak{S}\big{(}s+1,s\big{)}=D(s+1,s)\oplus
D\big{(}s+\frac{3}{2},s+\frac{1}{2}\big{)}$.

####  4.2.5 Superspin projection operators

Let us denote the space of tensor superfields of Lorentz type
$(\frac{m}{2},\frac{n}{2})$ on AdS4|4 by ${\cal V}_{(m,n)}$. For any integers
$m,n\geq 1$, the superspin projection operator
$\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}$ is defined by its action on ${\cal
V}_{(m,n)}$ according to the rule

$\displaystyle\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}:{\cal V}_{(m,n)}$
$\displaystyle\longrightarrow$ $\displaystyle{\cal V}_{(m,n)}~{},$ (4.146a)
$\displaystyle\Phi_{\alpha(m){\dot{\alpha}}(n)}$ $\displaystyle\longmapsto$
$\displaystyle\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}\Phi_{\alpha(m){\dot{\alpha}}(n)}=:\mbox{\boldmath$\Pi$}^{\perp}_{\alpha(m){\dot{\alpha}}(n)}(\Phi)~{}.$
(4.146b)

For fixed $m$ and $n$, the operator $\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}$ is
defined to satisfy the properties:

  1. 1.

Idempotence: The operator $\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}$ is
idempotent

$\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}\Phi_{\alpha(m){\dot{\alpha}}(n)}=\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}\Phi_{\alpha(m){\dot{\alpha}}(n)}~{}.\\\
$ (4.147a)

  2. 2.

TLAL: The operator $\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}$ maps
$\Phi_{\alpha(m){\dot{\alpha}}(n)}$ to a TLAL superfield

${\cal
D}^{\beta}\mbox{\boldmath$\Pi$}^{\perp}_{\beta\alpha(m-1){\dot{\alpha}}(n)}(\Phi)=0~{},\qquad{\bar{\cal
D}}^{\dot{\beta}}\mbox{\boldmath$\Pi$}^{\perp}_{\alpha(m){\dot{\beta}}{\dot{\alpha}}(n-1)}(\Phi)=0~{}.$
(4.147b)

The superspin projector $\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}$ maps a
superfield on ${\cal V}_{(m,n)}$ which satisfies the mass-shell equation
(4.92b), but is otherwise unconstrained, to an on-shell superfield
(4.92)191919 Off the mass-shell, $\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}$ maps
a superfield to a conformal supercurrent in AdS4|4 [218].

$\displaystyle{\cal
D}^{\beta}\mbox{\boldmath$\Pi$}^{\perp}_{\beta\alpha(m-1){\dot{\alpha}}(n)}(\Phi)=0~{},\qquad{\bar{\cal
D}}^{\dot{\beta}}\mbox{\boldmath$\Pi$}^{\perp}_{\alpha(m){\dot{\beta}}{\dot{\alpha}}(n-1)}(\Phi)=0~{},$
(4.148a)
$\displaystyle(\mathbb{Q}-M^{2})\mbox{\boldmath$\Pi$}^{\perp}_{\alpha(m){\dot{\alpha}}(n)}(\Phi)=0~{}.$
(4.148b)

In other words, the superspin projection operator
$\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}$ selects out the physical component of
an unconstrained superfield on the mass-shell (4.92b), which realises the
irreducible representation
$\mathfrak{S}\big{(}E_{0}(M),\frac{1}{2}(m+n+1)\big{)}$ of
$\mathfrak{osp}(1|4)$.

The superspin projection operators (4.147) take the following form

$\displaystyle\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}\Phi_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle:=\bigg{[}-8\prod_{t=1}^{n+1}\big{(}\mathbb{Q}-\lambda_{(t,m,n)}\mu\bar{\mu}\big{)}\bigg{]}^{-1}\mathbb{P}_{\alpha(m){\dot{\alpha}}(n)}(\Phi)~{},$
(4.149a)
$\displaystyle\widehat{\mbox{\boldmath$\Pi$}}{}^{\perp}_{(m,n)}\Phi_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle:=\bigg{[}\hskip
13.37282pt8\prod_{t=1}^{m+1}\big{(}\mathbb{Q}-\lambda_{(t,m,n)}\mu\bar{\mu}\big{)}\bigg{]}^{-1}\widehat{\mathbb{P}}_{\alpha(m){\dot{\alpha}}(n)}(\Phi)~{},$
(4.149b)

where $\mathbb{P}_{\alpha(m){\dot{\alpha}}(n)}(\Phi)$ and
$\widehat{\mathbb{P}}_{\alpha(m){\dot{\alpha}}(n)}(\Phi)$ are given by

