i}}{6}\mbox{\boldmath$\nabla$}^{\underline{2}}_{\beta}\lambda^{\alpha\beta}=-\frac{{\rm
i}}{12|\mu|}\mbox{\boldmath$\nabla$}^{\underline{2}}_{\beta}l^{\alpha\beta}~{}.$
(6.42e)

It follows from (6.41a) that the parameter $\lambda_{a}$ is a Killing vector
field

$\mbox{\boldmath$\nabla$}_{a}\lambda_{b}+\mbox{\boldmath$\nabla$}_{b}\lambda_{a}=0~{},$
(6.43)

and relations (6.41b) are Killing spinor equations.

####  6.1.3 Reduction from $(1,1)\to(1,0)$ AdS superspace

Given a tensor superfield $U(x,\theta_{I})$ on $(1,1)$ AdS superspace, where
indices have been suppressed, its bar-projection to ${\cal N}=1$ AdS
superspace is defined by the rule

$U|:=U(x,\theta_{I})|_{\theta_{\underline{2}}=0}$ (6.44)

in a special coordinate system which will be detailed below. Given the $(1,1)$
covariant derivative in the the real representation (6.35)

$\mbox{\boldmath$\nabla$}_{\cal
A}=(\mbox{\boldmath$\nabla$}_{a},\mbox{\boldmath$\nabla$}^{I}_{\alpha})=E_{{\cal
A}}{}^{{\cal M}}\frac{\partial}{\partial z^{{\cal
M}}}+\frac{1}{2}\Omega_{{\cal A}}{}^{bc}M_{bc}~{},$ (6.45)

we define its ${\cal N}=1$ projection by the rule

$\mbox{\boldmath$\nabla$}_{\cal A}|=E_{{\cal A}}{}^{{\cal
M}}|\frac{\partial}{\partial z^{{\cal M}}}+\frac{1}{2}\Omega_{{\cal
A}}{}^{bc}|M_{bc}~{}.$ (6.46)

We use the freedom to perform general coordinate and local Lorentz
transformations to impose the gauge

$\mbox{\boldmath$\nabla$}_{a}|=\nabla_{a}~{},\qquad\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha}|=\nabla_{\alpha}~{},$
(6.47)

where $\nabla_{A}=(\nabla_{a},\nabla_{\alpha})$ is the set of covariant
derivatives for AdS3|2, see eq. (5.113). In the chosen coordinate system, the
operator $\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha}$ does not involve
any partial derivatives with respect to $\theta_{\underline{2}}$. Thus for any
positive integer $k$, it follows that
$(\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha_{1}}\ldots\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha_{k}}U)|=\nabla_{\alpha_{1}}\ldots\nabla_{\alpha_{k}}U|$.

We now consider the ${\cal N}=1$ projection of the $(1,1)$ AdS Killing
supervector (6.38)

$\Lambda|=\xi^{a}\nabla_{a}+\xi^{\alpha}\nabla_{\alpha}+\epsilon^{\alpha}\mbox{\boldmath$\nabla$}^{\underline{2}}_{\alpha}|~{},$
(6.48)

where we have introduced the ${\cal N}=1$ superfields

$\xi^{a}:=\lambda^{a}|~{},\qquad\xi^{\alpha}:=\lambda^{\alpha}_{\underline{1}}|~{},\qquad\epsilon^{\alpha}:=\lambda^{\alpha}_{\underline{2}}|~{}.$
(6.49)

Additionally, we define the ${\cal N}=1$ projection of the Lorentz parameter
$l^{ab}$ to be

$\zeta^{ab}:=l^{ab}|~{}.$ (6.50)

It is important to note that the superfields
$(\xi^{a},\xi^{\alpha},\zeta^{ab})$ parametrise the infinitesimal isometries
of AdS3|2. Such transformations are generated by the Killing supervector
fields, $\xi=\xi^{a}\nabla_{a}+\xi^{\alpha}\nabla_{\alpha}$, satisfying the
${{\cal N}}=1$ Killing equation, eq. (5.117). Indeed, the relations (5.118)
and (5.119) automatically follow from the $(1,1)$ AdS Killing equations
(6.42), upon projection. The parameter $\epsilon_{\alpha}$, which generates
the second supersymmetry transformation, has the property

$\displaystyle\nabla_{a}\epsilon_{\alpha}$ $\displaystyle=$
$\displaystyle-|\mu|(\gamma_{a})_{\alpha}{}^{\beta}\epsilon_{\beta}~{}.$
(6.51)

Given the transformation law of a tensor superfield $U(x,\theta_{I})$ on
$(1,1)$ AdS superspace

$\delta_{\Lambda}U=\big{(}\Lambda+\frac{1}{2}l^{ab}M_{ab}\big{)}U~{},$ (6.52)

we find its projection to ${\cal N}=1$ AdS superspace to be

$\delta_{\Lambda}U|=\delta_{\xi}U|+\delta_{\epsilon}U|~{},$ (6.53)

where

$\displaystyle\delta_{\xi}U|$ $\displaystyle=$
$\displaystyle\big{(}\xi^{a}\nabla_{a}+\xi^{\alpha}\nabla_{\alpha}+\frac{1}{2}\zeta^{ab}M_{ab}\big{)}U|~{},$
(6.54a) $\displaystyle\delta_{\epsilon}U|$ $\displaystyle=$
$\displaystyle\epsilon^{\alpha}\big{(}\mbox{\boldmath$\nabla$}^{\underline{2}}_{\alpha}U\big{)}|~{}.$
(6.54b)

The first transformation (6.54a) coincides with the infinitesimal
transformation generated by a Killing supervector in AdS3|2, eq. (5.120).
Thus, $U|$ can be identified as a tensor superfield on ${\cal N}=1$ AdS
superspace. The other transformation (6.54b) corresponds to the second
supersymmetry transformation, which is generated by $\epsilon^{\alpha}$.

####  6.1.4 The (1,1) AdS supersymmetric actions in AdS3|2

Every supersymmetric field theory in $(1,1)$ AdS superspace can be reduced to
${\cal N}=1$ AdS superspace. In the following section, we explore the
necessary mathematical framework which will be employed to develop such a
reduction procedure. As presented in [172, 242, 243, 245], manifestly
supersymmetric actions in $(1,1)$ AdS superspace can be constructed by either:

  1. 1.

Integrating a real scalar Lagrangian ${\cal L}$ over the full $(1,1)$ AdS
superspace,

$\displaystyle\int{\rm d}^{3}x\,{\rm d}^{2}\theta\,{\rm
d}^{2}\bar{\theta}\,\mbox{\boldmath$E$}\,{\cal L}=\frac{1}{16}\int{\rm
d}^{3}x\,e\,({\mbox{\boldmath${\cal
D}$}}^{2}-16\bar{\mu})({\mbox{\boldmath${\cal D}$}}^{2}-4\mu){\cal
L}\Big{|}_{\theta=0}$ $\displaystyle=\int{\rm
d}^{3}x\,e\,\Big{(}\frac{1}{16}{\mbox{\boldmath${\cal
D}$}}^{\alpha}({{\mbox{\boldmath${\bar{\cal
D}}$}}}^{2}-6\mu){\mbox{\boldmath${\cal
D}$}}_{\alpha}-\frac{\mu}{4}{\mbox{\boldmath${\cal
D}$}}^{2}-\frac{\bar{\mu}}{4}{{\mbox{\boldmath${\bar{\cal
D}}$}}}^{2}+4\mu\bar{\mu}\Big{)}{\cal L}\Big{|}_{\theta=0}~{}.$ (6.55)

  2. 2.

Integrating a covariantly chiral Lagrangian ${\cal L}_{c}$ over the chiral
subspace,

$\int{\rm d}^{3}x\,{\rm d}^{2}\theta\,{\cal E}\,{\cal
L}_{c}=-\frac{1}{4}\int{\rm d}^{3}x\,e\,({\mbox{\boldmath${\cal
D}$}}^{2}-16\bar{\mu}){\cal
L}_{c}\Big{|}_{\theta=0}~{},\qquad{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{\alpha}{\cal L}_{c}=0~{}.$ (6.56)

The two supersymmetric invariants are related by the rule

$\displaystyle\int{\rm d}^{3}x\,{\rm d}^{2}\theta\,{\rm
d}^{2}\bar{\theta}\,\mbox{\boldmath$E$}\,{\cal L}=\int{\rm d}^{3}x\,{\rm
d}^{2}\theta\,{\cal E}\,{\cal L}_{c}~{},\qquad{\cal
L}_{c}:=-\frac{1}{4}({\mbox{\boldmath${\bar{\cal D}}$}}{}^{2}-4\mu){\cal
L}~{}.$ (6.57)

In $(1,1)$ AdS superspace, every chiral action can always be recast as an
integral over the full superspace

$\int{\rm d}^{3}x\,{\rm d}^{2}\theta\,{\cal E}\,{\cal
L}_{c}=\frac{1}{\mu}\int{\rm d}^{3}x\,{\rm d}^{2}\theta\,{\rm
d}^{2}\bar{\theta}\,\mbox{\boldmath$E$}\,{\cal L}_{c}~{}.$ (6.58)

We will use the notation ${\rm d}^{3|4}z:={\rm d}^{3}x\,{\rm d}^{2}\theta{\rm
d}^{2}\bar{\theta}$ to denote the full superspace measure.

Instead of reducing the above supersymmetric actions to components, we wish to
obtain a prescription which allows for their reduction to ${\cal N}=1$ AdS
superspace. The supersymmetric action in $\rm AdS^{3|2}$ is described by a
real scalar Lagrangian $L$

$S=\int{\rm d}^{3|2}z\,E\,L=\frac{1}{4}\int{\rm d}^{3}x\,e\,({\rm
i}\nabla^{2}+8|\mu|)L\,\big{|}_{\,\theta=0}~{}.$ (6.59)

The action (1) reduces to $\rm AdS^{3|2}$ as follows

${\mathbb{S}}=\int{\rm d}^{3|4}z\,\mbox{\boldmath$E$}\,{\cal L}=-\frac{{\rm
i}}{4}\int{\rm
d}^{3|2}z\,E\,\Big{(}(\mbox{\boldmath$\nabla$}^{\underline{2}})^{2}+8{\rm
i}|\mu|\Big{)}{\cal L}\big{|}~{}.$ (6.60)

By making use of the Killing equations (6.42) and (6.51), it can be shown that
the action (6.60) is invariant under $(1,1)$ AdS isometry transformations
given by equation (6.53).

###  6.2 Massless half-integer superspin: Transverse formulation

In $(1,1)$ AdS superspace, there exist two off-shell formulations for the
massless multiplet of half-integer superspin-$(s+\frac{1}{2})$, with $s\geq 2$
[125]. These two theories, which are called transverse and longitudinal, prove
to be dual to each other. In the following section, we develop the
$(1,1)\rightarrow(1,0)$ reduction procedure for the transverse formulation.

####  6.2.1 Transverse formulation

According to [125], the transverse formulation for the massless
superspin-$(s+\frac{1}{2})$ multiplet is described by the dynamical variables

${\cal
V}^{\perp}_{(s+\frac{1}{2})}=\big{\\{}\mathfrak{H}_{\alpha(2s)},\Gamma_{\alpha(2s-2)},\bar{\Gamma}_{\alpha(2s-2)}\big{\\}}~{},$
(6.61)

where the real superfield $\mathfrak{H}_{\alpha(2s)}$ is unconstrained, while
the complex superfield $\Gamma_{\alpha(2s-2)}$ is transverse linear (6.5a).
The superfields $\mathfrak{H}_{\alpha(2s)}$ and $\Gamma_{\alpha(2s-2)}$ are
defined modulo gauge transformations of the form

$\displaystyle\delta_{\lambda}\mathfrak{H}_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{(\alpha_{1}}\lambda_{\alpha_{2}\dots\alpha_{2s})}-{{\mbox{\boldmath${\cal
D}$}}}_{(\alpha_{1}}\bar{\lambda}_{\alpha_{2}\dots\alpha_{2s})}\equiv
g_{\alpha(2s)}+\bar{g}_{\alpha(2s)}~{},$ (6.62a)
$\displaystyle\delta_{\lambda}\Gamma_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}{{\mbox{\boldmath${\bar{\cal
D}}$}}}^{\beta}\big{(}{{\mbox{\boldmath${\cal
D}$}}}^{2}+2(2s-1)\bar{\mu}\big{)}\bar{\lambda}_{\beta\alpha(2s-2)}\equiv\frac{s}{2s+1}{{\mbox{\boldmath${\bar{\cal
D}}$}}}^{\beta}{\mbox{\boldmath${\cal
D}$}}^{\gamma}\bar{g}_{\beta\gamma\alpha{(2s-2)}}~{},$ (6.62b)

where the gauge parameter $\lambda_{\alpha(2s-1)}$ is complex unconstrained.
By construction, the complex gauge parameter
$g_{\alpha(2s)}:={{\mbox{\boldmath${\bar{\cal
D}}$}}}_{(\alpha_{1}}\lambda_{\alpha_{2}\dots\alpha_{2s})}$ is longitudinal
linear (6.7a). Note that the gauge transformation (6.62a) indicates that
$\mathfrak{H}_{\alpha(2s)}$ corresponds to the ${\cal N}=1$ superconformal
gauge prepotential [122].

Up to normalisation, there exists a unique quadratic action which is invariant
under the gauge transformations (6.62). This action takes the explicit form

$\displaystyle\mathbb{S}^{\perp}_{(s+\frac{1}{2})}[\mathfrak{H}_{\alpha(2s)},\Gamma_{\alpha(2s-2)},\bar{\Gamma}_{\alpha(2s-2)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3|4}z\,\mbox{\boldmath$E$}\,\bigg{\\{}2s(2s-1)\bar{\mu}\mu\mathfrak{H}^{\alpha(2s)}\mathfrak{H}_{\alpha(2s)}~{}$
$\displaystyle+\frac{1}{8}\mathfrak{H}^{\alpha(2s)}{\mbox{\boldmath${\cal
D}$}}^{\beta}\big{(}{{\mbox{\boldmath${\bar{\cal
D}}$}}}{}^{2}-6\mu\big{)}{\mbox{\boldmath${\cal
D}$}}_{\beta}\mathfrak{H}_{\alpha(2s)}+\mathfrak{H}^{\alpha(2s)}\big{(}{\mbox{\boldmath${\cal
D}$}}_{(\alpha_{1}}{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{\alpha_{2}}{\Gamma}_{\alpha_{3}\dots\alpha_{2s})}-{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{(\alpha_{1}}{\mbox{\boldmath${\cal
D}$}}_{\alpha_{2}}\bar{\Gamma}_{\alpha_{3}\dots\alpha_{2s})}\big{)}~{}$
$\displaystyle+\frac{2s-1}{s}\bar{\Gamma}^{\alpha(2s-2)}{\Gamma}_{\alpha(2s-2)}+\frac{2s+1}{2s}\big{(}{\Gamma}^{\alpha(2s-2)}{\Gamma}_{\alpha(2s-2)}+\bar{\Gamma}^{\alpha(2s-2)}\bar{\Gamma}_{\alpha(2s-2)}\big{)}\bigg{\\}}~{}.$
(6.63)

The above construction does not take into consideration the case where $s=1$,
since the transverse linear constraint (6.5a) is ill-defined for $n=0$.
However, corollary (6.6a) is consistent for $n=0$ and defines a covariantly
linear scalar superfield. The superfield $\Gamma$ and its complex conjugate
$\bar{\Gamma}$ can be interpreted as compensators, which have the
corresponding gauge transformations

$\displaystyle\delta_{\lambda}\mathfrak{H}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{(\alpha}\lambda_{\beta)}-{\mbox{\boldmath${\cal
D}$}}_{(\alpha}\bar{\lambda}_{\beta)}~{},$ (6.64a)
$\displaystyle\delta_{\lambda}\Gamma$ $\displaystyle=$
$\displaystyle-\frac{1}{4}{{\mbox{\boldmath${\bar{\cal
D}}$}}}^{\beta}({\mbox{\boldmath${\cal
D}$}}^{2}+2\bar{\mu})\bar{\lambda}_{\beta}~{},$ (6.64b)

as a result of (6.62). It is simple to show that transverse linear constraint
(6.5a) satisfied by $\Gamma$ is consistent with the variation (6.64b).
Choosing $s=1$ in (6.2.1) yields the linearised action for non-minimal $(1,1)$
AdS supergravity [242].

####  6.2.2 Reduction of gauge prepotentials to $\text{AdS}^{3|2}$

We wish to reduce the gauge prepotentials (6.61) to ${\cal N}=1$ AdS
superspace. We start by reducing the superconformal gauge multiplet
$\mathfrak{H}_{\alpha(2s)}$. Converting the longitudinal linear constraint of
$g_{\alpha(2s)}$ (6.7a) to the real representation (6.35) yields

$\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}g_{\alpha_{2}\ldots\alpha_{2s+1})}={\rm
i}\mbox{\boldmath$\nabla$}^{\underline{1}}{}_{(\alpha_{1}}g_{\alpha_{2}\ldots\alpha_{2s+1})}~{}.$
(6.65)

Taking a Taylor expansion of $g_{\alpha(2s)}(\theta^{I})$ about
$\theta^{\underline{2}}$, and using (6.65), we find the independent
$\theta^{\underline{2}}$ -components of $g_{\alpha(2s)}$ to be

$g_{\alpha(2s)}|~{},\qquad\mbox{\boldmath$\nabla$}^{\underline{2}\beta}g_{\beta\alpha(2s-1)}|~{}.$
(6.66)

The gauge transformation (6.62a) allows us to impose the gauge conditions

$\mathfrak{H}_{\alpha(2s)}|=0~{},\qquad\mbox{\boldmath$\nabla$}^{\underline{2}\beta}\mathfrak{H}_{\beta\alpha(2s-1)}|=0~{}.$
(6.67)

In this Wess-Zumino (WZ) gauge, we stay with the unconstrained real ${\cal
N}=1$ superfields

$\displaystyle H_{\alpha(2s+1)}:$ $\displaystyle=$ $\displaystyle{\rm
i}\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\mathfrak{H}_{\alpha_{2}\ldots\alpha_{2s+1})}|~{},$
(6.68a) $\displaystyle H_{\alpha(2s)}:$ $\displaystyle=$
$\displaystyle\frac{{\rm
i}}{4}(\mbox{\boldmath$\nabla$}^{\underline{2}})^{2}\mathfrak{H}_{\alpha(2s)}|~{}.$
(6.68b)

The residual gauge freedom which preserves the gauge conditions (6.67) is
described by the real ${\cal N}=1$ unconstrained superfields

$\displaystyle g_{\alpha(2s)}|$ $\displaystyle=-\frac{{\rm
i}}{2}\zeta_{\alpha(2s)}~{},$ $\displaystyle\bar{\zeta}_{\alpha(2s)}$
$\displaystyle=\zeta_{\alpha(2s)}~{},$ (6.69a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}\beta}g_{\beta\alpha(2s-1)}|$
$\displaystyle=\frac{2s+1}{2s}\zeta_{\alpha(2s-1)}~{},$
$\displaystyle\bar{\zeta}_{\alpha(2s-1)}$
$\displaystyle=\zeta_{\alpha(2s-1)}~{}.$ (6.69b)

From (6.69), we can readily determine the gauge transformations of the
superfields (6.68)

$\displaystyle\delta_{\zeta}H_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle{\rm
i}\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}\ldots\alpha_{2s+1})}~{},$ (6.70a)
$\displaystyle\delta_{\zeta}H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}\ldots\alpha_{2s})}~{}.$
(6.70b)

Next, we wish to reduce $\Gamma_{\alpha(2s-2)}$ to ${\cal N}=1$ AdS
superspace. The superfield $\Gamma_{\alpha(2s-2)}$ obeys the transverse linear
constraint (6.5a), which takes the following form in the real representation
(6.35)

$\mbox{\boldmath$\nabla$}^{\underline{2}\beta}\Gamma_{\beta\alpha(2s-3)}={\rm
i}\mbox{\boldmath$\nabla$}^{\underline{1}\beta}\Gamma_{\beta\alpha(2s-3)}~{}.$
(6.71)

It follows that $\Gamma_{\alpha(2s-2)}(\theta^{I})$ has two independent
$\theta^{\underline{2}}$ -components

$\Gamma_{\alpha(2s-2)}|\
,\qquad\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\Gamma_{\alpha_{2}\ldots\alpha_{2s-1})}|~{}.$
(6.72)

Utilising the gauge transformation of $\Gamma_{\alpha(2s-2)}$ (6.62b) and the
real representation (6.35), we find

$\displaystyle\delta_{\lambda}\Gamma_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{{\rm
i}s}{2s+1}\Big{(}\mbox{\boldmath$\nabla$}^{\underline{1}\beta}\mbox{\boldmath$\nabla$}^{\underline{2}\gamma}-\mbox{\boldmath$\nabla$}^{\beta\gamma}\Big{)}\bar{g}_{\beta\gamma\alpha(2s-2)}~{},$
(6.73a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\delta_{\lambda}\Gamma_{\alpha_{2}\ldots\alpha_{2s-1})}$
$\displaystyle=$
$\displaystyle\frac{s}{2s+1}\Big{(}\mbox{\boldmath$\nabla$}^{\underline{1}\beta}\mbox{\boldmath$\nabla$}_{(\alpha_{1}}{}^{\gamma}\bar{g}_{\alpha_{2}\ldots\alpha_{2s-1})\beta\gamma}-(4s+1)|\mu|\mbox{\boldmath$\nabla$}^{\underline{1}\beta}\bar{g}_{\beta\alpha(2s-1)}~{}$
$\displaystyle+$
$\displaystyle\mbox{\boldmath$\nabla$}_{\beta}{}^{\gamma}\mbox{\boldmath$\nabla$}^{\underline{1}\beta}\bar{g}_{\gamma\alpha(2s-1)}-{\rm
i}\mbox{\boldmath$\nabla$}_{\beta}{}^{\gamma}\mbox{\boldmath$\nabla$}^{\underline{2}\beta}\bar{g}_{\gamma\alpha(2s-1)}-\frac{1}{2}(\mbox{\boldmath$\nabla$}^{\underline{1}})^{2}\mbox{\boldmath$\nabla$}^{\underline{2}\beta}\bar{g}_{\beta\alpha(2s-1)}~{}$
$\displaystyle-$ $\displaystyle(2s+1){\rm
i}|\mu|\mbox{\boldmath$\nabla$}^{\underline{2}\beta}\bar{g}_{\beta\alpha(2s-1)}-{\rm
i}\mbox{\boldmath$\nabla$}^{\beta\gamma}\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\bar{g}_{\alpha_{2}\ldots\alpha_{2s-1})\beta\gamma}\Big{)}~{}.$

From (6.73), we can immediately read off the gauge transformations of the
complex ${\cal N}=1$ superfields (6.72)

