Since $y_{\alpha(2s-1)}$ and $\zeta_{\alpha(2s-1)}$ are of the same functional
form, it can be shown that the field $y_{\alpha(2s-1)}$ can be gauged away

$y_{\alpha(2s-1)}=0~{}.$ (5.95)

The residual gauge freedom is described by $\zeta_{\alpha(2s-1)}$, which is
further constrained by

$\big{(}\mathcal{F}\pm\rho_{(1,2s+1)}\big{)}\zeta_{\alpha(2s-1)}=0~{}.$ (5.96)

In the gauge (5.95), the equations of motion (5.90a) and (5.90b) reduce to

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\mathfrak{D}^{\beta\gamma}h_{\beta\gamma\alpha(2s-1)}~{},$
(5.97a) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\big{(}\mathcal{F}\pm\rho_{(1,2s+1)}\big{)}h_{\alpha(2s+1)}~{}.$
(5.97b)

Thus $h_{\alpha(2s+1)}$ is the only surviving field on-shell. In accordance
with (5.17), it follows from the above analysis that on-shell, the Fang-
Fronsdal action $S_{(s+\frac{1}{2},\pm)}^{\text{FF}}[h,y,z]$ (5.85) describes
a massless spin-$(s+\frac{1}{2})$ field $h_{\alpha(2s+1)}$ with helicity
$\pm(s+\frac{1}{2})$.

#####  5.1.5.2 Fronsdal action

The Fronsdal action in AdS4 [29] can be generalised to AdS3 as follows. Given
an integer $n\geq 4$, we introduce two real field variables $h_{\alpha(n)}$,
$y_{\alpha(n-4)}$, which are defined modulo gauge transformations of the form

$\displaystyle\delta_{\zeta}h_{\alpha(n)}$ $\displaystyle=$
$\displaystyle\mathfrak{D}_{\alpha(2)}\zeta_{\alpha(n-2)}~{},$ (5.98a)
$\displaystyle\delta_{\zeta}y_{\alpha(n-4)}$ $\displaystyle=$
$\displaystyle\frac{n-2}{n-1}\mathfrak{D}^{\beta\gamma}\zeta_{\beta\gamma\alpha(n-4)}~{},$
(5.98b)

where the gauge parameter $\zeta_{\alpha(n-2)}$ is real and unconstrained.
Modulo an overall constant, the gauge-invariant action is given by [117]

$\displaystyle S_{(\frac{n}{2})}^{\text{F}}[h,y]$ $\displaystyle=$
$\displaystyle\frac{{\rm i}^{n}}{2^{\lfloor n/2\rfloor+1}}\int{\rm
d}^{3}x\,e\,\bigg{\\{}h^{\alpha(n)}{\cal
Q}h_{\alpha(n)}-\frac{n}{4}\mathfrak{D}_{\beta\gamma}h^{\beta\gamma\alpha(n-2)}\mathfrak{D}^{\delta\lambda}h_{\delta\lambda\alpha(n-2)}~{}$
(5.99)
$\displaystyle-\frac{n-3}{2}y^{\alpha(n-4)}\mathfrak{D}^{\beta\gamma}\mathfrak{D}^{\delta\lambda}h_{\beta\gamma\delta\lambda\alpha(n-4)}-2n(n-2)|\mu|^{2}~{}h^{\alpha(n)}h_{\alpha(n)}~{}$
$\displaystyle-\frac{n-3}{n}\bigg{(}2y^{\alpha(n-4)}{\cal
Q}y_{\alpha(n-4)}-4(n^{2}+2)|\mu|^{2}y^{\alpha(n-4)}y_{\alpha(n-4)}~{}$
$\displaystyle+\frac{(n-4)(n-5)}{4(n-2)}\mathfrak{D}_{\beta\gamma}y^{\beta\gamma\alpha(n-6)}\mathfrak{D}^{\delta\lambda}y_{\delta\lambda\alpha(n-6)}\bigg{)}\bigg{\\}}~{},$

The model (5.99) is the $d=3$ counterpart of the Fronsdal action [29] provided
$n$ is even, $n=2s$, with $s\geq 2$.

Performing a similar component analysis as was done for the Fang-Fronsdal
action, it can be shown that the gauge symmetry (5.98) provides the freedom to
impose the gauge conditions

$y_{\alpha(2s-4)}=0~{},\qquad\mathfrak{D}^{\beta\gamma}h_{\beta\gamma\alpha(2s-2)}=0~{}.$
(5.100)

The residual gauge symmetry is described by $\zeta_{\alpha(2s-1)}$ constrained
by

$\mathfrak{D}^{\beta\gamma}\zeta_{\beta\gamma\alpha(2s-4)}=0~{},\qquad\big{(}\mathcal{Q}-\tau_{(1,2s)}|\mu|^{2}\big{)}\zeta_{\alpha(2s-1)}=0~{}.$
(5.101)

In the gauge (5.100), the equation of motion obtained by varying the action
(5.99) with respect to $h_{\alpha(2s)}$ reduces to

$0=\big{(}\mathcal{Q}-\tau_{(1,2s)}|\mu|^{2}\big{)}h_{\alpha(2s)}=\big{(}{\cal
F}-\rho_{(1,2s)}\big{)}\big{(}{\cal F}+\rho_{(1,2s)}\big{)}h_{\alpha(2s)}~{}.$
(5.102)

Thus, we are left with the remaining field $h_{\alpha(2s)}$ on-shell. In
accordance with (5.19), it follows from the above analysis that the Fronsdal
model $S_{(s)}^{\text{F}}$ describes two massless spin-$s$ modes, each with a
differing sign of helicity $\pm s$.

####  5.1.6 New topologically massive actions

New topologically massive models were recently computed in AdS3 by Kuzenko and
Ponds [117] for fields with arbitrary spin. These models are formulated solely
in terms of the gauge prepotentials $h_{\alpha(n)}$ and the associated Cotton
tensors $\mathfrak{C}_{\alpha(n)}(h)$. In particular, given an integer $n\geq
2$, the gauge-invariant NTM action for the field $h_{\alpha(n)}$ is [117]

$\displaystyle S_{\text{NTM}}^{(n)}[h]=\frac{\text{i}^{n}}{2^{\lceil
n/2\rceil+1}}\frac{1}{\rho}\int\text{d}^{3}x\,e\,h^{\alpha(n)}\big{(}\mathcal{F}-\sigma\rho\big{)}\mathfrak{C}_{\alpha(n)}(h)~{},$
(5.103)

where $\rho$ is some positive mass parameter and $\sigma:=\pm 1$. Making use
of the new representation (5.75) for the super-Cotton tensor leads to a
manifestly gauge invariant and factorised form for the action (5.103).

The equation of motion obtained by varying (5.103) with respect to the field
$h_{\alpha(n)}$ is

$\displaystyle
0=\big{(}\mathcal{F}-\sigma\rho\big{)}\mathfrak{C}_{\alpha(n)}(h)~{}.$ (5.104)

By analysing (5.104), it can be shown that on-shell, the action (5.103)
describes a propagating mode with pseudo-mass $\rho$, spin $\frac{n}{2}$ and
helicity $\frac{\sigma n}{2}$ given $\rho\neq\rho_{(t,2s)}$ (cf. (5.7)). For
the case $\rho=\rho_{(t,2s)}$, the model describes only pure gauge degrees of
freedom.

Next, we wish to compute the AdS3 extension of the NTM model (2.338), which
was recently derived in $\mathbb{M}^{3}$ [155]. Given an integer $s\geq 1$,
the action (2.338) may be readily extended to AdS3 as follows

$\displaystyle\widetilde{S}_{\text{NTM}}^{(2s)}[h]=\int{\rm
d}^{3}x\,e\,h^{\alpha(2s)}\big{(}{\cal
F}-\sigma\rho\big{)}\mathfrak{W}_{\alpha(2s)}(h)~{},$ (5.105)

where $\rho$ is a positive mass parameter, $\sigma:=\pm 1$, and
$\mathfrak{W}_{\alpha(2s)}(h)$ is the field strength,

$\mathfrak{W}_{\alpha(2s)}(h):=\prod_{t=1}^{s-1}\big{(}\mathcal{Q}-\tau_{(t,2s)}|\mu|^{2}\big{)}{\Pi}^{\perp}_{[2s]}h_{\alpha(2s)}~{}.$
(5.106)

Due to the properties of $\Pi^{\perp}_{[2s]}$, the action (5.105) is
manifestly gauge invariant and factorised. The descendent
$\mathfrak{W}_{\alpha(2s)}(h)$ may be obtained from
$\mathfrak{C}_{\alpha(2s)}(h)$ by stripping off $\mathcal{F}$:

$\displaystyle\mathfrak{C}_{\alpha(2s)}(h)=\frac{1}{2s}\mathcal{F}\mathfrak{W}_{\alpha(2s)}(h)~{}.$
(5.107)

A similar construction does not appear to be possible in the fermionic case.

The equation of motion obtained by varying (5.105) with respect to the field
$h_{\alpha(2s)}$ is

$\displaystyle 0=({\cal F}-\sigma\rho)\mathfrak{W}_{\alpha(2s)}(h)~{}.$
(5.108)

By analysing (5.108), it can be shown that on-shell, the model (5.105) has the
same particle content as the NTM model (5.103).

###  5.2 Three-dimensional ${\cal N}=1$ anti-de Sitter superspace

The main objective of this section is to construct the superspin projection
operators in ${\cal N}=1$ AdS superspace AdS3|2 and explore several of their
applications. These operators can be considered supersymmetric generalisations
of the AdS3 spin projection operators, which were derived in section 5.1.3.
They can also be considered AdS3 analogues of the superspin projectors in
$\mathbb{M}^{3|2}$, which were computed in section 3.3.1. Before studying
these operators, it is first necessary to outline the irreducible
representations of
$\mathfrak{osp}(1|2;{\mathbb{R}})\oplus\mathfrak{sl}(2,{\mathbb{R}})$.

####  5.2.1 Irreducible representations of the AdS3 superalgebra

The irreducible representations of the ${\cal N}=1$ AdS3 superalgebra
$\mathfrak{osp}(1|2;{\mathbb{R}})\oplus\mathfrak{sl}(2,{\mathbb{R}})$ are
described by their decomposition into irreducible representations of
$\mathfrak{so}(2,2)$, which were reviewed in section 5.1.1. We denote the
irreducible representations of
$\mathfrak{osp}(1|2;{\mathbb{R}})\oplus\mathfrak{sl}(2,{\mathbb{R}})$ by
$\mathfrak{S}(M,\sigma\frac{n}{2})$, where $M$ is the pseudo-mass,
$s:=\frac{n}{2}$ is the superspin and the parameter $\sigma:=\pm 1$ labels the
sign of the superhelicity $\kappa=\frac{1}{2}(n+\frac{1}{2})\sigma$. It can be
shown that the representation $\mathfrak{S}(M,\sigma\frac{n}{2})$ decomposes
into two irreducible representations of $\mathfrak{so}(2,2)$,

$\mathfrak{S}\Big{(}M,\sigma\frac{n}{2}\Big{)}=\mathfrak{D}\Big{(}\rho_{A},\sigma_{A}\frac{n}{2}\Big{)}\oplus\mathfrak{D}\Big{(}\rho_{B},\sigma_{B}\frac{n+1}{2}\Big{)}~{}.$
(5.109)

Here, the pseudo-masses $\rho_{A}$ and $\rho_{B}$ are given by

$\displaystyle\rho_{A}=\frac{n}{2n+1}\Big{|}\sigma
M-(n+2)|\mu|\Big{|}~{},\qquad\rho_{B}=\frac{n+1}{2n+1}\Big{|}\sigma
M+(n-1)|\mu|\Big{|}~{},$ (5.110)

and the corresponding signs of the helicities $\sigma_{A}$ and $\sigma_{B}$
are

$\displaystyle\sigma_{A}$ $\displaystyle=$ $\displaystyle\frac{\sigma
M-(n+2)|\mu|}{\big{|}\sigma
M-(n+2)|\mu|\big{|}}~{},\qquad\sigma_{B}=\frac{\sigma
M+(n-1)|\mu|}{\big{|}\sigma M+(n-1)|\mu|\big{|}}~{}.$ (5.111)

Here $|\mu|>0$ is a real constant parameter which determines the curvature of
AdS3|2.121212The parameter $|\mu|$ was denoted $\mathcal{S}$ in [124].
However, we prefer to make use of the notation $|\mu|$ since it is better
suited in the $(1,1)\to(1,0)$ superspace reduction scheme, which will be
considered later.  The representation $\mathfrak{S}(M,\sigma\frac{n}{2})$ is
unitary if the parameter $M$ obeys the unitarity bound

$M\geq 2(n-1)(n+1)|\mu|~{}.$ (5.112)

This bound ensures that both representations appearing in the decomposition
(5.109) are unitary.

####  5.2.2 Geometry of AdS3|2

Let us denote $z^{\cal M}=(x^{m},\theta^{\mu})$ the local coordinates which
parametrise AdS3|2.131313We remind the reader that AdS3|2 is the maximally
supersymmetric solution of three-dimensional ${\cal N}=1$ AdS supergravity
[41]. The geometric structure of AdS3|2 is described in terms of its covariant
derivatives[124]141414In the hope that no confusion arises, we use the same
notation for the vector covariant derivative in AdS3 and in AdS3|2.

$\nabla_{A}=(\nabla_{a},\nabla_{\alpha})=E_{A}{}^{M}{\partial}_{M}+\frac{1}{2}\Omega_{A}{}^{bc}M_{bc}~{}.$
(5.113)

Here $E_{A}{}^{M}$ is the inverse supervielbein and $\Omega_{A}{}^{bc}$ the
Lorentz connection. The covariant derivatives obey the following
(anti-)commutation relations

$\displaystyle\\{\nabla_{\alpha},\nabla_{\beta}\\}=2{\rm
i}\nabla_{\alpha\beta}-4{\rm i}|\mu|M_{\alpha\beta}~{},$ (5.114a)
$\displaystyle\
[\nabla_{\alpha\beta},\nabla_{\gamma}]=-2|\mu|\varepsilon_{\gamma(\alpha}\nabla_{\beta)}~{},\qquad\
[\nabla_{\alpha\beta},\nabla_{\gamma\delta}]=4|\mu|^{2}\big{(}\varepsilon_{\gamma(\alpha}M_{\beta)\delta}+\varepsilon_{\delta(\alpha}M_{\beta)\gamma}\big{)}~{},$
(5.114b)

It should be remarked that the geometry of ${\cal N}=1$ AdS superspace can
also be described by the graded commutation relations which are obtained from
(5.114) by replacing $|\mu|\to-|\mu|$. The two choices, $|\mu|$ and $-|\mu|$,
correspond to the so-called $(1,0)$ and $(0,1)$ AdS superspaces [124], which
are different realisations of AdS3|2. The $(1,0)$ and $(0,1)$ AdS superspaces
are naturally embedded in $(1,1)$ AdS superspace151515 As follows from (6.36),
in $(1,1)$ AdS superspace two subsets of covariant derivatives,
$({\mbox{\boldmath$\nabla$}}_{a},{\mbox{\boldmath$\nabla$}}_{\alpha}^{\underline{1}})$
and
$({\mbox{\boldmath$\nabla$}}_{a},{\mbox{\boldmath$\nabla$}}_{\alpha}^{\underline{2}})$,
form closed algebras of the type (5.114), with the curvature parameter being
$|\mu|$ and $-|\mu|$, respectively. and are related to each other by a parity
transformation.

We list several identities which prove indispensable for calculations:

$\displaystyle\nabla_{\alpha}\nabla_{\beta}$ $\displaystyle=$
$\displaystyle{\rm i}\nabla_{\alpha\beta}-2{\rm
i}|\mu|M_{\alpha\beta}+\frac{1}{2}\varepsilon_{\alpha\beta}\nabla^{2}~{},$
(5.115a) $\displaystyle\nabla^{\beta}\nabla_{\alpha}\nabla_{\beta}$
$\displaystyle=$ $\displaystyle 4{\rm
i}|\mu|\nabla_{\alpha}~{},\quad\\{\nabla^{2},\nabla_{\alpha}\\}=4{\rm
i}|\mu|\nabla_{\alpha}~{},$ (5.115b) $\displaystyle\nabla^{2}\nabla_{\alpha}$
$\displaystyle=$ $\displaystyle 2{\rm i}|\mu|\nabla_{\alpha}+2{\rm
i}\nabla_{\alpha\beta}\nabla^{\beta}-4{\rm
i}|\mu|\nabla^{\beta}M_{\alpha\beta}~{},$ (5.115c) $\displaystyle\qquad\
[\nabla_{\alpha}\nabla_{\beta},\nabla^{2}]$ $\displaystyle=$ $\displaystyle
0\quad\Longrightarrow\quad\ [\nabla_{\alpha\beta},\nabla^{2}]=0~{},$ (5.115d)
$\displaystyle\ [\nabla_{\alpha},\Box]$ $\displaystyle=$ $\displaystyle
2|\mu|\nabla_{\alpha\beta}\nabla^{\beta}+3|\mu|^{2}\nabla_{\alpha}~{}.$
(5.115e)

where we have denoted $\nabla^{2}=\nabla^{\alpha}\nabla_{\alpha}$ and
$\Box=\nabla^{a}\nabla_{a}=-\frac{1}{2}\nabla^{\alpha\beta}\nabla_{\alpha\beta}$.
These relations can be derived from the algebra of covariant derivatives
(5.114).

According to the general formalism of [105], the isometry transformations of
AdS3|2 are generated by the Killing supervector fields,

$\displaystyle\xi=\xi^{B}\nabla_{B}=\xi^{b}\nabla_{b}+\xi^{\beta}\nabla_{\beta}~{},$
(5.116)

which by definition solve the master equation [124]

$[\xi+\frac{1}{2}\zeta^{bc}M_{bc},\nabla_{A}]=0~{},$ (5.117)

for some Lorentz parameter $\zeta^{bc}=-\zeta^{cb}$. It can be shown that the
Killing equation (5.117) is equivalent to the set of relations [124] (see also
[117])

$\displaystyle\nabla_{(\alpha}\xi_{\beta\gamma)}$ $\displaystyle=$
$\displaystyle 0~{},\qquad\nabla_{\beta}\xi^{\beta\alpha}=-6{\rm
i}\xi^{\alpha}~{},$ (5.118a)
$\displaystyle\nabla_{(\alpha}\zeta_{\beta\gamma)}$ $\displaystyle=$
$\displaystyle 0~{},\qquad\nabla_{\beta}\zeta^{\beta\alpha}=-12{\rm
i}|\mu|\xi^{\alpha}~{},$ (5.118b) $\displaystyle\nabla_{\alpha}\xi_{\beta}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\zeta_{\alpha\beta}+|\mu|\xi_{\alpha\beta}~{},$
(5.118c)

which imply

$\displaystyle\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}$ $\displaystyle=$
$\displaystyle 0~{},$ (5.119a)
$\displaystyle\nabla_{\alpha\beta}\xi^{\beta}+3|\mu|\xi_{\alpha}$
$\displaystyle=$ $\displaystyle 0~{},$ (5.119b) $\displaystyle\big{(}{\rm
i}\nabla^{2}+12|\mu|\big{)}\xi_{\alpha}$ $\displaystyle=$ $\displaystyle
0~{}.$ (5.119c)

Thus, $\xi^{a}$ is a Killing vector. Given a tensor superfield $U(x,\theta)$
(with suppressed indices) on AdS3|2, its infinitesimal isometry transformation
law is given by

$\displaystyle\delta_{\xi}U=\big{(}\xi^{a}\nabla_{a}+\xi^{\alpha}\nabla_{\alpha}+\frac{1}{2}\zeta^{ab}M_{ab}\big{)}U~{}.$
(5.120)

Crucial to our analysis are the two independent Casimir operators of
$\mathfrak{osp}(1|2;{\mathbb{R}})\oplus\mathfrak{sl}(2,{\mathbb{R}})$. They
take the following form in the superfield representation [152, 117]

$\displaystyle\mathbb{Q}:$
$\displaystyle=-\frac{1}{4}\nabla^{2}\nabla^{2}+{\rm
i}|\mu|\nabla^{2}~{},\qquad$ $\displaystyle[\mathbb{Q},\nabla_{A}]=0~{},$
(5.121a) $\displaystyle\mathbb{F}:$ $\displaystyle=-\frac{{\rm
i}}{2}\nabla^{2}+2\nabla^{\alpha\beta}M_{\alpha\beta}~{},$
$\displaystyle[\mathbb{F},\nabla_{A}]=0~{}.$ (5.121b)

Making use of the identity

$\displaystyle-\frac{1}{4}\nabla^{2}\nabla^{2}$ $\displaystyle=$
$\displaystyle\Box-2{\rm
i}|\mu|\nabla^{2}+2|\mu|\nabla^{\alpha\beta}M_{\alpha\beta}-2|\mu|^{2}M^{\alpha\beta}M_{\alpha\beta}~{},$
(5.122)

allows us to express $\mathbb{Q}$ in terms of the AdS3|2 d’Alembertian $\Box$.
Let us denote by $\mathds{V}_{(n)}$ the space of real symmetric tensor
superfields $\Phi_{\alpha(n)}$ on AdS3|2. It can be shown that the Casimir
operators $\mathbb{Q}$ and $\mathbb{F}$ are related to each other on
$\mathds{V}_{(n)}$ via the relation

$\displaystyle\mathbb{F}^{2}\Phi_{\alpha(n)}$ $\displaystyle=$
$\displaystyle\Big{(}(2n+1)^{2}\mathbb{Q}+(2n+1)(2n^{2}+2n-1){\rm
i}|\mu|\nabla^{2}+4n^{2}(n+2)^{2}|\mu|^{2}\Big{)}\Phi_{\alpha(n)}$ (5.123)
$\displaystyle+4n(2n^{2}+n-2){\rm
i}|\mu|\nabla_{\alpha}\nabla^{\beta}\Phi_{\beta\alpha(n-1)}-4{\rm
i}n\nabla_{\alpha\beta}\nabla^{\beta}\nabla^{\gamma}\Phi_{\gamma\alpha(n-1)}~{}$
$\displaystyle+4n(n-1)\nabla_{\alpha(2)}\nabla^{\beta(2)}\Phi_{\beta(2)\alpha(n-2)}~{}.$

####  5.2.3 Irreducible superfield representations

We begin by reviewing aspects of on-shell superfields in AdS3|2, as presented
in [152]. Given an integer $n\geq 1$, a superfield $\Phi_{\alpha(n)}$ on
$\mathds{V}_{(n)}$ is said to be on-shell if it satisfies the two constraints

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\nabla^{\beta}\Phi_{\beta\alpha(n-1)}~{},$ (5.124a)
$\displaystyle 0$ $\displaystyle=$ $\displaystyle\big{(}\mathbb{F}-\sigma
M\big{)}\Phi_{\alpha(n)}~{},$ (5.124b)

where $\sigma:=\pm 1$ and $M\geq 0$ is a real parameter of unit mass
dimension. Such a superfield furnishes the irreducible representation
$\mathfrak{S}(M,\sigma\frac{n}{2})$ of
$\mathfrak{osp}(1|2;{\mathbb{R}})\oplus\mathfrak{sl}(2,{\mathbb{R}})$. Thus,
we say that an on-shell superfield carries pseudo-mass $M$, superspin
$\frac{n}{2}$ and superhelicity $\frac{1}{2}(n+\frac{1}{2})\sigma$.