$\displaystyle\mathbb{P}_{\alpha(m){\dot{\alpha}}(n)}(\Phi)$
$\displaystyle={\cal D}_{({\dot{\alpha}}_{1}}{}^{\beta_{1}}\dots\cal
D_{{\dot{\alpha}}_{n})}{}^{\beta_{n}}{\cal D}^{\gamma}({\bar{\cal
D}}^{2}-4\mu){\cal D}_{(\beta_{1}}{}^{{\dot{\beta}}_{1}}\dots\cal
D_{\beta_{n}}{}^{{\dot{\beta}}_{n}}{\cal
D}_{\gamma}\Phi_{\alpha_{1}\dots\alpha_{m}){\dot{\beta}}(n)}~{},$ (4.150a)
$\displaystyle\widehat{\mathbb{P}}_{\alpha(m){\dot{\alpha}}(n)}(\Phi)$
$\displaystyle={\cal D}_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\dots\cal
D_{\alpha_{m})}{}^{{\dot{\beta}}_{m}}{\bar{\cal D}}^{{\dot{\gamma}}}({\cal
D}^{2}-4\bar{\mu}){\cal D}_{({\dot{\beta}}_{1}}{}^{\beta_{1}}\dots\cal
D_{{\dot{\beta}}_{m}}{}^{\beta_{m}}{\bar{\cal
D}}_{\dot{\gamma}}\Phi_{\beta(m){\dot{\alpha}}_{1}\dots\dot{\alpha}_{n})}~{},$
(4.150b)

and $\lambda_{(t,m,n)}$ denotes the partially massless values (4.106). In the
flat superspace limit, the superspin projection operators (4.149) reduce to
those (2.205) in $\mathbb{M}^{4|4}$.

Although not immediately apparent, it can be shown that the two types of
superprojectors (4.149a) and (4.149b) coincide on ${\cal V}_{(m,n)}$

$\displaystyle\mbox{\boldmath$\Pi$}^{\perp}_{\alpha(m){\dot{\alpha}}(n)}(\Phi)=\widehat{\mbox{\boldmath$\Pi$}}{}^{\perp}_{\alpha(m){\dot{\alpha}}(n)}(\Phi)~{}.$
(4.151)

In accordance with this equivalence, we will only consider the superspin
projection operator $\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}$ on ${\cal
V}_{(m,n)}$. Furthermore, it can be shown that
$\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}$ acts like the identity operator on the
space of TLAL superfields $\Phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}$,

${\cal
D}^{\beta}\Phi^{\perp}_{\beta\alpha(m-1){\dot{\alpha}}(n)}=0~{},\quad{\bar{\cal
D}}^{{\dot{\beta}}}\Phi^{\perp}_{\alpha(m){\dot{\beta}}{\dot{\alpha}}(n-1)}=0\quad\Rightarrow\quad\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}\Phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}=\Phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}~{}.$
(4.152)

The superspin projection operators (4.149) encode all the information
necessary to deduce the partially massless supermultiplets in AdS4|4.
Specifically, the poles of these operators coincide with the partially
massless values (4.106). This observation, with the known results in AdS4 (see
section 4.1.3) leads one to conjecture that this relationship between
partially massless (super)fields and (super)spin projection operators is
universal to AdS space, for any dimension, and any number of supersymmetries.
In the subsequent chapter, we will probe the validity of this claim by
studying the (super)spin projection operators in AdS3. We would also like to
point out that (non-unitary) massive modes, characterised by
$\lambda_{(t,m,n)}$ with $\text{min}(m+1,n+1)<t\leq\text{max}(m+1,n+1)$, also
appear in the poles of the superprojectors (4.149).202020For $m>n$, the poles
of the superprojector $\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}$ only contain
partially massless modes $\lambda_{(t,m,n)}$ with $1\leq
t\leq\text{min}(m+1,n+1)$. One arrives at an analogous result for
$\widehat{\mbox{\boldmath$\Pi$}}^{\perp}_{(m,n)}$ when $n>m$.