$\displaystyle\delta_{\lambda}\Gamma_{\alpha(2s-2)}|$ $\displaystyle=$
$\displaystyle-\frac{{\rm
i}}{2}\nabla^{\beta}\zeta_{\beta\alpha(2s-2)}+\frac{s}{2(2s+1)}\nabla^{\beta\gamma}\zeta_{\beta\gamma\alpha(2s-2)}~{},$
(6.74a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\delta_{\lambda}\Gamma_{\alpha_{2}\ldots\alpha_{2s-1})}|$
$\displaystyle=$ $\displaystyle\frac{s}{2(2s+1)}\Big{(}{\rm
i}\nabla^{\beta}\nabla_{(\alpha_{1}}{}^{\gamma}\zeta_{\alpha_{2}\ldots\alpha_{2s-1})\beta\gamma}+{\rm
i}\nabla_{\beta}{}^{\gamma}\nabla^{\beta}\zeta_{\gamma\alpha(2s-1)}~{}$
$\displaystyle-$ $\displaystyle(4s+1){\rm
i}|\mu|\nabla^{\beta}\zeta_{\beta\alpha(2s-1)}+\frac{2s+1}{s}\big{(}(2s+1){\rm
i}|\mu|\zeta_{\alpha(2s-1)}~{}$ $\displaystyle+$ $\displaystyle{\rm
i}\nabla_{(\alpha_{1}}{}^{\beta}\zeta_{\alpha_{2}\ldots\alpha_{2s-1})\beta}+\frac{1}{2}\nabla^{2}\zeta_{\alpha(2s-1)}\big{)}\Big{)}~{}.$

Let us express the ${{\cal N}}=1$ superfields (6.72) in terms of their real
and imaginary parts,

$\displaystyle\Gamma_{\alpha(2s-2)}|$ $\displaystyle=$ $\displaystyle-
L_{\alpha(2s-2)}-{\rm i}sV_{\alpha(2s-2)}~{},$ (6.75a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\Gamma_{\alpha_{2}\ldots\alpha_{2s-1})}|$
$\displaystyle=$ $\displaystyle\frac{1}{2}\big{(}\Phi_{\alpha(2s-1)}+{\rm
i}\Omega_{\alpha(2s-1)}\big{)}~{}.$ (6.75b)

It then follows from the gauge transformations (6.70) and (6.74) that we are
in fact dealing with two different gauge theories. The first model is
formulated in terms of the real unconstrained gauge superfields

${\cal
V}^{\parallel}_{(s+\frac{1}{2})}=\\{H_{\alpha(2s+1)},L_{\alpha(2s-2)},\Phi_{\alpha(2s-1)}\\}~{},$
(6.76)

which are defined modulo gauge transformations of the form

$\displaystyle\delta_{\zeta}H_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle{\rm
i}\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}\ldots\alpha_{2s+1})}~{},$ (6.77a)
$\displaystyle\delta_{\zeta}L_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\frac{s}{2(2s+1)}\nabla^{\beta\gamma}\zeta_{\beta\gamma\alpha(2s-2)}~{},$
(6.77b) $\displaystyle\delta_{\zeta}\Phi_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle\frac{{\rm
i}s}{2s+1}\Big{(}\nabla^{\beta}\nabla_{(\alpha_{1}}{}^{\gamma}\zeta_{\alpha_{2}\ldots\alpha_{2s-1})\beta\gamma}-(4s+1)|\mu|\nabla^{\beta}\zeta_{\beta\alpha(2s-1)}~{}$
$\displaystyle-$
$\displaystyle\nabla^{\beta\gamma}\nabla_{\beta}\zeta_{\gamma\alpha(2s-1)}\Big{)}~{},$

where the gauge parameter $\zeta_{\alpha(2s)}$ is real unconstrained. The
other gauge theory is constructed in terms of the superfields

${\cal
V}^{\parallel}_{(s)}=\\{H_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-1)}\\}~{},$
(6.78)

which possess the gauge freedom

$\displaystyle\delta_{\zeta}H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}\ldots\alpha_{2s})}~{},$
(6.79a) $\displaystyle\delta_{\zeta}V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s}\nabla^{\beta}\zeta_{\beta\alpha(2s-2)}~{},$ (6.79b)
$\displaystyle\delta_{\zeta}\Omega_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}{}^{\beta}\zeta_{\alpha_{2}\ldots\alpha_{2s-1})\beta}-\frac{{\rm
i}}{2}\nabla^{2}\zeta_{\alpha(2s-1)}+(2s+1)|\mu|\zeta_{\alpha(2s-1)}~{},$
(6.79c)

where the parameter $\zeta_{\alpha(2s-1)}$ is real unconstrained.

Applying the superspace reduction procedure to the action (6.2.1) yields two
decoupled ${\cal N}=1$ supersymmetric theories, each formulated in terms of
the gauge fields (6.76) and (6.78) respectively

$\displaystyle\mathbb{S}^{\perp}_{(s+\frac{1}{2})}[\mathfrak{H}_{\alpha(2s)},\Gamma_{\alpha(2s-2)},\bar{\Gamma}_{\alpha(2s-2)}]$
$\displaystyle=$ $\displaystyle
S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)},\Phi_{\alpha(2s-1)}]~{}$
(6.80)
$\displaystyle+S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-1)}]~{}.$

Explicit expressions for these decoupled ${\cal N}=1$ supersymmetric actions
are given in the subsequent section.

####  6.2.3 Massless higher-spin ${\cal N}=1$ supermultiplets

The first of the decoupled ${\cal N}=1$ supersymmetric actions, which is
realised in terms of the superfields (6.76), takes the following form

$\displaystyle
S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)},\Phi_{\alpha(2s-1)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\Big{(}-\frac{{\rm
i}}{8}\Big{)}\int{\rm
d}^{3|2}z~{}E~{}\bigg{\\{}2H^{\alpha(2s+1)}\mathbb{Q}H_{\alpha(2s+1)}~{}$
$\displaystyle-8(s-2)(2s+1)|\mu|^{2}H^{\alpha(2s+1)}H_{\alpha(2s+1)}+(2s+1){\rm
i}|\mu|H^{\alpha(2s+1)}\nabla^{2}H_{\alpha(2s+1)}~{}$
$\displaystyle-4(2s+1)|\mu|H^{\beta\alpha(2s)}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s)}-{\rm
i}H^{\beta\alpha(2s)}\nabla^{2}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s)}-4H^{\beta\gamma\alpha(2s-1)}\nabla_{\beta\gamma}\Phi_{\alpha(2s-1)}~{}$
$\displaystyle-8{\rm
i}H^{\beta\gamma\delta\alpha(2s-2)}\nabla_{\beta\gamma}\nabla_{\delta}L_{\alpha(2s-2)}-16{\rm
i}L^{\alpha(2s-2)}\nabla^{\beta}\Phi_{\beta\alpha(2s-2)}+\frac{2}{s}\Phi^{\alpha(2s-1)}\Phi_{\alpha(2s-1)}~{}$
$\displaystyle+\frac{8{\rm
i}}{s(2s-1)}\Big{(}2(s-1)(4s-1)L^{\beta\alpha(2s-3)}\nabla_{\beta}{}^{\gamma}L_{\gamma\alpha(2s-3)}~{}$
$\displaystyle-(3s-1){\rm
i}L^{\alpha(2s-2)}\nabla^{2}L_{\alpha(2s-2)}-4s(4s^{2}-3s+1)|\mu|L^{\alpha(2s-2)}L_{\alpha(2s-2)}\Big{)}\bigg{\\}}~{}.$
(6.81)

Upon inspection, it is apparent that the superfield $\Phi_{\alpha(2s-1)}$ is
auxiliary. So by making use of its equation of motion,

$\Phi_{\alpha(2s-1)}=-s\nabla^{\beta\gamma}H_{\beta\gamma\alpha(2s-1)}-4{\rm
i}s\nabla_{(\alpha_{1}}L_{\alpha_{2}\ldots\alpha_{2s-1})}~{},$ (6.82)

we can eliminate the auxiliary field $\Phi_{\alpha(2s-1)}$ from the action
(6.2.3). It can be shown that the resulting action coincides with the action
for the massless superspin-$(s+\frac{1}{2})$ multiplet, which is given by
(5.2.6.1).

The other ${\cal N}=1$ action is formulated in terms of the gauge superfields
(6.78)

$\displaystyle
S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-1)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3|2}z~{}E~{}\bigg{\\{}\frac{{\rm
i}}{2}H^{\alpha(2s)}\nabla^{2}H_{\alpha(2s)}$
$\displaystyle+2|\mu|H^{\alpha(2s)}H_{\alpha(2s)}+2sH^{\beta\gamma\alpha(2s-2)}\nabla_{\beta\gamma}V_{\alpha(2s-2)}-H^{\beta\alpha(2s-1)}\nabla_{\beta}\Omega_{\alpha(2s-1)}~{}$
$\displaystyle+\frac{{\rm
i}}{2}\Omega^{\alpha(2s-1)}\Omega_{\alpha(2s-1)}+\frac{1}{2s-1}\Big{(}4s(s-1)^{2}V^{\beta\alpha(2s-3)}\nabla_{\beta}{}^{\gamma}V_{\gamma\alpha(2s-3)}~{}$
$\displaystyle+{\rm
i}s(2s^{2}-2s+1)V^{\alpha(2s-2)}\nabla^{2}V_{\alpha(2s-2)}+4s(2s^{3}-2s+1)|\mu|V^{\alpha(2s-2)}V_{\alpha(2s-2)}$
$\displaystyle+(2s-1)V^{\alpha(2s-2)}\nabla^{\beta}\Omega_{\beta\alpha(2s-2)}\Big{)}\bigg{\\}}~{}.$
(6.83)

The superfield $\Omega_{\alpha(2s-1)}$ is auxiliary, so upon elimination via
its equation of motion

$\Omega_{\alpha(2s-1)}=-{\rm i}\nabla^{\beta}H_{\beta\alpha(2s-1)}-{\rm
i}\nabla_{(\alpha_{1}}V_{\alpha_{2}\ldots\alpha_{2s-1})}~{},$ (6.84)

we find that the resulting action coincides with the action for the massless
superspin-$s$ multiplet (5.2.6.3).

####  6.2.4 Second supersymmetry transformations

As discussed in section 6.1.4, by construction, the ${\cal N}=1$ actions
(6.2.3) and (6.2.3) are invariant under the second supersymmetry
transformations (6.54b). For convenience, we recall the form of these
transformations

$\delta_{\epsilon}U|=\epsilon^{\alpha}(\mbox{\boldmath$\nabla$}^{\underline{2}}_{\alpha}U)|~{},$
(6.85)

where $U(x,\theta^{I})$ is a superfield in AdS(3|1,1), with indices
suppressed. The second supersymmetry transformation acts on the remaining
${\cal N}=1$ superfields (6.76) and (6.78) in the following fashion

$\displaystyle\delta_{\epsilon}\mathfrak{H}_{\alpha(2s)}|$ $\displaystyle=$
$\displaystyle-{\rm i}\epsilon^{\beta}H_{\beta\alpha(2s)}~{},$ (6.86a)
$\displaystyle\delta_{\epsilon}\mbox{\boldmath$\nabla$}^{\underline{2}\beta}\mathfrak{H}_{\beta\alpha(2s-1)}|$
$\displaystyle=$ $\displaystyle 2{\rm
i}\epsilon^{\beta}H_{\beta\alpha(2s-1)}~{},$ (6.86b)
$\displaystyle\delta_{\epsilon}H_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle-2\epsilon_{(\alpha_{1}}H_{\alpha_{2}\ldots\alpha_{2s+1})}~{},$
(6.86c) $\displaystyle\delta_{\epsilon}H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle-\frac{{\rm
i}}{2}\Big{(}(2s+1)|\mu|\epsilon^{\beta}H_{\beta\alpha(2s)}-\epsilon^{\beta}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s)}\Big{)}~{},$
(6.86d) $\displaystyle\delta_{\epsilon}L_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\frac{{\rm
i}}{2}\epsilon^{\beta}\Omega_{\beta\alpha(2s-2)}+\frac{2s}{2s-1}(s-1)\epsilon_{(\alpha_{1}}\nabla^{\beta}V_{\alpha_{2}\ldots\alpha_{2s-2})\beta}~{},~{}$
(6.86e) $\displaystyle\delta_{\epsilon}V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{{\rm
i}}{2s}\epsilon^{\beta}\Phi_{\beta\alpha(2s-2)}-\frac{2}{s(2s-1)}(s-1)\epsilon_{(\alpha_{1}}\nabla^{\beta}L_{\alpha_{2}\ldots\alpha_{2s-2})\beta}~{},$
(6.86f) $\displaystyle\delta_{\epsilon}\Phi_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s-1}\Big{(}2s(2s-1)\epsilon^{\beta}\nabla_{\beta(\alpha_{1}}V_{\alpha_{2}\ldots\alpha_{2s-1})}+{\rm
i}s\epsilon_{(\alpha_{1}}\nabla^{2}V_{\alpha_{2}\ldots\alpha_{2s-1})}~{}$
$\displaystyle-$ $\displaystyle
4s(s-1)\epsilon_{(\alpha_{1}}\nabla_{\alpha_{2}}{}^{\beta}V_{\alpha_{3}\ldots\alpha_{2s-1})\beta}+(2s-1)\epsilon_{(\alpha_{1}}\nabla^{\beta}\Omega_{\alpha_{2}\ldots\alpha_{2s-1})\beta}~{}$
$\displaystyle+$ $\displaystyle
4s(4s^{2}-3s+1)|\mu|\epsilon_{(\alpha_{1}}V_{\alpha_{2}\ldots\alpha_{2s-1})}\Big{)}~{},~{}$
$\displaystyle\delta_{\epsilon}\Omega_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2s-1}\Big{(}2(2s-1)\epsilon^{\beta}\nabla_{\beta(\alpha_{1}}L_{\alpha_{2}\ldots\alpha_{2s-1})}+{\rm
i}\epsilon_{(\alpha_{1}}\nabla^{2}L_{\alpha_{2}\ldots\alpha_{2s-1})}~{}$
$\displaystyle-$ $\displaystyle
4(s-1)\epsilon_{(\alpha_{1}}\nabla_{\alpha_{2}}{}^{\beta}L_{\alpha_{3}\ldots\alpha_{2s-1})\beta}+(2s-1)\epsilon_{(\alpha_{1}}\nabla^{\beta}\Phi_{\alpha_{2}\ldots\alpha_{2s-1})\beta}~{}$
$\displaystyle+$ $\displaystyle
4(4s^{2}-3s+1)|\mu|\epsilon_{(\alpha_{1}}L_{\alpha_{2}\ldots\alpha_{2s-1})}\Big{)}~{}.~{}$

It is apparent from (6.86a) and (6.86b) that the second supersymmetry
transformation (6.85) breaks the WZ gauge conditions (6.67), which we recall
for convenience

$\mathfrak{H}_{\alpha(2s)}|=0~{},\qquad\mbox{\boldmath$\nabla$}^{\underline{2}\beta}\mathfrak{H}_{\beta\alpha(2s-1)}|=0~{}.$
(6.87)

In order to resolve this, it is necessary to supplement the variation (6.85)
with the $\epsilon$-dependent gauge transformations:

$\displaystyle\delta_{g(\epsilon)}\mathfrak{H}_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle g_{\alpha(2s)}(\epsilon)+\bar{g}_{\alpha(2s)}(\epsilon)~{},$
(6.88a) $\displaystyle\delta_{g(\epsilon)}\Gamma_{\alpha(2s-2)}$
$\displaystyle=$ $\displaystyle\frac{s}{2s+1}{\mbox{\boldmath${\bar{\cal
D}}$}}^{\beta}{\mbox{\boldmath${\cal
D}$}}^{\gamma}\bar{g}_{\beta\gamma\alpha(2s-2)}(\epsilon)~{}.$ (6.88b)

The modified second supersymmetry transformations now take the following form

$\displaystyle\hat{\delta}_{\epsilon}\mathfrak{H}_{\alpha(2s)}$
$\displaystyle=$
$\displaystyle\delta_{\epsilon}\mathfrak{H}_{\alpha(2s)}+\delta_{g(\epsilon)}\mathfrak{H}_{\alpha(2s)}~{},$
(6.89a) $\displaystyle\hat{\delta}_{\epsilon}\Gamma_{\alpha(2s-2)}$
$\displaystyle=$
$\displaystyle\delta_{\epsilon}\Gamma_{\alpha(2s-2)}+\delta_{g(\epsilon)}\Gamma_{\alpha(2s-2)}~{}.$
(6.89b)

The sole purpose of introducing the $\epsilon$-dependent gauge transformations
(6.88) is to restore the original Wess-Zumino gauge (6.87). Fixing the form of
the ${{\cal N}}=1$ components of $g_{\alpha(2s)}(\epsilon)$ as

$\displaystyle g_{\alpha(2s)}(\epsilon)|$ $\displaystyle=$
$\displaystyle\frac{{\rm i}}{2}\epsilon^{\beta}H_{\beta\alpha(2s)}~{},$
(6.90a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}\beta}g_{\beta\alpha(2s-1)}(\epsilon)|$
$\displaystyle=$ $\displaystyle-{\rm
i}\epsilon^{\beta}H_{\beta\alpha(2s-1)}~{},$ (6.90b)

ensures that (6.89) takes us back to the original gauge (6.87). The modified
transformations (6.89) act on the ${{\cal N}}=1$ superfields (6.76) and (6.78)
in the following manner

$\displaystyle\hat{\delta}_{\epsilon}H_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle-2\epsilon_{(\alpha_{1}}H_{\alpha_{2}\ldots\alpha_{2s+1})}~{},$
(6.91a) $\displaystyle\hat{\delta}_{\epsilon}H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle-\frac{{\rm i}}{4(2s+1)}\Big{(}{\rm
i}\epsilon^{\beta}\nabla^{2}H_{\beta\alpha(2s)}+4s\epsilon_{(\alpha_{1}}\nabla^{\beta\gamma}H_{\alpha_{2}\ldots\alpha_{2s})\beta\gamma}$
$\displaystyle-$ $\displaystyle
2\epsilon^{\beta}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s)}+2(2s+1)(4s+1)|\mu|\epsilon^{\beta}H_{\beta\alpha(2s)}\Big{)}~{},$
$\displaystyle\hat{\delta}_{\epsilon}L_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{s}{2s+1}\epsilon^{\beta}\nabla^{\gamma}H_{\beta\gamma\alpha(2s-2)}+\frac{2s}{2s-1}(s-1)\epsilon_{(\alpha_{1}}\nabla^{\beta}V_{\alpha_{2}\ldots\alpha_{2s-2})\beta}$
$\displaystyle-$ $\displaystyle\frac{{\rm
i}}{2}\epsilon^{\beta}\Omega_{\beta\alpha(2s-2)}~{},$
$\displaystyle\hat{\delta}_{\epsilon}V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{{\rm
i}}{2(2s+1)}\epsilon^{\beta}\nabla^{\gamma\delta}H_{\beta\gamma\delta\alpha(2s-2)}-\frac{2}{s(2s-1)}(s-1)\epsilon_{(\alpha_{1}}\nabla^{\beta}L_{\alpha_{2}\ldots\alpha_{2s-2})\beta}~{}$
$\displaystyle+$ $\displaystyle\frac{{\rm
i}}{2s}\epsilon^{\beta}\Phi_{\beta\alpha(2s-2)}~{},$
$\displaystyle\hat{\delta}_{\epsilon}\Phi_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle-\frac{s}{2s+1}\epsilon^{\beta}\Big{(}2\nabla_{(\alpha_{1}}{}^{\gamma}H_{\alpha_{2}\ldots\alpha_{2s-1})\beta\gamma}-{\rm
i}\nabla^{2}H_{\beta\alpha(2s-1)}+4(s+1)|\mu|H_{\beta\alpha(2s-1)}\Big{)}$
$\displaystyle+$
$\displaystyle\frac{1}{2s-1}\Big{(}2s(2s-1)\epsilon^{\beta}\nabla_{\beta(\alpha_{1}}V_{\alpha_{2}\ldots\alpha_{2s-1})}+{\rm
i}s\epsilon_{(\alpha_{1}}\nabla^{2}V_{\alpha_{2}\ldots\alpha_{2s-1})}~{}$
$\displaystyle+$ $\displaystyle
4s(4s^{2}-3s+1)|\mu|\epsilon_{(\alpha_{1}}V_{\alpha_{2}\ldots\alpha_{2s-1})}+(2s-1)\epsilon_{(\alpha_{1}}\nabla^{\beta}\Omega_{\alpha_{2}\ldots\alpha_{2s-1})\beta}~{}$
$\displaystyle-$ $\displaystyle
4s(s-1)\epsilon_{(\alpha_{1}}\nabla_{\alpha_{2}}{}^{\beta}V_{\alpha_{3}\ldots\alpha_{2s-1})\beta}\Big{)}~{},$
$\displaystyle\hat{\delta}_{\epsilon}\Omega_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2s-1}\Big{(}2(2s-1)\epsilon^{\beta}\nabla_{\beta(\alpha_{1}}L_{\alpha_{2}\ldots\alpha_{2s-1})}+{\rm
i}\epsilon_{(\alpha_{1}}\nabla^{2}L_{\alpha_{2}\ldots\alpha_{2s-1})}~{}$
$\displaystyle+$ $\displaystyle
4(4s^{2}-3s+1)|\mu|\epsilon_{(\alpha_{1}}L_{\alpha_{2}\ldots\alpha_{2s-1})}+(2s-1)\epsilon_{(\alpha_{1}}\nabla^{\beta}\Phi_{\alpha_{2}\ldots\alpha_{2s-1})\beta}~{}$
$\displaystyle-$ $\displaystyle
4(s-1)\epsilon_{(\alpha_{1}}\nabla_{\alpha_{2}}{}^{\beta}L_{\alpha_{3}\ldots\alpha_{2s-1})\beta}\Big{)}+\frac{s}{2s+1}\epsilon^{\beta}\Big{(}2(2s-1)|\mu|\nabla^{\gamma}H_{\beta\gamma\alpha(2s-1)}~{}$
$\displaystyle-$
$\displaystyle\nabla_{(\alpha_{1}}{}^{\gamma}\nabla^{\delta}H_{\alpha_{2}\ldots\alpha_{2s-1})\beta\gamma\delta}+\nabla^{\gamma\delta}\nabla_{\gamma}H_{\beta\delta\alpha(2s-1)}\Big{)}~{}.$

###  6.3 Massless half-integer superspin: Longitudinal formulation

In this section, we describe the superspace reduction of the longitudinal
formulation for the massless multiplet of half-integer superspin, following
the prescription advocated in section 6.2.