It is necessary to study the component structure of an on-shell superfield
(5.124) in order to demonstrate that it furnishes the irreducible
representation $\mathfrak{S}(M,\sigma\frac{n}{2})$. An on-shell superfield
describes two non-vanishing independent component fields,

$A_{\alpha(n)}:=\Phi_{\alpha(n)}|~{},\qquad B_{\alpha(n+1)}={\rm
i}^{n+1}\nabla_{\alpha}\Phi_{\alpha(n)}|~{}.$ (5.125)

It follows from the superfield constraints (5.124) that the component fields
are transverse,

$\nabla^{\beta\gamma}A_{\beta\gamma\alpha(n-2)}=0~{},\qquad\nabla^{\beta\gamma}B_{\beta\gamma\alpha(n-1)}=0~{}.$
(5.126)

Furthermore, the component fields can be shown to satisfy the first-order
constraints,

$\displaystyle 0=\big{(}{\cal
F}-\sigma_{A}\rho_{A}\big{)}A_{\alpha(n)}~{},\qquad 0=\big{(}{\cal
F}-\sigma_{B}\rho_{B}\big{)}B_{\alpha(n+1)}~{},$ (5.127a)

where the pseudo-masses $\rho_{A}$ and $\rho_{B}$ take the form

$\displaystyle\rho_{A}=\frac{n}{2n+1}\Big{|}\sigma
M-(n+2)|\mu|\Big{|}~{},\qquad\rho_{B}=\frac{n+1}{2n+1}\Big{|}\sigma
M+(n-1)|\mu|\Big{|}~{},$ (5.128)

and the corresponding signs of the helicities $\sigma_{A}$ and $\sigma_{B}$
are given by

$\displaystyle\sigma_{A}$ $\displaystyle=$ $\displaystyle\frac{\sigma
M-(n+2)|\mu|}{\big{|}\sigma
M-(n+2)|\mu|\big{|}}~{},\qquad\sigma_{B}=\frac{\sigma
M+(n-1)|\mu|}{\big{|}\sigma M+(n-1)|\mu|\big{|}}~{}.$ (5.129)

In accordance with the definition (5.7) of an on-shell field in AdS3, it
follows from the above analysis that the component fields $A_{\alpha(n)}$ and
$B_{\alpha(n+1)}$ realise the irreducible representations
$\mathfrak{D}\big{(}\rho_{A},\sigma_{A}\frac{n}{2}\big{)}$ and
$\mathfrak{D}\big{(}\rho_{B},\sigma_{B}\frac{n+1}{2}\big{)}$ of
$\mathfrak{so}(2,2)$, respectively. Hence, the on-shell superfield (5.124)
furnishes the irreducible representation $\mathfrak{S}(M,\sigma\frac{n}{2})$
of $\mathfrak{osp}(1|2;{\mathbb{R}})\oplus\mathfrak{sl}(2,{\mathbb{R}})$ (cf.
eq. (5.109)).

A superfield satisfying the first condition (5.124a) is said to be transverse.
Any transverse superfield may be shown to satisfy the following relation

$-\frac{{\rm
i}}{2}\nabla^{2}\Phi_{\alpha(n)}=\nabla_{(\alpha_{1}}{}^{\beta}\Phi_{\alpha_{2}...\alpha_{n})\beta}+(n+2)|\mu|\Phi_{\alpha(n)}~{}.$
(5.130)

From (5.130) it follows that an on-shell superfield (5.124) satisfies

$-\frac{{\rm i}}{2}\nabla^{2}\Phi_{\alpha(n)}=\frac{1}{2n+1}\Big{(}\sigma
M+2n(n+2)|\mu|\Big{)}\Phi_{\alpha(n)}~{},$ (5.131)

and hence the second-order mass-shell equation

$\displaystyle 0$
$\displaystyle=\big{(}\mathbb{Q}-\lambda^{2}\big{)}\Phi_{\alpha(n)}~{},$
(5.132a) $\displaystyle\lambda^{2}:=\frac{1}{(2n+1)^{2}}\big{[}\sigma M+2n$
$\displaystyle(n+2)|\mu|\big{]}\big{[}\sigma M+2(n-1)(n+1)|\mu|\big{]}~{}.$
(5.132b)

The equations (5.124a) and (5.131) were introduced in [167]. On the other
hand, one may instead consider a superfield $\Phi_{\alpha(n)}$ satisfying
(5.124a) and (5.132a). In this case, using the identity (5.123), one can show
that (5.132a) becomes

$\displaystyle
0=\Big{(}\mathbb{F}-\sigma_{(-)}|M_{(-)}|\Big{)}\Big{(}\mathbb{F}-\sigma_{(+)}|M_{(+)}|\Big{)}\Phi_{\alpha(n)}~{},$
(5.133)

where we have defined $\sigma_{(\pm)}=\text{sgn}(M_{(\pm)})$ and

$\displaystyle
M_{(\pm)}:=-(2n^{2}+2n-1)|\mu|\pm(2n+1)\sqrt{\lambda^{2}+|\mu|^{2}}~{}.$
(5.134)

It follows that such a field furnishes the reducible representation

$\mathfrak{S}\Big{(}|M_{(-)}|,\sigma_{(-)}\frac{n}{2}\Big{)}\oplus\mathfrak{S}\Big{(}|M_{(+)}|,\sigma_{(+)}\frac{n}{2}\Big{)}~{}.$
(5.135)

#####  5.2.3.1 Massless superfields

In this section we will introduce the notion of massless on-shell
supermultiplets in AdS3|2. For $n\geq 3$, an on-shell superfield
$\Phi_{\alpha(n)}$ (5.124) is said to be a massless higher-spin superfield if
it does not propagate any physical degrees of freedom. In AdS3|2, there are
two distinct classes of massless higher-spin superfields which are
distinguished by the sign of their superhelicity.

Given $\sigma=-1$, an on-shell superfield (5.124) is said to be massless if it
carries the pseudo-mass

$M=M_{(0,n)}^{(-)}=2(n-1)(n+1)|\mu|~{}.$ (5.136)

It may be shown that the system of equations (5.124) with pseudo-mass (5.136)
is compatible with the gauge symmetry

$\delta_{\Lambda}\Phi_{\alpha(n)}={\rm
i}^{n}\nabla_{\alpha}\Lambda_{\alpha(n-1)}~{},$ (5.137)

given that the real gauge parameter $\Lambda_{\alpha(n-1)}$ is also on-shell

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\nabla^{\beta}\Lambda_{\beta\alpha(n-2)}~{},$ (5.138a)
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\big{(}\mathbb{F}+M^{(-)}_{(0,n)}\big{)}\Lambda_{\alpha(n-1)}~{}.$
(5.138b)

Given $\sigma=+1$, we say that an on-shell supermultiplet (5.124) is massless
if it carries pseudo-mass

$M=M^{(+)}_{(1,n)}=2n(n-1)|\mu|~{}.$ (5.139)

It can be shown that such a superfield is compatible with the gauge symmetry

$\delta_{\Lambda}\Phi_{\alpha(n)}=\nabla_{\alpha(2)}\Lambda_{\alpha(n-2)}~{},$
(5.140)

given that the real gauge parameter $\Lambda_{\alpha(n-2)}$ is also on-shell

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\nabla^{\beta}\Lambda_{\beta\alpha(n-2)}~{},$ (5.141a)
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\big{(}\mathbb{F}-M^{(+)}_{(1,n)}\big{)}\Lambda_{\alpha(n-2)}~{}.$
(5.141b)

Analysing the component structure of these massless higher-spin superfields,
it can be shown that, upon imposing the appropriate Wess-Zumino gauges, the
surviving component fields satisfy the massless on-shell conditions in AdS3
(see section 5.1.2.1). Thus, massless higher-spin superfields do not propagate
physical degrees of freedom in AdS3|2. Note that (5.137) is the gauge symmetry
commonly associated with massless (half-)integer superspin actions (see
section 5.2.6).

#####  5.2.3.2 Partially massless superfields

In AdS3|2 there exist two distinct types of on-shell partially massless
superfields [152], which are distinguished by the sign $\sigma$ of their
superhelicity. More specifically, they are described by an on-shell superfield
(5.124) whose pseudo-mass and parameter $\sigma$ assume the special
combinations [152]161616In [152], partially massless superfields carrying
negative $\sigma=-1$ and positive $\sigma=1$ sign of superhelicity were
referred to as type A and type B partially massless superfields respectively.

$\displaystyle\sigma=+1~{},\quad M$ $\displaystyle\equiv
M^{(+)}_{(t,n)}=2\big{[}n(n-2t+1)-(t-1)\big{]}|\mu|~{},$ $\displaystyle 1$
$\displaystyle\leq t\leq\lfloor n/2\rfloor~{},$ (5.142a)
$\displaystyle\sigma=-1~{},\quad M$ $\displaystyle\equiv
M^{(-)}_{(t,n)}=2\big{[}n(n-2t)-(t+1)\big{]}|\mu|~{},$ $\displaystyle 0$
$\displaystyle\leq t\leq\lceil n/2\rceil-1~{}.$ (5.142b)

The integer $t$ is called the (super)depth and the corresponding
supermultiplets are denoted by $\Phi^{(t,+)}_{\alpha(n)}$ and
$\Phi^{(t,-)}_{\alpha(n)}$ respectively. For $t>1$, the pseudo-mass
$M^{(+)}_{(t,n)}$ violates the unitarity bound (5.112). Further more, the
pseudo-mass $M^{(-)}_{(t,n)}$ also violates the unitarity bound for $t>0$.
Thus, partially massless representations are non-unitary.

It was shown in [152] that the gauge symmetry associated with positive and
negative superhelicity partially massless superfields of depth-$t$ is

$\displaystyle\delta_{\Lambda}\Phi^{(t,+)}_{\alpha(n)}$
$\displaystyle=\phantom{\rm{i}^{n}}\big{(}\mathcal{D}_{\alpha(2)}\big{)}^{t}\Lambda_{\alpha(n-2t)}~{},$
$\displaystyle 1\leq t\leq\lfloor n/2\rfloor~{},$ (5.143a)
$\displaystyle\delta_{\Lambda}\Phi^{(t,-)}_{\alpha(n)}$
$\displaystyle=\text{i}^{n}\big{(}\mathcal{D}_{\alpha(2)}\big{)}^{t}\mathcal{D}_{\alpha}\Lambda_{\alpha(n-2t-1)}~{},$
$\displaystyle 0\leq t\leq\lceil n/2\rceil-1~{}.$ (5.143b)

In particular, the system of equations (5.124) and (5.142) is only invariant
under these transformations for an on-shell real gauge parameter which carries
the same pseudo-mass as its parent field. Note that strictly massless
superfields with superhelicity sign $\sigma=+1$ and $\sigma=-1$ correspond to
partially massless superfields $\Phi^{(t,+)}_{\alpha(n)}$ and
$\Phi^{(t,-)}_{\alpha(n)}$ with minimal super-depth $t=1$ and $t=0$,
respectively.

Partially massless superfields satisfy the second order equations (5.132)

$0=\big{(}\mathbb{Q}-\lambda_{(t,n)}^{(+)}|\mu|^{2}\big{)}\Phi_{\alpha(n)}^{(t,+)}~{},\quad
0=\big{(}\mathbb{Q}-\lambda_{(t,n)}^{(-)}|\mu|^{2}\big{)}\Phi_{\alpha(n)}^{(t,-)}~{},$
(5.144)

where the dimensionless constants

$\lambda_{(t,n)}^{(+)}=4(n-t)(n-t+1)~{},\qquad\lambda_{(t,n)}^{(-)}=4t(t+1)~{},$
(5.145)

correspond to the partially massless values.

#####  5.2.3.3 Massive superfields

An on-shell superfield (5.124) is said to be massive if it carries pseudo-mass
satisfying

$M>M^{(-)}_{(0,n)}~{}.$ (5.146)

This pseudo-mass does not violate the unitarity bound (5.112), hence the
corresponding representation is unitary. We can also introduce the notion of a
(non-unitary) massive supermultiplet, which carries the defining pseudo-mass
$M<M^{(-)}_{(0,n)}$ such that $M\neq$ $M^{(-)}_{(t,n)}$ for $1\leq t\leq\lceil
n/2\rceil-1$ and $M\neq$ $M^{(+)}_{(t,n)}$ for $1\leq t\leq\lfloor
n/2\rfloor$. We will not be interested in (non-unitary) massive superfields in
this chapter.

####  5.2.4 Superspin projection operators

In this section we study the supersymmetric generalisations of the AdS3 spin
projection operators and their corresponding applications. For any integer
$n\geq 1$, the rank-$n$ superspin projection operator
$\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ is defined by its action on
$\mathds{V}_{(n)}$ according to the rule:

$\displaystyle\mbox{\boldmath$\Pi$}^{\perp}_{[n]}:\mathds{V}_{(n)}$
$\displaystyle\longrightarrow$ $\displaystyle\mathds{V}_{(n)}~{},$ (5.147a)
$\displaystyle\Phi_{\alpha(n)}$ $\displaystyle\longmapsto$
$\displaystyle\mbox{\boldmath$\Pi$}^{\perp}_{[n]}\Phi_{\alpha(n)}~{}=:\mbox{\boldmath$\Pi$}^{\perp}_{\alpha(n)}(\Phi)~{}.$
(5.147b)

For fixed $n$, the superspin projection operator
$\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ is defined by the following properties:

  1. 1.

Idempotence: The operator $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ squares to
itself,

$\mbox{\boldmath$\Pi$}^{\perp}_{[n]}\mbox{\boldmath$\Pi$}^{\perp}_{[n]}=\mbox{\boldmath$\Pi$}^{\perp}_{[n]}~{}.$
(5.148a)

  2. 2.

Transversality: The operator $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ maps
$\Phi_{\alpha(n)}$ to a transverse superfield,

$\nabla^{\beta}\mbox{\boldmath$\Pi$}^{\perp}_{\beta\alpha(n-1)}(\Phi)=0~{}.$
(5.148b)

  3. 3.

Surjectivity: Every transverse superfield $\Phi^{\perp}_{\alpha(n)}$ belongs
to the image of $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$,

$\mathcal{D}^{\beta}\Phi^{\perp}_{\beta\alpha(n-1)}=0~{}\quad\implies\quad\mbox{\boldmath$\Pi$}^{\perp}_{[n]}\Phi^{\perp}_{\alpha(n)}=\Phi^{\perp}_{\alpha(n)}~{}.$
(5.148c)

Let us consider a superfield $\Phi_{\alpha(n)}$ on $\mathds{V}_{(n)}$ which
satisfies the first-order differential constraint (5.124b). It follows that
the superspin projection operator $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ maps
$\Phi_{\alpha(n)}$ to an on-shell superfield (5.124),

$\displaystyle\nabla^{\beta}\mbox{\boldmath$\Pi$}^{\perp}_{\beta\alpha(n-1)}(\Phi)$
$\displaystyle=$ $\displaystyle 0~{},$ (5.149a)
$\displaystyle\big{(}\mathbb{F}-\sigma
M\big{)}\mbox{\boldmath$\Pi$}^{\perp}_{\alpha(n)}(\Phi)$ $\displaystyle=$
$\displaystyle 0~{}.$ (5.149b)

In other words, $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ extracts the physical
component from $\Phi_{\alpha(n)}$ which realises the irreducible
representation $\mathfrak{S}(M,\sigma\frac{n}{2})$ of
$\mathfrak{osp}(1|2;{\mathbb{R}})\oplus\mathfrak{sl}(2,{\mathbb{R}})$.

To obtain a superprojector, we introduce the operator
$\Delta^{\alpha}{}_{\beta}$ [117]

$\Delta^{\alpha}{}_{\beta}:=-\frac{{\rm
i}}{2}\nabla^{\alpha}\nabla_{\beta}-2|\mu|\delta^{\alpha}{}_{\beta}~{},\quad\nabla^{\beta}\Delta^{\alpha}{}_{\beta}=\Delta^{\alpha}{}_{\beta}\nabla_{\alpha}=0~{},$
(5.150)

and its corresponding extensions [152]

$\Delta^{\alpha}_{[j]}{}_{\beta}:=-\frac{{\rm
i}}{2}\nabla^{\alpha}\nabla_{\beta}-2j|\mu|\delta^{\alpha}{}_{\beta}~{}.$
(5.151)

Note that for the case $j=1$, (5.151) coincides with (5.150). It can be shown
that the operator (5.151) has the following properties

$\displaystyle[\Delta^{\alpha_{1}}_{[j]}{}_{\beta_{1}},\Delta^{\alpha_{2}}_{[k]}{}_{\beta_{2}}]$
$\displaystyle=$
$\displaystyle\varepsilon_{\beta_{1}\beta_{2}}|\mu|\big{(}\nabla^{\alpha(2)}-|\mu|M^{\alpha(2)}\big{)}-\varepsilon^{\alpha_{1}\alpha_{2}}|\mu|\big{(}\nabla_{\beta(2)}-|\mu|M_{\beta(2)}\big{)}~{},\hskip
28.45274pt$ (5.152a)
$\displaystyle\varepsilon^{\beta_{1}\beta_{2}}\Delta^{\alpha_{1}}_{[j]}{}_{\beta_{1}}\Delta^{\alpha_{2}}_{[j+1]}{}_{\beta_{2}}$
$\displaystyle=$
$\displaystyle-j\varepsilon^{\alpha_{1}\alpha_{2}}|\mu|\big{(}{\rm
i}\nabla^{2}+4(j+1)|\mu|^{2}\big{)}~{},$ (5.152b)
$\displaystyle\varepsilon_{\alpha_{1}\alpha_{2}}\Delta^{\alpha_{1}}_{[j+1]}{}_{\beta_{1}}\Delta^{\alpha_{2}}_{[j]}{}_{\beta_{2}}$
$\displaystyle=$ $\displaystyle
j\varepsilon_{\beta_{1}\beta_{2}}|\mu|\big{(}{\rm
i}\nabla^{2}+4(j+1)|\mu|^{2}\big{)}~{},$ (5.152c)
$\displaystyle\Delta^{\beta}_{[j]}{}_{\alpha}\Delta^{\gamma}_{[k]}{}_{\beta}$
$\displaystyle=$ $\displaystyle-\frac{{\rm
i}}{2}\nabla^{2}\Delta^{\gamma}_{[1]}{}_{\alpha}+(j+k-1){\rm
i}|\mu|\nabla^{\gamma}\nabla_{\alpha}+4jk|\mu|^{2}\delta_{\alpha}{}^{\gamma}~{},$
(5.152d) $\displaystyle\ [\Delta^{\alpha}_{[j]}{}_{\beta},\nabla^{2}]$
$\displaystyle=$ $\displaystyle 0~{},$ (5.152e)

for arbitrary integers $j$ and $k$.

Let us define the operator $\mathbb{T}_{[n]}$, which acts on
$\mathds{V}_{(n)}$ by the rule

$\mathbb{T}_{[n]}\Phi_{\alpha(n)}\equiv\mathbb{T}_{\alpha(n)}(\Phi)=\Delta^{\beta_{1}}_{[1]}{}_{(\alpha_{1}}\Delta^{\beta_{2}}_{[2]}{}_{\alpha_{2}}\cdots\Delta^{\beta_{n}}_{[n]}{}_{\alpha_{n})}\Phi_{\beta(n)}~{}.$
(5.153)

This operator maps $\Phi_{\alpha(n)}$ to a transverse superfield

$\nabla^{\beta}\mathbb{T}_{\beta\alpha(n-1)}(\Phi)=0~{}.$ (5.154)

To see this, one needs to open the symmetrisation in (5.153)

$\displaystyle\nabla^{\gamma}\mathbb{T}_{\gamma\alpha(n-1)}(\Phi)$
$\displaystyle=$
$\displaystyle\nabla^{\gamma}\Delta^{\beta_{1}}_{[1]}{}_{(\gamma}\Delta^{\beta_{2}}_{[2]}{}_{\alpha_{1}}\cdots\Delta^{\beta_{n}}_{[n]}{}_{\alpha_{n-1})}\Phi_{\beta(n)}~{}$
(5.155) $\displaystyle\propto$
$\displaystyle\nabla^{\gamma}\big{(}\Delta^{\beta_{1}}_{[1]}{}_{\gamma}\Delta^{\beta_{2}}_{[2]}{}_{\alpha_{1}}\cdots\Delta^{\beta_{n}}_{[n]}{}_{\alpha_{n-1}}+(n!-1)~{}\text{permutations}\big{)}\Phi_{\beta(n)}~{}.\hskip
28.45274pt$

By making use of (5.152b), it can be shown that the remaining $(n!-1)$ terms
can be expressed in the same form as the first. Then transversality follows
immediately as a consequence of property (5.150). However, $\mathbb{T}_{[n]}$
does not square to itself on $\mathds{V}_{(n)}$

$\displaystyle\mathbb{T}_{[n]}\mathbb{T}_{[n]}\Phi_{\alpha(n)}=\frac{1}{(2n+1)^{n}}\prod_{t=0}^{\lceil
n/2\rceil-1}\big{(}\mathbb{F}+M^{(-)}_{(t,n)}\big{)}\prod_{t=1}^{\lfloor
n/2\rfloor}\big{(}\mathbb{F}-M^{(+)}_{(t,n)}\big{)}\mathbb{T}_{[n]}\Phi_{\alpha(n)}~{},$
(5.156)

where $M^{(\pm)}_{(t,n)}$ denotes the pseudo-masses associated with a
partially massless superfield (5.142). We can immediately introduce the
dimensionless operator

$\displaystyle\mbox{\boldmath$\Pi$}^{\perp}_{[n]}\Phi_{\alpha(n)}:=(2n+1)^{n}\bigg{[}\prod_{t=0}^{\lceil
n/2\rceil-1}\big{(}\mathbb{F}+M^{(-)}_{(t,n)}\big{)}\prod_{t=1}^{\lfloor
n/2\rfloor}\big{(}\mathbb{F}-M^{(+)}_{(t,n)}\big{)}\bigg{]}^{-1}\mathbb{T}_{[n]}\Phi_{\beta(n)}~{},$
(5.157)

which is idempotent and transverse by construction. In addition, it can be
shown that the operator $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ acts as the
identity on the space of transverse superfields (5.148c). Hence,
$\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ satisfies properties (5.148) and can be
identified as a rank-$n$ superprojector on AdS3|2. So far, we have been unable
to obtain an expression for ${\mbox{\boldmath$\Pi$}}^{\perp}_{[n]}$ which is
purely in terms of the Casimir operators $\mathbb{F}$ and $\mathbb{Q}$.