In the case when $m>n=0$, the superprojectors analogous to (4.149) take the
form

$\displaystyle\mbox{\boldmath$\Pi$}^{\perp}_{(m)}\Phi_{\alpha(m)}\equiv\mbox{\boldmath$\Pi$}^{\perp}_{\alpha(m)}(\Phi)$
$\displaystyle:=\Big{(}\mathbb{Q}-\lambda_{(1,m,0)}\mu\bar{\mu}\Big{)}^{-1}\mathbb{P}_{\alpha(m)}(\Phi)~{},$
(4.153a)
$\displaystyle\widehat{\mbox{\boldmath$\Pi$}}{}^{\perp}_{(m)}\Phi_{\alpha(m)}\equiv\widehat{\mbox{\boldmath$\Pi$}}{}^{\perp}_{\alpha(m)}(\Phi)$
$\displaystyle:=\bigg{[}\prod_{t=1}^{m+1}\big{(}\mathbb{Q}-\lambda_{(t,m,0)}\mu\bar{\mu}\big{)}\bigg{]}^{-1}\widehat{\mathbb{P}}_{\alpha(m)}(\Phi)~{},$
(4.153b)

where $\mathbb{P}_{\alpha(m)}(\Phi)$ and
$\widehat{\mathbb{P}}_{\alpha(m)}(\Phi)$ are given by

$\displaystyle\mathbb{P}_{\alpha(m)}(\Phi)$ $\displaystyle=-\frac{1}{8}{\cal
D}^{\gamma}({\bar{\cal D}}^{2}-4\mu){\cal
D}_{(\gamma}\Phi_{\alpha_{1}\dots\alpha_{m})}~{},$ (4.154a)
$\displaystyle\widehat{\mathbb{P}}_{\alpha(m)}(\Phi)$
$\displaystyle=\frac{1}{8}{\cal
D}_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\dots\cal
D_{\alpha_{m})}{}^{{\dot{\beta}}_{m}}{\bar{\cal D}}^{{\dot{\gamma}}}({\cal
D}^{2}-4\bar{\mu}){\cal D}_{({\dot{\beta}}_{1}}{}^{\beta_{1}}\dots\cal
D_{{\dot{\beta}}_{m}}{}^{\beta_{m}}{\bar{\cal
D}}_{{\dot{\gamma}})}\Phi_{\beta(m)}~{}.$ (4.154b)

Note that we have introduced the notation
$\mbox{\boldmath$\Pi$}^{\perp}_{(m)}:=\mbox{\boldmath$\Pi$}{}^{\perp}_{(m,0)}$
and
$\widehat{\mbox{\boldmath$\Pi$}}{}^{\perp}_{(m)}:=\widehat{\mbox{\boldmath$\Pi$}}{}^{\perp}_{(m,0)}$
in (4.153). The operators (4.153) square to themselves but, in contrast to
(4.149), they project onto the subspace of LTAL superfields,

$\displaystyle\mathcal{D}^{\beta}\mbox{\boldmath$\Pi$}_{\beta\alpha(m-1)}(\Phi)$
$\displaystyle=0~{},\qquad~{}~{}~{}~{}\phantom{..}\mathcal{D}^{\beta}\widehat{\mbox{\boldmath$\Pi$}}{}^{\perp}_{\beta\alpha(m-1)}(\Phi)=0~{},$
(4.155a)
$\displaystyle\big{(}\bar{\mathcal{D}}^{2}-4\mu\big{)}\mbox{\boldmath$\Pi$}^{\perp}_{\alpha(m)}(\Phi)$
$\displaystyle=0~{},\qquad\big{(}\bar{\mathcal{D}}^{2}-4\mu\big{)}\widehat{\mbox{\boldmath$\Pi$}}{}^{\perp}_{\alpha(m)}(\Phi)=0~{}.~{}~{}~{}~{}~{}~{}~{}$
(4.155b)

Once again one may show that the two types of projectors coincide,
$\mbox{\boldmath$\Pi$}^{\perp}_{\alpha(m)}(\Phi)=\widehat{\mbox{\boldmath$\Pi$}}{}^{\perp}_{\alpha(m)}(\Phi)$.
In the case when $n>m=0$, the corresponding projectors
$\overline{\mbox{\boldmath$\Pi$}}{}^{\perp}_{{\dot{\alpha}}(n)}(\bar{\Phi})$
and
$\widehat{\overline{\mbox{\boldmath$\Pi$}}}{}^{\perp}_{{\dot{\alpha}}(n)}(\bar{\Phi})$
can be obtained by complex conjugation and similar comments apply. This time
however they project onto the subspace of simultaneously anti-linear and
transverse linear superfields.