####  6.3.1 Longitudinal formulation

For $s\geq 2$, the longitudinal formulation for the massless
superspin-$(s+\frac{1}{2})$ multiplet is described in terms of the variables

$\displaystyle{\cal
V}^{\|}_{(s+\frac{1}{2})}=\big{\\{}\mathfrak{H}_{\alpha(2s)},G_{\alpha(2s-2)},\bar{G}_{\alpha(2s-2)}\big{\\}}~{},$
(6.92)

Here, the real superfield $\mathfrak{H}_{\alpha(2s)}$ is identical to that of
(6.61), and the complex superfield ${G}_{\alpha(2s-2)}$ is longitudinal linear
(6.7a). The superfields (6.92) are defined modulo gauge transformations of the
form

$\displaystyle\delta_{\lambda}\mathfrak{H}_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{(\alpha_{1}}\lambda_{\alpha_{2}\dots\alpha_{2s})}-{{\mbox{\boldmath${\cal
D}$}}}_{(\alpha_{1}}\bar{\lambda}_{\alpha_{2}\dots\alpha_{2s})}\equiv
g_{\alpha(2s)}+\bar{g}_{\alpha(2s)}~{},$ (6.93a)
$\displaystyle\delta_{\lambda}{G}_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\big{(}{{\mbox{\boldmath${\bar{\cal
D}}$}}}{}^{2}-4s\mu\big{)}{\mbox{\boldmath${\cal
D}$}}^{\beta}\lambda_{\beta\alpha(2s-2)}+{\rm
i}(s-1){{\mbox{\boldmath${\bar{\cal
D}}$}}}_{(\alpha_{1}}{\mbox{\boldmath${\cal
D}$}}^{|\beta\gamma|}\lambda_{\alpha_{2}\dots\alpha_{2s-2})\beta\gamma}~{}$
(6.93b) $\displaystyle\equiv$
$\displaystyle\Big{(}\frac{s}{2s+1}{\mbox{\boldmath${\cal
D}$}}^{\beta}{\mbox{\boldmath${\bar{\cal D}}$}}{}^{\gamma}+{\rm
i}s{\mbox{\boldmath${\cal
D}$}}^{\beta\gamma}\Big{)}g_{\beta\gamma\alpha(2s-2)}~{},$

where $\lambda_{\alpha(2s-1)}$ is complex unconstrained and $g_{\alpha(2s)}$
is longitudinal linear, as in (6.62a).

Modulo an overall normalisation factor, the longitudinal formulation is
uniquely described by the gauge-invariant action [125]

$\displaystyle\mathbb{S}^{\|}_{(s+\frac{1}{2})}[\mathfrak{H}_{\alpha(2s)},G_{\alpha(2s-2)},\bar{G}_{\alpha(2s-2)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3|4}z\,\mbox{\boldmath$E$}~{}\bigg{\\{}\frac{1}{8}\mathfrak{H}^{\alpha(2s)}{\mbox{\boldmath${\cal
D}$}}^{\beta}({{\mbox{\boldmath${\bar{\cal
D}}$}}}{}^{2}-6\mu){\mbox{\boldmath${\cal
D}$}}_{\beta}\mathfrak{H}_{\alpha(2s)}$
$\displaystyle+2s(s-1)\mu\bar{\mu}\mathfrak{H}^{\alpha(2s)}\mathfrak{H}_{\alpha(2s)}-\frac{1}{16}\big{(}[{\mbox{\boldmath${\cal
D}$}}_{\beta},{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{\gamma}]\mathfrak{H}^{\beta\gamma\alpha(2s-2)}\big{)}[{\mbox{\boldmath${\cal
D}$}}^{\delta},{{\mbox{\boldmath${\bar{\cal
D}}$}}}{}^{\lambda}]\mathfrak{H}_{\delta\lambda\alpha(2s-2)}$
$\displaystyle+\frac{s}{2}{\mbox{\boldmath${\cal
D}$}}_{\beta\gamma}\mathfrak{H}^{\beta\gamma\alpha(2s-2)}{\mbox{\boldmath${\cal
D}$}}^{\delta\lambda}\mathfrak{H}_{\delta\lambda\alpha(2s-2)}$
$\displaystyle+\frac{2s-1}{2s}\Big{(}{\rm i}{\mbox{\boldmath${\cal
D}$}}_{\beta\gamma}\mathfrak{H}^{\beta\gamma\alpha(2s-2)}\left(G_{\alpha(2s-2)}-\bar{G}_{\alpha(2s-2)}\right)+\frac{1}{s}\bar{G}^{\alpha(2s-2)}G_{\alpha(2s-2)}\Big{)}$
$\displaystyle-\frac{2s+1}{4s^{2}}\big{(}G^{\alpha(2s-2)}G_{\alpha(2s-2)}+\bar{G}^{\alpha(2s-2)}\bar{G}_{\alpha(2s-2)}\big{)}\bigg{\\}}~{}.$
(6.94)

In the case where $s=1$, the compensator $G$ is covariantly chiral (6.8a). If
we introduce the field redefinition $G=3\sigma$ in action (6.3.1), along with
choosing $s=1$, then the corresponding model coincides with the linearised
action for minimal $(1,1)$ AdS supergravity [242]. In the case $s=1$, the
gauge transformations (6.93) become

$\displaystyle\delta_{\lambda}\mathfrak{H}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{(\alpha}\lambda_{\beta)}-{\mbox{\boldmath${\cal
D}$}}_{(\alpha}\bar{\lambda}_{\beta)}~{},$ (6.95)
$\displaystyle\delta_{\lambda}G$ $\displaystyle=$
$\displaystyle-\frac{1}{4}({{\mbox{\boldmath${\bar{\cal
D}}$}}}{}^{2}-4\bar{\mu}){\mbox{\boldmath${\cal
D}$}}^{\beta}\lambda_{\beta}~{}.$ (6.96)

It is an easy exercise to show that $\delta_{\lambda}G$ is covariantly chiral
(6.8a).

####  6.3.2 Reduction of gauge prepotentials to AdS3|2

The reduction of the superconformal gauge prepotential
$\mathfrak{H}_{\alpha(2s)}$ was addressed in section 6.2. We need only reduce
the compensator $G_{\alpha(2s-2)}$ to AdS3|2. We start by converting the
longitudinal linear constraint of $G_{\alpha(2s-2)}$ to the real basis (6.35)

$\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}G_{\alpha_{2}\ldots\alpha_{2s-1})}={\rm
i}\mbox{\boldmath$\nabla$}^{\underline{1}}{}_{(\alpha_{1}}G_{\alpha_{2}\ldots\alpha_{2s-1})}~{}.$
(6.97)

Taking a Taylor expansion of $G_{\alpha(2s-2)}(\theta^{I})$ about
$\theta^{\underline{2}}$, and using (6.97), we find the independent
$\theta^{\underline{2}}$ -components of $G_{\alpha(2s-2)}$ to be

$G_{\alpha(2s-2)}|~{},\qquad\mbox{\boldmath$\nabla$}^{\underline{2}\beta}G_{\beta\alpha(2s-3)}|~{}.$
(6.98)

Utilising the gauge transformations (6.93b) and the real representation
(6.35), we find

$\displaystyle\delta_{\lambda}G_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{{\rm
i}s}{2s+1}\big{(}2s\mbox{\boldmath$\nabla$}^{\beta\gamma}-\mbox{\boldmath$\nabla$}^{\underline{1}\beta}\mbox{\boldmath$\nabla$}^{\underline{2}\gamma}\big{)}g_{\beta\gamma\alpha(2s-2)}~{},$
(6.99a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}\beta}\delta_{\lambda}G_{\beta\alpha(2s-3)}$
$\displaystyle=$ $\displaystyle\frac{s}{2s+1}\big{(}2{\rm
i}s\mbox{\boldmath$\nabla$}^{\beta\gamma}\mbox{\boldmath$\nabla$}^{\underline{2}\delta}-\mbox{\boldmath$\nabla$}^{\beta\gamma}\mbox{\boldmath$\nabla$}^{\underline{1}\delta}\big{)}g_{\beta\gamma\delta\alpha(2s-3)}~{}.$
(6.99b)

We can use the residual gauge freedom (6.69) to compute the gauge
transformations of the complex ${\cal N}=1$ superfields (6.98)

$\displaystyle\delta_{\zeta}G_{\alpha(2s-2)}|$ $\displaystyle=$
$\displaystyle\frac{s^{2}}{2s+1}\nabla^{\beta\gamma}\zeta_{\beta\gamma\alpha(2s-2)}-\frac{{\rm
i}}{2}\nabla^{\beta}\zeta_{\beta\alpha(2s-2)}~{},$ (6.100a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}\beta}\delta_{\zeta}G_{\beta\alpha(2s-3)}|$
$\displaystyle=$ $\displaystyle{\rm
i}s\nabla^{\beta\gamma}\zeta_{\beta\gamma\alpha(2s-3)}+\frac{{\rm
i}s}{2(2s+1)}\nabla^{\beta\gamma}\nabla^{\delta}\zeta_{\beta\gamma\delta\alpha(2s-3)}~{}.$
(6.100b)

It is useful to separate the complex superfields (6.98) into its real and
imaginary parts

$\displaystyle G_{\alpha(2s-2)}|$ $\displaystyle=$
$\displaystyle-2sL_{\alpha(2s-2)}-{\rm i}sV_{\alpha(2s-2)}~{},$ (6.101a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}\beta}G_{\beta\alpha(2s-3)}|$
$\displaystyle=$ $\displaystyle\frac{1}{2}\big{(}\Phi_{\alpha(2s-3)}+{\rm
i}\Omega_{\alpha(2s-3)}\big{)}~{}.$ (6.101b)

It then becomes apparent from the gauge transformations (6.70) and (6.100)
that we are in fact dealing with two different gauge theories. The first gauge
theory is formulated in terms of the real dynamical superfields

${\cal
V}^{\parallel}_{(s+\frac{1}{2})}=\big{\\{}H_{\alpha(2s+1)},L_{\alpha(2s-2)},\Phi_{\alpha(2s-3)}\big{\\}}~{},$
(6.102)

which are defined modulo gauge transformations of the form

$\displaystyle\delta_{\zeta}H_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle{\rm
i}\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}\ldots\alpha_{2s+1})}~{},$ (6.103a)
$\displaystyle\delta_{\zeta}L_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\frac{s}{2(2s+1)}\nabla^{\beta\gamma}\zeta_{\beta\gamma\alpha(2s-2)}~{},$
(6.103b) $\displaystyle\delta_{\zeta}\Phi_{\alpha(2s-3)}$ $\displaystyle=$
$\displaystyle\frac{{\rm
i}s}{2s+1}\nabla^{\beta\gamma}\nabla^{\delta}\zeta_{\beta\gamma\delta\alpha(2s-3)}~{}.$
(6.103c)

The other gauge model is described in terms of the real superfields

${\cal
V}^{\parallel}_{(s)}=\big{\\{}H_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-3)}\big{\\}}~{},$
(6.104)

which possess the following gauge freedom

$\displaystyle\delta_{\zeta}H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}\ldots\alpha_{2s})}~{},$
(6.105a) $\displaystyle\delta_{\zeta}V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s}\nabla^{\beta}\zeta_{\beta\alpha(2s-2)}~{},$
(6.105b) $\displaystyle\delta_{\zeta}\Omega_{\alpha(2s-3)}$ $\displaystyle=$
$\displaystyle 2s\nabla^{\beta\gamma}\zeta_{\beta\gamma\alpha(2s-3)}~{}.$
(6.105c)

After carrying out the reduction to ${\cal N}=1$ AdS superspace, the action
(6.3.1) decouples into two ${\cal N}=1$ theories, which are formulated in
terms of the superfields (6.102) and (6.104), respectively

$\displaystyle\mathbb{S}^{\parallel}_{(s+\frac{1}{2})}[\mathfrak{H}_{\alpha(2s)},G_{\alpha(2s-2)},\bar{G}_{\alpha(2s-2)}]=$
$\displaystyle
S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)},\Phi_{\alpha(2s-3)}]~{}$
(6.106)
$\displaystyle+S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-3)}]~{}.$

In the following subsection, we provide the explicit forms of these decoupled
${\cal N}=1$ actions.

####  6.3.3 Massless higher-spin ${\cal N}=1$ supermultiplets

The first ${\cal N}=1$ gauge theory, which is realised in terms of the
superfield variables (6.102), is described by the action

$\displaystyle
S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)},\Phi_{\alpha(2s-3)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\Big{(}-\frac{{\rm
i}}{8}\Big{)}\int{\rm
d}^{3|2}z\,E~{}\bigg{\\{}2H^{\alpha(2s+1)}\mathbb{Q}H_{\alpha(2s+1)}~{}$
$\displaystyle+(2s+1){\rm
i}|\mu|H^{\alpha(2s+1)}\nabla^{2}H_{\alpha(2s+1)}-4(2s+1)|\mu|H^{\beta\alpha(2s)}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s)}-{\rm
i}H^{\beta\alpha(2s)}\nabla^{2}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s)}~{}$
$\displaystyle-8(s-2)(2s+1)|\mu|^{2}H^{\alpha(2s+1)}H_{\alpha(2s+1)}-2s\nabla_{\beta\gamma}H^{\beta\gamma\alpha(2s-1)}\nabla^{\delta\lambda}H_{\delta\lambda\alpha(2s-1)}~{}$
$\displaystyle-\frac{4}{2s-1}\Big{(}16{\rm
i}|\mu|(2s^{3}-2s+1)L^{\alpha(2s-2)}L_{\alpha(2s-2)}+4(2s^{2}-1)L^{\alpha(2s-2)}\nabla^{2}L_{\alpha(2s-2)}~{}$
$\displaystyle+16{\rm
i}(s-1)^{2}L^{\beta\alpha(2s-3)}\nabla_{\beta}{}^{\gamma}L_{\gamma\alpha(2s-3)}+2{\rm
i}(2s-1)^{2}\nabla_{\beta\gamma}H^{\beta\gamma\delta\alpha(2s-2)}\nabla_{\delta}L_{\alpha(2s-2)}~{}$
$\displaystyle+\frac{4{\rm
i}}{s}(s-1)L^{\beta\alpha(2s-3)}\nabla_{\beta}\Phi_{\alpha(2s-3)}+\frac{1}{s}(s-1)\Phi^{\alpha(2s-3)}\Phi_{\alpha(2s-3)}\Big{)}\bigg{\\}}~{}.$
(6.107)

The action (6.3.3) is invariant under the gauge transformations (6.103). We
can eliminate the auxiliary field $\Phi_{\alpha(2s-3)}$ from (6.3.3) by its
equation of motion

$\Phi_{\alpha(2s-3)}=-2{\rm i}\nabla^{\beta}L_{\beta\alpha(2s-3)}~{}.$ (6.108)

The resulting action, up to an overall factor, coincides with the massless
superspin-$(s+\frac{1}{2})$ multiplet (5.2.6.1).

The other ${\cal N}=1$ model, which is constructed in terms of the superfields
(6.104), is described by the action

$\displaystyle
S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-3)}]=-\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3|2}z~{}E~{}\bigg{\\{}(s-1)|\mu|H^{\alpha(2s)}H_{\alpha(2s)}~{}$
$\displaystyle-\frac{{\rm
i}}{4}H^{\alpha(2s)}\nabla^{2}H_{\alpha(2s)}+\frac{1}{2}H^{\beta\alpha(2s-1)}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s-1)}-(2s-1)H^{\alpha(2s)}\nabla_{(\alpha_{1}\alpha_{2}}V_{\alpha_{3}\ldots\alpha_{2s})}~{}$
$\displaystyle-\frac{1}{4(2s-1)}\Big{(}2{\rm
i}sV^{\alpha(2s-2)}\nabla^{2}V_{\alpha(2s-2)}-4(s-1)(4s-1)V^{\beta\alpha(2s-3)}\nabla_{\beta}{}^{\gamma}V_{\gamma\alpha(2s-3)}~{}$
$\displaystyle+8s(4s^{2}-3s+1)|\mu|V^{\alpha(2s-2)}V_{\alpha(2s-2)}+8(s-1)V^{\beta\alpha(2s-3)}\nabla_{\beta}\Omega_{\alpha(2s-3)}~{}$
$\displaystyle-\frac{{\rm
i}}{s^{2}}(s-1)\Omega^{\alpha(2s-3)}\Omega_{\alpha(2s-3)}\Big{)}\bigg{\\}}~{},$
(6.109)

which is invariant under the gauge transformations (6.105). It is evident that
the superfield $\Omega_{\alpha(2s-3)}$ is auxiliary, so upon elimination via
its equation of motion

$\Omega_{\alpha(2s-3)}=-4{\rm i}s^{2}\nabla^{\beta}V_{\beta\alpha(2s-3)}~{},$
(6.110)

the action (6.3.3) reduces to the massless superspin-$s$ multiplet (5.2.6.3).

####  6.3.4 Second supersymmetry transformations

Let us note that the reduced actions (6.106)

$\displaystyle\mathbb{S}^{\parallel}_{(s+\frac{1}{2})}[\mathfrak{H}_{\alpha(2s)},G_{\alpha(2s-2)},\bar{G}_{\alpha(2s-2)}]=$
$\displaystyle
S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)},\Phi_{\alpha(2s-3)}]~{}$
(6.111)
$\displaystyle+S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-3)}]~{},$

are invariant under the second supersymmetry transformations (6.54b). More
explicitly, the second supersymmetry transformation acts on the dynamical
superfields (6.102) and (6.104) by the following rule

$\displaystyle\delta_{\epsilon}\mathfrak{H}_{\alpha(2s)}|$ $\displaystyle=$
$\displaystyle-{\rm i}\epsilon^{\beta}H_{\beta\alpha(2s)}~{},$ (6.112a)
$\displaystyle\delta_{\epsilon}\mbox{\boldmath$\nabla$}^{\underline{2}\beta}\mathfrak{H}_{\beta\alpha(2s-1)}|$
$\displaystyle=$ $\displaystyle 2{\rm
i}\epsilon^{\beta}H_{\beta\alpha(2s-1)}~{},$ (6.112b)
$\displaystyle\delta_{\epsilon}H_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle-2\epsilon_{(\alpha_{1}}H_{\alpha_{2}\ldots\alpha_{2s+1})}~{},$
(6.112c) $\displaystyle\delta_{\epsilon}H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\frac{{\rm
i}}{2}\big{(}\epsilon^{\beta}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s)}-(2s+1)|\mu|\epsilon^{\beta}H_{\beta\alpha(2s)}\big{)}~{},$
(6.112d) $\displaystyle\delta_{\epsilon}L_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\epsilon^{\beta}\nabla_{(\beta}V_{\alpha_{1}\ldots\alpha_{2s-2})}+\frac{{\rm
i}(s-1)}{2s(2s-1)}\epsilon_{(\alpha_{1}}\Omega_{\alpha_{2}\ldots\alpha_{2s-2})}~{},$
(6.112e) $\displaystyle\delta_{\epsilon}V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle
2\epsilon^{\beta}\nabla_{(\beta}L_{\alpha_{1}\ldots\alpha_{2s-2})}-\frac{{\rm
i}(s-1)}{s(2s-1)}\epsilon_{(\alpha_{1}}\Phi_{\alpha_{2}\ldots\alpha_{2s-2})}~{},$
(6.112f) $\displaystyle\delta_{\epsilon}\Phi_{\alpha(2s-3)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s-1}\epsilon^{\beta}\Big{(}2s(2s-3)\nabla_{(\alpha_{1}}{}^{\gamma}V_{\alpha_{2}\ldots\alpha_{2s-3})\beta\gamma}-{\rm
i}s\nabla^{2}V_{\beta\alpha(2s-3)}~{}$ (6.112g) $\displaystyle+$
$\displaystyle
4s^{2}\nabla_{\beta}{}^{\gamma}V_{\gamma\alpha(2s-3)}-4s(4s^{2}-5s+2)|\mu|V_{\beta\alpha(2s-3)}~{}$
$\displaystyle-$ $\displaystyle
2(s-1)\nabla_{(\beta}\Omega_{\alpha_{1}\ldots\alpha_{2s-3})}\Big{)}~{},$
$\displaystyle\delta_{\epsilon}\Omega_{\alpha(2s-3)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s-1}\epsilon^{\beta}\Big{(}2{\rm
i}s\nabla^{2}L_{\beta\alpha(2s-3)}-4s(2s-3)\nabla_{(\alpha_{1}}{}^{\gamma}L_{\alpha_{2}\ldots\alpha_{2s-3})\beta\gamma}~{}$
(6.112h) $\displaystyle-$ $\displaystyle
8s^{2}\nabla_{\beta}{}^{\gamma}L_{\gamma\alpha(2s-3)}+8s(4s^{2}-5s+2)|\mu|L_{\beta\alpha(2s-3)}~{}$
$\displaystyle+$ $\displaystyle
2(s-1)\nabla_{(\beta}\Phi_{\alpha_{1}\ldots\alpha_{2s-3})}\Big{)}~{}.$

It is evident from the variations (6.112a) and (6.112b) that the second
supersymmetry transformations are not compatible with the WZ gauge conditions
(6.87).