An alternative form of the superprojector
$\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ can be derived, which instead makes
contact with the Casimir operator $\mathbb{Q}$. Let us introduce the
dimensionless operator

$\displaystyle\widehat{\mbox{\boldmath$\Pi$}}{}^{\perp}_{[n]}\Phi_{\alpha(n)}$
$\displaystyle=$ $\displaystyle\bigg{[}\prod_{t=0}^{n-1}\big{(}\mathbb{Q}+{\rm
i}t|\mu|\nabla^{2}\big{)}\bigg{]}^{-1}\widehat{\Delta}^{\beta_{1}}_{[1]}{}_{(\alpha_{1}}\widehat{\Delta}^{\beta_{2}}_{[2]}{}_{\alpha_{2}}...\widehat{\Delta}^{\beta_{n}}_{[n]}{}_{\alpha_{n})}\Phi_{\beta(n)}~{},$
(5.158)

where we denote $\widehat{\Delta}^{\beta}_{[j]}{}_{\alpha}$ as

$\widehat{\Delta}^{\beta}_{[j]}{}_{\alpha}:=-\frac{{\rm
i}}{2}\nabla^{2}{\Delta}^{\beta}_{[j]}{}_{\alpha}~{}.$ (5.159)

Making use of the properties of $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ and the
identity

$-\frac{{\rm
i}}{2}\nabla^{2}\Phi^{\perp}_{\alpha(n)}=\frac{1}{2n+1}\big{(}\mathbb{F}+2n(n+2)|\mu|\big{)}\Phi^{\perp}_{\alpha(n)}~{},$
(5.160)

where $\Phi^{\perp}_{\alpha(n)}$ is an arbitrary transverse superfield, it can
be shown that
$\widehat{\mbox{\boldmath$\Pi$}}{}^{\perp}_{[n]}\Phi_{\alpha(n)}$ satisfies
properties (5.148) and thus is also a superprojector on AdS3|2. In the flat-
superspace limit $|\mu|\rightarrow 0$, the superspin projection operator
$\widehat{\mbox{\boldmath$\Pi$}}{}^{\perp}_{[n]}$ coincides with the
superprojector (3.54) in $\mathbb{M}^{3|2}$. It can be shown that
${\mbox{\boldmath$\Pi$}}^{\perp}_{[n]}$ and
$\widehat{\mbox{\boldmath$\Pi$}}{}^{\perp}_{[n]}$ are equivalent on
$\mathds{V}_{(n)}$.

We recall that applying the AdS3 spin projection operator $\Pi^{\perp}_{[n]}$
to a field $\phi_{\alpha(n)}$ on the mass-shell (5.14b) yields the projection
$\Pi^{\perp}_{[n]}\phi_{\alpha(n)}$ which furnishes the reducible
representation (5.16). A single irreducible representation can be singled out
from the decomposition (5.16) via the application of the helicity projectors
(5.57). The significance of the condition (5.14b) is that it allows one to
resolve the poles in both types of projectors.

In the supersymmetric case, the equation analogous to (5.14b) which
$\Phi_{\alpha(n)}$ should satisfy is (5.132a). Applying
$\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ to such a $\Phi_{\alpha(n)}$, one
obtains the reducible representation (5.135). However, it appears that the
imposition of (5.132a) does not allow one to resolve the poles of the
superprojector in either of the forms (5.157) or (5.158). Therefore, rather
then imposing (5.132a), one must start with a superfield $\Phi_{\alpha(n)}$
obeying the first-order constraint (5.124b), which does allow for resolution
of the poles. In this case, after application of
$\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$, the superfield $\Phi_{\alpha(n)}$
already corresponds to an irreducible representation with fixed superhelicity,
relinquishing the need for superhelicity projectors. Thus, it suffices to
provide only the superspin projection operators
$\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$.

#####  5.2.4.1 Longitudinal projectors

For $n\geq 1$, let us define the orthogonal complement of
$\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ on $\mathds{V}_{(n)}$ by the rule

$\mbox{\boldmath$\Pi$}^{\parallel}_{[n]}\Phi_{\alpha(n)}=\big{(}\mathds{1}-\mbox{\boldmath$\Pi$}^{\perp}_{[n]}\big{)}\Phi_{\alpha(n)}~{}.$
(5.161)

By construction, the operators $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ and
$\mbox{\boldmath$\Pi$}^{\parallel}_{[n]}$ resolve the identity,
$\mathds{1}=\mbox{\boldmath$\Pi$}^{\parallel}_{[n]}+\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$,
and are orthogonal projectors

$\mbox{\boldmath$\Pi$}^{\perp}_{[n]}\mbox{\boldmath$\Pi$}^{\perp}_{[n]}=\mbox{\boldmath$\Pi$}^{\perp}_{[n]}~{},\qquad\mbox{\boldmath$\Pi$}^{\parallel}_{[n]}\mbox{\boldmath$\Pi$}^{\parallel}_{[n]}=\mbox{\boldmath$\Pi$}^{\parallel}_{[n]}~{},\qquad\mbox{\boldmath$\Pi$}^{\parallel}_{[n]}\mbox{\boldmath$\Pi$}^{\perp}_{[n]}=\mbox{\boldmath$\Pi$}^{\perp}_{[n]}\mbox{\boldmath$\Pi$}^{\parallel}_{[n]}=0~{}.$
(5.162)

It can be shown that $\mbox{\boldmath$\Pi$}^{\parallel}_{[n]}$ extracts the
longitudinal component of a superfield $\Phi_{\alpha(n)}$. A rank-$n$
superfield $\Phi^{\parallel}_{\alpha(n)}$ on $\mathds{V}_{(n)}$ is said to be
longitudinal if there exists a rank-$(n-1)$ superfield $\Phi_{\alpha(n-1)}$
such that $\Phi^{\parallel}_{\alpha(n)}$ can be expressed as
$\Phi^{\parallel}_{\alpha(n)}={\rm i}^{n}\nabla_{\alpha}\Phi_{\alpha(n-1)}$.
Thus, we find

$\mbox{\boldmath$\Pi$}^{\parallel}_{[n]}\Phi_{\alpha(n)}={\rm
i}^{n}\nabla_{\alpha}\Phi_{\alpha(n-1)}~{},$ (5.163)

for some unconstrained real superfield $\Phi_{\alpha(n-1)}$. In order to see
this, it proves beneficial to make use of the superprojector
$\widehat{\mbox{\boldmath$\Pi$}}{}^{\perp}_{[n]}$, and express the operator
$\widehat{\Delta}^{\beta}_{[j]}{}_{\alpha}$ in the form

$\widehat{\Delta}^{\beta}_{[j]}{}_{\alpha}:=-\frac{1}{4}\nabla_{\alpha}\nabla^{\beta}\nabla^{2}+\big{(}\mathbb{Q}+{\rm
i}(j-1)|\mu|\nabla^{2}\big{)}\delta_{\alpha}{}^{\beta}~{}.$ (5.164)

Let us consider the action of $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ on the
longitudinal superfield $\Phi^{\parallel}_{\alpha(n)}$. Opening the
symmetrisation present in $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ gives

$\displaystyle\mbox{\boldmath$\Pi$}^{\perp}_{[n]}\Phi^{\parallel}_{\alpha(n)}$
$\displaystyle=$ $\displaystyle{\rm
i}^{n}\Delta^{\beta_{1}}_{[1]}{}_{(\alpha_{1}}\Delta^{\beta_{2}}_{[2]}{}_{\alpha_{2}}...\Delta^{\beta_{n}}_{[n]}{}_{\alpha_{n})}\nabla_{(\beta_{1}}\Phi_{\beta_{2}...\beta_{n})}~{}$
$\displaystyle=$ $\displaystyle\frac{{\rm
i}^{n}}{n!}\Delta^{\beta_{1}}_{[n]}{}_{(\alpha_{1}}\Delta^{\beta_{2}}_{[n-1]}{}_{\alpha_{2}}...\Delta^{\beta_{n}}_{[1]}{}_{\alpha_{n})}\big{(}\nabla_{\beta_{n}}\Phi_{\beta_{1}...\beta_{n-1}}+(n!-1)~{}\text{permutations}\big{)}~{}.\hskip
28.45274pt$

Note that we have made use of the identity (5.152a) to rearrange the operators
$\Delta^{\beta}_{[j]}{}_{\alpha}$. Making use of the relation (5.152c) allows
us to express the other $(n!-1)$ permutations in the same form as the first.
Then due to the property (5.150), it follows that

$\Phi^{\parallel}_{\alpha(n)}={\rm
i}^{n}\nabla_{\alpha}\Phi_{\alpha(n-1)}\qquad\implies\qquad\mbox{\boldmath$\Pi$}^{\perp}_{[n]}\Phi^{\parallel}_{\alpha(n)}=0~{}.$
(5.166)

Thus, the superprojector $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ annihilates any
longitudinal superfield. Consequently,
$\mbox{\boldmath$\Pi$}^{\parallel}_{[n]}$ acts as the identity operator on the
space of rank-$n$ longitudinal superfields

$\Phi^{\parallel}_{\alpha(n)}={\rm
i}^{n}\nabla_{\alpha}\Phi_{\alpha(n-1)}\qquad\implies\qquad\mbox{\boldmath$\Pi$}^{\parallel}_{[n]}\Phi^{\parallel}_{\alpha(n)}=\Phi^{\parallel}_{\alpha(n)}~{}.$
(5.167)

Using the fact that the $\mbox{\boldmath$\Pi$}^{\parallel}_{[n]}$ and
$\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ resolve the identity, it follows that
one can decompose any superfield $\Phi_{\alpha(n)}$ in the following manner

$\Phi_{\alpha(n)}=\Phi^{\perp}_{\alpha(n)}+{\rm
i}^{n}\nabla_{\alpha}\Phi_{\alpha(n-1)}~{}.$ (5.168)

Here, $\Phi^{\perp}_{\alpha(n)}$ is transverse and $\Phi_{\alpha(n-1)}$ is
unconstrained. Repeating this prescription iteratively yields the
decomposition

$\displaystyle\Phi_{\alpha(n)}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\lfloor
n/2\rfloor}\big{(}\nabla_{\alpha(2)}\big{)}^{j}\Phi^{\perp}_{\alpha(n-2j)}+{\rm
i}^{n}\sum_{j=0}^{\lceil
n/2\rceil-1}\big{(}\nabla_{\alpha(2)}\big{)}^{j}\nabla_{\alpha}\Phi^{\perp}_{\alpha(n-2j-1)}~{}.$
(5.169a)

Here, the real superfields $\Phi^{\perp}_{\alpha(n-2j)}$ and
$\Phi^{\perp}_{\alpha(n-2j-1)}$ are transverse, except for $\Phi^{\perp}$.

####  5.2.5 Linearised higher-spin super-Cotton tensors

In this section we make use of the rank-$n$ superspin projection operators
$\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ to construct the linearised higher-spin
super-Cotton tensors in AdS3|2. These super-Cotton tensors were recently
derived in [152], where they were shown to take the explicit form

$\mathfrak{W}_{\alpha(n)}(H)=\Delta^{\beta_{1}}_{[1]}{}_{(\alpha_{1}}\Delta^{\beta_{2}}_{[2]}{}_{\alpha_{2}}\cdots\Delta^{\beta_{n}}_{[n]}{}_{\alpha_{n})}H_{\beta(n)}~{},$
(5.170)

which is a real primary descendent of the SCHS superfield $H_{\alpha(n)}$. The
latter is defined modulo gauge transformations of the form

$\delta_{\Lambda}H_{\alpha(n)}={\rm
i}^{n}\nabla_{\alpha}\Lambda_{\alpha(n-1)}~{},$ (5.171)

where the gauge parameter $\Lambda_{\alpha(n-1)}$ is a real unconstrained
superfield. The super-Cotton tensor (5.170) satisfies the defining properties:

  1.   2. 1.

$\mathfrak{W}_{\alpha(n)}(H)$ is transverse,

$\nabla^{\beta}\mathfrak{W}_{\beta\alpha(n-1)}(H)=0~{}.$ (5.172a)

  3. 2.

$\mathfrak{W}_{\alpha(n)}(H)$ is invariant under the gauge transformations
(5.171),

$\delta_{\Lambda}\mathfrak{W}_{\alpha(n)}(H)=0~{}.$ (5.172b)

The superprojectors (5.157) can be used to recast the super-Cotton tensors
(5.170) in the simple form

$\mathfrak{W}_{\alpha(n)}(H)=\frac{1}{(2n+1)^{n}}\prod_{t=0}^{\lceil
n/2\rceil-1}\big{(}\mathbb{F}+M^{(-)}_{(t,n)}\big{)}\prod_{t=1}^{\lfloor
n/2\rfloor}\big{(}\mathbb{F}-M^{(+)}_{(t,n)}\big{)}\mbox{\boldmath$\Pi$}^{\perp}_{[n]}H_{\alpha(n)}~{},$
(5.173)

where $M^{(\pm)}_{(t,n)}$ denotes the partially massless pseudo-masses
(5.142). In the flat-superspace limit, the super-Cotton tensor (5.173) reduces
to its counterpart (3.108) in $\mathbb{M}^{3|2}$. Expressing
$\mathfrak{W}_{\alpha(n)}(H)$ in the form (5.173) is beneficial for the
following reasons: (i) transversality of $\mathfrak{W}_{\alpha(n)}(H)$ is made
manifest on account of property (5.154); (ii) gauge invariance is also
manifest as a consequence of (5.166); and (iii) in the transverse gauge

$H_{\alpha(n)}\equiv
H^{\perp}_{\alpha(n)}~{},\qquad\nabla^{\beta}H^{\perp}_{\beta\alpha(n-1)}=0~{},$
(5.174)

it follows from (5.148c) that $\mathfrak{W}_{\alpha(n)}(H)$ factorises as
follows

$\mathfrak{W}_{\alpha(n)}(H^{\perp})=\frac{1}{(2n+1)^{n}}\prod_{t=0}^{\lceil
n/2\rceil-1}\big{(}\mathbb{F}+M^{(-)}_{(t,n)}\big{)}\prod_{t=1}^{\lfloor
n/2\rfloor}\big{(}\mathbb{F}-M^{(+)}_{(t,n)}\big{)}H^{\perp}_{\alpha(n)}~{}.$
(5.175)

From the above observations, it follows that the action [117, 86] for the
superconformal higher-spin prepotential $H_{\alpha(n)}$

$\displaystyle\mathbb{S}^{(n)}_{\rm SCHS}[H]=-\frac{{\rm
i}^{n}}{2^{\left\lfloor{n/2}\right\rfloor+1}}\int{\rm d}^{3}x{\rm
d}^{2}\theta\,E\,H^{\alpha(n)}\mathfrak{W}_{\alpha(n)}(H)~{},$ (5.176)

is manifestly gauge-invariant. In the transverse gauge (5.174), the kinetic
operator in (5.176) factorises into wave operators associated with partially
massless superfields of all depths, in accordance with (5.175).

####  5.2.6 Massless higher-spin theories

According to [127], for each superspin value $\hat{s}\geq 1$, where $\hat{s}$
is either integer $(\hat{s}=s)$ or half-integer $(\hat{s}=s+\frac{1}{2})$,
there exist two off-shell formulations for a massless ${\cal N}=1$
superspin-$\hat{s}$ multiplet in AdS3. These two formulations are referred to
as being either longitudinal or transverse. In this section, we review the
explicit formulation of these respective theories, which were derived in [127,
117].

An off-shell massless superspin-$\frac{n}{2}$ gauge theory in AdS3 can be
realised in terms of two superfields: a superconformal gauge prepotential (see
section 5.2.5); and a compensating multiplet. The longitudinal and transverse
formulations are distinguished by the compensating multiplet and its
corresponding gauge transformations. Given an integer $n\geq 4$, a massless
higher-spin theory is said to have integer superspin or half-integer superspin
if the associated superconformal prepotential $H_{\alpha(n)}$ has an even
$n=2s$ or odd $n=2s+1$ number of indices, respectively.

To study the dynamics of ${\cal N}=1$ supersymmetric field theories in AdS3, a
manifestly supersymmetric action principle is required. Such an action is
associated with a real scalar Lagrangian $L$ and has the form

$\displaystyle S=\int{\rm d}^{3}x\,{\rm d}^{2}\theta E\,L~{},\qquad
E^{-1}={\rm Ber}(E_{A}{}^{M})~{}.$ (5.177)

In what follows, we make use of the notation ${\rm d}^{3|2}z:={\rm d}^{3}x{\rm
d}^{2}\theta$. Making use of the ${\cal N}=1$ AdS transformation law
$\delta_{\xi}L=\xi L$ and the identities (5.118) and (5.119), it can be shown
that the action (5.177) is invariant under the isometry group of AdS3|2.

#####  5.2.6.1 Longitudinal formulation for the massless
superspin-$(s+\frac{1}{2})$ multiplet

The longitudinal formulation for the massless superspin-$(s+\frac{1}{2})$
multiplet is realised in terms of the real unconstrained variables

${\cal
V}^{\parallel}_{(s+\frac{1}{2})}=\big{\\{}H_{\alpha(2s+1)},L_{\alpha(2s-2)}\big{\\}}~{},$
(5.178)

which are defined modulo gauge transformations of the form

$\displaystyle\delta_{\zeta}H_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle{\rm i}\nabla_{\alpha}\zeta_{\alpha(2s)}~{},$ (5.179a)
$\displaystyle\delta_{\zeta}L_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\frac{s}{2(2s+1)}\nabla^{\beta\gamma}\zeta_{\beta\gamma\alpha(2s-2)}~{},$
(5.179b)

where the real gauge parameter $\zeta_{\alpha(2s)}$ is unconstrained. The
unique gauge-invariant action formulated in terms of the superfields (5.178)
takes the following form

$\displaystyle{S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3|2}z\,E\,\bigg{\\{}-\frac{{\rm
i}}{2}H^{\alpha(2s+1)}{\mathbb{Q}}H_{\alpha(2s+1)}}$ $\displaystyle-\frac{{\rm
i}}{8}\nabla_{\beta}H^{\beta\alpha(2s)}\nabla^{2}\nabla^{\gamma}H_{\gamma\alpha(2s)}+\frac{{\rm
i}s}{4}{\nabla}_{\beta\gamma}H^{\beta\gamma\alpha(2s-1)}{\nabla}^{\delta\lambda}H_{\delta\lambda\alpha(2s-1)}$
$\displaystyle+(2s-1)L^{\alpha(2s-2)}\nabla^{\beta\gamma}\nabla^{\delta}H_{\beta\gamma\delta\alpha(2s-2)}$
$\displaystyle+2(2s-1)\Big{(}L^{\alpha(2s-2)}({\rm
i}\nabla^{2}-4|\mu|)L_{\alpha(2s-2)}-\frac{{\rm
i}}{s}(s-1)\nabla_{\beta}L^{\beta\alpha(2s-3)}\nabla^{\gamma}L_{\gamma\alpha(2s-3)}\Big{)}$
$\displaystyle+|\mu|\Big{(}s\,\nabla_{\beta}H^{\beta\alpha(2s)}\nabla^{\gamma}H_{\gamma\alpha(2s)}+\frac{1}{2}(2s+1)H^{\alpha(2s+1)}(\nabla^{2}-4{\rm
i}|\mu|)H_{\alpha(2s+1)}\Big{)}\bigg{\\}}~{},$ (5.180)

where $\mathbb{Q}$ is the quadratic Casimir operator (5.121a). The action
(5.2.6.1) was first derived in [117]. In the flat-superspace limit
$|\mu|\rightarrow 0$, the action (5.2.6.1) coincides with the model derived in
(3.5.2) upon the field redefinition
$L_{\alpha(2s-2)}=-\frac{1}{2}Y_{\alpha(2s-2)}$.