#####  4.2.5.1 Orthogonal complement of
$\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}$

Given $m,n>0$, the orthogonal complement
$\mbox{\boldmath$\Pi$}^{\parallel}_{(m,n)}$ of
$\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}$ is given by

$\mbox{\boldmath$\Pi$}^{\parallel}_{(m,n)}=\mathds{1}-\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}~{},$
(4.156)

whereby construction, the operator $\mbox{\boldmath$\Pi$}^{\parallel}_{(m,n)}$
satisfies the following properties on $\mathcal{V}_{(m,n)}$

$\mbox{\boldmath$\Pi$}^{\parallel}_{(m,n)}\mbox{\boldmath$\Pi$}^{\parallel}_{(m,n)}=\mbox{\boldmath$\Pi$}^{\parallel}_{(m,n)}~{},\qquad\mbox{\boldmath$\Pi$}^{\parallel}_{(m,n)}\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}=\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}\mbox{\boldmath$\Pi$}^{\parallel}_{(m,n)}=0~{}.$
(4.157)

It may be shown that $\mbox{\boldmath$\Pi$}^{\parallel}_{(m,n)}$ acting on
$\Phi_{\alpha(m){\dot{\alpha}}(n)}$ can be expressed as the following sum

$\mbox{\boldmath$\Pi$}^{\parallel}_{(m,n)}\Phi_{\alpha(m){\dot{\alpha}}(n)}=\bar{\mathcal{D}}_{({\dot{\alpha}}_{1}}\Psi_{\alpha(m){\dot{\alpha}}_{2}\dots\dot{\alpha}_{n})}+\mathcal{D}_{(\alpha_{1}}\Omega_{\alpha_{2}\dots\alpha_{m}){\dot{\alpha}}(n)}~{},$
(4.158)

for some unconstrained superfields $\Psi_{\alpha(m){\dot{\alpha}}(n-1)}$ and
$\Omega_{\alpha(m-1){\dot{\alpha}}(n)}$. We see that
$\mbox{\boldmath$\Pi$}^{\parallel}_{(m,n)}$ projects
$\Phi_{\alpha(m){\dot{\alpha}}(n)}$ onto the union of spaces of longitudinal
linear (4.79) and longitudinal anti-linear (4.80) superfields.

Let us introduce the superfield
$\Phi^{\parallel}_{\alpha(m){\dot{\alpha}}(n)}=\bar{\mathcal{D}}_{{\dot{\alpha}}}\Psi_{\alpha(m){\dot{\alpha}}(n-1)}+\mathcal{D}_{\alpha}\Omega_{\alpha(m-1){\dot{\alpha}}(n)}$,
where $\Psi_{\alpha(m){\dot{\alpha}}(n-1)}$ and
$\Omega_{\alpha(m-1){\dot{\alpha}}(n)}$ are complex and unconstrained. We do
not assume $\Phi^{\parallel}_{\alpha(m){\dot{\alpha}}(n)}$ to be in the image
of $\mbox{\boldmath$\Pi$}^{\parallel}_{(m,n)}$. It can be shown that the
superspin projection operators $\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}$
annihilate the superfield $\Phi^{\parallel}_{\alpha(m){\dot{\alpha}}(n)}$

$\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}\Phi^{\parallel}_{\alpha(m){\dot{\alpha}}(n)}=0~{}.$
(4.159)

It follows from (4.156) and (4.159) that
$\mbox{\boldmath$\Pi$}^{\parallel}_{(m,n)}$ acts like the unit operator on
$\Phi^{\parallel}_{\alpha(m){\dot{\alpha}}(n)}$

$\mbox{\boldmath$\Pi$}^{\parallel}\Phi^{\parallel}_{\alpha(m){\dot{\alpha}}(n)}=\Phi^{\parallel}_{\alpha(m){\dot{\alpha}}(n)}~{}.$
(4.160)

These properties will be useful in the subsequent section.

If instead $m>0$ and $n=0$, the orthogonal complement may be written as the
sum of a longitudinal anti-linear superfield and a chiral superfield,

$\displaystyle\mbox{\boldmath$\Pi$}^{\parallel}_{(m)}\Phi_{\alpha(m)}=\mathcal{D}_{\alpha}\Omega_{\alpha(m-1)}+\Lambda_{\alpha(m)}~{},\qquad\bar{\mathcal{D}}_{{\dot{\beta}}}\Lambda_{\alpha(m)}=0~{},$
(4.161)

where $\Omega_{\alpha(m-1)}$ is unconstrained. and chiral
$\Lambda_{\alpha(m)}$ superfields. On the other hand, if $n>0$ and $m=0$, then
the orthogonal complement splits into the sum of a longitudinal linear
superfield and an anti-chiral superfield,

$\displaystyle\overline{\mbox{\boldmath$\Pi$}}^{\parallel}_{(n)}\bar{\Phi}_{{\dot{\alpha}}(n)}=\bar{\mathcal{D}}_{{\dot{\alpha}}}\bar{\Psi}_{{\dot{\alpha}}(n-1)}+\bar{\Lambda}_{{\dot{\alpha}}(n)}~{},\qquad\mathcal{D}_{\beta}\bar{\Lambda}_{{\dot{\alpha}}(n)}=0~{},$
(4.162)

where $\bar{\Lambda}_{{\dot{\alpha}}(n)}$ is unconstrained.