Using a similar approach as in subsection 6.2.4, the WZ gauge conditions
(6.87) in the longitudinal formulation can be restored, provided we accompany
the second supersymmetry transformations with the following
$\epsilon$-dependent gauge transformations:

$\displaystyle\delta_{g(\epsilon)}\mathfrak{H}_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle g_{\alpha(2s)}(\epsilon)+\bar{g}_{\alpha(2s)}(\epsilon)~{},$
(6.113a) $\displaystyle\delta_{g(\epsilon)}G_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\Big{(}\frac{s}{2s+1}{\mbox{\boldmath${\cal
D}$}}^{\beta}{\mbox{\boldmath${\bar{\cal D}}$}}^{\gamma}+{\rm
i}s{\mbox{\boldmath${\cal
D}$}}^{\beta\gamma}\Big{)}g_{\beta\gamma\alpha(2s-2)}(\epsilon)~{},$ (6.113b)

where

$\displaystyle g_{\alpha(2s)}(\epsilon)|$ $\displaystyle=$
$\displaystyle\frac{{\rm i}}{2}\epsilon^{\beta}H_{\beta\alpha(2s)}~{},$
(6.114a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}\beta}g_{\beta\alpha(2s-1)}(\epsilon)|$
$\displaystyle=$ $\displaystyle-{\rm
i}\epsilon^{\beta}H_{\beta\alpha(2s-1)}~{}.$ (6.114b)

The modified second supersymmetry transformations now read

$\displaystyle\hat{\delta}_{\epsilon}\mathfrak{H}_{\alpha(2s)}$
$\displaystyle=$
$\displaystyle\delta_{\epsilon}\mathfrak{H}_{\alpha(2s)}+\delta_{g(\epsilon)}\mathfrak{H}_{\alpha(2s)}~{},$
(6.115a) $\displaystyle\hat{\delta}_{\epsilon}G_{\alpha(2s-2)}$
$\displaystyle=$
$\displaystyle\delta_{\epsilon}G_{\alpha(2s-2)}+\delta_{g(\epsilon)}G_{\alpha(2s-2)}~{}.$
(6.115b)

The transformations (6.115) act on the ${\cal N}=1$ superfields (6.102) and
(6.104) as follows

$\displaystyle\hat{\delta}_{\epsilon}H_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle-2\epsilon_{(\alpha_{1}}H_{\alpha_{2}\ldots\alpha_{2s+1})}~{},$
(6.116a) $\displaystyle\hat{\delta}_{\epsilon}H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle-\frac{{\rm i}}{4(2s+1)}\Big{(}{\rm
i}\epsilon^{\beta}\nabla^{2}H_{\beta\alpha(2s)}+4s\epsilon_{(\alpha_{1}}\nabla^{\beta\gamma}H_{\alpha_{2}\ldots\alpha_{2s})\beta\gamma}$
$\displaystyle-$ $\displaystyle
2\epsilon^{\beta}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s)}+2(2s+1)(4s+1)|\mu|\epsilon^{\beta}H_{\beta\alpha(2s)}\Big{)}~{},$
$\displaystyle\hat{\delta}_{\epsilon}L_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\epsilon^{\beta}\nabla_{(\beta}V_{\alpha_{1}\ldots\alpha_{2s-2})}+\frac{{\rm
i}(s-1)}{2s(2s-1)}\epsilon_{(\alpha_{1}}\Omega_{\alpha_{2}\ldots\alpha_{2s-2})}$
$\displaystyle-$
$\displaystyle\frac{1}{2(2s+1)}\epsilon^{\beta}\nabla^{\gamma}H_{\beta\gamma\alpha(2s-2)}~{},$
$\displaystyle\hat{\delta}_{\epsilon}V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle
2\epsilon^{\beta}\nabla_{(\beta}L_{\alpha_{1}\ldots\alpha_{2s-2})}-\frac{{\rm
i}(s-1)}{s(2s-1)}\epsilon_{(\alpha_{1}}\Phi_{\alpha_{2}\ldots\alpha_{2s-2})}~{}$
$\displaystyle-$ $\displaystyle\frac{{\rm
i}s}{2s+1}\epsilon^{\beta}\nabla^{\gamma\delta}H_{\beta\gamma\delta\alpha(2s-2)}~{},$
$\displaystyle\hat{\delta}_{\epsilon}\Phi_{\alpha(2s-3)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s-1}\epsilon^{\beta}\Big{(}2s(2s-3)\nabla_{(\alpha_{1}}{}^{\gamma}V_{\alpha_{2}\ldots\alpha_{2s-3})\beta\gamma}-{\rm
i}s\nabla^{2}V_{\beta\alpha(2s-3)}~{}$ $\displaystyle+$ $\displaystyle
4s^{2}\nabla_{\beta}{}^{\gamma}V_{\gamma\alpha(2s-3)}-4s(4s^{2}-5s+2)|\mu|V_{\beta\alpha(2s-3)}~{}$
$\displaystyle-$ $\displaystyle
2(s-1)\nabla_{(\beta}\Omega_{\alpha_{1}\ldots\alpha_{2s-3})}\Big{)}+\frac{4s^{2}}{2s+1}\epsilon^{\beta}\nabla^{\gamma\delta}H_{\beta\gamma\delta\alpha(2s-3)}~{},$
$\displaystyle\hat{\delta}_{\epsilon}\Omega_{\alpha(2s-3)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s-1}\epsilon^{\beta}\Big{(}2{\rm
i}s\nabla^{2}L_{\beta\alpha(2s-3)}-4s(2s-3)\nabla_{(\alpha_{1}}{}^{\gamma}L_{\alpha_{2}\ldots\alpha_{2s-3})\beta\gamma}~{}$
$\displaystyle-$ $\displaystyle
8s^{2}\nabla_{\beta}{}^{\gamma}L_{\gamma\alpha(2s-3)}+8s(4s^{2}-5s+2)|\mu|L_{\beta\alpha(2s-3)}~{}$
$\displaystyle+$ $\displaystyle
2(s-1)\nabla_{(\beta}\Phi_{\alpha_{1}\ldots\alpha_{2s-3})}\Big{)}+\frac{s}{2s+1}\epsilon^{\beta}\nabla^{\gamma\delta}\nabla^{\lambda}H_{\beta\gamma\delta\lambda\alpha(2s-3)}~{}.$

###  6.4 Massless integer superspin: Longitudinal formulation

In [125], two dually equivalent off-shell formulations, called transverse and
longitudinal, for the massless multiplets of integer superspin were developed
in $(1,1)$ AdS superspace. In the following section, we reduce the
longitudinal model to AdS3|2.

####  6.4.1 Longitudinal formulation

Given an integer $s\geq 1$, the longitudinal formulation for the massless
superspin-$s$ multiplet is realised in terms of the following dynamical
variables

$\displaystyle{\cal
V}^{\|}_{(s)}=\big{\\{}U_{\alpha(2s-2)},G_{\alpha(2s)},\bar{G}_{\alpha(2s)}\big{\\}}~{}.$
(6.117)

Here, the real superfield $U_{\alpha(2s-2)}$ is unconstrained, while the
complex superfield $G_{\alpha(2s)}$ is longitudinal linear (6.7a). The
superfields $U_{\alpha(2s-2)}$ and $G_{\alpha(2s)}$ are defined modulo gauge
transformations of the form

$\displaystyle\delta_{L}U_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle{\mbox{\boldmath${\cal
D}$}}^{\beta}L_{\beta\alpha(2s-2)}-{{\mbox{\boldmath${\bar{\cal
D}}$}}}{}^{\beta}\bar{L}_{\beta\alpha(2s-2)}\equiv\bar{\gamma}_{\alpha(2s-2)}+{\gamma}_{\alpha(2s-2)}~{},$
(6.118a) $\displaystyle\delta_{L}G_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{(\alpha_{1}}\big{(}{\mbox{\boldmath${\cal
D}$}}^{2}-2(2s+1)\bar{\mu}\big{)}L_{\alpha_{2}\dots\alpha_{2s})}\equiv{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{(\alpha_{1}}{\mbox{\boldmath${\cal
D}$}}_{\alpha_{2}}\bar{\gamma}_{\alpha_{3}\dots\alpha_{2s})}~{}.$ (6.118b)

Here the gauge parameter $L_{\alpha(2s-1)}$ is complex and unconstrained,
while ${\gamma}^{\alpha(2s-2)}:={{\mbox{\boldmath${\bar{\cal
D}}$}}}_{\beta}\bar{L}^{\beta\alpha(2s-2)}$ is transverse linear (6.5a).

Modulo an overall normalisation factor, the longitudinal formulation for the
massless superspin-$s$ multiplet is described by the action

$\displaystyle\mathbb{S}_{(s)}^{\|}[U_{\alpha(2s-2)},G_{\alpha(2s)},\bar{G}_{\alpha(2s)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3|4}z\,\mbox{\boldmath$E$}\,\bigg{\\{}\frac{1}{8}U^{\alpha(2s-2)}{\mbox{\boldmath${\cal
D}$}}^{\gamma}({{\mbox{\boldmath${\bar{\cal
D}}$}}}{}^{2}-6\mu){\mbox{\boldmath${\cal D}$}}_{\gamma}U_{\alpha(2s-2)}$
$\displaystyle+\frac{s}{2s+1}U^{\alpha(2s-2)}\Big{(}{\mbox{\boldmath${\cal
D}$}}^{\beta}{{\mbox{\boldmath${\bar{\cal
D}}$}}}{}^{\gamma}G_{\beta\gamma\alpha(2s-2)}-{{\mbox{\boldmath${\bar{\cal
D}}$}}}{}^{\beta}{{\mbox{\boldmath${\cal
D}$}}}^{\gamma}\bar{G}_{\beta\gamma\alpha(2s-2)}\Big{)}+\frac{s}{2s-1}\bar{G}^{\alpha(2s)}G_{\alpha(2s)}$
$\displaystyle+\frac{s}{2(2s+1)}\Big{(}G^{\alpha(2s)}G_{\alpha(2s)}+\bar{G}^{\alpha(2s)}\bar{G}_{\alpha(2s)}\Big{)}+2s(s+1)\mu\bar{\mu}U^{\alpha(2s-2)}U_{\alpha(2s-2)}\bigg{\\}}~{},$
(6.119)

which is invariant under the gauge transformations (6.118).

####  6.4.2 Reduction of gauge prepotentials to AdS3|2

We wish to reduce the gauge superfields (6.117) to AdS3|2. We start by
reducing the superfield $U_{\alpha(2s-2)}$. Converting the transverse linear
constraint (6.5a) of $\gamma_{\alpha(2s-2)}$ to the real basis (6.35) gives

$\mbox{\boldmath$\nabla$}^{\underline{2}\beta}\gamma_{\beta\alpha(2s-3)}={\rm
i}\mbox{\boldmath$\nabla$}^{\underline{1}\beta}\gamma_{\beta\alpha(2s-3)}~{}.$
(6.120)

Taking a Taylor expansion of $\gamma_{\alpha(2s-2)}(\theta^{I})$ about
$\theta^{\underline{2}}$, then using (6.120), we find the independent
$\theta^{\underline{2}}$ -components of $\gamma_{\alpha(2s-2)}$ to be

$\gamma_{\alpha(2s-2)}|,\qquad\qquad\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\gamma_{\alpha_{2}\ldots\alpha_{2s-1})}|~{}.$
(6.121)

The gauge transformation (6.118a) allows us to impose the gauge condition

$U_{\alpha(2s-2)}|=0~{}.$ (6.122)

It must be noted that the gauge condition (6.122) is less restrictive than
those proposed in appendix B of [125], where the analogous reduction procedure
was carried out in $\mathbb{M}^{3|2}$. This was done in order to ensure that
one of the decoupled ${\cal N}=1$ actions coincides with (5.2.6.4). It can be
shown that if one chooses to impose the same gauge conditions as detailed in
[125], i.e

$U_{\alpha(2s-2)}|=0~{},\qquad\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}U_{\alpha_{2}\ldots\alpha_{2s-1})}|=0~{},$
(6.123)

then one will obtain the ${\cal N}=1$ action (5.2.6.4) after reduction, rather
than (5.2.6.4).

The residual gauge symmetry which preserves the gauge condition (6.122) is
described by the real unconstrained ${\cal N}=1$ superfield

$\gamma_{\alpha(2s-2)}|=-\frac{{\rm
i}}{2}\zeta_{\alpha(2s-2)}~{},\qquad\bar{\zeta}_{\alpha(2s-2)}=\zeta_{\alpha(2s-2)}~{}.$
(6.124)

We now wish to reduce the superfield $\bar{G}_{\alpha(2s)}$ to AdS3|2.
Converting the longitudinal linear constraint of $\bar{G}_{\alpha(2s)}$ (6.7a)
to the real basis yields

$\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\bar{G}_{\alpha_{2}\ldots\alpha_{2s+1})}=-{\rm
i}\mbox{\boldmath$\nabla$}^{\underline{1}}{}_{(\alpha_{1}}\bar{G}_{\alpha_{1}\ldots\alpha_{2s+1})}~{}.$
(6.125)

Performing the superspace reduction to AdS3|2, it follows from constraint
(6.125) that $\bar{G}_{\alpha(2s)}$ has two independent
$\theta^{\underline{2}}$ -components:

$\bar{G}_{\alpha(2s)}|~{},\qquad\mbox{\boldmath$\nabla$}^{\underline{2}\beta}\bar{G}_{\beta\alpha(2s-1)}|~{}.$
(6.126)

Making use of the gauge transformations (6.118b) and the real representation
(6.35), we find

$\displaystyle\delta_{L}\bar{G}_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle{\rm
i}\big{(}\mbox{\boldmath$\nabla$}^{\underline{1}}{}_{(\alpha_{1}}\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{\alpha_{2}}+\mbox{\boldmath$\nabla$}_{(\alpha_{1}\alpha_{2}}\big{)}\gamma_{\alpha_{3}\ldots{\alpha_{2s})}}~{},$
(6.127a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}\beta}\delta_{L}\bar{G}_{\beta\alpha(2s-1)}$
$\displaystyle=$ $\displaystyle\frac{{\rm
i}}{2s}\big{(}2s\mbox{\boldmath$\nabla$}_{(\alpha_{1}}{}^{\beta}\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{\alpha_{2}}\gamma_{\alpha_{3}\ldots\alpha_{2s-1})\beta}+(4s^{2}-1)|\mu|\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\gamma_{\alpha_{2}\ldots\alpha_{2s-1})}~{}$
$\displaystyle+$ $\displaystyle 2{\rm
i}(s+1)\mbox{\boldmath$\nabla$}_{\beta(\alpha_{1}}\mbox{\boldmath$\nabla$}^{\underline{1}\beta}\gamma_{\alpha_{2}\ldots\alpha_{2s-1})}+6{\rm
i}s(2s+1)|\mu|\mbox{\boldmath$\nabla$}^{\underline{1}}{}_{(\alpha_{1}}\gamma_{\alpha_{2}\ldots\alpha_{2s-1})}~{}$
$\displaystyle+$ $\displaystyle\frac{{\rm
i}}{2}(2s+1)(\mbox{\boldmath$\nabla$}^{\underline{1}})^{2}\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\gamma_{\alpha_{2}\ldots\alpha_{2s-1})}-2{\rm
i}(s-1)\mbox{\boldmath$\nabla$}_{(\alpha_{1}}{}^{\beta}\mbox{\boldmath$\nabla$}^{\underline{1}}{}_{\alpha_{2}}\gamma_{\alpha_{3}\ldots\alpha_{2s-1})\beta}\big{)}~{}.$

Using the residual gauge freedom (6.124), we can directly compute the gauge
transformations of the ${\cal N}=1$ superfields (6.126) from (6.127)

$\displaystyle\delta_{L}\bar{G}_{\alpha(2s)}|$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}\nabla_{(\alpha_{1}\alpha_{2}}\zeta_{\alpha_{3}\ldots\alpha_{2s})}+2{\rm
i}\nabla_{(\alpha_{1}}\big{(}\tau_{\alpha_{2}\ldots\alpha_{2s})}+{\rm
i}\tilde{\tau}_{\alpha_{2}\dots\alpha_{2s})}\big{)}\Big{)}~{},$ (6.128a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}\beta}\delta_{L}\bar{G}_{\beta\alpha(2s-1)}|$
$\displaystyle=$ $\displaystyle\frac{{\rm
i}}{2s}\Big{(}2s(2s+1)|\mu|\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}\ldots\alpha_{2s-1})}+(s+1)\nabla_{\beta(\alpha_{1}}\nabla^{\beta}\zeta_{\alpha_{2}\ldots\alpha_{2s-1})}~{}$
$\displaystyle+$ $\displaystyle(4s^{2}-1)|\mu|\big{(}\tau_{\alpha(2s-1)}+{\rm
i}\tilde{\tau}_{\alpha(2s-1)}\big{)}+\frac{{\rm
i}}{2}(2s+1)\nabla^{2}\big{(}\tau_{\alpha(2s-1)}+{\rm
i}\tilde{\tau}_{\alpha(2s-1)}\big{)}~{}$ $\displaystyle+$
$\displaystyle(2s-1)\nabla_{(\alpha_{1}}{}^{\beta}\big{(}\tau_{\alpha_{2}\ldots\alpha_{2s-1})\beta}+{\rm
i}\tilde{\tau}_{\alpha_{2}\ldots\alpha_{2s-1})\beta}\big{)}-(s-1)\nabla_{(\alpha_{1}}{}^{\beta}\nabla_{\alpha_{2}}\zeta_{\alpha_{3}\ldots\alpha_{2s-1})\beta}\Big{)}~{},$

where we have defined

$\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\gamma_{\alpha_{2}\ldots\alpha_{2s-1})}|=\tau_{\alpha{(2s-1)}}+{\rm
i}\tilde{\tau}_{\alpha{(2s-1)}}~{}.$ (6.129)

It proves useful to decompose the complex ${\cal N}=1$ superfields (6.126)
into real and imaginary parts as follows

$\displaystyle\bar{G}_{\alpha(2s)}|$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\big{(}H_{\alpha(2s)}-2{\rm
i}\tilde{H}_{\alpha(2s)}\big{)}~{},$ (6.130a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}\beta}\bar{G}_{\beta\alpha(2s-1)}|$
$\displaystyle=$ $\displaystyle\Phi_{\alpha(2s-1)}-2{\rm
i}\Omega_{\alpha(2s-1)}~{}.$ (6.130b)

The gauge transformations of these four real ${\cal N}=1$ superfields
immediately follow from gauge transformations (6.128)

$\displaystyle\delta H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\rho_{\alpha_{2}\ldots\alpha_{2s})}~{},$
(6.131a) $\displaystyle\delta\tilde{H}_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\tau_{\alpha_{2}\ldots\alpha_{2s})}~{},$
(6.131b) $\displaystyle\delta\Phi_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{8s}\Big{(}2(2s-1)\nabla_{(\alpha_{1}}{}^{\beta}\rho_{\alpha_{2}\ldots\alpha_{2s-1})\beta}+2(4s^{2}-1)|\mu|\rho_{\alpha(2s-1)}$
$\displaystyle+(2s+1){\rm i}\nabla^{2}\rho_{\alpha(2s-1)}-8{\rm
i}(4s^{2}-1)|\mu|\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}\ldots\alpha_{2s-1})}\Big{)}~{},$
$\displaystyle\delta\Omega_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{8s}\Big{(}2(2s-1)\nabla_{(\alpha_{1}}{}^{\beta}\tau_{\alpha_{2}\ldots\alpha_{2s-1})\beta}+2(4s^{2}-1)|\mu|\tau_{\alpha(2s-1)}~{}$
$\displaystyle+{\rm i}(2s+1)\nabla^{2}\tau_{\alpha(2s-1)}\Big{)}~{},$

where we have introduced the gauge parameter redefinition

$\rho_{\alpha(2s-1)}:=2\tilde{\tau}_{\alpha(2s-1)}+{\rm
i}\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}\ldots\alpha_{2s-1})}~{}.$ (6.132)

We are now left with the remaining unconstrained real ${\cal N}=1$ superfields

$\displaystyle{\cal U}_{\beta;\alpha(2s-2)}$ $\displaystyle:=$
$\displaystyle{\rm
i}\mbox{\boldmath$\nabla$}^{\underline{2}}_{\beta}U_{\alpha(2s-2)}|~{},$
(6.133a) $\displaystyle V_{\alpha(2s-2)}$ $\displaystyle:=$
$\displaystyle-\frac{{\rm
i}}{8s}(\mbox{\boldmath$\nabla$}^{\underline{2}})^{2}U_{\alpha(2s-2)}|~{}.$
(6.133b)

Utilising the gauge transformations (6.118a), in conjunction with the residual
gauge symmetry (6.124) and definition (6.129), we find that the superfields
(6.133) possess the following gauge freedom

$\displaystyle\delta{\cal U}_{\beta;\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\rho_{\beta\alpha(2s-2)}+{\rm
i}\nabla_{\beta}\zeta_{\alpha(2s-2)}~{},$ (6.134a) $\displaystyle\delta
V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s}\nabla^{\beta}\tau_{\beta\alpha(2s-2)}~{}.$ (6.134b)

It then becomes apparent from gauge transformations (6.131) and (6.134) that
we are dealing with two different gauge theories. The first of these gauge
theories is formulated in terms of superfields

${\cal V}^{\perp}_{(s)}=\big{\\{}H_{\alpha(2s)},{\cal
U}_{\beta;\alpha(2s-2)},\Phi_{\alpha(2s-1)}\big{\\}}~{},$ (6.135)

which are defined modulo gauge transformations of the form

$\displaystyle\delta H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\rho_{\alpha_{2}\ldots\alpha_{2s})}~{},$
(6.136a) $\displaystyle\delta{\cal U}_{\beta;\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\rho_{\beta\alpha(2s-2)}+{\rm
i}\nabla_{\beta}\zeta_{\alpha(2s-2)}~{},$ (6.136b)
$\displaystyle\delta\Phi_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{8s}\Big{(}2(2s-1)\nabla_{(\alpha_{1}}{}^{\beta}\rho_{\alpha_{2}\ldots\alpha_{2s-1})\beta}+(2s+1){\rm
i}\nabla^{2}\rho_{\alpha(2s-1)}$
$\displaystyle+2(2s-1)(2s+1)|\mu|\big{(}\rho_{\alpha(2s-1)}-4\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}\ldots\alpha_{2s-1})}\big{)}\Big{)}~{}.$

The other gauge model is described in terms of the superfields

${\cal
V}^{\parallel}_{(s)}=\big{\\{}\tilde{H}_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-1)}\big{\\}}~{},$
(6.137)

which possess the following gauge freedom

$\displaystyle\delta\tilde{H}_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\tau_{\alpha_{2}\ldots\alpha_{2s})}~{},$
(6.138a) $\displaystyle\delta V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s}\nabla^{\beta}\tau_{\beta\alpha(2s-2)}~{},$ (6.138b)
$\displaystyle\delta\Omega_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{8s}\Big{(}2(2s-1)\nabla_{(\alpha_{1}}{}^{\beta}\tau_{\alpha_{2}\ldots\alpha_{2s-1})\beta}+{\rm
i}(2s+1)\nabla^{2}\tau_{\alpha(2s-1)}~{}$
$\displaystyle+2(2s-1)(2s+1)|\mu|\tau_{\alpha(2s-1)}\Big{)}~{}.$

Applying the reduction prescription to the action (6.4.1), we obtain two
decoupled ${\cal N}=1$ supersymmetric actions, which are formulated in terms
of superfields (6.135) and (6.137), respectively

$\displaystyle\mathbb{S}^{\parallel}_{(s)}[U_{\alpha(2s-2)},G_{\alpha(2s)},\bar{G}_{\alpha(2s)}]$
$\displaystyle=$ $\displaystyle S^{\perp}_{(s)}[H_{\alpha(2s)},{\cal
U}_{\beta;\alpha(2s-2)},\Phi_{\alpha(2s-1)}]~{}$ (6.139) $\displaystyle+$
$\displaystyle
S^{\parallel}_{(s)}[\tilde{H}_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-1)}]~{}.$

The explicit form of the ${\cal N}=1$ decoupled actions are provided in the
following subsection.

Let us point out that there also appeared a new off-shell formulation for the
massless integer superspin multiplet in $(1,1)$ AdS superspace in [125]. This
formulation proves to be a generalised version of the longitudinal action
(6.119), as the gauge-invariant action involves not only
$U_{\alpha(2s-2)},\Psi_{\alpha(2s-1)}$ and $\bar{\Psi}_{\alpha(2s-1)}$, but
also new compensating superfields. Furthermore, the prepotential
$\Psi_{\alpha(2s-1)}$ (associated to the longitudinal linear field strength
$G_{\alpha(2s)}$) enjoys a larger gauge symmetry, which is that of the
superconformal complex superspin-$s$ multiplet [125]. Upon reduction to
${{\cal N}}=1$ AdS superspace, we found that this new formulation decoupled
into two ${{\cal N}}=1$ supersymmetric higher-spin models which coincide with
the right-hand side of eq. (6.139).