#####  5.2.6.2 Transverse formulation for the massless
superspin-$(s+\frac{1}{2})$ multiplet

The transverse formulation for the massless superspin-$(s+\frac{1}{2})$
multiplet is constructed in terms of the real superfields

${\cal
V}^{\perp}_{(s+\frac{1}{2})}=\big{\\{}H_{\alpha(2s+1)},\Upsilon_{\beta;\alpha(2s-2)}\big{\\}}~{},$
(5.181)

where $\Upsilon_{\beta;\alpha(2s-2)}$ is a reducible superfield pertaining to
the representation $2\otimes(2s-1)$ of $\mathsf{SL}(2,\mathbb{R})$. The
superfield $\Upsilon_{\beta;\alpha(2s-2)}$ can be decomposed into irreducible
components by the following rule

$\Upsilon_{\beta;\alpha(2s-2)}=\Upsilon_{(\beta;\alpha_{1}...\alpha_{2s-2})}+\frac{1}{2s-1}\sum_{k=1}^{2s-2}\varepsilon_{\beta\alpha_{k}}\Upsilon^{\gamma;}{}_{\gamma\alpha_{1}...\hat{\alpha}_{k}...\alpha_{2s-2}}~{}.$
(5.182)

where the hatted index of
$\Upsilon^{\gamma;}{}_{\gamma\alpha_{1}...\hat{\alpha}_{k}...\alpha_{2s-2}}$
is omitted. The superfields (5.181) are defined modulo gauge transformations
of the form

$\displaystyle\delta_{\zeta}H_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle{\rm i}\nabla_{\alpha}\zeta_{\alpha(2s)}~{},$ (5.183a)
$\displaystyle\delta_{\zeta,\eta}\Upsilon_{\beta;\alpha(2s-2)}$
$\displaystyle=$ $\displaystyle\frac{{\rm
i}}{2s+1}\Big{(}\nabla^{\gamma}\zeta_{\gamma\beta\alpha(2s-2)}+(2s+1)\nabla_{\beta}\eta_{\alpha(2s-2)}\Big{)}~{},$
(5.183b)

where the gauge parameters $\zeta_{\alpha(2s)}$ and $\eta_{\alpha(2s-2)}$ are
real unconstrained. The unique action which is invariant under the gauge
transformations (5.183) takes the form

$\displaystyle
S^{\perp}_{(s+\frac{1}{2})}[{H}_{\alpha(2s+1)},\Upsilon_{\beta;\alpha(2s-2)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3|2}z\,E\,\bigg{\\{}-\frac{{\rm
i}}{2}H^{\alpha(2s+1)}{\mathbb{Q}}H_{\alpha(2s+1)}$ $\displaystyle-\frac{{\rm
i}}{8}\nabla_{\beta}H^{\beta\alpha(2s)}\nabla^{2}\nabla^{\gamma}H_{\gamma\alpha(2s)}+\frac{{\rm
i}}{8}{\nabla}_{\beta\gamma}H^{\beta\gamma\alpha(2s-1)}{\nabla}^{\delta\lambda}H_{\delta\lambda\alpha(2s-1)}$
$\displaystyle-\frac{{\rm
i}}{4}(2s-1)\Omega^{\beta;\,\alpha(2s-2)}\nabla^{\gamma\delta}H_{\gamma\delta\beta\alpha(2s-2)}+{\rm
i}s(2s-1)|\mu|\,\Upsilon^{\beta;\,\alpha(2s-2)}\Omega_{\beta;\,\alpha(2s-2)}$
$\displaystyle-\frac{{\rm
i}}{8}(2s-1)\Big{(}\Omega^{\beta;\,\alpha(2s-2)}\Omega_{\beta;\,\alpha(2s-2)}-2(s-1)\Omega_{\beta;}\,^{\beta\alpha(2s-3)}\Omega^{\gamma;}\,_{\gamma\alpha(2s-3)}\Big{)}$
$\displaystyle+|\mu|\Big{(}H^{\alpha(2s+1)}\big{(}\nabla^{2}-4{\rm
i}|\mu|\big{)}H_{\alpha(2s+1)}+\frac{1}{2}\
\nabla_{\beta}H^{\beta\alpha(2s)}\nabla^{\gamma}H_{\gamma\alpha(2s)}\Big{)}\bigg{\\}}~{},$
(5.184)

where $\Omega_{\beta;\alpha(2s-2)}$ corresponds to the real ${\cal N}=1$ field
strength

$\displaystyle\Omega_{\beta;\,\alpha(2s-2)}=-{\rm
i}\big{(}\nabla^{\gamma}\nabla_{\beta}-4{\rm
i}|\mu|\delta_{\beta}\,^{\gamma}\big{)}{\Upsilon}_{\gamma;\,\alpha(2s-2)}~{},\qquad\nabla^{\beta}\Omega_{\beta;\,\alpha(2s-2)}=0~{}.$
(5.185)

The above theory was recently introduced in [127].

#####  5.2.6.3 Longitudinal formulation for the massless superspin-$s$
multiplet

The longitudinal formulation for the massless superspin-$s$ multiplet is
realised in terms of the real unconstrained variables

${\cal
V}^{\parallel}_{(s)}=\big{\\{}H_{\alpha(2s)},V_{\alpha(2s-2)}\big{\\}}~{},$
(5.186)

which are defined modulo gauge transformations of the form

$\displaystyle\delta_{\zeta}H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{\alpha}\zeta_{\alpha(2s-1)}~{},$ (5.187a)
$\displaystyle\delta_{\zeta}V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s}\nabla^{\beta}\zeta_{\beta\alpha(2s-2)}~{},$
(5.187b)

where the gauge parameter $\zeta_{\alpha(2s-1)}$ is real unconstrained. The
unique action which is invariant under the gauge transformations (5.187) takes
the following form

$\displaystyle
S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3|2}z\,E\,\bigg{\\{}\frac{1}{2}H^{\alpha(2s)}\big{(}{\rm
i}\nabla^{2}+4|\mu|\big{)}H_{\alpha(2s)}$ $\displaystyle-\frac{{\rm
i}}{2}\nabla_{\beta}H^{\beta\alpha(2s-1)}\nabla^{\gamma}H_{\gamma\alpha(2s-1)}-(2s-1)V^{\alpha(2s-2)}\nabla^{\beta\gamma}H_{\beta\gamma\alpha(2s-2)}$
$\displaystyle+(2s-1)\Big{(}\frac{1}{2}V^{\alpha(2s-2)}\big{(}{\rm
i}\nabla^{2}+8s|\mu|\big{)}V_{\alpha(2s-2)}+{\rm
i}(s-1)\nabla_{\beta}V^{\beta\alpha(2s-3)}\nabla^{\gamma}V_{\gamma\alpha(2s-3)}\Big{)}\bigg{\\}}~{}.$

Up to an overall normalisation factor, the action (5.2.6.3) coincides with the
off-shell ${\cal N}=1$ action for the massless superspin-$s$ multiplet in
[117]. In the flat-superspace limit, the action (5.2.6.3) reduces to (3.144)
in $\mathbb{M}^{3|2}$.

#####  5.2.6.4 Transverse formulation for the massless superspin-$s$ multiplet

The transverse formulation for the massless superspin-$s$ multiplet is
constructed in terms of the real unconstrained superfields

${\cal
V}^{\perp}_{(s)}=\big{\\{}H_{\alpha(2s)},\Psi_{\beta;\alpha(2s-2)}\big{\\}}~{},$
(5.189)

which are defined modulo gauge transformations of the form

$\displaystyle\delta_{\zeta}H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{\alpha}\zeta_{\alpha(2s-1)}~{},$ (5.190a)
$\displaystyle\delta_{\zeta,\eta}\Psi_{\beta;\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\zeta_{\beta\alpha(2s-2)}+{\rm
i}\nabla_{\beta}\eta_{\alpha(2s-2)}~{},$ (5.190b)

where the gauge parameters $\zeta_{\alpha(2s-1)}$ and $\eta_{\alpha(2s-2)}$
are real unconstrained. The unique gauge-invariant action is given by

$\displaystyle
S^{\perp}_{(s)}[H_{\alpha(2s)},\Psi_{\beta;\alpha(2s-2)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3|2}z\,E\,\bigg{\\{}\frac{1}{2}H^{\alpha(2s)}({\rm
i}\nabla^{2}+8s|\mu|)H_{\alpha(2s)}$ $\displaystyle-{\rm
i}s\nabla_{\beta}H^{\beta\alpha(2s-1)}\nabla^{\gamma}H_{\gamma\alpha(2s-1)}-(2s-1){\cal
W}^{\beta;\,\alpha(2s-2)}\nabla^{\gamma}H_{\gamma\beta\alpha(2s-2)}$
$\displaystyle-\frac{{\rm i}}{2}(2s-1)\Big{(}{\cal
W}^{\beta;\,\alpha(2s-2)}{\cal W}_{\beta;\,\alpha(2s-2)}+\frac{s-1}{s}{\cal
W}_{\beta;}\,^{\beta\alpha(2s-3)}{\cal
W}^{\gamma;}\,_{\gamma\alpha(2s-3)}\Big{)}$ $\displaystyle-2{\rm
i}(2s-1)|\mu|\Psi^{\beta;\,\alpha(2s-2)}{\cal
W}_{\beta;\,\alpha(2s-2)}\bigg{\\}}~{},$ (5.191)

where ${\cal W}_{\beta;\,\alpha(2s-2)}$ denotes the real ${\cal N}=1$ field
strength

$\displaystyle{\cal W}_{\beta;\,\alpha(2s-2)}=-{\rm
i}\big{(}\nabla^{\gamma}\nabla_{\beta}-4{\rm
i}|\mu|\delta_{\beta}{}^{\gamma}\big{)}{\Psi}_{\gamma;\,\alpha(2s-2)}~{},\qquad\nabla^{\beta}{\cal
W}_{\beta;\,\alpha(2s-2)}=0~{}.$ (5.192)

For $s>1$, the $\zeta$-gauge freedom (5.190b) can be used to impose the gauge
condition

$\displaystyle\Psi_{(\alpha_{1};\,\alpha_{2}\dots\alpha_{2s-1})}=0\quad\Longleftrightarrow\quad\Psi_{\beta;\,\alpha(2s-2)}=\sum_{k=1}^{2s-2}\varepsilon_{\beta\alpha_{k}}\varphi_{\alpha_{1}\dots\hat{\alpha}_{k}\dots\alpha_{2s-2}}~{},$
(5.193)

for some superfield $\varphi_{\alpha(2s-3)}$. The residual gauge freedom is
characterised by

$\displaystyle\zeta_{\alpha(2s-1)}={\rm
i}\nabla_{\alpha}\eta_{\alpha(2s-2)}~{},$ (5.194)

which means that $\eta_{\alpha(2s-2)}$ is the only independent gauge
parameter. As a consequence, the model can be reformulated in terms of the
following gauge superfields

$\displaystyle\big{\\{}H_{\alpha(2s)},~{}\varphi_{\alpha(2s-3)}\big{\\}}~{},$
(5.195)

which are defined modulo gauge transformations of the form

$\displaystyle\delta_{\eta}H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle-\nabla_{\alpha(2)}\eta_{\alpha(2s-2)}~{},$ (5.196)
$\displaystyle\delta_{\eta}\varphi_{\alpha(2s-3)}$ $\displaystyle=$
$\displaystyle{\rm i}\nabla^{\beta}\eta_{\beta\alpha(2s-3)}~{}.$ (5.197)

The corresponding gauge-invariant action is given by

$\displaystyle
S^{\perp}_{(s)}[H_{\alpha(2s)},\varphi_{\alpha(2s-3)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3|2}z~{}E~{}\bigg{\\{}\frac{1}{2}H^{\alpha(2s)}({\rm
i}\nabla^{2}+8s|\mu|)H_{\alpha(2s)}~{}$ $\displaystyle-{\rm
i}s\nabla_{\beta}H^{\beta\alpha(2s-1)}\nabla^{\gamma}H_{\gamma\alpha(2s-1)}-2(s-1)\varphi^{\alpha(2s-3)}\nabla^{\beta}\nabla^{\gamma\delta}H_{\beta\gamma\delta\alpha(2s-3)}~{}$
$\displaystyle+\frac{1}{s}(s-1)\Big{(}8{\rm
i}(2s-1)|\mu|^{2}\varphi^{\alpha(2s-3)}\varphi_{\alpha(2s-3)}+2(2s-3)|\mu|\varphi^{\alpha(2s-3)}\nabla^{2}\varphi_{\alpha(2s-3)}$
$\displaystyle-2{\rm
i}\varphi^{\alpha(2s-3)}\mathbb{Q}\varphi_{\alpha(2s-3)}+\frac{{\rm
i}}{2(2s-1)}(2s-3)\nabla_{\beta}\varphi^{\beta\alpha(2s-4)}\nabla^{2}\nabla^{\gamma}\varphi_{\gamma\alpha(2s-4)}~{}$
$\displaystyle+\frac{4}{2s-1}(2s-3)(3s-2)|\mu|\nabla_{\beta}\varphi^{\beta\alpha(2s-4)}\nabla^{\gamma}\varphi_{\gamma\alpha(2s-4)}~{}$
$\displaystyle-\frac{{\rm
i}}{2s-1}(2s-3)(s-2)\nabla_{\beta\gamma}\varphi^{\beta\gamma\alpha(2s-5)}\nabla^{\delta\lambda}\varphi_{\delta\lambda\alpha(2s-5)}\Big{)}\bigg{\\}}~{}.$
(5.198)

The above theory was recently proposed in [127]. In the flat-superspace limit,
the action (5.2.6.4) coincides with the ${\cal N}=1$ model given in [125].

####  5.2.7 Component structure of ${\cal N}=1$ supersymmetric higher-spin
actions in AdS3

In this section, we study the component structure of the longitudinal and
transverse formulations for the (half-)integer superspin multiplets, which
were reviewed in section 5.2.6. Any supersymmetric action in $\rm AdS^{3|2}$
can be reduced to components by the rule

$\displaystyle S=\frac{1}{4}\int{\rm d}^{3}x\,e\,({\rm
i}\nabla^{2}+8|\mu|)L\,\big{|}_{\,\theta=0}~{}.$ (5.199)

In what follows, we will denote the torsion-free covariant derivative on AdS3
by ${\mathfrak{D}}_{a}$. It is related to the vector covariant derivative
$\nabla_{a}$ in (5.113) by the simple rule
${\mathfrak{D}}_{a}:=\nabla_{a}|_{\theta=0}$, provided an appropriate Wess-
Zumino gauge is chosen.

#####  5.2.7.1 Transverse formulation for the superspin-$(s+\frac{1}{2})$
multiplet

The transverse formulation for the massless superspin-$(s+\frac{1}{2})$
multiplet is described by the action (5.2.6.2), which is invariant under the
gauge transformations (5.183). The gauge freedom (5.183) can be used to impose
the following Wess-Zumino gauge

$H_{\alpha(2s+1)}|=0~{},\qquad\nabla^{\beta}H_{\beta\alpha(2s)}|=0~{},\qquad\Upsilon_{\beta;\alpha(2s-2)}|=0~{},\qquad\nabla^{\beta}\Upsilon_{\beta;\alpha(2s-2)}|=0~{}.$
(5.200)

The residual gauge freedom preserving the gauge conditions (5.200) is given by

$\displaystyle\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{2s+1})}|$
$\displaystyle=$ $\displaystyle 0~{},$ (5.201a)
$\displaystyle\nabla^{2}\zeta_{\alpha(2s)}|$ $\displaystyle=$
$\displaystyle-\frac{2{\rm
i}s}{s+1}\big{(}\nabla_{(\alpha_{1}}{}^{\beta}\zeta_{\alpha_{2}...\alpha_{2s})\beta}+2(s+1)|\mu|\zeta_{\alpha(2s)}\big{)}|~{},$
(5.201b) $\displaystyle\nabla_{\beta}\eta_{\alpha(2s-2)}|$ $\displaystyle=$
$\displaystyle\nabla_{(\beta}\eta_{\alpha_{1}\ldots\alpha_{2s-2})}|=-\frac{1}{2s+1}\nabla^{\gamma}\zeta_{\beta\gamma\alpha(2s-2)}|~{},$
(5.201c) $\displaystyle\nabla^{2}\eta_{\alpha(2s-2)}|$ $\displaystyle=$
$\displaystyle-\frac{{\rm
i}}{2s+1}\nabla^{\beta\gamma}\zeta_{\beta\gamma\alpha(2s-2)}|~{}.$ (5.201d)

This implies that there are three real independent gauge parameters at the
component level, which we define as

$\displaystyle\xi_{\alpha(2s)}:=\zeta_{\alpha(2s)}|~{},\quad\lambda_{\alpha(2s-1)}:=-\frac{{\rm
i}s}{2s+1}\nabla^{\beta}\zeta_{\beta\alpha(2s-1)}|~{},\quad\rho_{\alpha(2s-2)}:=-\eta_{\alpha(2s-2)}|~{}.$
(5.202)

The next task is to identify the remaining independent component fields of
$H_{\alpha(2s+1)}$ and $\Upsilon_{\beta;\alpha(2s-2)}$ in the Wess-Zumino
gauge (5.200).

Let us first consider the fermionic sector. We begin by decomposing the
superfield $\Upsilon_{\beta;\alpha(2s-2)}$ into irreducible components

$\Upsilon_{\beta;\alpha(2s-2)}:=Y_{\beta\alpha(2s-2)}+\sum_{k=1}^{2s-2}\varepsilon_{\beta\alpha_{k}}Z_{\alpha_{1}...\hat{\alpha}_{k}...\alpha_{2s-2}}~{},$
(5.203)

where we have introduced the irreducible superfields

$\displaystyle Y_{\alpha(2s-1)}$ $\displaystyle:=$
$\displaystyle\Upsilon_{(\alpha_{1};\alpha_{2}\dots\alpha_{2s-1})}~{},$
(5.204a) $\displaystyle Z_{\alpha(2s-3)}$ $\displaystyle:=$
$\displaystyle\frac{1}{2s-1}\Upsilon^{\beta;}{}_{\beta\alpha(2s-3)}~{}.$
(5.204b)

We find that the remaining independent fermionic fields are given by

$\displaystyle h_{\alpha(2s+1)}$ $\displaystyle:=$ $\displaystyle\frac{{\rm
i}}{4}\nabla^{2}H_{\alpha(2s+1)}|~{},$ (5.205a) $\displaystyle
y_{\alpha(2s-1)}$ $\displaystyle:=$ $\displaystyle\frac{{\rm
i}}{8}\nabla^{2}Y_{\alpha(2s-1)}|~{},$ (5.205b) $\displaystyle
z_{\alpha(2s-3)}$ $\displaystyle:=$ $\displaystyle\frac{{\rm
i}s}{2}(2s-1)\nabla^{2}Z_{\alpha(2s-3)}|~{},$ (5.205c)

Their gauge transformations are given by

$\displaystyle\delta_{\lambda}h_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle\mathfrak{D}_{\alpha(2)}\lambda_{\alpha(2s-1)}~{},$ (5.206a)
$\displaystyle\delta_{\lambda}y_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s+1}\mathfrak{D}_{(\alpha_{1}}{}^{\beta}\lambda_{\alpha_{2}...\alpha_{2s-1})\beta}+|\mu|\lambda_{\alpha(2s-1)}~{},$
(5.206b) $\displaystyle\delta_{\lambda}z_{\alpha(2s-3)}$ $\displaystyle=$
$\displaystyle\mathfrak{D}^{\beta\gamma}\lambda_{\beta\gamma\alpha(2s-3)}~{}.$
(5.206c)

Upon component reduction of the action (5.2.6.2), we find that the fermionic
sector coincides with the Fang-Fronsdal-type action
$S_{(s+\frac{1}{2},+)}^{\text{FF}}[h,y,z]$, which is given by eq. (5.85).

We now turn to the bosonic sector. To start with, we point out that the fourth
condition in (5.200) is equivalent to

$\nabla_{\beta}\Upsilon_{\gamma;\alpha(2s-2)}|=\nabla_{\gamma}\Upsilon_{\beta;\alpha(2s-2)}|~{}.$
(5.207)

We then choose to define the remaining independent bosonic fields as follows

$\displaystyle h_{\alpha(2s+2)}$ $\displaystyle:=$
$\displaystyle-\nabla_{\alpha}H_{\alpha(2s+1)}|~{},$ (5.208a)
$\displaystyle\phi_{\beta\gamma;\alpha(2s-2)}$ $\displaystyle:=$
$\displaystyle\nabla_{\beta}\Upsilon_{\gamma;\alpha(2s-2)}|=\nabla_{(\beta}\Upsilon_{\gamma);\alpha(2s-2)}|~{}.$
(5.208b)

Performing the component reduction on the action (5.2.6.2), we arrive at the
bosonic action

$\displaystyle S_{\text{bos}}[h,\phi]$ $\displaystyle=$
$\displaystyle\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3}x\,e\,\bigg{\\{}-\frac{1}{4}h^{\alpha(2s+2)}\mathcal{Q}h_{\alpha(2s+2)}+\frac{3}{16}\mathfrak{D}_{\beta\gamma}h^{\beta\gamma\alpha(2s)}\mathfrak{D}^{\delta\rho}h_{\delta\rho\alpha(2s)}~{}$
(5.209)
$\displaystyle+\frac{1}{4}(2s-1)F^{(\beta\gamma;\alpha(2s-2))}\mathfrak{D}^{\delta\lambda}h_{\beta\gamma\delta\lambda\alpha(2s-2)}-\frac{1}{4}(2s-1)F^{\beta\gamma;\alpha(2s-2)}F_{\beta\gamma;\alpha(2s-2)}~{}$
$\displaystyle+\frac{1}{2}(s-1)(2s-1)F^{\beta\gamma;\alpha(2s-3)}{}_{\beta}F_{\delta\gamma;\alpha(2s-3)}{}^{\delta}$
$\displaystyle-\frac{1}{2}(2s-1)|\mu|\Big{(}h^{\alpha(2s+2)}\mathfrak{D}_{(\alpha_{1}}{}^{\beta}h_{\alpha_{2}\dots\alpha_{2s+2})\beta}-(s-1)\phi^{\beta\gamma;\,\alpha(2s-2)}\mathfrak{D}^{\delta\lambda}h_{\delta\lambda\beta\gamma\alpha(2s-2)}$
$\displaystyle-2F^{\beta\gamma;\,\alpha(2s-2)}\phi_{\beta\gamma;\,\alpha(2s-2)}+8(s-1)F^{\beta\gamma;\,\alpha(2s-3)}\,_{\beta}\phi_{\delta\gamma;\,\alpha(2s-3)}{}^{\delta}$
$\displaystyle-2(s-1)(2s-3)F_{\beta\gamma;}{}^{\beta\gamma\alpha(2s-4)}\phi^{\delta\lambda;}{}_{\delta\lambda\alpha(2s-4)}\Big{)}$
$\displaystyle+|\mu|^{2}(s-1)(2s-1)\Big{(}\frac{3s}{(s-1)(2s-1)}\,h^{\alpha(2s+2)}h_{\alpha(2s+2)}$
$\displaystyle+(s+1)\phi^{\beta\gamma;\,\alpha(2s-2)}\phi_{\beta\gamma;\,\alpha(2s-2)}+2(2s^{2}-s+3)\phi^{\beta\gamma;\,\alpha(2s-3)}\,_{\beta}\phi_{\gamma\delta;\,\alpha(2s-3)}{}^{\delta}$
$\displaystyle+(s-2)(2s-3)\phi_{\beta\gamma;}{}^{\beta\gamma\alpha(2s-4)}\phi^{\delta\lambda;}{}_{\delta\lambda\alpha(2s-4)}\Big{)}\bigg{\\}}~{},$

where we have introduced the field strength $F_{\beta\gamma;\alpha(2s-2)}$,

$F_{\beta\gamma;\alpha(2s-2)}:=\mathfrak{D}_{(\beta}{}^{\delta}\phi_{\gamma)\delta;\alpha(2s-2)}~{}.$
(5.210)