Using the fact that $\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}$ and
$\mbox{\boldmath$\Pi$}^{\parallel}_{(m,n)}$ resolve the identity operator, one
can decompose an arbitrary superfield $\Phi_{\alpha(m){\dot{\alpha}}(n)}$ as
follows

$\displaystyle\Phi_{\alpha(m){\dot{\alpha}}(n)}=\Phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}+\bar{\mathcal{D}}_{{\dot{\alpha}}}\zeta_{\alpha(m){\dot{\alpha}}(n-1)}+\mathcal{D}_{\alpha}\xi_{\alpha(m-1){\dot{\alpha}}(n)}~{},$
(4.163)

where $\Phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}$ is TLAL and is hence
irreducible, whilst $\zeta_{\alpha(m){\dot{\alpha}}(n-1)}$ and
$\xi_{\alpha(m-1){\dot{\alpha}}(n)}$ are unconstrained and thus reducible. It
is apparent from this decomposition that the superspin projection operator
$\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}$ selects out the component
$\Phi^{\perp}_{\alpha(m){\dot{\alpha}}(n)}$ which describes the pure superspin
state with maximal superspin $s=\frac{1}{2}(m+n+1)$, as a consequence of the
identities (4.152) and (4.159).

One can repeat the decomposition (4.163) on the lower-rank unconstrained
superfields in (4.163). After reiterating this procedure a finite number of
times, one eventually arrives at a decomposition of
$\Phi_{\alpha(m){\dot{\alpha}}(n)}$ solely in terms of irreducible
superfields.

In the $m>n$ case, the result of this process is

$\displaystyle\Phi_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=\sum_{t=0}^{n}(\mathcal{D}_{\alpha{\dot{\alpha}}})^{t}\Phi^{\perp}_{\alpha(m-t){\dot{\alpha}}(n-t)}+\sum_{t=0}^{n-1}\big{[}\mathcal{D}_{\alpha},\bar{\mathcal{D}}_{{\dot{\alpha}}}\big{]}(\mathcal{D}_{\alpha{\dot{\alpha}}})^{t}\Psi_{\alpha(m-t-1){\dot{\alpha}}(n-t-1)}$
$\displaystyle+\sum_{t=0}^{n}(\mathcal{D}_{\alpha{\dot{\alpha}}})^{t}\mathcal{D}_{\alpha}\Xi_{\alpha(m-t-1){\dot{\alpha}}(n-t)}+\sum_{t=0}^{n-1}(\mathcal{D}_{\alpha{\dot{\alpha}}})^{t}\bar{\mathcal{D}}_{{\dot{\alpha}}}\Omega_{\alpha(m-t){\dot{\alpha}}(n-t-1)}$
$\displaystyle+(\mathcal{D}_{\alpha{\dot{\alpha}}})^{n}\Big{(}\Lambda_{\alpha(m-n)}+\mathcal{D}_{\alpha}\Lambda_{\alpha(m-n-1)}\Big{)}~{},$
(4.164)

for some irreducible complex superfields $\Phi,\Psi,\Xi$ and $\Omega$ whose
properties are summarised in table 4.2.2. The superfields
$\Lambda_{\alpha(m-n)}$ and $\Lambda_{\alpha(m-n-1)}$ in (4.164) are chiral,

$\displaystyle\bar{\mathcal{D}}_{{\dot{\alpha}}}\Lambda_{\alpha(m-n)}=0~{},\qquad\bar{\mathcal{D}}_{{\dot{\alpha}}}\Lambda_{\alpha(m-n-1)}=0~{}.$
(4.165)

| $0\leq t\leq n-2$ | $t=n-1$ |  $t=n$
---|---|---|---
$\Phi_{\alpha(m-t){\dot{\alpha}}(n-t)}$ | TLAL | TLAL | LTAL
$\Psi_{\alpha(m-t-1){\dot{\alpha}}(n-t-1)}$ | TLAL | TLAL | –
$\Xi_{\alpha(m-t-1){\dot{\alpha}}(n-t)}$ | TLAL | TLAL | LTAL
$\Omega_{\alpha(m-t){\dot{\alpha}}(n-t-1)}$ | TLAL | LTAL | –
Table 4.2.2: Properties of the superfields appearing in (4.164).