####  6.4.3 Massless higher-spin ${\cal N}=1$ supermultiplets

The first ${\cal N}=1$ supersymmetric action, which is described in terms of
the superfields (6.135), takes the form

$\displaystyle S^{\perp}_{(s)}[H_{\alpha(2s)},{\cal
U}_{\beta;\alpha(2s-2)},\Phi_{\alpha(2s-1)}]=\bigg{(}-\frac{1}{2}\bigg{)}^{s}\frac{1}{4}\int{\rm
d}^{3|2}z\,E\,\bigg{\\{}-{\rm i}\,{\cal U}^{\beta;\,\alpha(2s-2)}\Box{\cal
U}_{\beta;\,\alpha(2s-2)}$ $\displaystyle-\frac{1}{2}{\cal
U}^{\beta;\,\alpha(2s-2)}\nabla^{2}\nabla_{\beta}{}^{\gamma}{\cal
U}_{\gamma;\,\alpha(2s-2)}+\frac{1}{2}(2s-3)|\mu|{\cal
U}^{\beta;\,\alpha(2s-2)}\nabla^{2}{\cal U}_{\beta;\,\alpha(2s-2)}$
$\displaystyle+2(s-1)|\mu|{\cal
U}_{\beta;}{}^{\beta\alpha(2s-3)}\nabla^{2}{\cal
U}^{\gamma;}{}_{\gamma\alpha(2s-3)}+{\rm i}(4s^{2}-1)|\mu|^{2}{\cal
U}^{\beta;\,\alpha(2s-2)}{\cal U}_{\beta;\,\alpha(2s-2)}$ $\displaystyle-8{\rm
i}(s-1)|\mu|^{2}{\cal U}_{\beta;}{}^{\beta\alpha(2s-3)}{\cal
U}^{\gamma;}{}_{\gamma\alpha(2s-3)}$ $\displaystyle-\frac{2s}{2s+1}{\cal
U}^{\beta;\,\alpha(2s-2)}\Big{(}-2\nabla^{\gamma\delta}\nabla_{\gamma}H_{\beta\delta\alpha(2s-2)}+\nabla^{\gamma\delta}\nabla_{\beta}H_{\gamma\delta\alpha(2s-2)}+\nabla^{2}\Phi_{\beta\alpha(2s-2)}$
$\displaystyle+2{\rm
i}\,\nabla_{\beta}{}^{\gamma}\Phi_{\gamma\alpha(2s-2)}-4s|\mu|\nabla^{\gamma}H_{\gamma\beta\alpha(2s-2)}+2{\rm
i}(2s+1)|\mu|\Phi_{\beta\alpha(2s-2)}\Big{)}$
$\displaystyle-\frac{s}{(2s+1)^{2}(2s-1)}\Big{(}-{\rm
i}(3s+1)H^{\alpha(2s)}\nabla^{2}H_{\alpha(2s)}$
$\displaystyle+2s(4s+1)H^{\beta\alpha(2s-1)}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s-1)}+16s^{2}H^{\alpha(2s)}\nabla_{(\alpha_{1}}\Phi_{\alpha_{2}\dots\alpha_{2s})}$
$\displaystyle-8{\rm
i}s\Phi^{\alpha(2s-1)}\Phi_{\alpha(2s-1)}-4s(4s^{2}+3s+1)|\mu|H^{\alpha(2s)}H_{\alpha(2s)}\Big{)}\bigg{\\}}~{}.$
(6.140)

It is apparent that the superfield $\Phi_{\alpha(2s-1)}$ is auxiliary, thus,
upon elimination via its equation of motion,

$\displaystyle\Phi_{\alpha(2s-1)}$ $\displaystyle=$ $\displaystyle-\frac{{\rm
i}}{8s}\Big{(}(4s^{2}-1)\nabla^{2}{\cal U}_{\alpha(2s-1)}-2{\rm
i}(4s^{2}-1)\nabla_{\beta(\alpha_{1}}{\cal
U}^{\beta;}{}_{\alpha_{2}\ldots\alpha_{2s-1})}~{}$ $\displaystyle+2{\rm
i}(2s+1)(4s^{2}-1)|\mu|{\cal
U}_{\alpha(2s-1)}+8s^{2}\nabla^{\beta}H_{\beta\alpha(2s-1)}\Big{)}~{},$

we arrive at the resulting action

$\displaystyle S^{\perp}_{(s)}[H_{\alpha(2s)},{\cal
U}_{\beta;\alpha(2s-2)}]=\bigg{(}-\frac{1}{2}\bigg{)}^{s}\frac{1}{4}\int{\rm
d}^{3|2}z\,E\,\bigg{\\{}-\frac{{\rm
i}s(s-1)}{2s-1}H^{\alpha(2s)}\nabla^{2}H_{\alpha(2s)}$
$\displaystyle-\frac{2s^{2}}{2s-1}H^{\beta\alpha(2s-1)}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s-1)}-\frac{4s^{2}(s-1)}{2s-1}|\mu|H^{\alpha(2s)}H_{\alpha(2s)}$
$\displaystyle-2s\,{\cal
U}^{\delta\alpha(2s-2)}\nabla^{\beta\gamma}\Big{(}\nabla_{(\delta}H_{\alpha_{1}\dots\alpha_{2s-2})\beta\gamma}-2\nabla_{\beta}H_{\gamma\delta\alpha(2s-2)}\Big{)}$
$\displaystyle+\frac{4s(s-1)}{2s-1}{\cal
U}_{\lambda;}{}^{\lambda\alpha(2s-3)}\nabla^{\beta\gamma}\nabla^{\delta}H_{\beta\gamma\delta\alpha(2s-3)}-8s^{2}|\mu|\,{\cal
U}^{\beta;\,\alpha(2s-2)}\nabla^{\gamma}H_{\beta\gamma\alpha(2s-2)}$
$\displaystyle+{\cal U}^{\alpha(2s-1)}\bigg{(}-2{\rm i}s\Box{\cal
U}_{\alpha(2s-1)}-s\nabla^{2}\nabla^{\beta}{}_{(\alpha_{1}}{\cal
U}_{\alpha_{2}\dots\alpha_{2s-1})\beta}$ $\displaystyle-{\rm
i}(s-1)\nabla_{(\alpha_{1}\alpha_{2}}\nabla^{\beta\gamma}{\cal
U}_{\alpha_{3}\dots\alpha_{2s-1})\beta\gamma}+2{\rm
i}\frac{(s-1)(2s-3)}{2s-1}\nabla_{(\alpha_{1}\alpha_{2}}\nabla_{\alpha_{3}}{}^{\beta}{\cal
U}^{\gamma;}{}_{\alpha_{4}\dots\alpha_{2s-1})\beta\gamma}$
$\displaystyle+\frac{(s-1)(2s+1)}{2s-1}\nabla^{2}\nabla_{(\alpha_{1}\alpha_{2}}{\cal
U}^{\beta;}{}_{\alpha_{3}\dots\alpha_{2s-1})\beta}-s(2s+1)|\mu|\nabla^{2}{\cal
U}_{\alpha(2s-1)}$ $\displaystyle-4{\rm
i}\,s(2s-1)|\mu|\nabla^{\beta}{}_{(\alpha_{1}}{\cal
U}_{\alpha_{2}\dots\alpha_{2s-1})\beta}+2{\rm
i}(s-1)(2s+1)|\mu|\nabla_{(\alpha_{1}\alpha_{2}}{\cal
U}^{\beta;}{}_{\alpha_{3}\dots\alpha_{2s-1})\beta}$ $\displaystyle-2{\rm
i}s(2s-1)(2s+1)|\mu|^{2}{\cal U}_{\alpha(2s-1)}\bigg{)}$ $\displaystyle+{\cal
U}_{\lambda;}{}^{\lambda\alpha(2s-3)}\bigg{(}2{\rm
i}\frac{(s-1)(6s-5)}{(2s-1)^{2}}\Box{\cal U}^{\beta;}{}_{\beta\alpha(2s-3)}$
$\displaystyle-\frac{(s-1)(2s-3)}{(2s-1)^{2}}\Big{(}2{\rm
i}(s-2)\nabla^{\beta\gamma}\nabla_{(\alpha_{1}\alpha_{2}}{\cal
U}^{\delta;}{}_{\alpha_{3}\dots\alpha_{2s-3})\beta\gamma\delta}+\nabla^{2}\nabla^{\beta}{}_{(\alpha_{1}}{\cal
U}^{\gamma;}{}_{\alpha_{2}\dots\alpha_{2s-3})\beta\gamma}\Big{)}$
$\displaystyle+\frac{s-1}{2s-1}|\mu|\Big{(}(2s+1)\nabla^{2}{\cal
U}^{\beta;}{}_{\beta\alpha(2s-3)}-2{\rm i}(4s^{2}+16s-17)|\mu|\,{\cal
U}^{\beta;}{}_{\beta\alpha(2s-3)}\Big{)}\bigg{)}\bigg{\\}}~{}.$ (6.142)

Note that we have introduced the notation ${\cal U}_{\alpha(2s-1)}={\cal
U}_{(\alpha_{1};\alpha_{2}\ldots\alpha_{2s-1})}$. It is a tedious but
straightforward exercise to verify that the above ${{\cal N}}=1$ action
coincides with the action for the massless superspin-$s$ multiplet (5.2.6.4).
To prove this, one needs to introduce the ${{\cal N}}=1$ field strength ${\cal
W}_{\beta;\,\alpha(2s-2)}$ associated with the prepotential ${\cal
U}_{\beta;\alpha(2s-2)}$,

$\displaystyle{\cal W}_{\beta;\,\alpha(2s-2)}=-{\rm
i}\big{(}\nabla^{\gamma}\nabla_{\beta}-4{\rm
i}|\mu|\delta_{\beta}{}^{\gamma}\big{)}\,{\cal
U}_{\gamma;\,\alpha(2s-2)}~{},\qquad\nabla^{\beta}{\cal
W}_{\beta;\,\alpha(2s-2)}=0~{}.$ (6.143)

The action (6.4.3) can then be written in the compact form

$\displaystyle S^{\perp}_{(s)}[H_{\alpha(2s)},{\cal
U}_{\beta;\alpha(2s-2)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\bigg{(}\frac{2s}{2s-1}\bigg{)}\int{\rm
d}^{3|2}z\,E\,\bigg{\\{}\frac{1}{2}H^{\alpha(2s)}({\rm
i}\nabla^{2}+8s|\mu|)H_{\alpha(2s)}$ $\displaystyle-{\rm
i}s\nabla_{\beta}H^{\beta\alpha(2s-1)}\nabla^{\gamma}H_{\gamma\alpha(2s-1)}-(2s-1){\cal
W}^{\beta;\,\alpha(2s-2)}\nabla^{\gamma}H_{\beta\gamma\alpha(2s-2)}$
$\displaystyle-\frac{{\rm i}}{2}(2s-1)\Big{(}{\cal
W}^{\beta;\,\alpha(2s-2)}{\cal W}_{\beta;\,\alpha(2s-2)}+\frac{s-1}{s}{\cal
W}_{\beta;}\,^{\beta\alpha(2s-3)}{\cal
W}^{\gamma;}\,_{\gamma\alpha(2s-3)}\Big{)}$ $\displaystyle-2{\rm
i}(2s-1)|\mu|\,{\cal U}^{\beta;\,\alpha(2s-2)}{\cal
W}_{\beta;\,\alpha(2s-2)}\bigg{\\}}~{},$ (6.144)

which is exactly the action describing the massless superspin-$s$ multiplet
(5.2.6.4), modulo an overall normalisation factor.

The other decoupled ${\cal N}=1$ action, which is formulated in terms of the
dynamical variables (6.137), assumes the form

$\displaystyle
S^{\parallel}_{(s)}[\tilde{H}_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-1)}]=\bigg{(}-\frac{1}{2}\bigg{)}^{s}\int{\rm
d}^{3|2}z\,E\,\bigg{\\{}2s^{2}V^{\alpha(2s-2)}\big{(}{\rm
i}\nabla^{2}+4|\mu|\big{)}V_{\alpha(2s-2)}~{}$
$\displaystyle-\frac{4s^{2}}{2s+1}V^{\alpha(2s-2)}\big{(}{\nabla}^{\beta\gamma}\tilde{H}_{\beta\gamma\alpha(2s-2)}+2{\nabla}^{\beta}\Omega_{\beta\alpha(2s-2)}\big{)}$
$\displaystyle+\frac{s}{(2s+1)^{2}(2s-1)}\Big{(}(2s+1)^{2}\tilde{H}^{\alpha(2s)}({\rm
i}\nabla^{2}+4|\mu|)\tilde{H}_{\alpha(2s)}-8s\tilde{H}^{\alpha(2s)}\nabla_{(\alpha_{1}}\Omega_{\alpha_{2}\dots\alpha_{2s})}$
$\displaystyle-4{\rm
i}s(s+1)\nabla_{\beta}\tilde{H}^{\beta\alpha(2s-1)}\nabla^{\gamma}\tilde{H}_{\gamma\alpha(2s-1)}+16{\rm
i}s^{2}\Omega^{\alpha(2s-1)}\Omega_{\alpha(2s-1)}\Big{)}\bigg{\\}}~{}.$
(6.145)

It is clear from the action (6.4.3) that the superfield
$\Omega_{\alpha(2s-1)}$ is auxiliary. Integrating it out by using its equation
of motion,

$\Omega_{\alpha(2s-1)}=\frac{{\rm
i}}{4s}\Big{(}(4s^{2}-1)\nabla_{(\alpha_{1}}V_{\alpha_{2}\ldots\alpha_{2s-1})}-\nabla^{\beta}\tilde{H}_{\beta\alpha(2s-1)}\Big{)}~{},$
(6.146)

we find that the resulting action, modulo a normalisation factor, coincides
with the action for the massless superspin-$s$ multiplet (5.2.6.3).

####  6.4.4 Second supersymmetry transformations

Let us compute the second supersymmetry transformations (6.85) for the
superfields (6.135) and (6.137)

$\displaystyle\delta_{\epsilon}U_{\alpha(2s-2)}|$ $\displaystyle=$
$\displaystyle-{\rm i}\epsilon^{\beta}{\cal U}_{\beta;\alpha(2s-2)}~{},$
(6.147a) $\displaystyle\delta_{\epsilon}{\cal U}_{\beta;\alpha(2s-2)}$
$\displaystyle=$ $\displaystyle 4s\epsilon_{\beta}V_{\alpha(2s-2)}~{},$
(6.147b) $\displaystyle\delta_{\epsilon}V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\frac{{\rm
i}}{4s}\Big{(}\epsilon^{\beta}\nabla_{\beta}{}^{\gamma}{\cal
U}_{\gamma;\alpha(2s-2)}-(2s-1)|\mu|\epsilon^{\beta}{\cal
U}_{\beta;\alpha(2s-2)}~{}$ $\displaystyle-$ $\displaystyle
4(s-1)|\mu|\epsilon_{(\alpha_{1}}{\cal
U}^{\beta;}{}_{\alpha_{2}\ldots\alpha_{2s-2})\beta}\Big{)}~{},$
$\displaystyle\delta_{\epsilon}H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle-2\epsilon^{\beta}\nabla_{(\beta}\tilde{H}_{\alpha_{1}\ldots\alpha_{2s})}-\frac{8{\rm
i}s}{2s+1}\epsilon_{(\alpha_{1}}\Omega_{\alpha_{2}\ldots\alpha_{2s})}~{},$
(6.147d) $\displaystyle\delta_{\epsilon}\tilde{H}_{\alpha(2s)}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\epsilon^{\beta}\nabla_{(\beta}H_{\alpha_{1}\ldots\alpha_{2s})}+\frac{2{\rm
i}s}{2s+1}\epsilon_{(\alpha_{1}}\Phi_{\alpha_{2}\ldots\alpha_{2s})}~{},$
(6.147e) $\displaystyle\delta_{\epsilon}\Phi_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2(2s+1)}\epsilon^{\beta}\Big{(}{\rm
i}\nabla^{2}\tilde{H}_{\beta\alpha(2s-1)}-2(4s+3)\nabla_{\beta}{}^{\gamma}\tilde{H}_{\gamma\alpha(2s-1)}~{}$
$\displaystyle-$ $\displaystyle
2(2s-1)\nabla_{(\alpha_{1}}{}^{\gamma}\tilde{H}_{\alpha_{2}\ldots\alpha_{2s-1})\beta\gamma}+2(4s^{2}+3s+1)|\mu|\tilde{H}_{\beta\alpha(2s-1)}~{}$
$\displaystyle+$ $\displaystyle
8s\nabla_{(\beta}\Omega_{\alpha_{1}\ldots\alpha_{2s-1})}\Big{)}~{},$
$\displaystyle\delta_{\epsilon}\Omega_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{8(2s+1)}\epsilon^{\beta}\Big{(}{\rm
i}\nabla^{2}H_{\beta\alpha(2s-1)}-2(4s+3)\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s-1)}~{}$
$\displaystyle-$ $\displaystyle
2(2s-1)\nabla_{(\alpha_{1}}{}^{\gamma}H_{\alpha_{2}\ldots\alpha_{2s-1})\beta\gamma}+4(4s^{2}+3s+1)|\mu|H_{\beta\alpha(2s-1)}~{}$
$\displaystyle+$ $\displaystyle
8s\nabla_{(\beta}\Phi_{\alpha_{1}\ldots\alpha_{2s-1})}\Big{)}~{}.$

It is clear from the variation (6.147a) that the second supersymmetry
transformation (6.85) breaks the WZ gauge (6.122). To restore this gauge
condition, it is necessary to supplement the second supersymmetry
transformation with the $\epsilon$-dependent gauge transformations:

$\displaystyle\delta_{\gamma(\epsilon)}U_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\gamma_{\alpha(2s-2)}(\epsilon)+\bar{\gamma}_{\alpha(2s-2)}(\epsilon)~{},$
(6.148a) $\displaystyle\delta_{\gamma(\epsilon)}G_{\alpha(2s)}$
$\displaystyle=$ $\displaystyle{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{(\alpha_{1}}{\mbox{\boldmath${\cal
D}$}}_{\alpha_{2}}\bar{\gamma}_{\alpha_{3}\dots\alpha_{2s})}(\epsilon)$
(6.148b) $\displaystyle=$ $\displaystyle{\rm
i}\Big{(}\mbox{\boldmath$\nabla$}^{\underline{1}}{}_{(\alpha_{1}}\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{\alpha_{2}}+\mbox{\boldmath$\nabla$}_{(\alpha_{1}\alpha_{2}}\Big{)}\gamma_{\alpha_{3}\ldots{\alpha_{2s})}}(\epsilon)~{},$

where

$\gamma_{\alpha(2s-2)}(\epsilon)|=\frac{{\rm i}}{2}\epsilon^{\beta}{\cal
U}_{\beta;\alpha(2s-2)}~{}.$ (6.149)

The modified second supersymmetry transformations take the form

$\displaystyle\hat{\delta}_{\epsilon}U_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\delta_{\epsilon}U_{\alpha(2s-2)}+\delta_{\gamma(\epsilon)}U_{\alpha(2s-2)}~{},$
(6.150a) $\displaystyle\hat{\delta}_{\epsilon}G_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\delta_{\epsilon}G_{\alpha(2s)}+\delta_{\gamma(\epsilon)}G_{\alpha(2s)}~{}.$
(6.150b)

They act on the ${{\cal N}}=1$ gauge superfields (6.135) and (6.137) in the
following manner

$\displaystyle\hat{\delta}_{\epsilon}{\cal U}_{\beta;\alpha(2s-2)}$
$\displaystyle=$ $\displaystyle 4s\epsilon_{\beta}V_{\alpha(2s-2)}~{},$
(6.151a) $\displaystyle\hat{\delta}_{\epsilon}V_{\alpha(2s-2)}$
$\displaystyle=$ $\displaystyle-\frac{{\rm
i}}{8s}\Big{(}2\epsilon^{\beta}\nabla_{\beta}{}^{\gamma}{\cal
U}_{\gamma;\alpha(2s-2)}+{\rm i}\epsilon^{\beta}\nabla^{2}{\cal
U}_{\beta;\alpha(2s-2)}$ $\displaystyle-$ $\displaystyle
8(s-1)|\mu|\epsilon_{(\alpha_{1}}{\cal
U}^{\beta;}{}_{\alpha_{2}\ldots\alpha_{2s-2})\beta}+2(2s+1)|\mu|\epsilon^{\beta}{\cal
U}_{\beta;\alpha(2s-2)}\Big{)}~{},$
$\displaystyle\hat{\delta}_{\epsilon}H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle-2\epsilon^{\beta}\nabla_{(\beta}\tilde{H}_{\alpha_{1}\ldots\alpha_{2s})}-\frac{8{\rm
i}s}{2s+1}\epsilon_{(\alpha_{1}}\Omega_{\alpha_{2}\ldots\alpha_{2s})}~{},$
$\displaystyle\hat{\delta}_{\epsilon}\tilde{H}_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\epsilon^{\beta}\nabla_{(\beta}H_{\alpha_{1}\ldots\alpha_{2s})}+\frac{2{\rm
i}s}{2s+1}\epsilon_{(\alpha_{1}}\Phi_{\alpha_{2}\ldots\alpha_{2s})}-\frac{{\rm
i}}{2}\epsilon_{\beta}\nabla_{(\alpha_{1}\alpha_{2}}{\cal
U}^{\beta;}{}_{\alpha_{3}\ldots\alpha_{2s})}$ $\displaystyle+$
$\displaystyle{\rm i}|\mu|\epsilon_{(\alpha_{1}}{\cal
U}_{\alpha_{2}\ldots\alpha_{2s})}~{},$
$\displaystyle\hat{\delta}_{\epsilon}\Phi_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2(2s+1)}\epsilon^{\beta}\Big{(}{\rm
i}\nabla^{2}\tilde{H}_{\beta\alpha(2s-1)}-2(2s-1)\nabla_{(\alpha_{1}}{}^{\gamma}\tilde{H}_{\alpha_{2}\ldots\alpha_{2s-1})\beta\gamma}~{}$
$\displaystyle-$ $\displaystyle
2(4s+3)\nabla_{\beta}{}^{\gamma}\tilde{H}_{\gamma\alpha(2s-1)}+2(4s^{2}+3s+1)|\mu|\tilde{H}_{\beta\alpha(2s-1)}~{}$
$\displaystyle+$ $\displaystyle
8s\nabla_{(\beta}\Omega_{\alpha_{1}\ldots\alpha_{2s-1})}\Big{)}~{},$
$\displaystyle\hat{\delta}_{\epsilon}\Omega_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{8(2s+1)}\epsilon^{\beta}\Big{(}{\rm
i}\nabla^{2}H_{\beta\alpha(2s-1)}-2(2s-1)\nabla_{(\alpha_{1}}{}^{\gamma}H_{\alpha_{2}\ldots\alpha_{2s-1})\beta\gamma}~{}$
$\displaystyle-$ $\displaystyle
2(4s+3)\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s-1)}+4(4s^{2}+3s+1)|\mu|H_{\beta\alpha(2s-1)}~{}$
$\displaystyle+$ $\displaystyle
8s\nabla_{(\beta}\Phi_{\alpha_{1}\ldots\alpha_{2s-1})}\Big{)}~{}$
$\displaystyle+$
$\displaystyle\frac{1}{4s}\Big{(}(s+1)\epsilon_{\beta}\nabla_{\gamma(\alpha_{1}}\nabla^{\gamma}{\cal
U}^{\beta;}{}_{\alpha_{2}\ldots\alpha_{2s-1})}-(s-1)\epsilon_{\beta}\nabla_{(\alpha_{1}}{}^{\gamma}\nabla_{\alpha_{2}}{\cal
U}^{\beta;}{}_{\alpha_{3}\ldots\alpha_{2s-1})\gamma}~{}$ $\displaystyle+$
$\displaystyle 2(s-1)|\mu|\epsilon_{(\alpha_{1}}\nabla_{\alpha_{2}}{\cal
U}^{\beta;}{}_{\alpha_{3}\ldots\alpha_{2s-1})\beta}+(2s+1)|\mu|\epsilon^{\beta}\nabla_{\beta}{\cal
U}_{\alpha(2s-1)}$ $\displaystyle+$
$\displaystyle(6s^{2}+3s+2)|\mu|\epsilon_{\beta}\nabla_{(\alpha_{1}}{\cal
U}^{\beta;}{}_{\alpha_{2}\ldots\alpha_{2s-1})}\Big{)}~{}.$

###  6.5 Massless integer superspin: Transverse formulation

We now apply the same reduction procedure to the off-shell transverse
formulation for the massless multiplet of integer superspin [125].