The bosonic action (5.209) is invariant under the gauge transformations

$\displaystyle\delta_{\xi}h_{\alpha(2s+2)}$ $\displaystyle=$
$\displaystyle\mathfrak{D}_{\alpha(2)}\xi_{\alpha(2s)}~{},$ (5.211a)
$\displaystyle\delta_{\xi,\rho}\phi_{\beta\gamma;\alpha(2s-2)}$
$\displaystyle=$
$\displaystyle\mathfrak{D}_{\beta\gamma}\rho_{\alpha(2s-2)}+2(s-1)|\mu|\big{(}\varepsilon_{\beta(\alpha_{1}}\rho_{\alpha_{2}...\alpha_{2s-2})\gamma}+\varepsilon_{\gamma(\alpha_{1}}\rho_{\alpha_{2}...\alpha_{2s-2})\beta}\big{)}~{}$
(5.211b)
$\displaystyle-\frac{1}{(2s+1)(s+1)}\Big{(}(s+2)\mathfrak{D}_{(\beta}{}^{\delta}\xi_{\gamma)\delta\alpha(2s-2)}+(s-1)\mathfrak{D}_{(\alpha_{1}}{}^{\delta}\xi_{\alpha_{2}...\alpha_{2s-2})\beta\gamma\delta}$
$\displaystyle+2(s+1)(2s+1)|\mu|\xi_{\beta\gamma\alpha(2s-2)}\Big{)}~{}.$

From (5.211b), it can be shown that $F_{\beta\gamma;\alpha(2s-2)}$ transforms
in the following manner

$\displaystyle\delta_{\xi,\rho}F_{\beta\gamma;\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-2(s-1)|\mu|\Big{(}\varepsilon_{(\alpha_{1}|(\beta}\mathfrak{D}_{\gamma)|}{}^{\delta}\rho_{\alpha_{2}...\alpha_{2s-2})\delta}+\mathfrak{D}_{(\beta|(\alpha_{1}}\rho_{\alpha_{2}...\alpha_{2s-2})|\gamma)}~{}$
$\displaystyle-4|\mu|\varepsilon_{(\beta|(\alpha_{1}}\rho_{\alpha_{2}...\alpha_{2s-2})|\gamma)}\Big{)}$
$\displaystyle-\frac{1}{(2s+1)(s+1)}\Big{(}\frac{1}{2}(s+2)\mathfrak{D}_{\beta\gamma}\mathfrak{D}^{\delta\rho}\xi_{\delta\rho\alpha(2s-2)}+(2s+1){\cal
Q}\xi_{\gamma\beta\alpha(2s-2)}~{}$
$\displaystyle+(s-1)\mathfrak{D}_{(\beta|(\alpha_{1}}\mathfrak{D}^{\delta\rho}\xi_{\alpha_{2}...\alpha_{2s-2})|\gamma)\delta\rho}+2(s+1)(2s+1)|\mu|\mathfrak{D}_{(\beta}{}^{\delta}\xi_{\gamma)\alpha(2s-2)\delta}~{}$
$\displaystyle-4(s-1)(s+1)(2s+1)|\mu|^{2}\xi_{\beta\gamma\alpha(2s-2)}\Big{)}~{}~{}.$

Inspecting the gauge transformation (5.2.7.1), one sees that the field
strength $F_{\beta\gamma;\alpha(2s-2)}$ is invariant under $\rho$-gauge
transformations in the flat-superspace limit.

The complete component action of our transverse half-integer model takes the
form

$\displaystyle S^{\perp}_{(s+\frac{1}{2})}[{H},\Upsilon]=S_{\rm
bos}[h,\phi]+S_{(s+\frac{1}{2},+)}^{\text{FF}}[h,y,z]~{}.$ (5.213)

By applying the duality transformation described in subsection 2.3.5.3, it can
be shown that the flat-space counterpart of (5.209) is the dual formulation
(2.333) of the flat-space counterpart of Fronsdal action $S^{\rm F}_{(s+1)}$
(5.99) provided we fix the coefficients $A$ and $B$ as

$\displaystyle A=-\frac{1}{2}(s-1)~{},\qquad B=\frac{3}{4s}(s-1)~{}.$ (5.214)

That is, the dual action given in (2.333) coincides with the bosonic action
(5.209) in Minkowski space

$S_{\rm bos}[h,\phi]\Big{|}_{|\mu|=0}=S_{\rm A}^{(s+1)}[h,\phi]~{}.$ (5.215)

However, it must be noted that this duality does not hold in the presence of a
non-vanishing AdS curvature, since (5.209) cannot be written solely in terms
of the field strength $F_{\beta\gamma;\,\alpha(2s-2)}$.

#####  5.2.7.2 Longitudinal formulation for the superspin-$(s+\frac{1}{2})$
multiplet

The longitudinal formulation for the massless superspin-$(s+\frac{1}{2})$
multiplet is described by the action (5.2.6.1), which is invariant under the
gauge transformations (5.179). The component structure of this model was
studied in[117] only in the flat-superspace limit. Here we extend these
results to AdS3.

The gauge freedom (5.183) can be used to impose the following Wess-Zumino
gauge

$H_{\alpha(2s+1)}|=0~{},\qquad\nabla^{\beta}H_{\beta\alpha(2s)}|=0~{}.$
(5.216)

The residual gauge freedom which preserves the conditions (5.216) is described
by

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{2s+1})}|~{},$
(5.217a) $\displaystyle\nabla^{2}\zeta_{\alpha(2s)}|$ $\displaystyle=$
$\displaystyle-\frac{2{\rm
i}s}{s+1}\big{(}\nabla_{(\alpha_{1}}{}^{\beta}\zeta_{\alpha_{2}...\alpha_{2s})\beta}+2(s+1)|\mu|\zeta_{\alpha(2s)}\big{)}|~{}.$
(5.217b)

These conditions imply that there are only two independent gauge parameters,
which we define as follows:

$\xi_{\alpha(2s)}:=\zeta_{\alpha(2s)}|~{},\qquad\lambda_{\alpha(2s-1)}:=-\frac{{\rm
i}s}{2s+1}\nabla^{\beta}\zeta_{\beta\alpha(2s-1)}|~{}.$ (5.218)

Upon imposing the gauge (5.216), we are left with the following component
fields:

$\displaystyle h_{\alpha(2s+1)}$ $\displaystyle:=$ $\displaystyle\frac{{\rm
i}}{4}\nabla^{2}H_{\alpha(2s+1)}|~{},$ (5.219a) $\displaystyle
h_{\alpha(2s+2)}$ $\displaystyle:=$
$\displaystyle-\nabla_{\alpha}H_{\alpha(2s+1)}|~{},$ (5.219b) $\displaystyle
y_{\alpha(2s-2)}$ $\displaystyle:=$ $\displaystyle-4L_{\alpha(2s-2)}|~{}$
(5.219c) $\displaystyle y_{\alpha(2s-1)}$ $\displaystyle:=$
$\displaystyle(2s+1){\rm i}\nabla_{\alpha}L_{\alpha(2s-2)}|~{},$ (5.219d)
$\displaystyle z_{\alpha(2s-3)}$ $\displaystyle:=$ $\displaystyle 2{\rm
i}\nabla^{\beta}L_{\beta\alpha(2s-3)}|~{},$ (5.219e) $\displaystyle
F_{\alpha(2s-2)}$ $\displaystyle:=$ $\displaystyle\frac{{\rm
i}}{4}\nabla^{2}L_{\alpha(2s-2)}|~{}.$ (5.219f)

Performing a component reduction on the higher-spin model (5.2.6.1), we find
that the theory decouples into a bosonic and a fermionic sector, respectively.
The bosonic action is given by

$\displaystyle S_{\text{bos}}[h,y,F]=\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3}x\,e\,\bigg{\\{}-\frac{1}{4}h^{\alpha(2s+2)}\mathcal{Q}h_{\alpha(2s+2)}+\frac{1}{8}(s+1)\mathfrak{D}_{\beta\gamma}h^{\beta\gamma\alpha(2s)}\mathfrak{D}^{\delta\lambda}h_{\delta\lambda\alpha(2s)}~{}$
$\displaystyle+\frac{1}{8}(2s-1)y^{\alpha(2s-2)}\mathfrak{D}^{\beta\gamma}\mathfrak{D}^{\delta\lambda}h_{\beta\gamma\delta\lambda\alpha(2s-2)}+\frac{1}{16s}(2s-1)(s+1)y^{\alpha(2s-2)}\mathcal{Q}y_{\alpha(2s-2)}~{}$
$\displaystyle+\frac{1}{s}(s-1)(2s-1)F^{\beta\alpha(2s-3)}\mathfrak{D}_{\beta}{}^{\gamma}y_{\gamma\alpha(2s-3)}+\frac{4}{s}(s+1)(2s-1)F^{\alpha(2s-2)}F_{\alpha(2s-2)}~{}$
$\displaystyle+\frac{1}{4}(s-1)(2s-1)|\mu|y^{\beta\alpha(2s-3)}\mathfrak{D}_{\beta}{}^{\gamma}y_{\gamma\alpha(2s-3)}+2(s-1)(2s-1)|\mu|F^{\alpha(2s-2)}y_{\alpha(2s-2)}~{}$
$\displaystyle+2s(s+1)|\mu|^{2}h^{\alpha(2s+2)}h_{\alpha(2s+2)}-\frac{1}{2}(2s-1)(s+1)|\mu|^{2}y^{\alpha(2s-2)}y_{\alpha(2s-2)}\bigg{\\}}~{}.$
(5.220)

The bosonic field $F_{\alpha(2s-2)}$ is auxiliary, so upon elimination via its
equation of motion

$F_{\alpha(2s-2)}=-\frac{1}{8(s+1)}(s-1)\mathfrak{D}_{(\alpha_{1}}{}^{\beta}y_{\alpha_{2}...\alpha_{2s-2})\beta}-\frac{s}{4}|\mu|y_{\alpha(2s-2)}~{},$
(5.221)

we find that the resulting action coincides with the massless spin-$(s+1)$
action, $S_{(s+1)}^{\text{F}}$, which is obtained from (5.99) by setting
$n=2s+2$.

It can be shown that the fermionic action emerging from the reduction
procedure coincides with the Fang-Fronsdal type action
$S^{\text{FF}}_{(s+\frac{1}{2},-)}$ (5.85) Therefore, our component actions
take the form

$\displaystyle
S^{\parallel}_{(s+\frac{1}{2})}[H,L]=S_{(s+1)}^{\text{F}}[h,y]+S^{\text{FF}}_{(s+\frac{1}{2},-)}[h,y,z]~{}.$
(5.222)

Comparing (5.222) with (5.213), it is seen that the fermionic sector resulting
from the reduction of the longitudinal model is given by
$S^{\text{FF}}_{(s+\frac{1}{2},+)}[h,y,z]$. In flat-space, the two bosonic
actions, $S_{(s+1)}^{\text{F}}[h,y]$ and $S_{\rm bos}[h,\phi]$, are related to
each other by a duality transformation as described in section 2.3.5.3. The
fermionic actions are now given by $S^{\text{FF}}_{(s+\frac{1}{2})}[h,y,z]$,
thus they coincide identically.

#####  5.2.7.3 Transverse formulation for the superspin-$s$ multiplet

The transverse formulation for the massless superspin-$s$ multiplet is
described by the action (5.2.6.4) and is invariant under the gauge
transformations (5.190). The gauge freedom (5.190) can be used to impose the
following Wess-Zumino gauge conditions:

$H_{\alpha(2s)}|=0~{},\qquad\nabla^{\beta}H_{\beta\alpha(2s-1)}|=0~{},\qquad\Psi_{\beta;\alpha(2s-2)}|=0~{},\qquad\nabla^{\beta}\Psi_{\beta;\alpha(2s-2)}|=0~{}.$
(5.223)

The residual gauge symmetry preserving (5.223) is given by

$\displaystyle\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{2s})}|$
$\displaystyle=$ $\displaystyle 0~{},$ (5.224a)
$\displaystyle\nabla^{2}\zeta_{\alpha(2s-1)}|$ $\displaystyle=$
$\displaystyle-\frac{2{\rm
i}(2s-1)}{2s+1}\big{(}\nabla_{(\alpha_{1}}{}^{\beta}\zeta_{\alpha_{2}...\alpha_{2s-1})\beta}+(2s+1)|\mu|\zeta_{\alpha(2s-1)}\big{)}|~{},$
(5.224b) $\displaystyle\nabla_{\beta}\eta_{\alpha(2s-2)}|$ $\displaystyle=$
$\displaystyle\nabla_{(\beta}\eta_{\alpha_{1}\ldots\alpha_{2s-2})}|=-{\rm
i}\zeta_{\beta\alpha(2s-2)}|~{},$ (5.224c)
$\displaystyle\nabla^{2}\eta_{\alpha(2s-2)}|$ $\displaystyle=$
$\displaystyle-{\rm i}\nabla^{\beta}\zeta_{\beta\alpha(2s-2)}|~{}.$ (5.224d)

Thus, there are three real independent gauge parameters, which we define as

$\displaystyle\xi_{\alpha(2s-1)}:=\zeta_{\alpha(2s-1)}|~{},\qquad\rho_{\alpha(2s-2)}:=-\eta_{\alpha(2s-2)}|~{},$
(5.225a)
$\displaystyle\lambda_{\alpha(2s-2)}:=-\frac{2s-1}{4s}\nabla^{\beta}\zeta_{\beta\alpha(2s-2)}|~{}.$
(5.225b)

We now wish to find the remaining independent component fields of
$H_{\alpha(2s)}$ and $\Psi_{\beta;\alpha(2s-2)}$ in the Wess-Zumino gauge
(5.223). The last Wess-Zumino gauge condition in (5.223) yields the relation

$\nabla_{\beta}\Psi_{\gamma;\alpha(2s-2)}=\nabla_{\gamma}\Psi_{\beta;\alpha(2s-2)}~{}.$
(5.226)

Thus, we choose to describe the bosonic sector in terms of the independent
fields

$\displaystyle X_{\alpha(2s)}$ $\displaystyle:=$ $\displaystyle\frac{{\rm
i}}{4}\nabla^{2}H_{\alpha(2s)}|~{},$ (5.227a)
$\displaystyle\phi_{\beta\gamma;\alpha(2s-2)}$ $\displaystyle:=$
$\displaystyle\nabla_{\beta}\Psi_{\gamma;\alpha(2s-2)}|=\nabla_{(\beta}\Psi_{\gamma);\alpha(2s-2)}|~{}.$
(5.227b)

Performing a component reduction on the action (5.2.6.2), we obtain the
bosonic action

$\displaystyle
S_{\text{bos}}[\phi,X]=-2(2s-1)\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3}x~{}e~{}\bigg{\\{}\frac{s-1}{2s-1}X^{\alpha(2s)}X_{\alpha(2s)}+F^{\beta\gamma;\alpha(2s-2)}X_{\beta\gamma\alpha(2s-2)}~{}$
$\displaystyle+\frac{1}{2}F^{\beta\gamma;\alpha(2s-2)}F_{\beta\gamma;\alpha(2s-2)}+\frac{1}{2s}(s-1)F^{\beta\gamma;\alpha(2s-3)}{}_{\beta}F_{\gamma\delta;\alpha(2s-3)}{}^{\delta}~{}$
$\displaystyle+2(s-1)|\mu|\phi^{\beta\gamma;\alpha(2s-2)}X_{\beta\gamma\alpha(2s-2)}+(2s-1)|\mu|F^{\beta\gamma;\alpha(2s-2)}\phi_{\beta\gamma;\alpha(2s-2)}~{}$
$\displaystyle+2(s-1)|\mu|F^{\beta\gamma;\alpha(2s-3)}{}_{\beta}\phi_{\gamma\delta;\alpha(2s-3)}{}^{\delta}+\frac{2}{s}(s-1)^{2}|\mu|F^{\beta\gamma;\alpha(2s-3)}{}_{\beta}\phi_{\delta(\gamma;\alpha_{1}\ldots\alpha_{2s-3})}{}^{\delta}~{}$
$\displaystyle+2s(s-1)|\mu|^{2}\phi^{\beta\gamma;\alpha(2s-2)}\phi_{\beta\gamma;\alpha(2s-2)}+2s(s-1)|\mu|^{2}\phi^{\beta\gamma;\alpha(2s-3)}{}_{\beta}\phi_{\gamma\delta;\alpha(2s-3)}{}^{\delta}~{}$
$\displaystyle+\frac{2}{s}(s-1)^{2}(s-2)|\mu|^{2}\phi^{\beta\gamma;\alpha(2s-3)}{}_{\beta}\phi_{\delta(\gamma;\alpha_{1}\ldots\alpha_{2s-3})}{}^{\delta}\bigg{\\}}~{},$
(5.228)

where we have introduced the field strength $F_{\beta\gamma;\alpha(2s-2)}$,

$F_{\beta\gamma;\alpha(2s-2)}:=\mathfrak{D}_{(\beta}{}^{\delta}\phi_{\gamma)\delta;\alpha(2s-2)}~{}.$
(5.229)

It can be shown that the action (5.2.7.3) is invariant under the gauge
transformations

$\displaystyle\delta_{\lambda}X_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\mathfrak{D}_{\alpha(2)}\lambda_{\alpha(2s-2)}~{},$ (5.230a)
$\displaystyle\delta_{\lambda,\rho}\phi_{\beta\gamma;\alpha(2s-2)}$
$\displaystyle=$
$\displaystyle\mathfrak{D}_{\beta\gamma}\rho_{\alpha(2s-2)}+4(s-1)|\mu|\varepsilon_{(\beta|(\alpha_{1}}\rho_{\alpha_{2}...\alpha_{2s-2})|\gamma)}~{}$
$\displaystyle+\frac{4}{2s-1}(s-1)\varepsilon_{(\beta|(\alpha_{1}}\lambda_{\alpha_{2}...\alpha_{2s-2})|\gamma)}~{}.$

The field $X_{\alpha(2s)}$ is auxiliary, so upon elimination via its equation
of motion

$X_{\alpha(2s)}=-\frac{2s-1}{2(s-1)}\big{(}F_{\alpha(2s)}+2(s-1)|\mu|\phi_{(\alpha_{1}\alpha_{2};\,\alpha_{3}\dots\alpha_{2s})}\big{)}~{},$
(5.231)

we obtain

$\displaystyle
S_{\text{bos}}[\phi]=\frac{2s-1}{s(2s-2)}\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3}x~{}e~{}\bigg{\\{}s\,F^{\beta\gamma;\alpha(2s-2)}F_{\beta\gamma;\alpha(2s-2)}~{}$
$\displaystyle+2s(s-1)F^{\beta\gamma;\alpha(2s-3)}{}_{\beta}F_{\gamma\delta;\alpha(2s-3)}{}^{\delta}+(2s-3)(s-1)F_{\beta\gamma;}\,^{\beta\gamma\alpha(2s-4)}F^{\delta\lambda;}\,_{\delta\lambda\alpha(2s-4)}~{}$
$\displaystyle-4(s-1)^{2}|\mu|^{2}\Big{(}s\,\phi^{\beta\gamma;\alpha(2s-2)}\phi_{\beta\gamma;\alpha(2s-2)}+2\,\phi^{\beta\gamma;\alpha(2s-3)}{}_{\beta}\phi_{\delta\gamma;\alpha(2s-3)}{}^{\delta}$
$\displaystyle-(2s-3)\phi_{\beta\gamma;}\,^{\beta\gamma\alpha(2s-4)}\phi^{\delta\lambda;}\,_{\delta\lambda\alpha(2s-4)}\Big{)}\bigg{\\}}~{}.$
(5.232)

We can express the field $\phi_{\beta\gamma;\alpha(2s-2)}$ in terms of its
irreducible components

$\displaystyle h_{\alpha(2s)}$ $\displaystyle:=$
$\displaystyle\phi_{(\alpha_{1}\alpha_{2};\alpha_{3}...\alpha_{2s})}~{},$
(5.233a) $\displaystyle y_{\alpha(2s-2)}$ $\displaystyle:=$
$\displaystyle\phi^{\beta}{}_{(\alpha_{1};\alpha_{2}...\alpha_{2s-2})\beta}~{},$
(5.233b) $\displaystyle y_{\alpha(2s-4)}$ $\displaystyle:=$
$\displaystyle\frac{2(s-1)}{2s-1}\phi^{\beta\gamma;}{}_{\beta\gamma\alpha(2s-4)}~{}.$
(5.233c)

The gauge freedom (5.230) can be used to gauge away $y_{\alpha(2s-2)}$. It
follows that the remaining fields, (5.233a) and (5.233c), have the
corresponding gauge transformations

$\displaystyle\delta_{\rho}h_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\mathfrak{D}_{\alpha(2)}\rho_{\alpha(2s-2)}~{},$ (5.234a)
$\displaystyle\delta_{\rho}y_{\alpha(2s-4)}$ $\displaystyle=$
$\displaystyle\frac{2(s-1)}{2s-1}\mathfrak{D}^{\beta\gamma}\rho_{\beta\gamma\alpha(2s-4)}~{}.$
(5.234b)

Recasting (5.2.7.3) in terms of the irreducible fields (5.233a) and (5.233c),
the resulting action coincides with, up to an overall factor, with the
massless spin-$s$ action (5.99).

The study of the fermionic sector requires the decomposition of the reducible
superfield $\Psi_{\beta;\alpha(2s-2)}$ into irreducible parts. This procedure
is completely analogous to that of the prepotential
$\Upsilon_{\beta;\alpha(2s-2)}$ (5.203). In this case, we find that the
remaining independent fermionic fields are given by

$\displaystyle h_{\alpha(2s+1)}$ $\displaystyle:=$ $\displaystyle-{\rm
i}\nabla_{\alpha}H_{\alpha(2s)}|~{},$ (5.235a) $\displaystyle
y_{\alpha(2s-1)}$ $\displaystyle:=$ $\displaystyle\frac{{\rm
i}}{4}\nabla^{2}Y_{\alpha(2s-1)}|~{},$ (5.235b) $\displaystyle
z_{\alpha(2s-3)}$ $\displaystyle:=$ $\displaystyle\frac{{\rm
i}}{2}(2s-1)\nabla^{2}Z_{\alpha(2s-3)}|~{}.$ (5.235c)

Upon the component reduction of the action (5.2.6.4), it can be shown that the
fermionic action obtained coincides with the Fang-Fronsdal-type action,
$S_{(s+\frac{1}{2},-)}^{\text{FF}}$ (5.85). As a result, we find that the
action corresponding to the transverse formulation of the massless
superspin-$s$ multiplet in AdS3 decomposes into

$\displaystyle
S^{\perp}_{(s)}[H,\Psi]=\frac{2s-1}{s-1}S_{(s)}^{\text{F}}[h,y]+\,S_{(s+\frac{1}{2},-)}^{\text{FF}}[h,y,z]~{}.$
(5.236)

#####  5.2.7.4 Longitudinal formulation for the superspin-$s$ multiplet

The longitudinal formulation for the massless superspin-$s$ multiplet is
described by the action (5.2.6.3) and is invariant under gauge transformations
(5.187). This formulation corresponds to the massless first-order model, whose
component reduction in the flat-superspace limit has been studied in [117].