If instead $m=n=s$, then we may further impose the reality condition

$\displaystyle
H_{\alpha(s){\dot{\alpha}}(s)}:=\Phi_{\alpha(s){\dot{\alpha}}(s)}=\bar{H}_{\alpha(s){\dot{\alpha}}(s)}~{}.$
(4.166)

In contrast to (4.164), $H_{\alpha(s){\dot{\alpha}}(s)}$ now decomposes as

$\displaystyle H_{\alpha(s){\dot{\alpha}}(s)}$
$\displaystyle=\sum_{t=0}^{s}(\mathcal{D}_{\alpha{\dot{\alpha}}})^{t}\Phi_{\alpha(s-t){\dot{\alpha}}(s-t)}+\sum_{t=0}^{s-1}\big{[}\mathcal{D}_{\alpha},\bar{\mathcal{D}}_{{\dot{\alpha}}}\big{]}(\mathcal{D}_{\alpha{\dot{\alpha}}})^{t}\Psi_{\alpha(s-t-1){\dot{\alpha}}(s-t-1)}$
$\displaystyle+\sum_{t=0}^{s-1}(\mathcal{D}_{\alpha{\dot{\alpha}}})^{t}\Big{(}\bar{\mathcal{D}}_{{\dot{\alpha}}}\Xi_{\alpha(s-t){\dot{\alpha}}(s-t-1)}+\text{c.c.}\Big{)}+(\mathcal{D}_{\alpha{\dot{\alpha}}})^{s}\big{(}\Lambda+\bar{\Lambda}\big{)}~{},$
(4.167)

for some irreducible complex superfield
$\Xi_{\alpha(s-t){\dot{\alpha}}(s-t-1)}$ and irreducible real superfields

$\displaystyle\Phi_{\alpha(s-t){\dot{\alpha}}(s-t)}=\bar{\Phi}_{\alpha(s-t){\dot{\alpha}}(s-t)}~{},\qquad\Psi_{\alpha(s-t-1){\dot{\alpha}}(s-t-1)}=\bar{\Psi}_{\alpha(s-t-1){\dot{\alpha}}(s-t-1)}~{},$
(4.168)

whose properties are described in table 4.2.3. The scalar $\sigma$ in (4.167)
is chiral, $\bar{\mathcal{D}}_{{\dot{\alpha}}}\sigma=0$.

| $0\leq t\leq s-2$ | $t=s-1$ |  $t=s$
---|---|---|---
$\Phi_{\alpha(s-t){\dot{\alpha}}(s-t)}$ | TLAL | TLAL | LAL
$\Psi_{\alpha(s-t-1){\dot{\alpha}}(s-t-1)}$ | TLAL | LAL | –
$\Xi_{\alpha(s-t){\dot{\alpha}}(s-t-1)}$ | TLAL | LTAL | –
Table 4.2.3: Properties of the superfields appearing in (4.167).

The above decompositions were used, albeit without derivation, in [54] for the
covariant quantisation of the massless supersymmetric higher-spin models in
AdS4 [110].

####  4.2.6 Superconformal higher-spin theory

For integers $m,n\geq 1$,212121For the superconformal gauge multiplets with
either $m=0$ or $n=0$ see [219]. the SCHS action in AdS4|4 is described in
terms of the gauge prepotential $H_{\alpha(m){\dot{\alpha}}(n)}$, which is
defined modulo gauge transformations [85] (see also [86])

$\delta_{\Lambda,\Omega}H_{\alpha(m){\dot{\alpha}}(n)}=\bar{\mathcal{D}}_{({\dot{\alpha}}_{1}}\Lambda_{\alpha(m){\dot{\alpha}}_{2}\dots\dot{\alpha}_{n})}+\mathcal{D}_{(\alpha_{1}}\Omega_{\alpha_{2}\dots\alpha_{m}){\dot{\alpha}}(n)}~{},$
(4.169)

where the gauge parameters $\Lambda_{\alpha(m){\dot{\alpha}}(n-1)}$ and
$\Omega_{\alpha(m-1){\dot{\alpha}}(n)}$ are unconstrained.