####  6.5.1 Transverse formulation

For any integer $s\geq 1$, the transverse formulation for the massless
superspin-$s$ multiplet is realised in terms of the following superfields

${\cal
V}^{\perp}_{(s)}=\big{\\{}U_{\alpha(2s-2)},\Gamma_{\alpha(2s)},\bar{\Gamma}_{\alpha(2s)}\big{\\}}~{}.$
(6.152)

The real superfield $U_{\alpha(2s-2)}$ is the same as in (6.117) and the
complex superfield $\Gamma_{\alpha(2s)}$ is transverse linear (6.5a). Modulo
an overall factor, the transverse formulation for the massless superspin-$s$
multiplet is described by the action111111The transverse formulation for the
massless superspin-$s$ multiplet was derived in [125] by performing a
superfield duality transformation to the longitudinal action (6.119).

$\displaystyle\mathbb{S}^{\perp}_{(s)}[U_{\alpha(2s-2)},\Gamma_{\alpha(2s)},\bar{\Gamma}_{\alpha(2s)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\int\text{d}^{3|4}z~{}\mbox{\boldmath$E$}~{}\bigg{\\{}2s(s+1)\mu\bar{\mu}U^{\alpha(2s-2)}U_{\alpha(2s-2)}$
$\displaystyle+\frac{1}{8}U^{\alpha(2s-2)}{\mbox{\boldmath${\cal
D}$}}^{\beta}\big{(}{{\mbox{\boldmath${\bar{\cal
D}}$}}}{}^{2}-6\mu\big{)}{\mbox{\boldmath${\cal
D}$}}_{\beta}U_{\alpha(2s-2)}+{\rm i}{\mbox{\boldmath${\cal
D}$}}^{(\alpha_{1}\alpha_{2}}U^{\alpha_{3}\ldots\alpha_{2s})}\big{(}\Gamma_{\alpha(2s)}-\bar{\Gamma}_{\alpha(2s)}\big{)}~{}$
$\displaystyle-\frac{2s-1}{16(2s+1)}\Big{(}8s{\mbox{\boldmath${\cal
D}$}}^{(\alpha_{1}\alpha_{2}}U^{\alpha_{3}\ldots\alpha_{2s})}{\mbox{\boldmath${\cal
D}$}}_{(\alpha_{1}\alpha_{2}}U_{\alpha_{3}\ldots\alpha_{2s})}~{}$
$\displaystyle+[{\mbox{\boldmath${\cal
D}$}}^{(\alpha_{1}},{{\mbox{\boldmath${\bar{\cal
D}}$}}}{}^{\alpha_{2}}]U^{\alpha_{3}\ldots\alpha_{2s})}[{\mbox{\boldmath${\cal
D}$}}_{(\alpha_{1}},\bar{{\mbox{\boldmath${\cal
D}$}}}_{\alpha_{2}}]U_{\alpha_{3}\ldots\alpha_{2s})}\Big{)}~{}~{}$
$\displaystyle-\frac{2}{2s-1}\bar{\Gamma}^{\alpha(2s)}\Gamma_{\alpha(2s)}+\frac{1}{2s+1}\big{(}\Gamma^{\alpha(2s)}\Gamma_{\alpha(2s)}+\bar{\Gamma}^{\alpha(2s)}\bar{\Gamma}_{\alpha(2s)}\big{)}\bigg{\\}}~{},$
(6.153)

which is invariant under the gauge transformations

$\displaystyle\delta_{L}U_{\alpha(2s-2)}$
$\displaystyle={\mbox{\boldmath${\cal
D}$}}^{\beta}L_{\beta\alpha(2s-2)}-{{\mbox{\boldmath${\bar{\cal
D}}$}}}{}^{\beta}\bar{L}_{\beta\alpha(2s-2)}\equiv\bar{\gamma}_{\alpha(2s-2)}+\gamma_{\alpha(2s-2)}~{},$
(6.154a) $\displaystyle\delta_{L}\Gamma_{\alpha(2s)}$
$\displaystyle=-\frac{1}{4}({{\mbox{\boldmath${\bar{\cal
D}}$}}}{}^{2}+4s\mu){\mbox{\boldmath${\cal
D}$}}_{(\alpha_{1}}\bar{L}_{\alpha_{2}\ldots\alpha_{2s})}+\frac{{\rm
i}}{2}(2s+1){{\mbox{\boldmath${\bar{\cal
D}}$}}}{}^{\beta}{\mbox{\boldmath${\cal
D}$}}_{(\beta\alpha_{1}}\bar{L}_{\alpha_{2}\ldots\alpha_{2s})}$ (6.154b)
$\displaystyle\equiv\frac{1}{2}{\mbox{\boldmath${\cal
D}$}}_{(\alpha_{1}}{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{\alpha_{2}}\gamma_{\alpha_{3}\ldots\alpha_{2s})}-\frac{{\rm
i}}{2}(2s-1){\mbox{\boldmath${\cal
D}$}}_{(\alpha_{1}\alpha_{2}}\gamma_{\alpha_{3}\ldots\alpha_{2s})}~{},$

where $L_{\alpha(2s-1)}$ is complex unconstrained and $\gamma_{\alpha(2s-2)}$
is transverse linear, as in (6.118a).

#####  6.5.1.1 Reduction of gauge prepotentials to AdS3|2

We begin by reducing the real superfield $U_{\alpha(2s-2)}$. Following the
prescription employed in section 6.4, we have the freedom to impose the gauge
condition

$U_{\alpha(2s-2)}|=0~{}.$ (6.155)

The residual gauge symmetry which preserves this gauge is described by the
real unconstrained ${\cal N}=1$ superfield

$\gamma_{\alpha(2s-2)}|=-\frac{{\rm
i}}{2}\zeta_{\alpha(2s-2)}~{},\qquad\zeta_{\alpha(2s-2)}=\bar{\zeta}_{\alpha(2s-2)}~{}.$
(6.156)

As a result of choosing the gauge condition (6.155), we are left with the
remaining independent ${\cal N}=1$ superfields with respect to
$U_{\alpha(2s-2)}$

$\displaystyle{\cal U}_{\beta;\alpha(2s-2)}:$ $\displaystyle=$
$\displaystyle{\rm
i}\mbox{\boldmath$\nabla$}^{\underline{2}}_{\beta}U_{\alpha(2s-2)}|~{},$
(6.157a) $\displaystyle V_{\alpha(2s-2)}:$ $\displaystyle=$
$\displaystyle-\frac{{\rm
i}}{8s}(\mbox{\boldmath$\nabla$}^{\underline{2}})^{2}U_{\alpha(2s-2)}|~{}.$
(6.157b)

Using (6.154a) enables the computation of the corresponding gauge
transformations

$\displaystyle\delta{\cal U}_{\beta;\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-2s\rho_{\beta\alpha(2s-2)}+{\rm
i}\nabla_{\beta}\zeta_{\alpha(2s-2)}+{\rm
i}(2s-1)\nabla_{(\beta}\zeta_{\alpha(2s-2))}~{},$ (6.158a)
$\displaystyle\delta V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s}\nabla^{\beta}\tau_{\beta\alpha(2s-2)}~{},$ (6.158b)

where we have introduced the ${\cal N}=1$ gauge parameters
$\tau_{\alpha(2s-1)}$ and $\tilde{\tau}_{\alpha(2s-1)}$

$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\gamma_{\alpha_{2}\ldots\alpha_{2s-1})}|=\tau_{\alpha(2s-1)}+{\rm
i}s\tilde{\tau}_{\alpha(2s-1)}~{},$ (6.159)

and the gauge parameter redefinition

$\rho_{\alpha(2s-1)}:=\tilde{\tau}_{\alpha(2s-1)}+{\rm
i}\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}\ldots\alpha_{2s-1})}~{}.$ (6.160)

It is immediately obvious that the gauge transformation of ${\cal
U}_{\beta;\alpha(2s-2)}$ does not coincide with its analogue from the
longitudinal formulation (6.134a). Addressing this, we introduce the following
${{\cal N}}=1$ superfield redefinition

$\tilde{{\cal U}}_{\beta;\alpha(2s-2)}:={\cal
U}_{\beta;\alpha(2s-2)}-\frac{2s-1}{2s}{\cal U}_{\beta\alpha(2s-2)}~{}.$
(6.161)

By using the gauge transformation (6.158a), it can be shown that $\tilde{{\cal
U}}_{\beta;\alpha(2s-2)}$ possesses the desired gauge freedom

$\delta\tilde{{\cal U}}_{\beta;\alpha(2s-2)}=-\rho_{\beta\alpha(2s-2)}+{\rm
i}\nabla_{\beta}\zeta_{\alpha(2s-2)}~{}.$ (6.162)

Next, we wish to go about reducing $\Gamma_{\alpha(2s)}$ to ${\cal N}=1$ AdS
superspace. We begin by converting the transverse linear constraint
$\Gamma_{\alpha(2s)}$(6.5a) to the real representation (6.35)

$\mbox{\boldmath$\nabla$}^{\underline{2}\beta}\Gamma_{\beta\alpha(2s-1)}={\rm
i}\mbox{\boldmath$\nabla$}^{\underline{1}\beta}\Gamma_{\beta\alpha(2s-1)}~{}.$
(6.163)

Taking a Taylor expansion of $\Gamma_{\alpha(2s)}(\theta^{I})$ about
$\theta^{\underline{2}}$, and using the constraint (6.163), we find that the
independent $\theta^{\underline{2}}$ -components of $\Gamma_{\alpha(2s)}$ are

$\Gamma_{\alpha(2s)}|~{},\qquad\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\Gamma_{\alpha_{2}\ldots\alpha_{2s+1})}|~{}.$
(6.164)

Using the gauge transformations (6.154b) and the real representation (6.35),
we find

$\displaystyle\delta\Gamma_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle-\frac{{\rm
i}}{2}\mbox{\boldmath$\nabla$}^{\underline{1}}{}_{(\alpha_{1}}\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{\alpha_{2}}\gamma_{\alpha_{3}\ldots\alpha_{2s})}-{\rm
i}s\mbox{\boldmath$\nabla$}_{(\alpha_{1}\alpha_{2}}\gamma_{\alpha_{3}\ldots\alpha_{2s})}~{},$
(6.165a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\delta\Gamma_{\alpha_{2}\ldots\alpha_{2s+1})}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\mbox{\boldmath$\nabla$}_{(\alpha_{1}\alpha_{2}}\mbox{\boldmath$\nabla$}^{\underline{1}}{}_{\alpha_{3}}\gamma_{\alpha_{4}\ldots\alpha_{2s+1})}-{\rm
i}s\mbox{\boldmath$\nabla$}_{(\alpha_{1}\alpha_{2}}\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{\alpha_{3}}\gamma_{\alpha_{4}\ldots\alpha_{2s+1})}~{}.$
(6.165b)

Making use of (6.165) and the gauge freedom (6.124) allows us to directly
compute the gauge transformations of the ${\cal N}=1$ complex superfields
(6.164)

$\displaystyle\delta\Gamma_{\alpha(2s)}|$ $\displaystyle=$
$\displaystyle-\frac{{\rm
i}}{2}\nabla_{(\alpha_{1}}\tau_{\alpha_{2}\ldots\alpha_{2s})}+\frac{s}{2}\nabla_{(\alpha_{1}}\tilde{\tau}_{\alpha_{2}\ldots\alpha_{2s})}-\frac{s}{2}\nabla_{(\alpha_{1}\alpha_{2}}\zeta_{\alpha_{3}\ldots\alpha_{2s})}~{},$
(6.166a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\delta\Gamma_{\alpha_{2}\ldots\alpha_{2s+1})}|$
$\displaystyle=$ $\displaystyle\frac{{\rm
i}}{4}\nabla_{(\alpha_{1}\alpha_{2}}\nabla_{\alpha_{3}}\zeta_{\alpha_{4}\ldots\alpha_{2s+1})}-{\rm
i}s\nabla_{(\alpha_{1}\alpha_{2}}\tau_{\alpha_{3}\ldots\alpha_{2s+1})}$
$\displaystyle+$ $\displaystyle
s^{2}\nabla_{(\alpha_{1}\alpha_{2}}\tilde{\tau}_{\alpha_{3}\ldots\alpha_{2s+1})}~{}.$

Let us split the complex ${\cal N}=1$ superfields (6.164) into real and
imaginary parts

$\displaystyle\Gamma_{\alpha(2s)}|=\frac{1}{2}\big{(}sH_{\alpha(2s)}-{\rm
i}\tilde{H}_{\alpha(2s)}\big{)}~{},$ (6.167a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\Gamma_{\alpha_{2}\ldots\alpha_{2s+1})}|=\Phi_{\alpha(2s+1)}-{\rm
i}s\Omega_{\alpha(2s+1)}~{}.$ (6.167b)

The gauge transformations of these four real ${\cal N}=1$ superfields can be
read off from the gauge transformations (6.165)

$\displaystyle\delta H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\rho_{\alpha_{2}\ldots\alpha_{2s})}~{},$
(6.168a) $\displaystyle\delta\tilde{H}_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\tau_{\alpha_{2}\ldots\alpha_{2s})}~{},$
(6.168b) $\displaystyle\delta\Phi_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle
s^{2}\nabla_{(\alpha_{1}\alpha_{2}}\rho_{\alpha_{3}\ldots\alpha_{2s+1})}-\frac{{\rm
i}}{4}(4s^{2}-1)\nabla_{(\alpha_{1}\alpha_{2}}\nabla_{\alpha_{3}}\zeta_{\alpha_{4}\ldots\alpha_{2s+1})}~{},$
(6.168c) $\displaystyle\delta\Omega_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}\alpha_{2}}\tau_{\alpha_{3}\ldots\alpha_{2s+1})}~{}.$
(6.168d)

It then follows from gauge transformations (6.158b), (6.162) and (6.168) that
we are indeed dealing with two different gauge theories. The first of these
models is formulated in terms of the superfield variables

${\cal V}^{\perp}_{(s)}=\big{\\{}H_{\alpha(2s)},\tilde{{\cal
U}}_{\beta;\alpha(2s-2)},\Phi_{\alpha(2s+1)}\big{\\}}~{},$ (6.169)

which are defined modulo gauge transformations

$\displaystyle\delta_{\rho}H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\rho_{\alpha_{2}\ldots\alpha_{2s})}~{},$
(6.170a) $\displaystyle\delta_{\rho,\zeta}\tilde{{\cal
U}}_{\beta;\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\rho_{\beta\alpha(2s-2)}+{\rm
i}\nabla_{\beta}\zeta_{\alpha(2s-2)}~{},$ (6.170b)
$\displaystyle\delta_{\rho,\zeta}\Phi_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle
s^{2}\nabla_{(\alpha_{1}\alpha_{2}}\rho_{\alpha_{3}\ldots\alpha_{2s+1})}-\frac{{\rm
i}}{4}(4s^{2}-1)\nabla_{(\alpha_{1}\alpha_{2}}\nabla_{\alpha_{3}}\zeta_{\alpha_{4}\ldots\alpha_{2s+1})}~{}.$
(6.170c)

The other gauge theory is described in terms of the superfields

${\cal
V}^{\parallel}_{(s)}=\big{\\{}\tilde{H}_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s+1)}\big{\\}}~{},$
(6.171)

which possess the following gauge freedom

$\displaystyle\delta_{\tau}\tilde{H}_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\tau_{\alpha_{2}\ldots\alpha_{2s})}~{},$
(6.172a) $\displaystyle\delta_{\tau}V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s}\nabla^{\beta}\tau_{\beta\alpha(2s-2)}~{},$ (6.172b)
$\displaystyle\delta_{\tau}\Omega_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}\alpha_{2}}\tau_{\alpha_{3}\ldots\alpha_{2s+1})}~{}.$
(6.172c) From the perspective of ${\cal N}=1$ AdS superspace, the reduction of
the action (6.5.1) gives rise to two decoupled supersymmetric actions, which
are formulated in terms of superfields (6.169) and (6.171), respectively
$\displaystyle\mathbb{S}^{\perp}_{(s)}[U_{\alpha(2s-2)},\Gamma_{\alpha(2s)},\bar{\Gamma}_{\alpha(2s)}]$
$\displaystyle=$ $\displaystyle S^{\perp}_{(s)}[H_{\alpha(2s)},\tilde{{\cal
U}}_{\beta;\alpha(2s-2)},\Phi_{\alpha(2s+1)}]~{}$ (6.173)
$\displaystyle+S^{\parallel}_{(s)}[\tilde{H}_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s+1)}]~{}.$

We provide the explicit realisations of these two ${\cal N}=1$ decoupled
actions in the following subsection.

####  6.5.2 Massless higher-spin ${\cal N}=1$ supermultiplets

The first of the decoupled ${\cal N}=1$ supersymmetric actions, which is
formulated in terms of the superfields (6.169), takes the form

$\displaystyle S^{\perp}_{(s)}[H_{\alpha(2s)},\tilde{{\cal
U}}_{\beta;\alpha(2s-2)},\Phi_{\alpha(2s+1)}]=\bigg{(}-\frac{1}{2}\bigg{)}^{s}\frac{1}{4}\int{\rm
d}^{3|2}z~{}E~{}\bigg{\\{}\tilde{{\cal U}}^{\alpha(2s-1)}\Big{(}-2{\rm
i}s(4s-1)\Box\tilde{{\cal U}}_{\alpha(2s-1)}~{}$
$\displaystyle-s\nabla^{2}\nabla^{\beta}{}_{(\alpha_{1}}\tilde{{\cal
U}}_{\alpha_{2}\dots\alpha_{2s-1})\beta}+4{\rm
i}s(s-1)^{2}\nabla_{(\alpha_{1}\alpha_{2}}\nabla^{\beta\gamma}\tilde{{\cal
U}}_{\alpha_{3}\dots\alpha_{2s-1})\beta\gamma}-s(2s+1)|\mu|\nabla^{2}\tilde{{\cal
U}}_{\alpha(2s-1)}~{}$ $\displaystyle+\frac{2{\rm
i}(s-1)(2s-3)}{2s-1}\nabla_{(\alpha_{1}\alpha_{2}}\nabla_{\alpha_{3}}{}^{\beta}\tilde{{\cal
U}}^{\gamma;}{}_{\alpha_{4}\dots\alpha_{2s-1})\beta\gamma}+\frac{(s-1)(2s+1)}{2s-1}\nabla^{2}\nabla_{(\alpha_{1}\alpha_{2}}\tilde{{\cal
U}}^{\beta;}{}_{\alpha_{3}\dots\alpha_{2s-1})\beta}~{}$ $\displaystyle-4{\rm
i}s(2s-1)|\mu|\nabla_{(\alpha_{1}}{}^{\beta}\,\tilde{{\cal
U}}_{\alpha_{2}\dots\alpha_{2s-1})\beta}+2{\rm
i}(s-1)(2s+1)|\mu|\nabla_{(\alpha_{1}\alpha_{2}}\tilde{{\cal
U}}^{\beta;}{}_{\alpha_{3}\dots\alpha_{2s-1})\beta}$ $\displaystyle+2{\rm
i}s(2s-1)(2s+1)(4s-3)|\mu|^{2}\,\tilde{{\cal U}}_{\alpha(2s-1)}\Big{)}~{}$
$\displaystyle+\tilde{{\cal U}}_{\beta;}{}^{\beta\alpha(2s-3)}\bigg{(}2{\rm
i}\frac{(s-1)(6s-5)}{(2s-1)^{2}}\Box\tilde{{\cal
U}}^{\gamma;}{}_{\gamma\alpha(2s-3)}+\frac{(s-1)(2s+1)}{(2s-1)}|\mu|\nabla^{2}\tilde{{\cal
U}}^{\gamma;}{}_{\gamma\alpha(2s-3)}~{}$
$\displaystyle-\frac{(s-1)(2s-3)}{(2s-1)^{2}}\Big{(}2{\rm
i}(s-2)\nabla^{\gamma\delta}\nabla_{(\alpha_{1}\alpha_{2}}\tilde{{\cal
U}}^{\lambda;}{}_{\alpha_{3}\dots\alpha_{2s-3})\gamma\delta\lambda}+\nabla^{2}\nabla^{\gamma}{}_{(\alpha_{1}}\tilde{{\cal
U}}^{\delta;}{}_{\alpha_{2}\dots\alpha_{2s-3})\gamma\delta}\Big{)}$
$\displaystyle-\frac{2{\rm
i}(s-1)}{2s-1}(4s^{2}+16s-17)|\mu|^{2}\,\tilde{{\cal
U}}^{\gamma;}{}_{\gamma\alpha(2s-3)}\bigg{)}$
$\displaystyle-\frac{8s}{(2s+1)^{2}(2s-1)}\Big{(}s(2s^{3}-2s-1)|\mu|H^{\alpha(2s)}H_{\alpha(2s)}+s^{2}(s+1)H^{\beta\alpha(2s-1)}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s-1)}$
$\displaystyle+\frac{{\rm
i}s}{4}(2s^{2}-1)H^{\alpha(2s)}\nabla^{2}H_{\alpha(2s)}+(2s+1)H^{\alpha(2s)}\nabla^{\beta}\Phi_{\beta\alpha(2s)}+2{\rm
i}(2s+1)\Phi^{\alpha(2s+1)}\Phi_{\alpha(2s+1)}\Big{)}$
$\displaystyle+\frac{8s}{2s+1}\tilde{{\cal
U}}^{\alpha(2s-1)}\Big{(}s\nabla^{\beta\gamma}\nabla_{\beta}H_{\gamma\alpha(2s-1)}-s\nabla^{\beta\gamma}\nabla_{(\alpha_{1}}H_{\alpha_{2}\ldots\alpha_{2s-1})\beta\gamma}+s(2s+1)|\mu|\nabla^{\beta}H_{\beta\alpha(2s-1)}$
$\displaystyle+{\rm
i}(2s+1)\nabla^{\beta\gamma}\Phi_{\beta\gamma\alpha(2s-1)}\Big{)}+\frac{s(s-1)}{2s-1}\tilde{{\cal
U}}_{\beta;}{}^{\beta\alpha(2s-3)}\nabla^{\beta\gamma}\nabla^{\delta}H_{\beta\gamma\delta\alpha(2s-3)}\bigg{\\}}~{},$
(6.174)

which is invariant under the gauge transformations (6.170). It is apparent
that the superfield $\Phi_{\alpha(2s+1)}$ is auxiliary, so upon elimination
via its equation of motion,

$\Phi_{\alpha(2s+1)}=-\frac{1}{4}\Big{(}(4s^{2}-1)\nabla_{(\alpha_{1}\alpha_{2}}\tilde{{\cal
U}}_{\alpha_{3}\ldots\alpha_{2s+1})}+{\rm
i}\nabla_{(\alpha_{1}}H_{\alpha_{2}\ldots\alpha_{2s+1})}\Big{)}~{},$ (6.175)

one arrives at the action which coincides with (6.4.3), up to an overall
normalisation factor. Recall that in section 6.4.3, the action (6.4.3) was
shown to coincide with the action for the massless superspin-$s$ multiplet
(5.2.6.4).