The gauge freedom (5.187) can be used to impose the following Wess-Zumino
gauge

$H_{\alpha(2s)}|=0~{},\qquad\nabla^{\beta}H_{\beta\alpha(2s-1)}|=0~{},\qquad
V_{\alpha(2s-2)}|=0~{}.$ (5.237)

The residual gauge symmetry which preserves the gauge transformations (5.237)
is

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{2s})}|~{},$
(5.238a) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\nabla^{\beta}\zeta_{\beta\alpha(2s-2)}|~{},$ (5.238b)
$\displaystyle\nabla^{2}\zeta_{\alpha(2s-1)}|$ $\displaystyle=$
$\displaystyle-\frac{2{\rm
i}}{2s+1}(2s-1)\big{(}\nabla_{(\alpha_{1}}{}^{\beta}\zeta_{\alpha_{2}...\alpha_{2s-1})\beta}+(2s+1)|\mu|\zeta_{\alpha(2s-1)}\big{)}|~{}.$
(5.238c)

The relations (5.238) indicate that there is only one independent gauge
parameter, which we choose to define as

$\xi_{\alpha(2s-1)}:=\zeta_{\alpha(2s-1)}|~{}.$ (5.239)

Thus, we are left with the remaining independent component fields in the gauge
(5.216):

$\displaystyle h_{\alpha(2s+1)}$ $\displaystyle:=$ $\displaystyle-{\rm
i}\nabla_{\alpha}H_{\alpha(2s)}|~{},$ (5.240a) $\displaystyle h_{\alpha(2s)}$
$\displaystyle:=$ $\displaystyle\frac{{\rm
i}}{4}\nabla^{2}H_{\alpha(2s)}|~{},$ (5.240b) $\displaystyle y_{\alpha(2s-2)}$
$\displaystyle:=$ $\displaystyle\frac{{\rm
i}}{4}\nabla^{2}V_{\alpha(2s-2)}|~{}$ (5.240c) $\displaystyle
y_{\alpha(2s-1)}$ $\displaystyle:=$ $\displaystyle\frac{{\rm
i}}{2}\nabla_{\alpha}V_{\alpha(2s-2)}|~{},$ (5.240d) $\displaystyle
z_{\alpha(2s-3)}$ $\displaystyle:=$ $\displaystyle-2{\rm
i}s\nabla^{\beta}V_{\beta\alpha(2s-3)}|~{}.$ (5.240e)

It is easy to see that the gauge transformation laws for the bosonic fields
are

$\displaystyle\delta h_{\alpha(2s)}=0~{},\qquad\delta y_{\alpha(2s-2)}=0~{}.$
(5.241)

Upon reduction of the higher-spin model (5.2.6.1), it can be shown that these
bosonic fields appear without derivatives and hence, they are pure auxiliary
fields. Indeed, we find that

$\displaystyle S^{\parallel}_{(s)}[H,V]$ $\displaystyle=$
$\displaystyle\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3}x\,e\,\Big{\\{}h^{\alpha(2s)}h_{\alpha(2s)}+2s(2s-1)y^{\alpha(2s-2)}y_{\alpha(2s-2)}\Big{\\}}$
(5.242) $\displaystyle+S^{\text{FF}}_{(s+\frac{1}{2},+)}\,[h,y,z]~{}.$

Unlike the reduction of the transverse superspin-$s$ multiplet described by
(5.236), here the fermionic action comes with a positive sign in the
$|\mu|$-dependent terms, $S^{\text{FF}}_{(s+\frac{1}{2},+)}$.

####  5.2.8 Topologically massive theories

Topologically massive higher-spin theories were constructed in AdS3|2 in [117,
127]. In accordance with [127, 117], there exist four off-shell formulations
for topologically massive higher-spin ${{\cal N}}=1$ multiplets in AdS3. For a
positive integer $s$, there are two off-shell gauge-invariant models for a
topologically massive superspin-$(s+\frac{1}{2})$ multiplet in AdS3:

$\displaystyle
S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)}|m]$
$\displaystyle=$ $\displaystyle\kappa
S^{(s+\frac{1}{2})}_{\text{SCHS}}[H_{\alpha(2s+1)}]$
$\displaystyle+m^{2s-1}S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)}]~{},$
$\displaystyle
S^{\perp}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},\Upsilon_{\beta;\alpha(2s-2)}|m]$
$\displaystyle=$ $\displaystyle\kappa
S^{(s+\frac{1}{2})}_{\text{SCHS}}[H_{\alpha(2s+1)}]$
$\displaystyle+m^{2s-1}S^{\perp}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},\Upsilon_{\beta;\alpha(2s-2)}]~{}.$

Here $\kappa$ is a dimensionless parameter and $m$ is a real massive
parameter. For a topologically massive superspin-$s$ multiplet in AdS3, we
have the following models:

$\displaystyle S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)}|m]$
$\displaystyle=$
$\displaystyle{S}_{\rm{SCHS}}^{(s)}[H_{\alpha(2s)}]+m^{2s-1}S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)}]~{},$
(5.244a) $\displaystyle
S^{\perp}_{(s)}[H_{\alpha(2s)},{\Psi}_{\beta;\,\alpha(2s-2)}|m]$
$\displaystyle=$
$\displaystyle{S}_{\rm{SCHS}}^{(s)}[H_{\alpha(2s)}]+m^{2s-1}S^{\perp}_{(s)}[H_{\alpha(2s)},{\Psi}_{\beta;\,\alpha(2s-2)}]~{}.$
(5.244b)

Note that the massive actions (5.243) and (5.244a) were derived by Kuzenko and
Ponds in [117], while the remaining models, (5.243) and (5.244b), were derived
by Hutomo and Kuzenko in [127]. It is an instructive exercise to show that the
actions (5.243) and (5.244) describe massive dynamics on-shell, in accordance
with (5.124).

It is worth noting that the new topologically massive action (5.103) in AdS3
was recently extended to AdS3|2 in [117, 152]. For $n\geq 1$, the gauge-
invariant NTM action is given by [117, 152]

$S^{(n)}_{\text{NTM}}[H_{\alpha(n)}]=-\frac{{\rm
i}^{n}}{2^{\lfloor\frac{n}{2}\rfloor+1}}\frac{1}{M}\int{\rm
d}^{3|2}z~{}E~{}\mathfrak{W}^{\alpha(n)}(H)\big{(}\mathbb{F}-\sigma
M\big{)}H_{\alpha(n)}~{},$ (5.245)

where $\sigma=\pm 1$, $H_{\alpha(n)}$ is the SCHS gauge superfield and
$\mathfrak{W}_{\alpha(n)}(H)$ is the linearised super-Cotton tensor (5.170).

The equation of motion obtained by varying (5.245) with respect to
$H_{\alpha(n)}$ is

$0=\big{(}\mathbb{F}-\sigma M\big{)}\mathfrak{W}_{\alpha(n)}(H)~{}.$ (5.246)

In accordance with (5.246) and the fact that $\mathfrak{W}_{\alpha(n)}(H)$ is
transverse (5.172), it follows from (5.124) that the field strength
$\mathfrak{W}_{\alpha(n)}(H)$ describes a massive superfield with pseudo-mass
$M$, superspin $\frac{n}{2}$ and superhelicity
$\frac{1}{2}(n+\frac{1}{2})\sigma$.

###  5.3 Summary of results

The main goal of this chapter was to compute the spin projection operators in
three-dimensional anti-de Sitter (super)space and study their corresponding
applications. In section 5.1.3, we computed the AdS3 spin projection operators
$\widehat{\Pi}^{\perp}_{(n)}$, for $n\geq 2$. These operators were computed
for the first time in [4] and are given by eqs. (5.35) and (5.46) for the
bosonic $(n=2s)$ and fermionic $(n=2s+1)$ cases, respectively. We also derived
an equivalent171717Recall that the spin projection operators
$\widehat{\Pi}^{\perp}_{(n)}$ and ${\Pi}^{\perp}_{(n)}$ are only equivalent on
the space ${\cal V}_{(n)}$. spin projection operator ${\Pi}^{\perp}_{(n)}$
which is written solely in terms of the Casimir operators of
$\mathfrak{so}(2,2)$ [4]. These bosonic $(n=2s)$ and fermionic $(n=2s+1)$
projectors are given by eqs. (5.47) and (5.36), respectively.

The spin projection operator ${\Pi}^{\perp}_{(n)}$ maps a field
$\phi_{\alpha(n)}$ satisfying the first-order equation (5.7b), but otherwise
unconstrained, to an on-shell superfield (5.7) which realises the irreducible
representation $D(E_{0},\sigma|s|)$ of $\mathfrak{so}(2,2)$. If instead the
field $\phi_{\alpha(n)}$ obeys the Klein-Gordon equation (5.14b), then the
projected field furnishes the reducible representation
${\mathfrak{D}}(\rho,-\frac{n}{2})\oplus{\mathfrak{D}}(\rho,\frac{n}{2})$ of
$\mathfrak{so}(2,2)$. In order to isolate the component with fixed helicity,
one needs to make use of the helicity projectors $\mathbb{P}^{(\pm)}_{[n]}$
[4]. They are given by eq. (5.57).

The spin projectors had many fruitful applications. Similar to the case in
AdS4, we demonstrated that these projectors are connected to partially
massless fields in AdS3. In particular, we established a correspondence
between partially massless on-shell fields $\phi^{(t)}_{\alpha(n)}$ with depth
$t$ and the poles of ${\Pi}^{\perp}_{(n)}$, which are determined by
$\tau_{(t,n)}|\mu|^{2}$, with $2\leq t\leq\lfloor n/2\rfloor$. In section
5.1.3.4, the spin projection operators ${\Pi}^{\perp}_{(n)}$ were used to
decompose any field $\phi_{\alpha(n)}$ into its irreducible components (5.66)
[4]. They were also used construct the lower-spin extractors, which single out
the spin $\frac{1}{2}(n-2j)$ component from the decomposition (5.66) [4]. In
section 5.1.4 we made use of ${\Pi}^{\perp}_{(n)}$ to obtain new
representations for the linearised higher-spin super Cotton tensors (5.74) and
their corresponding conformal actions in AdS3 [4]. The significance of these
new realisations is that the properties of gauge invariance, transversality
and factorisation are all made manifest.

In section 5.1.5 we reviewed free massless higher-spin field theories in AdS3.
The corresponding actions were shown not to propagate any physical degrees of
freedom on-shell, both at the level of superfields and component fields. We
also studied new topologically massive theories in section 5.1.6. In
particular, new realisations for the NTM massive models (5.103) were found
which immediately appear in a factorised form [4]. We also computed the AdS3
extension of the new NTM model for bosonic fields, which was recently derived
in $\mathbb{M}^{3}$ [155]. This is given by eq. (5.105) [4].

In section 5.2, we generalised many of the results of the previous section to
AdS3|2. In section 5.2.4 we derived the superspin projection operators
$\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ in AdS3|2 for the first time [4]. These
are given by eq. (5.157). These operators map a superfield $\Phi_{\alpha(n)}$
satisfying the first-order equation (5.124b) to an on-shell superfield
(5.124), which realises the irreducible representation
$\mathfrak{S}(M,\sigma\frac{n}{2})$ of
$\mathfrak{osp}(1|2;{\mathbb{R}})\oplus\mathfrak{sl}(2,{\mathbb{R}})$. We
demonstrated that the poles of $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ are
intimately related to partially massless superfields, supporting our claim
that this is a universal feature of anti-de Sitter space.

In section 5.2.4.1 we made use of the superspin projection operators to
decompose an arbitrary superfield into irreducible components [4]. This
decomposition is given by eq. (5.169). In section 5.2.5, the superprojectors
were also used to recast the linearised super-Cotton tensors, and their
corresponding SCHS actions, which were recently derived in [152]. These new
representations are given by eq. (5.173). Recasting the higher-spin super-
Cotton tensors in terms of $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ makes the
properties of gauge-invariance and transversality manifest.

In section 5.2.6 we reviewed the structure of the four series of off-shell
massless higher-spin ${\cal N}=1$ supersymmetric models in AdS3. These models
were derived in [127, 117]. In section 5.2.7 we studied the corresponding
component structure of these ${\cal N}=1$ actions for the first time [2]. At
the component level, it was shown that these actions decoupled into a bosonic
and a fermionic sector. The bosonic sector coincided with either the Fronsdal
theory or a pure auxiliary sector, while the fermionic part always coincided
with a Fang-Fronsdal action. In section 5.2.8 we reviewed the topologically
massive higher-spin theories of [117, 127], which are built from the SCHS
actions of section 5.2.5 and the massless theories of section 5.2.6.

## Appendix

##  Appendix 5.A Generating function formalism

We employ the generating function formalism which was developed in [152].
Within this framework, a one-to-one correspondence between a homogenous
polynomial $\phi_{(n)}(\Upsilon)$ of degree $n$ and a rank-$n$ spinor field
$\phi_{\alpha(n)}$ is established via the rule

$\phi_{(n)}(\Upsilon):=\Upsilon^{\alpha_{1}}\cdots\Upsilon^{\alpha_{n}}\phi_{\alpha(n)}~{}.$
(5.247)

Here, we have introduced the commuting real auxiliary variables
$\Upsilon^{\alpha}$, which are inert under the action of the Lorentz
generators $M_{\alpha\beta}$.

Making use of the auxiliary fields $\Upsilon^{\alpha}$, and their
corresponding partial derivatives,
$\partial_{\beta}:=\frac{\partial}{\partial\Upsilon^{\beta}}$, we can realise
the AdS3 derivatives as index-free operators on the the space of homogenous
polynomials of degree $n$. We introduce the differential operators which
increase and decrease the degree of homogeneity by $2$, $0$ and $-2$
respectively:

$\nabla_{(2)}:=\Upsilon^{\alpha}\Upsilon^{\beta}\nabla_{\alpha\beta}~{},\quad\nabla_{(0)}:=\Upsilon^{\alpha}\nabla_{\alpha}{}^{\beta}\partial_{\beta},\quad\nabla_{(-2)}:=\nabla^{\alpha\beta}\partial_{\alpha}\partial_{\beta}.$
(5.248)

Note that the action $\nabla_{(0)}$ is equivalent to that of the Casimir
operator ${\cal F}$. Making use of the algebra (5.4), one can derive the
important identities

$\displaystyle\big{[}\nabla_{(2)},\nabla^{\phantom{.}t}_{(-2)}\big{]}\phi_{(n)}$
$\displaystyle=$ $\displaystyle 4t(n-t+2)\big{(}{\cal Q}-\tau_{(t,n+2)}{\cal
S}^{2}\big{)}\nabla_{(-2)}^{t-1}\phi_{(n)}~{},$ (5.249a)
$\displaystyle\big{[}\nabla_{(-2)},\nabla^{\phantom{.}t}_{(2)}\big{]}\phi_{(n)}$
$\displaystyle=$ $\displaystyle-4t(n+t)\big{(}{\cal Q}-\tau_{(t,n+2t)}{\cal
S}^{2}\big{)}\nabla_{(2)}^{t-1}\phi_{(n)}~{},$ (5.249b)
$\displaystyle\nabla^{t}_{(2)}\nabla^{t}_{(-2)}\phi_{(n)}$ $\displaystyle=$
$\displaystyle\prod_{j=0}^{t-1}\Big{(}{\cal
F}^{2}-\big{(}n-2j\big{)}^{2}\big{(}{\cal Q}-(n-2j-2)(n-2j+2){\cal
S}^{2}\big{)}\Big{)}\phi_{(n)}~{},\hskip 38.41139pt$ (5.249c)

via induction on $t$. Here ${\cal Q}$ and ${\cal F}$ are the quadratic Casimir
operators (5.5) and $\tau_{(t,n)}$ are the partially massless values (5.25).

##  Chapter 6 Higher-spin gauge models with $(1,1)$ supersymmetry in AdS3

There exist several incarnations of ${\cal N}$-extended AdS supersymmetry
[241], which are known as $(p,q)$ AdS supersymmetry types, where the integers
$p\geq q\geq 0$ are such that ${\cal N}=p+q$. In principle, field theories
possessing $(p,q)$ AdS supersymmetry may be realised in $(p,q)$ AdS superspace
${\rm AdS}^{(3|p,q)}$ [124], which can be understood as a maximally
supersymmetric solution of $(p,q)$ AdS supergravity [241].111In order to
realise field theories with the $(p,q)$ AdS supersymmetry in ${\rm
AdS}^{(3|p,q)}$ for $p+q\leq 4$, one can employ the off-shell supergravity
methods developed in [172, 242, 243].

Specifically, within the supergravity framework of [124, 172], the superspace
${\rm AdS}^{(3|p,q)}$ originates as a maximally symmetric supergeometry
possessing covariantly constant torsion and curvature generated by a real
symmetric torsion $S^{IJ}=S^{JI}$. Here the indices $I,J$ of the structure
group $\mathsf{SO}({\cal N})$ take values from 1 to ${\cal N}$. The torsion
$S^{IJ}$ is nonsingular with signature determined by the parameters $p$ and
$q={\cal N}-p$. Since the isometry group of ${\rm AdS}^{(3|p,q)}$ is
${\mathsf{OSp}}(p|2;{\mathbb{R}})\times{\mathsf{OSp}}(q|2;{\mathbb{R}})$ and
$S^{IJ}$ is invariant under its compact subgroup
$\mathsf{SO}(p)\times{\mathsf{SO}}(q)$, the global realisation of $(p,q)$ AdS
superspace is222As detailed in [124], the superspaces (6.1) are conformally
flat. The superconformal flatness of AdS superspaces in diverse dimensions,
including the $d=3$ case, was studied in [244].

$\displaystyle{\rm
AdS}^{(3|p,q)}=\frac{{\mathsf{OSp}}(p|2;{\mathbb{R}})\times{\mathsf{OSp}}(q|2;{\mathbb{R}})}{{\mathsf{SL}}(2,{\mathbb{R}})\times{\mathsf{SO}}(p)\times{\mathsf{SO}}(q)}~{}.$
(6.1)

Consider a $(p,q)$ supersymmetric field theory formulated in ${\rm
AdS}^{(3|p,q)}$, with $p+q\geq 3$ and $p\geq q$. As argued in [245], it can
always be recast as a supersymmetric theory realised in $(2,0)$ AdS
superspace, with $(p+q-2)$ supersymmetries being hidden. In the case $p+q\geq
3$ and $p\geq q>0$, every supersymmetric field theory in ${\rm
AdS}^{(3|p,q)}$, can be reformulated as a theory in $(1,1)$ AdS
superspace.333Such reformulations were developed in [245] for general $(p,q)$
supersymmetric nonlinear $\sigma$-models in AdS3, with $p+q\leq 4$. In
accordance with this discussion, the comparison of higher-spin field theories
living in arbitrary $(p,q)$ superspaces for allowed values of $p$ and $q$, can
be boiled down to the problem of comparing supersymmetric field theories
possessing $(1,1)$ and $(2,0)$ AdS supersymmetry.

The main motivation of this chapter is to investigate the differences between
massless higher-spin supersymmetric field theories realised on ${\rm
AdS}^{(3|2,0)}$ and ${\rm AdS}^{(3|1,1)}$. Specifically, the off-shell
structure of the half-integer superspin multiplets with $(2,0)$ AdS
supersymmetry [126] drastically differs to their $(1,1)$ counterparts
[125].444It is pertinent to mention here that the $(2,0)$ and $(1,1)$ AdS
supersymmetries support different supercurrent multiplets [242]. Direct
comparison of these theories is difficult in a manifestly supersymmetric
setting since they are each formulated in different superspaces, ${\rm
AdS}^{(3|2,0)}$ and ${\rm AdS}^{(3|1,1)}$, respectively. However, both
families of higher-spin theories can be reformulated in the same $(1,0)$ AdS
superspace, upon where the precise difference between the $(2,0)$ and $(1,1)$
higher-spin supermultiplets can be elucidated.

The off-shell massless higher-spin multiplets in ${\rm AdS}^{(3|2,0)}$ [126]
were reduced to AdS3|2 in [127]. This chapter is devoted to applying the
$(1,1)\to(1,0)$ AdS reduction prescription to the off-shell higher-spin
multiplets in AdS(3|1,1) (both massless and massive), which were computed in
[125] by Hutomo, Kuzenko and Ogburn.555The massless higher-spin theories
proposed in [125, 122] have natural counterparts in four dimensions [108, 109,
110], see also [105] for a review. We will demonstrate that every massless
higher-spin theory possessing $(1,1)$ or $(2,0)$ AdS supersymmetry decomposes
into a sum of two off-shell $(1,0)$ supermultiplets which belong to the four
series of inequivalent higher-spin gauge models constructed in [127].

This chapter is based on the publication [2] and is organised as follows. In
section 6.1, a formalism to reduce every field theory in AdS(3|1,1) to AdS3|2
is developed. As an aside, we also initiate a program to formulate the
superspin projection operators in AdS(3|1,1). The formalism developed in
section 6.1 is then applied throughout sections 6.2$-$6.5 to carry out the
$(1,1)\to(1,0)$ AdS superspace reduction of all known massless higher-spin
models with $(1,1)$ AdS supersymmetry. These are detailed in sections
6.2$-$6.3 for the half-integer superspin-$(s+\frac{1}{2})$ case, and in
sections 6.4$-$6.5 for the integer superspin-$s$ case. In addition to
summarising our results in section 6.7, we also detail key differences between
the $(2,0)$ and $(1,1)$ higher-spin (massless and massive) supermultiplets. In
appendix 6.A, we present some identities which prove indispensable when
performing the $(1,1)\to(1,0)$ AdS superspace reduction in sections 6.2$-$6.5.