Associated with $H_{\alpha(m){\dot{\alpha}}(n)}$ and its complex conjugate
$\bar{H}_{\alpha(n){\dot{\alpha}}(m)}$ are the the linearised higher-spin
super-Weyl tensors [85]

$\displaystyle\mathfrak{W}_{\alpha(m+n+1)}(H)$
$\displaystyle=-\frac{1}{4}\big{(}\bar{\mathcal{D}}^{2}-4\mu\big{)}\mathcal{D}_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\mathcal{D}_{\alpha_{n}}{}^{{\dot{\beta}}_{n}}\mathcal{D}_{\alpha_{n+1}}H_{\alpha_{n+2}\dots\alpha_{m+n+1}){\dot{\beta}}(n)}~{},$
(4.170a) $\displaystyle\mathfrak{W}_{\alpha(m+n+1)}(\bar{H})$
$\displaystyle=-\frac{1}{4}\big{(}\bar{\mathcal{D}}^{2}-4\mu\big{)}\mathcal{D}_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\mathcal{D}_{\alpha_{m}}{}^{{\dot{\beta}}_{m}}\mathcal{D}_{\alpha_{m+1}}\bar{H}_{\alpha_{m+2}\dots\alpha_{m+n+1}){\dot{\beta}}(m)}~{},$
(4.170b)

which are invariant under the gauge transformations (4.169)

$\displaystyle\delta_{\zeta,\xi}\mathfrak{W}_{\alpha(m+n+1)}(H)=0~{},\qquad\delta_{\zeta,\xi}\mathfrak{W}_{\alpha(m+n+1)}(\bar{H})=0~{}.$
(4.171)

For the cases $m=n=s$ and $m=n+1=s$, the SCHS gauge prepotentials
$H_{\alpha(m){\dot{\alpha}}(n)}$ first appeared in [110] as dynamical
variables in the massless superspin-$s$ and superspin-$(s+\frac{1}{2})$
multiplets, respectively. Additionally, their corresponding gauge-invariant
field strengths (4.170) were also given in [110].

The gauge-invariant action $S_{\text{SCHS}}^{(m,n)}[H,\bar{H}]$, which
describes the dynamics of $H_{\alpha(m){\dot{\alpha}}(n)}$ and its conjugate
$\bar{H}_{\alpha(n){\dot{\alpha}}(m)}$, is typically written as a functional
over the chiral subspace of the full superspace [85, 86]. The specific feature
of AdS4|4 is the identity [220]

$\displaystyle\int{\rm d}^{4}x{\rm d}^{2}\theta\,{\cal E}\,{\cal L}_{\rm
c}=\int{\rm d}^{4}x{\rm d}^{2}\theta{\rm
d}^{2}{\bar{\theta}}\,\frac{E}{\mu}\,{\cal L}_{\rm c}~{},\qquad\bar{\cal
D}_{\dot{\alpha}}{\cal L}_{\rm c}=0~{},$ (4.172)

which relates the integration over the chiral subspace to that over the full
superspace; here ${\cal E}$ denotes the chiral integration measure and
$E^{-1}={\rm Ber}(E_{A}{}^{M})$. Keeping in mind (4.172), the gauge-invariant
SCHS222222For details concerning the superconformal properties of the SCHS
action (4.173), and its constituents, we direct the reader to the original
work [85] (see also [86]).  action is given by [85]

$\displaystyle
S_{\text{SCHS}}^{(m,n)}[H,\bar{H}]=\frac{1}{2}\text{i}^{m+n}\int\text{d}^{4|4}z\,\frac{E}{\mu}\,\mathfrak{W}^{\alpha(m+n+1)}(H)\mathfrak{W}_{\alpha(m+n+1)}(\bar{H})+{\rm
c.c.}~{},$ (4.173)

where we have denoted ${\rm d}^{4|4}z={\rm d}^{4}x{\rm d}^{2}\theta{\rm
d}^{2}{\bar{\theta}}$.

Upon integrating by parts, the action (4.173) may be written in the suggestive
forms

$\displaystyle S_{\text{SCHS}}^{(m,n)}[H,\bar{H}]$
$\displaystyle=\text{i}^{m+n}\int\text{d}^{4|4}z\,E\,\bar{H}^{\alpha(n){\dot{\alpha}}(m)}\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)+\text{c.c.
}$ (4.174a)
$\displaystyle=\text{i}^{m+n}\int\text{d}^{4|4}z\,E\,\bar{H}^{\alpha(n){\dot{\alpha}}(m)}\widehat{\mathfrak{B}}_{\alpha(n){\dot{\alpha}}(m)}(H)+\rm{c.c.}$
(4.174b)

They are suggestive because, in addition to being gauge invariant, the
linearised higher-spin super-Bach tensors [221]