The second ${\cal N}=1$ action in (6.173), which is formulated in terms of the
superfields (6.171), takes the form

$\displaystyle
S^{\parallel}_{(s)}[\tilde{H}_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s+1)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3|2}z~{}E~{}\bigg{\\{}-2sV^{\alpha(2s-2)}\nabla^{\beta\gamma}\tilde{H}_{\beta\gamma\alpha(2s-2)}\hskip
36.98866pt$ (6.176)
$\displaystyle+2s(s-1)V^{\beta\alpha(2s-3)}\nabla_{\beta}{}^{\gamma}V_{\gamma\alpha(2s-3)}+{\rm
i}s^{2}V^{\alpha(2s-2)}\nabla^{2}V_{\alpha(2s-2)}+4s^{2}(s+1)|\mu|V^{\alpha(2s-2)}V_{\alpha(2s-2)}$
$\displaystyle-\frac{1}{2(2s+1)^{2}(2s-1)}\Big{(}4{\rm
i}s^{2}(2s+1)\Omega^{\alpha(2s+1)}\Omega_{\alpha(2s+1)}+2s(4s+1)\tilde{H}^{\beta\alpha(2s-1)}\nabla_{\beta}{}^{\gamma}\tilde{H}_{\gamma\alpha(2s-1)}~{}$
$\displaystyle-{\rm
i}s\tilde{H}^{\alpha(2s)}\nabla^{2}\tilde{H}_{\alpha(2s)}+8s^{2}(2s+1)\tilde{H}^{\alpha(2s)}\nabla^{\beta}\Omega_{\beta\alpha(2s)}-4s(4s^{2}+3s+1)|\mu|\tilde{H}^{\alpha(2s)}\tilde{H}_{\alpha(2s)}\Big{)}\bigg{\\}}~{}.$

The superfield $\Omega_{\alpha(2s+1)}$ is auxiliary, so upon integrating it
out via its equation of motion,

$\Omega_{\alpha(2s+1)}=-{\rm
i}\nabla_{(\alpha_{1}}\tilde{H}_{\alpha_{2}\ldots\alpha_{2s+1})}~{},$ (6.177)

one finds that up to an overall factor, the resulting action coincides with
the action for the massless superspin-$s$ multiplet (5.2.6.3).

####  6.5.3 Second supersymmetry transformations

We begin by computing the second supersymmetry transformations for the real
${\cal N}=1$ fields (6.169) and (6.171)

$\displaystyle\delta_{\epsilon}U_{\alpha(2s-2)}|$ $\displaystyle=$
$\displaystyle-2{\rm i}s\epsilon^{\beta}\tilde{{\cal
U}}_{\beta\alpha(2s-2)}+\frac{2{\rm
i}(s-1)}{2s-1}\epsilon_{(\alpha_{1}}\tilde{{\cal
U}}^{\beta;}{}_{\alpha_{2}\ldots\alpha_{2s-2})\beta}~{},$ (6.178a)
$\displaystyle\delta_{\epsilon}\tilde{{\cal U}}_{\beta;\alpha(2s-2)}$
$\displaystyle=$ $\displaystyle
2(2s-1)\epsilon_{\beta}V_{\alpha(2s-2)}-4(s-1)\epsilon_{(\alpha_{1}}V_{\alpha_{2}\ldots\alpha_{2s-2})\beta}~{},$
(6.178b) $\displaystyle\delta_{\epsilon}V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{{\rm
i}}{2s}\Big{(}\frac{s-1}{2s-1}\epsilon^{\beta}\nabla_{\beta(\alpha_{1}}\tilde{{\cal
U}}^{\gamma;}{}_{\alpha_{2}\ldots\alpha_{2s-2})\gamma}-s\epsilon^{\beta}\nabla_{\beta}{}^{\gamma}\tilde{{\cal
U}}_{\gamma\alpha(2s-2)}~{}$ $\displaystyle+$ $\displaystyle
s(2s-1)|\mu|\epsilon^{\beta}\tilde{{\cal
U}}_{\beta\alpha(2s-2)}+(s-1)|\mu|\epsilon_{(\alpha_{1}}\tilde{{\cal
U}}^{\beta;}{}_{\alpha_{2}\ldots\alpha_{2s-2})\beta}\Big{)}~{},$
$\displaystyle\delta_{\epsilon}H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle-2{\rm
i}\epsilon^{\beta}\Omega_{\beta\alpha(2s)}-\frac{2}{2s+1}\epsilon_{(\alpha_{1}}\nabla^{\beta}\tilde{H}_{\alpha_{2}\ldots\alpha_{2s})\beta}~{},$
(6.178d) $\displaystyle\delta_{\epsilon}\tilde{H}_{\alpha(2s)}$
$\displaystyle=$ $\displaystyle 2{\rm
i}\epsilon^{\beta}\Phi_{\beta\alpha(2s)}+\frac{2s^{2}}{2s+1}\epsilon_{(\alpha_{1}}\nabla^{\beta}H_{\alpha_{2}\ldots\alpha_{2s})\beta}~{},~{}$
(6.178e) $\displaystyle\delta_{\epsilon}\Phi_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\epsilon^{\beta}\nabla_{\beta(\alpha_{1}}\tilde{H}_{\alpha_{2}\ldots\alpha_{2s+1})}-s\epsilon_{(\alpha_{1}}\nabla^{\beta}\Omega_{\alpha_{2}\ldots\alpha_{2s+1})\beta}~{}$
$\displaystyle+$ $\displaystyle\frac{1}{4(2s+1)}\Big{(}{\rm
i}\epsilon_{(\alpha_{1}}\nabla^{2}\tilde{H}_{\alpha_{2}\ldots\alpha_{2s+1})}-4s\epsilon_{(\alpha_{1}}\nabla_{\alpha_{2}}{}^{\beta}\tilde{H}_{\alpha_{3}\ldots\alpha_{2s+1})\beta}~{}$
$\displaystyle+$ $\displaystyle
4(4s^{2}+5s+2)|\mu|\epsilon_{(\alpha_{1}}\tilde{H}_{\alpha_{2}\ldots\alpha_{2s+1})}\Big{)}~{},$
$\displaystyle\delta_{\epsilon}\Omega_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\epsilon^{\beta}\nabla_{\beta(\alpha_{1}}H_{\alpha_{2}\ldots\alpha_{2s+1})}+\frac{1}{s}\epsilon_{(\alpha_{1}}\nabla^{\beta}\Phi_{\alpha_{2}\ldots\alpha_{2s+1})\beta}~{}$
$\displaystyle-$ $\displaystyle\frac{1}{4(2s+1)}\Big{(}{\rm
i}\epsilon_{(\alpha_{1}}\nabla^{2}H_{\alpha_{2}\ldots\alpha_{2s+1})}-4s\epsilon_{(\alpha_{1}}\nabla_{\alpha_{2}}{}^{\beta}H_{\alpha_{3}\ldots\alpha_{2s+1})\beta}~{}$
$\displaystyle+$ $\displaystyle
4(4s^{2}+5s+2)|\mu|\epsilon_{(\alpha_{1}}H_{\alpha_{2}\ldots\alpha_{2s+1})}\Big{)}~{}.$

It is clear from the variation (6.178a) that the second supersymmetry
transformation (6.85) breaks the WZ gauge condition (6.155). Thus, we need to
supplement the second supersymmetry transformation with the
$\epsilon$-dependent gauge transformations:

$\displaystyle\delta_{\gamma(\epsilon)}U_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\gamma_{\alpha(2s-2)}(\epsilon)+\bar{\gamma}_{\alpha(2s-2)}(\epsilon)~{},$
(6.179a) $\displaystyle\delta_{\gamma(\epsilon)}\Gamma_{\alpha(2s)}$
$\displaystyle=$ $\displaystyle\frac{1}{2}{{\mbox{\boldmath${\cal
D}$}}}_{(\alpha_{1}}{\mbox{\boldmath${\bar{\cal
D}}$}}_{\alpha_{2}}{\gamma}_{\alpha_{3}\dots\alpha_{2s})}(\epsilon)-\frac{{\rm
i}}{2}(2s-1){\mbox{\boldmath${\cal
D}$}}_{(\alpha_{1}\alpha_{2}}\gamma_{\alpha_{3}\dots\alpha_{2s})}(\epsilon)~{},$
(6.179b)

where

$\gamma_{\alpha(2s-2)}(\epsilon)|={\rm i}s\epsilon^{\beta}\tilde{{\cal
U}}_{\beta\alpha(2s-2)}-\frac{{\rm
i}(s-1)}{2s-1}\epsilon_{(\alpha_{1}}\tilde{{\cal
U}}^{\beta;}{}_{\alpha_{2}\ldots\alpha_{2s-2})\beta}~{}.$ (6.180)

The modified second supersymmetry transformations have the form

$\displaystyle\hat{\delta}_{\epsilon}U_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\delta_{\epsilon}U_{\alpha(2s-2)}+\delta_{\gamma(\epsilon)}U_{\alpha(2s-2)}~{},$
(6.181a) $\displaystyle\hat{\delta}_{\epsilon}\Gamma_{\alpha(2s)}$
$\displaystyle=$
$\displaystyle\delta_{\epsilon}\Gamma_{\alpha(2s)}+\delta_{\gamma(\epsilon)}\Gamma_{\alpha(2s)}~{}.$
(6.181b)

They act on the ${{\cal N}}=1$ gauge superfields (6.169) and (6.171) as
follows

$\displaystyle\hat{\delta}_{\epsilon}\tilde{{\cal U}}_{\beta;\alpha(2s-2)}$
$\displaystyle=$ $\displaystyle
2(2s-1)\epsilon_{\beta}V_{\alpha(2s-2)}-4(s-1)\epsilon_{(\alpha_{1}}V_{\alpha_{2}\ldots\alpha_{2s-2})\beta}~{},$
(6.182a) $\displaystyle\hat{\delta}_{\epsilon}V_{\alpha(2s-2)}$
$\displaystyle=$ $\displaystyle-\frac{{\rm
i}}{4}\Big{(}2\epsilon^{\beta}\nabla_{\beta}{}^{\gamma}\tilde{{\cal
U}}_{\gamma\alpha(2s-2)}+{\rm i}\epsilon^{\beta}\nabla^{2}\tilde{{\cal
U}}_{\beta\alpha(2s-2)}~{}$ $\displaystyle+$ $\displaystyle
2(2s+1)|\mu|\epsilon^{\beta}\tilde{{\cal
U}}_{\beta\alpha(2s-2)}-\frac{(s-1)}{s(2s-1)}\big{(}2\epsilon^{\beta}\nabla_{\beta(\alpha_{1}}\tilde{{\cal
U}}^{\gamma;}{}_{\alpha_{2}\ldots\alpha_{2s-2})\gamma}~{}$ $\displaystyle+$
$\displaystyle{\rm i}\epsilon_{(\alpha_{1}}\nabla^{2}\tilde{{\cal
U}}^{\beta;}{}_{\alpha_{2}\ldots\alpha_{2s-2})\beta}+2(6s-1)|\mu|\epsilon_{(\alpha_{1}}\tilde{{\cal
U}}^{\beta;}{}_{\alpha_{2}\ldots\alpha_{2s-2})\beta}\big{)}\Big{)}~{},$
$\displaystyle\hat{\delta}_{\epsilon}H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle-2{\rm
i}\epsilon^{\beta}\Omega_{\beta\alpha(2s)}-\frac{2}{2s+1}\epsilon_{(\alpha_{1}}\nabla^{\beta}\tilde{H}_{\alpha_{2}\ldots\alpha_{2s})\beta}~{},$
(6.182c) $\displaystyle\hat{\delta}_{\epsilon}\tilde{H}_{\alpha(2s)}$
$\displaystyle=$ $\displaystyle 2{\rm
i}\epsilon^{\beta}\Phi_{\beta\alpha(2s)}+\frac{2s^{2}}{2s+1}\epsilon_{(\alpha_{1}}\nabla^{\beta}H_{\alpha_{2}\ldots\alpha_{2s})\beta}+2{\rm
i}s^{2}\epsilon^{\beta}\nabla_{(\alpha_{1}\alpha_{2}}\tilde{{\cal
U}}_{\alpha_{3}\ldots\alpha_{2s})\beta}~{}$ $\displaystyle-$
$\displaystyle\frac{2{\rm
i}s}{2s-1}(s-1)\epsilon_{(\alpha_{1}}\nabla_{\alpha_{2}\alpha_{3}}\tilde{{\cal
U}}^{\beta;}{}_{\alpha_{4}\ldots\alpha_{2s})\beta}+4{\rm
i}s^{2}|\mu|\epsilon_{(\alpha_{1}}\tilde{{\cal
U}}_{\alpha_{2}\ldots\alpha_{2s})}~{},$
$\displaystyle\hat{\delta}_{\epsilon}\Phi_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\epsilon^{\beta}\nabla_{\beta(\alpha_{1}}\tilde{H}_{\alpha_{2}\ldots\alpha_{2s+1})}-s\epsilon_{(\alpha_{1}}\nabla^{\beta}\Omega_{\alpha_{2}\ldots\alpha_{2s+1})\beta}~{}$
$\displaystyle+$ $\displaystyle\frac{1}{4(2s+1)}\Big{(}{\rm
i}\epsilon_{(\alpha_{1}}\nabla^{2}\tilde{H}_{\alpha_{2}\ldots\alpha_{2s+1})}-4s\epsilon_{(\alpha_{1}}\nabla_{\alpha_{2}}{}^{\beta}\tilde{H}_{\alpha_{3}\ldots\alpha_{2s+1})\beta}~{}$
$\displaystyle+$ $\displaystyle
4(4s^{2}+5s+2)|\mu|\epsilon_{(\alpha_{1}}\tilde{H}_{\alpha_{2}\ldots\alpha_{2s+1})}\Big{)}~{},$
$\displaystyle\hat{\delta}_{\epsilon}\Omega_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\epsilon^{\beta}\nabla_{\beta(\alpha_{1}}H_{\alpha_{2}\ldots\alpha_{2s+1})}+\frac{1}{s}\epsilon_{(\alpha_{1}}\nabla^{\beta}\Phi_{\alpha_{2}\ldots\alpha_{2s+1})\beta}~{}$
$\displaystyle-$ $\displaystyle\frac{1}{4(2s+1)}\Big{(}{\rm
i}\epsilon_{(\alpha_{1}}\nabla^{2}H_{\alpha_{2}\ldots\alpha_{2s+1})}-4s\epsilon_{(\alpha_{1}}\nabla_{\alpha_{2}}{}^{\beta}H_{\alpha_{3}\ldots\alpha_{2s+1})\beta}~{}$
$\displaystyle+$ $\displaystyle
4(4s^{2}+5s+2)|\mu|\epsilon_{(\alpha_{1}}H_{\alpha_{2}\ldots\alpha_{2s+1})}\Big{)}-\frac{1}{2}\epsilon^{\beta}\nabla_{(\alpha_{1}\alpha_{2}}\nabla_{\alpha_{3}}\tilde{{\cal
U}}_{\alpha_{4}\ldots\alpha_{2s+1})\beta}$ $\displaystyle+$
$\displaystyle\frac{1}{2s(2s-1)}(s-1)\epsilon_{(\alpha_{1}}\nabla_{\alpha_{2}\alpha_{3}}\nabla_{\alpha_{4}}\tilde{{\cal
U}}^{\beta;}{}_{\alpha_{5}\ldots\alpha_{2s+1})\beta}-|\mu|\epsilon_{(\alpha_{1}}\nabla_{\alpha_{2}}\tilde{{\cal
U}}_{\alpha_{3}\ldots\alpha_{2s+1})}~{}.$

###  6.6 Topologically massive theories

In this section we study the $(1,1)\to(1,0)$ AdS reduction of the off-shell
topologically massive higher-spin supersymmetric theories, which were first
given in [125]. In particular, we will only be interested in employing the
reduction prescription to the massive longitudinal and transverse half-integer
theories.

The $(1,1)$ massless higher-spin theories presented in sections 6.2 \- 6.5 do
not propagate any physical degrees of freedom.121212This follows from the fact
that these theories describe two massless higher-spin actions in AdS3|2, which
were shown explicitly in section 5.2.6 to propagate no physical degrees of
freedom on-shell. However, they can be deformed in order to generate off-shell
topologically massive higher-spin supersymmetric theories. The gauge-invariant
actions for such massive multiplets are obtained by adding a superconformal
and massless higher-spin action together, following the philosophy of
topologically massive theories [118, 119, 120, 121].

Given a positive integer $n$, the superconformal higher-spin action in
AdS(3|1,1) [122, 125, 117, 1] is given by

$\mathbb{S}^{(n)}_{\text{SCHS}}[\mathfrak{H}_{\alpha(n)}]=-\frac{{\rm
i}^{n}}{2^{\lfloor n/2\rfloor+1}}\int{\rm
d}^{3|4}z\,\mbox{\boldmath$E$}\,\mathfrak{H}^{\alpha(n)}\mathfrak{W}_{\alpha(n)}(\mathfrak{H})~{},$
(6.183)

where $\mathfrak{H}_{\alpha(n)}$ is the superconformal gauge multiplet, and
$\mathfrak{W}_{\alpha(n)}(\mathfrak{H})$ is the higher-spin super-Cotton
tensor in AdS(3|1,1). The action (6.183) is invariant under the gauge
transformations

$\delta_{\lambda}\mathfrak{H}_{\alpha(n)}={\mbox{\boldmath${\bar{\cal
D}}$}}_{(\alpha_{1}}\lambda_{\alpha_{2}\ldots\alpha_{n})}-(-1)^{n}{\mbox{\boldmath${\cal
D}$}}_{(\alpha_{1}}\bar{\lambda}_{\alpha_{2}\ldots\alpha_{n})}~{},$ (6.184)

where the gauge parameter $\lambda_{\alpha(n-1)}$ is complex unconstrained. To
check that the action (6.183) is gauge invariant, one needs to make use of the
properties of the super-Cotton tensor
$\mathfrak{W}_{\alpha(n)}(\mathfrak{H})$. In particular,
$\mathfrak{W}_{\alpha(n)}(\mathfrak{H})$ is gauge invariant
$\delta_{\lambda}\mathfrak{W}_{\alpha(n)}(\mathfrak{H})=0$, and is
simultaneously transverse linear (6.5a) and transverse anti-linear
(6.5b).131313Recall that the closed form expression for
$\mathfrak{W}_{\alpha(n)}(\mathfrak{H})$ in $\mathbb{M}^{3|4}$ is given by
(3.113). It was generalised to an arbitrary conformally flat superspace in
[86] by using the formalism of ${\cal N}=2$ conformal superspace [159].
However, it has yet to be degauged to AdS(3|1,1).

In $(1,1)$ AdS superspace, let us consider the two dually equivalent off-shell
formulations for a massive superspin-$(s+\frac{1}{2})$ multiplet [125]141414
As pointed out in [125], it is expected that the topologically massive actions
(6.185) and (6.185) describe the on-shell massive supermultiplets in
AdS(3|1,1) [167]. Recall that a massive on-shell supermultiplet is realised in
terms of a superfield on $\mbox{\boldmath$\mathcal{V}$}_{(n)}$ which satisfies
the on-shell conditions (6.29).

$\displaystyle\mathbb{S}^{\perp}_{(s+\frac{1}{2})}[\mathfrak{H}_{\alpha(2s)},\Gamma_{\alpha(2s-2)},\bar{\Gamma}_{\alpha(2s-2)}|m]$
$\displaystyle=$
$\displaystyle\kappa\mathbb{S}^{(2s)}_{\text{SCHS}}[\mathfrak{H}_{\alpha(2s)}]~{}$
$\displaystyle+m^{2s-1}\mathbb{S}^{\perp}_{(s+\frac{1}{2})}[\mathfrak{H}_{\alpha(2s)},\Gamma_{\alpha(2s-2)},\bar{\Gamma}_{\alpha(2s-2)}]~{},$
$\displaystyle\mathbb{S}^{\parallel}_{(s+\frac{1}{2})}[\mathfrak{H}_{\alpha(2s)},G_{\alpha(2s-2)},\bar{G}_{\alpha(2s-2)}|m]$
$\displaystyle=$
$\displaystyle\kappa\mathbb{S}^{(2s)}_{\text{SCHS}}[\mathfrak{H}_{\alpha(2s)}]~{}$
$\displaystyle+m^{2s-1}\mathbb{S}^{\parallel}_{(s+\frac{1}{2})}[\mathfrak{H}_{\alpha(2s)},G_{\alpha(2s-2)},\bar{G}_{\alpha(2s-2)}]~{},$

where $\kappa$ is dimensionless and $m$ has dimension of mass. The
superconformal action
$\mathbb{S}^{(2s)}_{\text{SCHS}}[\mathfrak{H}_{\alpha(2s)}]$ is given by
(6.183), for the case $n=2s$. The transverse
$\mathbb{S}^{\perp}_{(s+\frac{1}{2})}[\mathfrak{H}_{\alpha(2s)},\Gamma_{\alpha(2s-2)},\bar{\Gamma}_{\alpha(2s-2)}]$
and longitudinal
$\mathbb{S}^{\parallel}_{(s+\frac{1}{2})}[\mathfrak{H}_{\alpha(2s)},G_{\alpha(2s-2)},\bar{G}_{\alpha(2s-2)}]$
actions are given by (6.2.1) and (6.3.1), respectively.