###  6.1  (1,1) $\rightarrow$ (1,0) AdS superspace reduction

In this section we review key aspects of $(1,1)$ AdS superspace [242, 125,
124] which are crucial in the development of a consistent reduction procedure
for field theories from AdS(3|1,1) to AdS3|2. In addition, we also comment on
the form of the superspin projection operators in AdS(3|1,1). Specifically, we
provide closed form expressions for the superprojectors which act on the space
of rank-$1$ and rank-$2$ tensor superfields in ${\rm AdS}^{(3|1,1)}$.

####  6.1.1 (1,1) AdS superspace: Complex basis

We begin by summarising salient facts concerning $(1,1)$ AdS superspace [242,
125] and superfield representations of the associated isometry group
${{\mathsf{OSp}}(1|2;{\mathbb{R}})}\times{{\mathsf{OSp}}(1|2;{\mathbb{R}})}$,
following the presentation in [125]. The geometry of $(1,1)$ AdS superspace
can be described using either a real or complex basis for the spinor covariant
derivatives. In this section we consider the formulation in terms of the
complex basis.

Three-dimensional $(1,1)$ AdS superspace is parametrised by the local complex
superspace coordinates $z^{{\cal M}}=(x^{m},\theta^{\mu},\bar{\theta}_{\mu})$,
where $m=0,1,2$ and $\mu=1,2$.666The fermionic coordinates
$(\theta^{\mu},\bar{\theta}_{\mu})$ are related to each other via complex
conjugation $\overline{\theta^{\mu}}={\bar{\theta}}^{\mu}$.  The geometry of
${\rm AdS}^{(3|1,1)}$ is described in terms of the covariant derivatives

$\displaystyle{{\mbox{\boldmath${\cal D}$}}}_{{\cal
A}}=\big{(}{\mbox{\boldmath${\cal D}$}}_{a},{\mbox{\boldmath${\cal
D}$}}_{\alpha},{{\mbox{\boldmath${\bar{\cal D}}$}}}{}^{\alpha}\big{)}=E_{{\cal
A}}+\Omega_{{{\cal A}}}~{},$ (6.2)

where $E_{{\cal A}}$ and $\Omega_{{\cal A}}$ denote the inverse supervielbein
and Lorentz connection,

$\displaystyle E_{{\cal A}}=E_{{\cal A}}{}^{{\cal M}}\frac{\partial}{\partial
z^{{\cal M}}}~{},\qquad\Omega_{{\cal A}}=\frac{1}{2}\Omega_{{\cal
A}}{}^{bc}M_{bc}=\frac{1}{2}\Omega_{{\cal
A}}{}^{\beta\gamma}M_{\beta\gamma}~{}.$ (6.3)

The covariant derivatives ${{\mbox{\boldmath${\cal D}$}}}_{{\cal A}}$ satisfy
the following algebra

$\displaystyle\qquad\\{{\mbox{\boldmath${\cal
D}$}}_{\alpha},{\mbox{\boldmath${\bar{\cal D}}$}}_{\beta}\\}=-2\rm
i{\mbox{\boldmath${\cal D}$}}_{\alpha\beta}~{},$ (6.4a)
$\displaystyle\qquad\\{{\mbox{\boldmath${\cal
D}$}}_{\alpha},{\mbox{\boldmath${\cal
D}$}}_{\beta}\\}=-4\bar{\mu}\,M_{\alpha\beta}~{},\qquad\hskip
23.33147pt\\{{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{\alpha},{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{\beta}\\}=4\mu\,M_{\alpha\beta}~{},$ (6.4b)
$\displaystyle\qquad[{\mbox{\boldmath${\cal
D}$}}_{\alpha\beta},{\mbox{\boldmath${\cal D}$}}_{\gamma}]=-2\rm
i\bar{\mu}\,\varepsilon_{\gamma(\alpha}{\mbox{\boldmath${\bar{\cal
D}}$}}_{\beta)}~{},\qquad\,\,[{\mbox{\boldmath${\cal
D}$}}_{\alpha\beta},{{\mbox{\boldmath${\bar{\cal D}}$}}}_{\gamma}]=2\rm
i\mu\,\varepsilon_{\gamma(\alpha}{\mbox{\boldmath${\cal D}$}}_{\beta)}~{},$
(6.4c) $\displaystyle\quad\,\,\,\,[{\mbox{\boldmath${\cal
D}$}}_{\alpha\beta},{\mbox{\boldmath${\cal
D}$}}_{\gamma\delta}]=4\bar{\mu}\mu\big{(}\varepsilon_{\gamma(\alpha}M_{\beta)\delta}+\varepsilon_{\delta(\alpha}M_{\beta)\gamma}\big{)}~{},$
(6.4d)

where $\mu\neq 0$ is a complex parameter which determines the curvature of
${\rm AdS}^{(3|1,1)}$. The phase of $\mu=|\mu|{\rm e}^{{\rm i}\varphi}$ can be
given any fixed value by a re-definition ${\mbox{\boldmath${\cal
D}$}}_{\alpha}\to{\rm e}^{{\rm i}\rho}{\mbox{\boldmath${\cal D}$}}_{\alpha}$
and ${\mbox{\boldmath${\bar{\cal D}}$}}_{\alpha}\to{\rm e}^{-{\rm
i}\rho}{\mbox{\boldmath${\bar{\cal D}}$}}_{\alpha}$, with $\rho$ constant. In
appendix 6.A, we collate a list of identities involving the covariant
derivatives $\mbox{\boldmath${\cal D}$}_{{\cal A}}$ which will be helpful in
the subsequent analysis.

Let us compile a dictionary of important constrained superfields in AdS(3|1,1)
which will appear frequently in the subsequent analysis. Given an integer
$n\geq 1$, a superfield $\Gamma_{\alpha(n)}$ on
$\mbox{\boldmath$\mathcal{V}$}_{(n)}$ is called transverse linear if it
satisfies the following constraint

${\mbox{\boldmath${\bar{\cal
D}}$}}{}^{\beta}\Gamma_{\beta\alpha(n-1)}=0\qquad\Longrightarrow\qquad({\mbox{\boldmath${\bar{\cal
D}}$}}{}^{2}-2(n+2)\mu)\Gamma_{\alpha(n)}=0~{}.$ (6.5a) On the other hand, a
superfield ${\Gamma}_{\alpha(n)}$ on $\mbox{\boldmath$\mathcal{V}$}_{(n)}$ is
said to transverse anti-linear if it obeys ${\mbox{\boldmath${\cal
D}$}}^{\beta}{\Gamma}_{\beta\alpha(n-1)}=0\qquad\Longrightarrow\qquad({\mbox{\boldmath${\cal
D}$}}^{2}-2(n+2)\bar{\mu}){\Gamma}_{\alpha(n)}=0~{}.$ (6.5b)

In the $n=0$ case, the constraints (6.5) are clearly not defined. However
their corollaries (6.5a) and (6.5b) are perfectly consistent,

$\displaystyle\big{(}{\mbox{\boldmath${\bar{\cal
D}}$}}{}^{2}-4\mu\big{)}{\Gamma}=0~{},$ (6.6a)
$\displaystyle\big{(}{\mbox{\boldmath${\cal
D}$}}^{2}-4\bar{\mu}\big{)}{\Gamma}=0~{},$ (6.6b)

and defines a covariantly linear (6.6a) and anti-linear (6.6b) scalar
superfield, respectively.

Given $n\geq 1$, a superfield ${G}_{\alpha(n)}$ on
$\mbox{\boldmath$\mathcal{V}$}_{(n)}$ is said to be longitudinal linear if it
obeys the following first-order constraint

${\mbox{\boldmath${\bar{\cal
D}}$}}_{(\alpha_{1}}{G}_{\alpha_{2}\dots\alpha_{n+1})}=0\qquad\Longrightarrow\qquad\big{(}{\mbox{\boldmath${\bar{\cal
D}}$}}{}^{2}+2n\mu\big{)}{G}_{\alpha(n)}=0~{}.$ (6.7a) Similarly, a superfield
${G}_{\alpha(n)}$ on $\mbox{\boldmath$\mathcal{V}$}_{(n)}$ is said to be
longitudinal anti-linear if it satisfies the condition ${\mbox{\boldmath${\cal
D}$}}_{(\alpha_{1}}{G}_{\alpha_{2}\dots\alpha_{n+1})}=0\qquad\Longrightarrow\qquad\big{(}{\mbox{\boldmath${\cal
D}$}}^{2}+2n\bar{\mu}\big{)}{G}_{\alpha(n)}=0~{}.$ (6.7b)

In the scalar case, $n=0$, the conditions (6.7a) and (6.7b) reduce to the
chiral and anti-chiral constraints, respectively

$\displaystyle{\mbox{\boldmath${\bar{\cal D}}$}}_{\alpha}{G}$ $\displaystyle=$
$\displaystyle 0~{},$ (6.8a) $\displaystyle{\mbox{\boldmath${\cal
D}$}}_{\alpha}{G}$ $\displaystyle=$ $\displaystyle 0~{}.$ (6.8b)

The transverse (6.5a) and longitudinal (6.7a) linear constraints may be solved
in terms of the complex unconstrained prepotentials ${\Lambda}_{\alpha(n+1)}$
and ${\Omega}_{\alpha(n-1)}$ as follows777Taking the complex conjugate of
(6.9) yields the solution to the transverse (6.6b) and longitudinal (6.7b)
anti-linear constraints in terms of complex unconstrained prepotentials.

$\displaystyle{\Gamma}_{\alpha(n)}$ $\displaystyle=$
$\displaystyle{\mbox{\boldmath${\bar{\cal
D}}$}}{}^{\beta}{\Lambda}_{\beta\alpha(n)}~{},$ (6.9a)
$\displaystyle{G}_{\alpha(n)}$ $\displaystyle=$
$\displaystyle{\mbox{\boldmath${\bar{\cal
D}}$}}_{\alpha}{\Omega}_{\alpha(n-1)}~{}.$ (6.9b)

The superfields ${\Lambda}_{\alpha(n+1)}$ and ${\Omega}_{\alpha(n-1)}$ are
defined modulo gauge transformations of the form

$\displaystyle\delta_{\xi}{\Lambda}_{\alpha(n+1)}$ $\displaystyle=$
$\displaystyle{\mbox{\boldmath${\bar{\cal
D}}$}}{}^{\beta}{\xi}_{\beta\alpha(n+1)}~{},$ (6.10a)
$\displaystyle\delta_{\zeta}{\Omega}_{\alpha(n-1)}$ $\displaystyle=$
$\displaystyle{\mbox{\boldmath${\bar{\cal
D}}$}}_{\alpha}{\zeta}_{\alpha(n-2)}~{},$ (6.10b)

where the gauge parameters ${\xi}_{\alpha(n+2)}$ and ${\zeta}_{\alpha(n-2)}$
are complex unconstrained.

Let us consider a superfield $\Phi_{\alpha(n)}$ on
$\mbox{\boldmath$\mathcal{V}$}_{(n)}$ which is simultaneously transverse
linear and transverse anti-linear (6.5). It can be shown that the
$\boldsymbol{\Delta}$ operator (6.23) preserves the TLAL nature of
$\Phi_{\alpha(n)}$

${\mbox{\boldmath${\cal D}$}}^{\beta
I}\Phi_{\beta\alpha(n-1)}=0\qquad\Longrightarrow\qquad{\mbox{\boldmath${\cal
D}$}}^{\beta I}\boldsymbol{\Delta}\Phi_{\beta\alpha(n-1)}=0~{}.$ (6.11)

Here, we have introduced the notation ${\mbox{\boldmath${\cal D}$}}^{\beta I}$
which represents both ${\mbox{\boldmath${\cal D}$}}^{\beta}$ and
${\mbox{\boldmath${\bar{\cal D}}$}}{}^{\beta}$.

Given a positive integer $n$, an arbitrary complex tensor superfield
${V}_{\alpha(n)}$ can be uniquely represented as a sum of transverse linear
and longitudinal linear multiplets [125]

$\displaystyle{V}_{\alpha(n)}=$ $\displaystyle-$
$\displaystyle\frac{1}{2\mu(n+2)}{\mbox{\boldmath${\bar{\cal
D}}$}}^{\gamma}{\mbox{\boldmath${\bar{\cal
D}}$}}_{(\gamma}{V}_{\alpha_{1}\dots\alpha_{n})}-\frac{1}{2\mu(n+1)}{\mbox{\boldmath${\bar{\cal
D}}$}}_{(\alpha_{1}}{\mbox{\boldmath${\bar{\cal
D}}$}}{}^{|\gamma|}{V}_{\alpha_{2}\dots\alpha_{n})\gamma}~{},~{}~{}~{}$ (6.12)

There exists projection operators $\mathcal{P}^{\perp}_{(n)}$ and
$\mathcal{P}^{\parallel}_{(n)}$ which map
$\mbox{\boldmath$\mathcal{V}$}_{(n)}$ to the the space of transverse linear
(6.5a) and longitudinal linear (6.7a) superfields respectively [125]. These
operators have the form888The projectors (6.13) are the three-dimensional
cousins of the AdS4|4 projectors (4.85) of [212].

$\displaystyle\mathcal{P}^{\perp}_{(n)}$ $\displaystyle=$
$\displaystyle\frac{1}{4(n+1)\mu}({\mbox{\boldmath${\bar{\cal
D}}$}}{}^{2}+2n\mu)~{},$ (6.13a) $\displaystyle\mathcal{P}^{\parallel}_{(n)}$
$\displaystyle=$
$\displaystyle-\frac{1}{4(n+1)\mu}({\mbox{\boldmath${\bar{\cal
D}}$}}{}^{2}-2(n+2)\mu)~{},$ (6.13b)

and satisfy the projector properties

$\displaystyle\mathcal{P}^{\perp}_{(n)}\mathcal{P}^{\perp}_{(n)}=\mathcal{P}^{\perp}_{(n)}~{},\quad\mathcal{P}^{\parallel}_{(n)}\mathcal{P}^{\parallel}_{(n)}=\mathcal{P}^{\parallel}_{(n)}~{},\quad\mathcal{P}^{\perp}_{(n)}\mathcal{P}^{\parallel}_{(n)}=\mathcal{P}^{\parallel}_{(n)}\mathcal{P}^{\perp}_{(n)}=0~{},\quad\mathcal{P}^{\perp}_{(n)}+\mathcal{P}^{\parallel}_{(n)}=\mathds{1}~{}.\hskip
5.69046pt$ (6.14)

Note that the projectors (6.13) are not superspin projection operators, as the
corresponding independent superfields obtained via a $(1,1)\to(1,0)$
superfield reduction are unconstrained, and thus cannot furnish irreducible
representations of the ${\cal N}=1$ AdS3 superalgebra. The properties (6.14)
imply that any superfield $V_{\alpha(n)}$ on
$\mbox{\boldmath$\mathcal{V}$}_{(n)}$ can be uniquely represented as the sum
of transverse linear and longitudinal linear superfields,

$V_{\alpha(n)}=\Gamma_{\alpha(n)}+G_{\alpha(n)}~{},$ (6.15)

which is consistent with (6.12).

The isometry transformations of ${\rm AdS}^{(3|1,1)}$ are generated by real
supervector fields ${\lambda}^{{\cal A}}E_{{\cal A}}$ which solve the Killing
equation

$\displaystyle\Big{[}{\Lambda}+\frac{1}{2}l^{ab}M_{ab},{\mbox{\boldmath${\cal
D}$}}_{{\cal C}}\Big{]}=0~{},$ (6.16)

where

$\displaystyle\Lambda=\lambda^{{\cal A}}{\mbox{\boldmath${\cal D}$}}_{{\cal
A}}=\lambda^{a}{\mbox{\boldmath${\cal
D}$}}_{a}+\lambda^{\alpha}{\mbox{\boldmath${\cal
D}$}}_{\alpha}+\bar{\lambda}_{\alpha}{\mbox{\boldmath${\bar{\cal
D}}$}}{}^{\alpha}~{},\qquad\bar{\lambda}^{a}=\lambda^{a}~{},$ (6.17)

and $l^{ab}$ is some local Lorentz parameter. As shown in [242], this equation
implies that the parameters $\lambda^{\alpha}$ and $l^{ab}$ can be uniquely
determined in terms of the vector $\lambda_{\alpha\beta}$,

$\displaystyle\lambda_{\alpha}=\frac{{\rm i}}{6}{\mbox{\boldmath${\bar{\cal
D}}$}}{}^{\beta}\lambda_{\alpha\beta}~{},\qquad
l_{\alpha\beta}=2{\mbox{\boldmath${\cal D}$}}_{(\alpha}\lambda_{\beta)}~{},$
(6.18)

where the vector parameter obeys the equation

$\displaystyle{\mbox{\boldmath${\cal
D}$}}_{(\alpha}\lambda_{\beta\gamma)}=0\quad\Longleftrightarrow\quad{\mbox{\boldmath${\bar{\cal
D}}$}}_{(\alpha}\lambda_{\beta\gamma)}=0~{}.$ (6.19)

A specific feature of ${\rm AdS}^{(3|1,1)}$ is that any two of the three
parameters $\\{\lambda_{{\alpha\beta}},\lambda_{\alpha},l_{\alpha\beta}\\}$
can be expressed in terms of the third parameter. In particular,

$\displaystyle\lambda_{\alpha\beta}=\frac{{\rm
i}}{\mu}{\mbox{\boldmath${\bar{\cal
D}}$}}_{(\alpha}\lambda_{\beta)}~{},\qquad\lambda_{\alpha}=\frac{1}{12\bar{\mu}}{\mbox{\boldmath${\cal
D}$}}^{\beta}l_{\alpha\beta}~{}.$ (6.20)

From (6.18) and (6.20) we deduce

$\displaystyle{\mbox{\boldmath${\bar{\cal
D}}$}}_{\alpha}\lambda^{\alpha}={\mbox{\boldmath${\cal
D}$}}^{\alpha}\lambda_{\alpha}=0~{}.$ (6.21)

The solution to these equations is given in [242].

The Casimir operators of the ${\rm AdS}^{(3|1,1)}$ isometry algebra
${\mathfrak{osp}(1|2;{\mathbb{R}})}\oplus{\mathfrak{osp}(1|2;{\mathbb{R}})}$
take the following form in the superfield representation999The Casimir
operator (6.22a) is reminiscent of its counterpart (4.76) in AdS4|4.

$\displaystyle\mathbb{Q}$
$\displaystyle:=\Box+\frac{1}{2}(\mu{\mbox{\boldmath${\cal
D}$}}^{2}+\bar{\mu}{\mbox{\boldmath${\bar{\cal
D}}$}}{}^{2})-2|\mu|^{2}M^{\beta\gamma}M_{\beta\gamma}~{},$
$\displaystyle[\mathbb{Q},{\mbox{\boldmath${\cal D}$}}_{{\cal A}}]=0~{},$
(6.22a) $\displaystyle\mathbb{F}$ $\displaystyle:=2{\mbox{\boldmath${\cal
D}$}}^{\beta\gamma}M_{\beta\gamma}+2\boldsymbol{\Delta}~{},$
$\displaystyle[\mathbb{F},{\mbox{\boldmath${\cal D}$}}_{{\cal A}}]=0~{},$
(6.22b)

where $\Box:=-\frac{1}{2}{\mbox{\boldmath${\cal
D}$}}^{\alpha\beta}{\mbox{\boldmath${\cal D}$}}_{\alpha\beta}$ and
$\boldsymbol{\Delta}$ is the real scalar operator which we define as

$\boldsymbol{\boldsymbol{\Delta}}:=\frac{{\rm i}}{2}{\mbox{\boldmath${\cal
D}$}}^{\beta}{\mbox{\boldmath${\bar{\cal D}}$}}_{\beta}=\frac{{\rm
i}}{2}{\mbox{\boldmath${\bar{\cal D}}$}}{}^{\beta}{\mbox{\boldmath${\cal
D}$}}_{\beta}~{}.$ (6.23)

The operator $\boldsymbol{\Delta}$ can be shown to satisfy the property

$[\boldsymbol{\Delta},{\mbox{\boldmath${\cal
D}$}}^{\gamma}{\mbox{\boldmath${\bar{\cal
D}}$}}_{\beta}+{\mbox{\boldmath${\bar{\cal
D}}$}}{}^{\gamma}{\mbox{\boldmath${\cal D}$}}_{\beta}]=0~{}.$ (6.24)

Note that the overall normalisation factor of the Casimir operator (6.22b) was
chosen so that it coincides with its $\mathbb{M}^{3|4}$ counterpart (3.10) in
the flat-superspace limit $\mu\to 0$. Note that (3.10) needs to be converted
to the complex basis.