$\displaystyle\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)$
$\displaystyle=\frac{1}{2}\mathcal{D}_{({\dot{\alpha}}_{1}}{}^{\beta_{1}}\cdots\mathcal{D}_{{\dot{\alpha}}_{m})}{}^{\beta_{m}}\mathcal{D}^{\beta_{m+1}}\mathfrak{W}_{\alpha(n)\beta(m+1)}(H)~{},$
(4.175a) $\displaystyle\widehat{\mathfrak{B}}_{\alpha(n){\dot{\alpha}}(m)}(H)$
$\displaystyle=\frac{1}{2}\mathcal{D}_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\mathcal{D}_{\alpha_{n})}{}^{{\dot{\beta}}_{n}}\bar{\mathcal{D}}^{{\dot{\beta}}_{n+1}}\overline{\mathfrak{W}}_{{\dot{\alpha}}(m){\dot{\beta}}(n+1)}(H)~{},$
(4.175b)

are simultaneously transverse linear and transverse anti-linear

$\displaystyle\mathcal{D}^{\beta}\mathfrak{B}_{\beta\alpha(n-1){\dot{\alpha}}(m)}(H)$
$\displaystyle=0~{},\qquad\bar{\mathcal{D}}^{{\dot{\beta}}}\mathfrak{B}_{\alpha(n){\dot{\beta}}{\dot{\alpha}}(m-1)}(H)=0~{},$
(4.176a)
$\displaystyle\mathcal{D}^{\beta}\widehat{\mathfrak{B}}_{\beta\alpha(n-1){\dot{\alpha}}(m)}(H)$
$\displaystyle=0~{},\qquad\bar{\mathcal{D}}^{{\dot{\beta}}}\widehat{\mathfrak{B}}_{\alpha(n){\dot{\beta}}{\dot{\alpha}}(m-1)}(H)=0~{}.$
(4.176b)

It can be shown that the super-Bach tensors (4.175) coincide

$\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)=\widehat{\mathfrak{B}}_{\alpha(n){\dot{\alpha}}(m)}(H)~{}.$
(4.177)

Proving this equivalence is non-trivial and is closely related to the
superprojector coincidence relation (4.151). As a by-product, it follows that
the SCHS actions (4.174a) and (4.174b) are also equivalent.

The linearised higher-spin super-Bach tensors (4.175) can be written in a
manifestly TLAL form by recasting them in terms of the unique TLAL
superprojectors (4.149)

$\displaystyle\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)$
$\displaystyle=\prod_{t=1}^{n+1}\big{(}\mathbb{Q}-\lambda_{(t,m,n)}\mu\bar{\mu}\big{)}\big{(}{\cal
D}_{{\dot{\alpha}}}{}^{\beta}\big{)}^{m-n}\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}H_{\alpha(n)\beta(m-n){\dot{\alpha}}(n)}~{},$
(4.178a) $\displaystyle\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)$
$\displaystyle=\prod_{t=1}^{m+1}\big{(}\mathbb{Q}-\lambda_{(t,m,n)}\mu\bar{\mu}\big{)}\big{(}{\cal
D}_{\alpha}{}^{\dot{\beta}}\big{)}^{n-m}\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}H_{\alpha(m){\dot{\beta}}(n-m){\dot{\alpha}}(m)}~{},$
(4.178b)

for the cases $m\geq n$ and $n\geq m$, respectively. It follows from (4.175)
and (4.178) that the higher-spin super-Weyl tensors (4.170) can also be recast
in terms of the superspin projection operator (4.149a).

We can immediately make use of these new representations for the higher-spin
Bach tensors to recast the SCHS action (4.174a) in terms of the superprojector
(4.149a)

$\displaystyle S_{\text{SCHS}}^{(m,n)}[H,\bar{H}]=\text{i}^{m+n}$
$\displaystyle\int\text{d}^{4|4}z\,E\,\bar{H}^{\alpha(n){\dot{\alpha}}(m)}\prod_{t=1}^{n+1}\big{(}\mathbb{Q}-\lambda_{(t,m,n)}\mu\bar{\mu}\big{)}$
$\displaystyle\times(\mathcal{D}_{{\dot{\alpha}}}{}^{\beta})^{m-n}\mbox{\boldmath$\Pi$}^{\perp}_{(m,n)}H_{\alpha(n)\beta(m-n){\dot{\alpha}}(n)}+\text{c.c.}~{},\hskip
38.98013ptm\geq n~{},$ (4.179a) $\displaystyle
S_{\text{SCHS}}^{(m,n)}[H,\bar{H}]=\text{i}^{m+n}$
$\displaystyle\int\text{d}^{4|4}z\,E\,\bar{H}^{\alpha(n){\dot{\alpha}}(m)}\prod_{t=1}^{m+1}\big{(}\mathbb{Q}-\lambda_{(t,m,n)}\mu\bar{\mu}\big{)}$