Let us denote by ${\mbox{\boldmath$\mathcal{V}$}}_{(n)}$ the space of
symmetric complex rank-$n$ tensor superfields $\Phi_{\alpha(n)}$ on
AdS(3|1,1). The Casimir operators (6.22) are related to each other on
$\mbox{\boldmath$\mathcal{V}$}_{(n)}$ as follows

$\displaystyle\mathbb{F}^{2}\Phi_{\alpha(n)}=$ $\displaystyle
4(2n+1)\boldsymbol{\Delta}^{2}\Phi_{\alpha(n)}+4n^{2}\mathbb{Q}\Phi_{\alpha(n)}+4{\rm
i}n\boldsymbol{\Delta}\big{(}\mbox{\boldmath${\cal
D}$}_{\alpha}\mbox{\boldmath${\bar{\cal
D}}$}{}^{\beta}+\mbox{\boldmath${\bar{\cal D}}$}_{\alpha}\mbox{\boldmath${\cal
D}$}^{\beta}\big{)}\Phi_{\beta\alpha(n-1)}$ (6.25)
$\displaystyle+4n(n-1){\mbox{\boldmath${\cal
D}$}}_{\alpha(2)}{\mbox{\boldmath${\cal
D}$}}^{\beta(2)}\Phi_{\beta(2)\alpha(n-2)}-2n^{2}\big{(}\mu{\mbox{\boldmath${\cal
D}$}}^{2}+\bar{\mu}{\mbox{\boldmath${\bar{\cal
D}}$}}{}^{2}\big{)}\Phi_{\alpha(n)}$
$\displaystyle-4n^{2}(n-2)(n+2)|\mu|^{2}\Phi_{\alpha(n)}~{}.$

In general, the operator $\boldsymbol{\Delta}$ is not invertible, as seen by
the following relation

$\mathbb{Q}=\frac{1}{16}\\{{\mbox{\boldmath${\cal
D}$}}^{2},{\mbox{\boldmath${\bar{\cal
D}}$}}{}^{2}\\}+\boldsymbol{\Delta}^{2}-\frac{1}{4}\big{(}\mu{\mbox{\boldmath${\cal
D}$}}^{2}+\bar{\mu}{\mbox{\boldmath${\bar{\cal
D}}$}}{}^{2}\big{)}-|\mu|^{2}M^{\beta\gamma}M_{\beta\gamma}~{}.$ (6.26)

Let us introduce the operators

${\cal P}_{(0)}:=\frac{1}{\mathbb{Q}}\boldsymbol{\Delta}^{2}~{},\quad{\cal
P}_{(+)}:=\frac{1}{16\mathbb{Q}}\big{(}{\mbox{\boldmath${\bar{\cal
D}}$}}{}^{2}-4\mu\big{)}{\mbox{\boldmath${\cal D}$}}^{2}~{},\quad{\cal
P}_{(-)}:=\frac{1}{16\mathbb{Q}}\big{(}{\mbox{\boldmath${\cal
D}$}}^{2}-4\bar{\mu}\big{)}{\mbox{\boldmath${\bar{\cal D}}$}}{}^{2}~{},$
(6.27)

which can be extracted from the identity (6.26). The operators (6.27) can be
shown to be orthogonal projectors

$\sum_{i}{\cal P}_{(i)}=\mathds{1}~{},\qquad{\cal P}_{(i)}{\cal
P}_{(j)}=\delta_{ij}{\cal P}_{(i)}~{},$ (6.28)

for $i=\\{0,+,-\\}$. When restricted to the space of scalar superfields, the
projectors ${\cal P}_{(+)}$ and ${\cal P}_{(-)}$ map to the subspace of chiral
(6.8a) and anti-chiral (6.8b) superfields, respectively, while ${\cal
P}_{(0)}$ projects onto the space of linear and anti-linear superfields (6.6).

#####  6.1.1.1 On-shell superfields

Given a positive integer $n$, a superfield $\Phi_{\alpha(n)}$ on
$\mbox{\boldmath$\mathcal{V}$}_{(n)}$ is said to be on-shell if it obeys the
conditions [167]

$\displaystyle{\mbox{\boldmath${\cal D}$}}^{\beta I}\Phi_{\beta\alpha(n-1)}$
$\displaystyle=$ $\displaystyle 0~{},$ (6.29a)
$\displaystyle\mbox{\boldmath$\Delta$}\Phi_{\alpha(n)}$ $\displaystyle=$
$\displaystyle\sigma\mbox{\boldmath$M$}\Phi_{\alpha(n)}~{}.$ (6.29b)

Here, the real parameter $M$ is the pseudo-mass which carries mass dimension
one and $\sigma:=\pm 1$. In the flat-superspace limit, the on-shell conditions
(6.29) reduce to those (3.16) in $\mathbb{M}^{3|4}$ when converted to the
complex basis. It follows from the constraints (6.29) that an on-shell
superfield also satisfies the condition

$\big{(}\mathbb{F}-2(n+1)\sigma\mbox{\boldmath$M$}\big{)}\Phi_{\alpha(n)}=0~{}.$
(6.30)

Using the identity (6.25) in conjunction with (6.30), it can be shown that an
on-shell superfield (6.29) is an eigenvector of the Casimir operator (6.22a),

$\big{(}\mathbb{Q}-\mbox{\boldmath$\rho$}^{2}\big{)}\Phi_{\alpha(n)}=0~{},\qquad\mbox{\boldmath$\rho$}^{2}:=\mbox{\boldmath$M$}^{2}+n(n+2)|\mu|^{2}~{}.$
(6.31)

It follows from (6.29b) and (6.31) that the Casimir operators (6.22) are
proportional to the unit operator. Hence an on-shell superfield (6.29)
realises an irreducible representation of
${\mathfrak{osp}(1|2;{\mathbb{R}})}\oplus{\mathfrak{osp}(1|2;{\mathbb{R}})}$,
which we denote by
$\mbox{\boldmath$\mathfrak{S}$}(\mbox{\boldmath$M$},\sigma\frac{n}{2})$. We
say that an on-shell superfield (6.29) carries pseudo-mass $M$, spin
$|s|:=\frac{n}{2}$ and superhelicity $s=\frac{1}{2}(n+1)\sigma$.

Recall in anti-de Sitter (super)space in three and four dimensions, we
classified the on-shell superfields into two classes: i) partially massless
superfields which carry certain pseudo-masses that give rise to a higher depth
gauge symmetry;101010Recall that strictly massless (super)fields correspond to
the partially massless (super)fields with the maximal gauge symmetry i.e. in
general the lowest depth gauge transformation. ii) massive superfields which
possess a pseudo-mass which is not compatible with a gauge symmetry.
Currently, such a dictionary for on-shell supermultiplets is not available in
AdS(3|1,1). In particular, the pseudo masses characteristic of partially
massless superfields are unknown.

In order to complete the picture of on-shell superfields in AdS(3|1,1), it is
necessary to compute the partially massless values. As heavily emphasised
throughout this thesis, the information concerning partially massless pseudo-
masses is encoded within the poles of the (super)spin projection operators. To
conclude this subsection, we will initiate the program to compute the
superspin projection operators in AdS(3|1,1).

#####  6.1.1.2 Superspin projection operators

For any integer $n\geq 1$, we define the rank-$n$ superspin projection
operator $\mbox{\boldmath$\Pi$}_{[n]}^{\perp}$ by its action on
$\mbox{\boldmath$\mathcal{V}$}_{(n)}$ by the rule

$\displaystyle\mbox{\boldmath$\Pi$}^{\perp}_{[n]}:\mbox{\boldmath$\mathcal{V}$}_{(n)}$
$\displaystyle\longrightarrow$
$\displaystyle\mbox{\boldmath$\mathcal{V}$}_{(n)},$ (6.32a)
$\displaystyle\Phi_{\alpha(n)}$ $\displaystyle\longmapsto$
$\displaystyle\mbox{\boldmath$\Pi$}^{\perp}_{[n]}\Phi_{\alpha(n)}=:\mbox{\boldmath$\Pi$}^{\perp}_{\alpha(n)}(\Phi)~{}.$
(6.32b) The operator $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ satisfies the
defining properties:

  1. 1.

Idempotence: The operator $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ squares to
itself,

$\mbox{\boldmath$\Pi$}^{\perp}_{[n]}\mbox{\boldmath$\Pi$}^{\perp}_{[n]}=\mbox{\boldmath$\Pi$}^{\perp}_{[n]}~{}.$
(6.32c)

  2. 2.

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

${\mbox{\boldmath${\cal D}$}}^{\beta
I}\mbox{\boldmath$\Pi$}^{\perp}_{\beta\alpha(n-1)}(\Phi)=0~{}.$ (6.32d)

  3. 3.

Surjectivity: Every transverse superfield $\Phi^{\perp}_{\alpha(n)}$ belongs
to the image of $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$,

${\mbox{\boldmath${\cal D}$}}^{\beta
I}\Phi^{\perp}_{\beta\alpha(n-1)}=0\qquad\Longrightarrow\qquad\mbox{\boldmath$\Pi$}^{\perp}_{[n]}\Phi^{\perp}_{\alpha(n)}=\Phi^{\perp}_{\alpha(n)}~{}.$
(6.32e)

Let us consider a superfield $\Phi_{\alpha(n)}$ which satisfies the first-
order differential constraint (6.29b). By virtue of the properties (6.1.1.2),
it follows that the superspin projection operator
$\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ maps such a superfield to the space of
on-shell superfields (6.29)

$\displaystyle{\mbox{\boldmath${\cal D}$}}^{\beta
I}\mbox{\boldmath$\Pi$}^{\perp}_{\beta\alpha(n-1)}(\Phi)$ $\displaystyle=$
$\displaystyle 0~{},$ (6.33a)
$\displaystyle\mbox{\boldmath$\Delta$}\mbox{\boldmath$\Pi$}^{\perp}_{\alpha(n)}(\Phi)$
$\displaystyle=$
$\displaystyle\sigma\mbox{\boldmath$M$}\mbox{\boldmath$\Pi$}^{\perp}_{\alpha(n)}(\Phi)~{}.$
(6.33b)

In order to quench our apparent addiction of deriving ${\cal N}$-extended
superprojectors in diverse backgrounds and dimensions, we will initiate the
program of formulating the superprojectors (6.32) in AdS(3|1,1). This will be
done by applying a minimal uplift prescription to the ${\cal N}=2$ Minkowski
superprojectors (3.73), which involves promoting the $\mathbb{M}^{3|4}$
covariant derivatives in (3.73) to covariant derivatives in AdS(3|1,1). This
will ensure that the superprojectors have the correct form in the flat-
superspace limit. As of yet, we have only been able to compute the
superprojectors $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ for the cases $n=1$ and
$n=2$. They take the explicit form

$\displaystyle\mbox{\boldmath$\Pi$}^{\perp}_{[1]}\Phi_{\alpha}$
$\displaystyle=$
$\displaystyle\frac{1}{2\big{(}\mathbb{Q}-4|\mu|^{2}\big{)}}\Big{(}\boldsymbol{\Delta}\big{(}{\mbox{\boldmath${\cal
D}$}}_{\alpha}{}^{\beta}+\boldsymbol{\Delta}\delta_{\alpha}{}^{\beta}\big{)}-\frac{1}{4}\big{(}\bar{\mu}{\mbox{\boldmath${\bar{\cal
D}}$}}{}^{2}+\mu{\mbox{\boldmath${\cal
D}$}}^{2}-4|\mu|^{2}\big{)}\delta_{\alpha}{}^{\beta}\Big{)}\Phi_{\beta}~{},\hskip
22.76228pt$ (6.34a)
$\displaystyle\mbox{\boldmath$\Pi$}^{\perp}_{[2]}\Phi_{\alpha(2)}$
$\displaystyle=$
$\displaystyle\frac{1}{4\big{(}\mathbb{Q}-8|\mu|^{2}\big{)}\big{(}\mathbb{Q}-12|\mu|^{2}\big{)}}\boldsymbol{\Delta}\Big{(}{\mbox{\boldmath${\cal
D}$}}_{(\alpha_{1}}{}^{\beta_{1}}+\boldsymbol{\Delta}\delta_{(\alpha_{1}}{}^{\beta_{1}}\Big{)}$
(6.34b)
$\displaystyle\times\Big{(}\boldsymbol{\Delta}\big{(}{\mbox{\boldmath${\cal
D}$}}_{\alpha_{2})}{}^{\beta_{2}}+\boldsymbol{\Delta}\delta_{\alpha_{2})}{}^{\beta_{2}}\big{)}-\frac{1}{2}\big{(}\bar{\mu}{\mbox{\boldmath${\bar{\cal
D}}$}}{}^{2}+\mu{\mbox{\boldmath${\cal
D}$}}^{2}\big{)}\delta_{\alpha_{2})}{}^{\beta_{2}}\Big{)}\Phi_{\beta(2)}~{}.$

In the flat superspace limit $\mu\rightarrow 0$, the superspin projection
operators (6.34a) and (6.34b) coincide with the superprojectors (3.73) in
$\mathbb{M}^{3|4}$, for the cases $n=1$ and $n=2$, respectively. Computing the
higher-rank generalisations of the superspin projection operators (6.34) in
AdS(3|1,1) remains an open problem.

####  6.1.2 (1,1) AdS superspace: Real basis

It proves beneficial to realise the $(1,1)$ AdS covariant derivatives in a
real basis when performing the reduction procedure to ${\cal N}=1$ AdS
superspace. In accordance with [124], the algebra of covariant derivatives
(6.4) can be converted to the real basis by (i) making the convenient choice
$\mu=-{\rm i}|\mu|$; and (ii) replacing the complex operators
${\mbox{\boldmath${\cal D}$}}_{\alpha},{\mbox{\boldmath${\bar{\cal
D}}$}}_{\alpha}$ with
${\mbox{\boldmath$\nabla$}}^{I}_{\alpha}=({\mbox{\boldmath$\nabla$}}^{\underline{1}}_{\alpha},{\mbox{\boldmath$\nabla$}}^{\underline{2}}_{\alpha})$
defined by

$\displaystyle\qquad{\mbox{\boldmath${\cal
D}$}}_{\alpha}=\frac{1}{\sqrt{2}}({\mbox{\boldmath$\nabla$}}^{\underline{1}}_{\alpha}-{\rm
i}{\mbox{\boldmath$\nabla$}}^{\underline{2}}_{\alpha})~{},\qquad{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{\alpha}=-\frac{1}{\sqrt{2}}({\mbox{\boldmath$\nabla$}}^{\underline{1}}_{\alpha}+{\rm
i}{\mbox{\boldmath$\nabla$}}^{\underline{2}}_{\alpha})~{}.$ (6.35)

Furthermore, we introduce the real coordinates $z^{{\cal
M}}=(x^{m},\theta^{\mu}_{I})$ which are used to parametrise ${\rm
AdS}^{(3|1,1)}$. Choosing to define
${\mbox{\boldmath$\nabla$}_{a}}={\mbox{\boldmath${\cal D}$}}_{a}$, it can be
shown that the algebra of $(1,1)$ AdS covariant derivatives assumes the
following form in the real basis (6.35)

$\displaystyle\qquad\\{{\mbox{\boldmath$\nabla$}}^{\underline{1}}_{\alpha},{\mbox{\boldmath$\nabla$}}^{\underline{2}}_{\beta}\\}$
$\displaystyle=0~{},$ (6.36a)
$\displaystyle\qquad\\{{\mbox{\boldmath$\nabla$}}^{\underline{1}}_{\alpha},{\mbox{\boldmath$\nabla$}}^{\underline{1}}_{\beta}\\}$
$\displaystyle=2{\rm i}{\mbox{\boldmath$\nabla$}}_{\alpha\beta}-4{\rm
i}|\mu|M_{\alpha\beta}~{},$
$\displaystyle\qquad\\{{\mbox{\boldmath$\nabla$}}^{\underline{2}}_{\alpha},{\mbox{\boldmath$\nabla$}}^{\underline{2}}_{\beta}\\}$
$\displaystyle=2{\rm i}{\mbox{\boldmath$\nabla$}}_{\alpha\beta}+4{\rm
i}|\mu|M_{\alpha\beta}~{},$ (6.36b)
$\displaystyle\qquad[{\mbox{\boldmath$\nabla$}}_{a},{\mbox{\boldmath$\nabla$}}^{\underline{1}}_{\alpha}]$
$\displaystyle=|\mu|(\gamma_{a})_{\alpha}{}^{\beta}{\mbox{\boldmath$\nabla$}}^{\underline{1}}_{\beta}~{},$
$\displaystyle\qquad[{\mbox{\boldmath$\nabla$}}_{a},{\mbox{\boldmath$\nabla$}}^{\underline{2}}_{\alpha}]$
$\displaystyle=-|\mu|(\gamma_{a})_{\alpha}{}^{\beta}{\mbox{\boldmath$\nabla$}}^{\underline{2}}_{\beta}~{},$
(6.36c)
$\displaystyle\qquad[{\mbox{\boldmath$\nabla$}}_{a},{\mbox{\boldmath$\nabla$}}_{b}]$
$\displaystyle=-4|\mu|^{2}M_{ab}~{}.$ (6.36d)

We collect an array of identities involving the $(1,1)$ AdS covariant
derivatives in section 6.A, which will be useful in deriving many of the
results in this chapter.

It follows from (6.36) that the operators ${\mbox{\boldmath$\nabla$}}_{a}$ and
${\mbox{\boldmath$\nabla$}}_{\alpha}^{\underline{1}}$ possess the property:

  1. 1.

These operators form a closed algebra given by

$\\{{\mbox{\boldmath$\nabla$}}_{\alpha}^{\underline{1}},{\mbox{\boldmath$\nabla$}}^{\underline{1}}_{\beta}\\}=2{\rm
i}{\mbox{\boldmath$\nabla$}}_{\alpha\beta}-4{\rm i}|\mu|M_{\alpha\beta}~{},$
(6.37a)
$\quad[{\mbox{\boldmath$\nabla$}}_{a},{\mbox{\boldmath$\nabla$}}_{\beta}^{\underline{1}}]=|\mu|(\gamma_{a})_{\beta}{}^{\gamma}{\mbox{\boldmath$\nabla$}}^{\underline{1}}_{\gamma}~{},\quad[{\mbox{\boldmath$\nabla$}}_{a},{\mbox{\boldmath$\nabla$}}_{b}]=-4|\mu|^{2}M_{ab}~{}.$
(6.37b)

Relations (6.37) coincide with the algebra (5.114) of covariant derivatives of
AdS3|2.

This property implies that AdS3|2 is naturally realised as a surface embedded
in $(1,1)$ AdS superspace. One can make an appropriate choice in the real
Grassmann variables
$\theta^{\mu}_{I}=(\theta^{\mu}_{\underline{1}},\theta^{\mu}_{\underline{2}})$
such that AdS3|2 can be identified as the surface defined by
$\theta^{\mu}_{\underline{2}}=0$ in ${\rm AdS}^{(3|1,1)}$. These properties
enable the consistent reduction of any field theory in ${\rm AdS}^{(3|1,1)}$
to AdS3|2.

We now wish to recast the fundamental properties of the Killing supervector
fields of $(1,1)$ AdS superspace (6.17) in the real representation (6.35). The
isometries of $(1,1)$ AdS superspace are generated by the $(1,1)$ AdS Killing
supervector fields,

$\Lambda:=\lambda^{\cal A}\mbox{\boldmath$\nabla$}_{\cal
A}=\lambda^{a}\mbox{\boldmath$\nabla$}_{a}+\lambda^{\alpha}_{I}\mbox{\boldmath$\nabla$}^{I}_{\alpha}~{},\qquad
I=\underline{1},\underline{2}~{},$ (6.38)

which are defined to satisfy the Killing equation

$[\Lambda+\frac{1}{2}l^{ab}M_{ab},\mbox{\boldmath$\nabla$}_{\cal A}]=0~{},$
(6.39)

for some real Lorentz parameter $l^{ab}=-l^{ba}$. It can be shown that
equation (6.39) is equivalent to the set of equations

$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha}\lambda^{\underline{1}}_{\beta}$
$\displaystyle=$
$\displaystyle\frac{1}{2}l_{\alpha\beta}+|\mu|\lambda_{\alpha\beta}~{},\qquad\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha}\lambda^{\underline{2}}_{\beta}=0~{},$
(6.40a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}}_{\alpha}\lambda^{\underline{2}}_{\beta}$
$\displaystyle=$
$\displaystyle\frac{1}{2}l_{\alpha\beta}-|\mu|\lambda_{\alpha\beta}~{},\qquad\mbox{\boldmath$\nabla$}^{\underline{2}}_{\alpha}\lambda^{\underline{1}}_{\beta}=0~{},$
(6.40b)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha}l_{\beta\gamma}$
$\displaystyle=$ $\displaystyle 8{\rm
i}|\mu|\varepsilon_{\alpha(\beta}\lambda^{\underline{1}}{}_{\gamma)}~{},\qquad~{}\mbox{\boldmath$\nabla$}^{\underline{2}}_{\alpha}l_{\beta\gamma}=-8{\rm
i}|\mu|\varepsilon_{\alpha(\beta}\lambda^{\underline{2}}{}_{\gamma)}~{},$
(6.40c) $\displaystyle\mbox{\boldmath$\nabla$}^{I}_{\alpha}\lambda_{a}$
$\displaystyle=$ $\displaystyle 2{\rm
i}(\gamma_{a})_{\alpha\beta}\lambda^{I\beta}~{},$ (6.40d)

and

$\displaystyle\mbox{\boldmath$\nabla$}_{a}\lambda_{b}$ $\displaystyle=$
$\displaystyle l_{ab}~{},$ (6.41a)
$\displaystyle\mbox{\boldmath$\nabla$}_{a}\lambda^{\underline{1}}_{\alpha}$
$\displaystyle=$
$\displaystyle|\mu|(\gamma_{a})_{\alpha}{}^{\beta}\lambda^{\underline{1}}_{\beta}~{},\qquad\mbox{\boldmath$\nabla$}_{a}\lambda^{\underline{2}}_{\alpha}=-|\mu|(\gamma_{a})_{\alpha}{}^{\beta}\lambda^{\underline{2}}_{\beta}~{},$
(6.41b) $\displaystyle\mbox{\boldmath$\nabla$}_{a}l_{bc}$ $\displaystyle=$
$\displaystyle 8|\mu|^{2}\eta_{a[b}\lambda_{c]}~{}.$ (6.41c)

Equations (6.40) and (6.41) can be recast in the equivalent form

$\displaystyle\mbox{\boldmath$\nabla$}^{I}_{(\alpha}\lambda_{\beta\gamma)}$
$\displaystyle=$ $\displaystyle 0~{},\hskip
85.35826pt\mbox{\boldmath$\nabla$}^{I}_{(\alpha}l_{\beta\gamma)}=0~{},$
(6.42a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{1}}_{(\alpha}\lambda^{\underline{1}}_{\beta)}$
$\displaystyle=$
$\displaystyle\frac{1}{2}l_{\alpha\beta}+|\mu|\lambda_{\alpha\beta}~{},\qquad\mbox{\boldmath$\nabla$}^{\underline{2}}_{(\alpha}\lambda^{\underline{2}}_{\beta)}=\frac{1}{2}l_{\alpha\beta}-|\mu|\lambda_{\alpha\beta}~{},$
(6.42b)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{1}}_{(\alpha}\lambda^{\underline{2}}_{\beta)}$
$\displaystyle=$
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}}_{(\alpha}\lambda^{\underline{1}}_{\beta)}=0,\qquad~{}~{}\mbox{\boldmath$\nabla$}^{\alpha(I}\lambda^{J)}_{\alpha}=0~{},$
(6.42c) $\displaystyle\lambda^{\underline{1}\alpha}$ $\displaystyle=$
$\displaystyle\frac{{\rm
i}}{6}\mbox{\boldmath$\nabla$}^{\underline{1}}_{\beta}\lambda^{\alpha\beta}=\frac{{\rm
i}}{12|\mu|}\mbox{\boldmath$\nabla$}^{\underline{1}}_{\beta}l^{\alpha\beta}~{},$
(6.42d) $\displaystyle\lambda^{\underline{2}\alpha}$ $\displaystyle=$
$\displaystyle\frac{{\rm