We now turn to describing the $(1,1)\to(1,0)$ AdS reduction of the massive
models (6.185). The reduction of the massless transverse (6.2.1) and
longitudinal (6.3.1) formulations was achieved in sections 6.2 and 6.3,
respectively. Thus, we need only carry out the reduction for the
superconformal action.

The superconformal gauge multiplet $\mathfrak{H}_{\alpha(2s)}$ was reduced in
subsection 6.2.2. Upon reduction, the superfield $\mathfrak{H}_{\alpha(2s)}$
is equivalent to two unconstrained real ${\cal N}=1$ superfields (6.68),
defined modulo gauge transformations of the form (6.70). Since
$\mathfrak{W}_{\alpha(2s)}$ is a real transverse linear superfield, it follows
that $\mathfrak{W}_{\alpha(2s)}$ is equivalent to two real ${\cal N}=1$
superfields, defined as:

$W_{\alpha(2s)}(H)=-\mathfrak{W}_{\alpha(2s)}(\mathfrak{H})|~{},\qquad
W_{\alpha(2s+1)}(H)=\frac{{\rm
i}}{2}\nabla_{(\alpha_{1}}\mathfrak{W}_{\alpha_{2}\ldots\alpha_{2s+1})}(\mathfrak{H})|~{}.$
(6.186)

Here each of the ${{\cal N}}=1$ superfields $W_{\alpha(2s)}(H)$ and
$W_{\alpha(2s+1)}(H)$ are transverse linear and gauge invariant.

Applying the reduction procedure to the superconformal action (6.183) yields
two decoupled ${\cal N}=1$ supersymmetric theories

$\mathbb{S}^{(2s)}_{\text{SCHS}}[\mathfrak{H}_{\alpha(2s)}]=-\frac{(-1)^{s}}{2^{s+1}}\int{\rm
d}^{3|2}z\,E\,\big{\\{}H^{\alpha(2s)}W_{\alpha(2s)}+{\rm
i}H^{\alpha(2s+1)}W_{\alpha(2s+1)}\big{\\}}~{}.$ (6.187)

These two gauge-invariant theories can be identified as the ${\cal N}=1$
superconformal actions in AdS3 (5.245), for the cases $n=2s$ and $n=2s+1$,
respectively. Recalling the reduction results (6.192a), it is apparent that
both (6.185) and (6.185) decouple into two off-shell topologically massive
${\cal N}=1$ theories. The first
$S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)}|m]$
coincides with (5.243), while the second model
$S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)}|m]$ can be identified
with (5.244a).

One can also construct a new topologically massive model which leads directly
to the equations (6.29). The corresponding action is given by

$\displaystyle{\mathbb{S}}^{(n)}_{\rm{NTM}}[{\mathfrak{H}}_{\alpha(n)}]=-\frac{{\rm
i}^{n}}{2^{\left\lfloor{n/2}\right\rfloor+1}}\frac{\kappa}{M}\int{\rm
d}^{3|4}z\,\mbox{\boldmath$E$}\,{\mathfrak{W}}^{\alpha(n)}({\mathfrak{H}})\big{(}\mathbb{F}-\sigma
M\big{)}{\mathfrak{H}}_{\alpha(n)}~{}.$ (6.188)

In the flat-superspace limit, the action reduces to (3.170) in the case ${\cal
N}=2$. It is easy to see that the equation of motion with respect to
$\mathfrak{H}_{\alpha(n)}$ is

$0=\big{(}\mbox{\boldmath$\Delta$}-\sigma
M\big{)}{\mathfrak{W}}_{\alpha(n)}~{}.$ (6.189)

Since the field strength ${\mathfrak{W}}_{\alpha(n)}(\mathfrak{H})$ is TLAL
and satisfies the constraint (6.189), it follows from (6.29) that on-shell,
the action (6.188) describes a massive superfield with mass $M$ and
superhelicity $\frac{1}{2}(n+1)\sigma$.

###  6.7 Discussion and summary of results

We are now in the position to compare massless and massive theories with
$(1,1)$ AdS supersymmetry to their $(2,0)$ counterparts. Before doing this, it
is worth summarising the main results of this chapter.

In section 6.1, we initiated the program to derive the superspin projection
operators in AdS(3|1,1). These superprojectors map an arbitrary superfield in
AdS(3|1,1) satisfying the constraint (6.29b) to an on-shell superfield (6.29).
Making use of the minimal uplift method, we computed the explicit form of the
rank-$1$ and rank-$2$ superprojectors for the first time. They are given by
eqs. (6.34a) and (6.34b), respectively. Important to the construction of these
superspin projection operators were the Casimir operators of the ${\rm
AdS}^{(3|1,1)}$ isometry algebra. These Casimir operators (6.22) were derived
in the superfield representation for the first time in this thesis.

Sections 6.2 and 6.3 detail the $(1,1)\rightarrow(1,0)$ AdS superspace
reduction of the transverse and longitudinal formulations for the massless
half-integer superspin multiplets, respectively [2]. When reduced to ${\cal
N}=1$ AdS superspace, we found that the actual difference between the
transverse and longitudinal models lies in the structure of auxiliary
superfields. Let us refer to eq. (6.80) which shows that the transverse
formulation gives rise to two decoupled ${\cal N}=1$ supersymmetric theories
[2]:

$\displaystyle\mathbb{S}^{\perp}_{(s+\frac{1}{2})}[\mathfrak{H}_{\alpha(2s)},\Gamma_{\alpha(2s-2)},\bar{\Gamma}_{\alpha(2s-2)}]$
$\displaystyle=$ $\displaystyle
S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)},\Phi_{\alpha(2s-1)}]~{}$
(6.190)
$\displaystyle+S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-1)}]~{}.$

In the longitudinal case, from eq. (6.106) we have that [2]

$\displaystyle\mathbb{S}^{\parallel}_{(s+\frac{1}{2})}[\mathfrak{H}_{\alpha(2s)},G_{\alpha(2s-2)},\bar{G}_{\alpha(2s-2)}]=$
$\displaystyle
S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)},\Phi_{\alpha(2s-3)}]~{}$
(6.191)
$\displaystyle+S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-3)}]~{}.$

We further showed that the superfields
$\Phi_{\alpha(2s-1)},\Omega_{\alpha(2s-1)},\Phi_{\alpha(2s-3)},\Omega_{\alpha(2s-3)}$
are all auxiliary. Once they are eliminated, each formulation then leads to
the same ${{\cal N}}=1$ supersymmetric higher-spin actions [2]:

Transverse:
$\mathbb{S}^{\perp}_{(s+\frac{1}{2})}[\mathfrak{H}_{\alpha(2s)},\Gamma_{\alpha(2s-2)},\bar{\Gamma}_{\alpha(2s-2)}]$$S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)}]~{},$$S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)}]$
(6.192a) Longitudinal:
$\mathbb{S}^{\parallel}_{(s+\frac{1}{2})}[\mathfrak{H}_{\alpha(2s)},G_{\alpha(2s-2)},\bar{G}_{\alpha(2s-2)}]$$S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)}]~{}.$$S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)}]$
(6.192b)

Sections 6.4 and 6.5 concern the $(1,1)\rightarrow(1,0)$ AdS superspace
reduction of the longitudinal and transverse formulations for the massless
integer superspin multiplets, respectively. In $(1,1)$ AdS superspace, they
are dual to each other. As in the half-integer case, we demonstrated that upon
reduction to ${\cal N}=1$ AdS superspace, these formulations differ by the
structure of the auxiliary superfields. As shown in eq. (6.139), the
longitudinal model is equivalent to [2]

$\displaystyle\mathbb{S}^{\parallel}_{(s)}[U_{\alpha(2s-2)},G_{\alpha(2s)},\bar{G}_{\alpha(2s)}]$
$\displaystyle=$ $\displaystyle S^{\perp}_{(s)}[H_{\alpha(2s)},{\cal
U}_{\beta;\alpha(2s-2)},\Phi_{\alpha(2s-1)}]~{}$ (6.193)
$\displaystyle+S^{\parallel}_{(s)}[\tilde{H}_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-1)}]~{}.$

On the other hand, eq. (6.173) shows that the transverse model yields [2]

$\displaystyle\mathbb{S}^{\perp}_{(s)}[U_{\alpha(2s-2)},\Gamma_{\alpha(2s)},\bar{\Gamma}_{\alpha(2s)}]$
$\displaystyle=$ $\displaystyle S^{\perp}_{(s)}[H_{\alpha(2s)},\tilde{{\cal
U}}_{\beta;\alpha(2s-2)},\Phi_{\alpha(2s+1)}]~{}$ (6.194)
$\displaystyle+S^{\parallel}_{(s)}[\tilde{H}_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s+1)}]~{}.$

The superfields
$\Phi_{\alpha(2s-1)},\Omega_{\alpha(2s-1)},\Phi_{\alpha(2s+1)},\Omega_{\alpha(2s+1)}$
are all auxiliary and thus can be integrated out from their corresponding
actions. We showed that the resulting ${{\cal N}}=1$ actions are the same in
both formulations [2]:

Transverse:
$\mathbb{S}^{\perp}_{(s)}[U_{\alpha(2s-2)},\Gamma_{\alpha(2s)},\bar{\Gamma}_{\alpha(2s)}]$$S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)}]~{},$$S^{\perp}_{(s)}[H_{\alpha(2s)},\Psi_{\beta;\,\alpha(2s-2)}]$
(6.195a) Longitudinal:
$\mathbb{S}^{\parallel}_{(s)}[U_{\alpha(2s-2)},G_{\alpha(2s)},\bar{G}_{\alpha(2s)}]$$S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)}]~{}.$$S^{\perp}_{(s)}[H_{\alpha(2s)},\Psi_{\beta;\,\alpha(2s-2)}]$
(6.195b)

We are now in a position to compare the above $(1,1)\rightarrow(1,0)$ AdS
reduction results with the ${{\cal N}}=1$ supersymmetric higher-spin gauge
theories obtained via $(2,0)\to(1,0)$ AdS reduction [127]. There exist two
off-shell formulations for massless half-integer superspin multiplets with
$(2,0)$ AdS supersymmetry, which are not dual to each other.151515Off-shell
constructions for the massless integer superspin $(2,0)$ multiplets remains an
open problem. They are known as type II and type III series [126]. Upon
reduction to ${{\cal N}}=1$ superspace, the type II series yields

Type II:
$\mathbb{S}^{\text{{II}}}_{(s+\frac{1}{2})}[\mathfrak{H}_{\alpha(2s)},\mathfrak{L}_{\alpha(2s-2)}]$$S^{\perp}_{(s)}[H_{\alpha(2s)},\Psi_{\beta;\alpha(2s-2)}]~{}.$$S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)}]$
(6.196)

On the other hand, the type III series leads to

Type
III:$\mathbb{S}^{\text{{III}}}_{(s+\frac{1}{2})}[\mathfrak{H}_{\alpha(2s)},\mathfrak{V}_{\alpha(2s-2)}]$$S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)}]~{}.$$S^{\perp}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},\Upsilon_{\beta;\alpha(2s-2)}]$
(6.197)

We point out that unlike the situation for the $(1,1)$ AdS multiplets, the
${\cal N}=1$ models obtained from the $(2,0)\to(1,0)$ AdS reduction do not
involve any auxiliary superfields. Additionally, reductions of the $(1,1)$ AdS
multiplets only produce three of the four series of off-shell ${\cal N}=1$
multiplets, with the exception of the transverse half-integer model,
$S^{\perp}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},\Upsilon_{\beta;\,\alpha(2s-2)}]$.
The latter can only be derived via $(2,0)\to(1,0)$ AdS reduction of the type
III series. This formulation corresponds to the linearised supergravity model
(for $s=1$), which does not admit a non-linear extension [242].

In section 6.6 we studied the $(1,1)\to(1,0)$ AdS reduction of the two dually
equivalent off-shell formulations for a massive superspin-$(s+\frac{1}{2})$
multiplet, which are described by the actions (6.185) [125]. It was shown in
[2] that these two gauge-invariant models decouple into two off-shell
topologically massive ${\cal N}=1$ theories. The first
$S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)}|m]$
coincides with (5.243), while the second model
$S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)}|m]$ can be identified
with (5.244a). Thus, the $(1,1)\to(1,0)$ AdS reduction of the massive actions
(6.185) is only able to generate two out of the four gauge-invariant
formulations for topologically massive ${\cal N}=1$ supermultiplets.

In $(2,0)$ AdS superspace, the following off-shell gauge-invariant massive
models were proposed in [126]:

$\displaystyle\mathbb{S}_{(s+\frac{1}{2})}^{\rm
II}[{\mathfrak{H}}_{\alpha(2s)},{\mathfrak{L}}_{\alpha(2s-2)}|m]$
$\displaystyle=$ $\displaystyle\kappa{\mathbb{S}}^{(2s)}_{\rm
SCHS}[{\mathfrak{H}}_{\alpha(2s)}]+m^{2s-1}\mathbb{S}^{\rm
II}_{(s+\frac{1}{2})}[{\mathfrak{H}}_{\alpha(2s)},{\mathfrak{L}}_{\alpha(2s-2)}]~{},$
(6.198a) $\displaystyle\mathbb{S}_{(s+\frac{1}{2})}^{\rm
III}[{\mathfrak{H}}_{\alpha(2s)},{\mathfrak{V}}_{\alpha(2s-2)}|m]$
$\displaystyle=$ $\displaystyle\kappa{\mathbb{S}}^{(2s)}_{\rm
SCHS}[{\mathfrak{H}}_{\alpha(2s)}]+m^{2s-1}\mathbb{S}^{\rm
III}_{(s+\frac{1}{2})}[{\mathfrak{H}}_{\alpha(2s)},{\mathfrak{V}}_{\alpha(2s-2)}]~{}.\hskip
14.22636pt$ (6.198b)

Here ${\mathbb{S}}^{(2s)}_{\rm{SCHS}}[{\mathfrak{H}}_{\alpha(2s)}]$ is the
superconformal higher-spin action [125],

$\displaystyle{\mathbb{S}}^{(2s)}_{\rm{SCHS}}[{\mathfrak{H}}_{\alpha(2s)}]=-\frac{(-1)^{s}}{2^{s+1}}\int{\rm
d}^{3|4}z\,\mbox{\boldmath$E$}\,{\mathfrak{H}}^{\alpha(2s)}{\mathfrak{W}}_{\alpha(2s)}({\mathfrak{H}})~{},$
(6.199)

where
${\mathfrak{W}}_{\alpha(2s)}({\mathfrak{H}})=\bar{\mathfrak{W}}_{\alpha(2s)}({\mathfrak{H}})$
is the higher-spin super-Cotton tensor of AdS(3|2,0).

Recalling the reduction result (6.196), it is clear that (6.198a) decouples
into two off-shell massive ${{\cal N}}=1$ models:
$S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)}|m]$ and
$S^{\perp}_{(s)}[H_{\alpha(2s)},{\Psi}_{\beta;\,\alpha(2s-2)}|m]$. On the
other hand, it follows directly from (6.197) that the reduction of (6.198b)
leads to
$S^{\perp}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},\Upsilon_{\beta;\,\alpha(2s-2)}|m]$
and $S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)}|m]$. Thus, the
$(2,0)\to(1,0)$ AdS reduction of massless half-integer actions (6.198) allows
us to recover all four off-shell gauge-invariant formulations for
topologically massive ${{\cal N}}=1$ supermultiplets, which are described by
(5.243) and (5.244).

## Appendix

##  Appendix 6.A $(1,1)$ AdS superspace toolkit

In this appendix, we provide a summary of essential identities for (1,1) AdS
covariant derivatives in both the complex ${{\mbox{\boldmath${\cal
D}$}}}_{{\cal A}}=({\mbox{\boldmath${\cal D}$}}_{a},{\mbox{\boldmath${\cal
D}$}}_{\alpha},{{\mbox{\boldmath${\bar{\cal D}}$}}}{}^{\alpha})$ and real
$\mbox{\boldmath$\nabla$}_{A}=(\mbox{\boldmath$\nabla$}_{a},\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha},\mbox{\boldmath$\nabla$}^{\underline{2}}_{\alpha})$
basis.

Making use of the algebra in the complex basis (6.4), one can derive the
following useful identities:

$\displaystyle{\mbox{\boldmath${\cal D}$}}_{\alpha}{\mbox{\boldmath${\cal
D}$}}_{\beta}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{1}{2}\varepsilon_{\alpha\beta}{\mbox{\boldmath${\cal
D}$}}^{2}-2{\bar{\mu}}\,M_{\alpha\beta}~{},\quad\qquad\,\,\,{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{\alpha}{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{\beta}=-\frac{1}{2}\varepsilon_{\alpha\beta}{{\mbox{\boldmath${\bar{\cal
D}}$}}}{}^{2}+2\mu\,{M}_{\alpha\beta}~{},$ (6.200a)
$\displaystyle{\mbox{\boldmath${\cal D}$}}_{\alpha}{\mbox{\boldmath${\cal
D}$}}^{2}\\!$ $\displaystyle=$
$\displaystyle\\!4\bar{\mu}\,{\mbox{\boldmath${\cal
D}$}}^{\beta}M_{\alpha\beta}+4{\bar{\mu}}\,{\mbox{\boldmath${\cal
D}$}}_{\alpha}~{},\quad\qquad{\mbox{\boldmath${\cal
D}$}}^{2}{\mbox{\boldmath${\cal
D}$}}_{\alpha}=-4\bar{\mu}\,{\mbox{\boldmath${\cal
D}$}}^{\beta}M_{\alpha\beta}-2\bar{\mu}\,{\mbox{\boldmath${\cal
D}$}}_{\alpha}~{},$ (6.200b) $\displaystyle{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{\alpha}{{\mbox{\boldmath${\bar{\cal D}}$}}}{}^{2}\\!$
$\displaystyle=$ $\displaystyle\\!4\mu\,{{\mbox{\boldmath${\bar{\cal
D}}$}}}{}^{\beta}{M}_{\alpha\beta}+4\mu\,{\mbox{\boldmath${\bar{\cal
D}}$}}_{\alpha}~{},\quad\qquad{{\mbox{\boldmath${\bar{\cal
D}}$}}}^{2}{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{\alpha}=-4\mu\,{{\mbox{\boldmath${\bar{\cal
D}}$}}}{}^{\beta}{M}_{\alpha\beta}-2\mu\,{\mbox{\boldmath${\bar{\cal
D}}$}}_{\alpha}~{},$ (6.200c) $\displaystyle\ [{\mbox{\boldmath${\bar{\cal
D}}$}}{}^{2},{\mbox{\boldmath${\cal D}$}}_{\alpha}]\\!$ $\displaystyle=$
$\displaystyle\\!4{\rm i}{\mbox{\boldmath${\cal
D}$}}_{\alpha\beta}{\mbox{\boldmath${\bar{\cal
D}}$}}{}^{\beta}+6\mu\,{\mbox{\boldmath${\cal D}$}}_{\alpha}=4\rm
i{\mbox{\boldmath${\bar{\cal D}}$}}{}^{\beta}{\mbox{\boldmath${\cal
D}$}}_{\beta\alpha}-6\mu\,{\mbox{\boldmath${\cal D}$}}_{\alpha}~{},$ (6.200d)
$\displaystyle\ [{\mbox{\boldmath${\cal D}$}}^{2},{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{\alpha}]\\!$ $\displaystyle=$ $\displaystyle\\!-4{\rm
i}{\mbox{\boldmath${\cal D}$}}_{\beta\alpha}{\mbox{\boldmath${\cal
D}$}}^{\beta}+6\bar{\mu}\,{{\mbox{\boldmath${\bar{\cal D}}$}}}_{\alpha}=-4\rm
i{\mbox{\boldmath${\cal D}$}}^{\beta}{\mbox{\boldmath${\cal
D}$}}_{\beta\alpha}-6\bar{\mu}\,{{\mbox{\boldmath${\bar{\cal
D}}$}}}_{\alpha}~{}$ (6.200e)

Here, we have made use of the shorthand notation ${\mbox{\boldmath${\cal
D}$}}^{2}={\mbox{\boldmath${\cal D}$}}^{\alpha}{\mbox{\boldmath${\cal
D}$}}_{\alpha}$ and ${{\mbox{\boldmath${\bar{\cal
D}}$}}}^{2}={{\mbox{\boldmath${\bar{\cal
D}}$}}}_{\alpha}{{\mbox{\boldmath${\bar{\cal D}}$}}}^{\alpha}$. These
relations imply the identity

$\displaystyle{\mbox{\boldmath${\cal
D}$}}^{\alpha}({\mbox{\boldmath${\bar{\cal
D}}$}}^{2}-6\mu){\mbox{\boldmath${\cal
D}$}}_{\alpha}={\mbox{\boldmath${\bar{\cal
D}}$}}_{\alpha}({\mbox{\boldmath${\cal
D}$}}^{2}-6\bar{\mu}){\mbox{\boldmath${\bar{\cal D}}$}}{}^{\alpha}~{}.$
(6.201)

Utilising the algebra in the real basis (6.36) allows for the computation of
the important relations:

$\displaystyle\mbox{\boldmath$\nabla$}^{{\underline{1}}}_{\alpha}\mbox{\boldmath$\nabla$}^{\underline{1}}_{\beta}$
$\displaystyle=$ $\displaystyle{\rm
i}\mbox{\boldmath$\nabla$}_{\alpha\beta}-2{\rm
i}|\mu|M_{\alpha\beta}+\frac{1}{2}\varepsilon_{\alpha\beta}\mbox{\boldmath$($}\mbox{\boldmath$\nabla$}^{\underline{1}})^{2}~{},$
(6.202a)
$\displaystyle\mbox{\boldmath$\nabla$}^{{\underline{1}}\beta}\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha}\mbox{\boldmath$\nabla$}^{\underline{1}}_{\beta}$
$\displaystyle=$ $\displaystyle 4{\rm
i}|\mu|\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha}~{},\qquad\
[\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha},\mbox{\boldmath$\nabla$}_{\beta\gamma}]=2|\mu|\varepsilon_{\alpha(\beta}\mbox{\boldmath$\nabla$}^{\underline{1}}{}_{\gamma)}~{},$
(6.202b)
$\displaystyle\\{\mbox{\boldmath$($}\mbox{\boldmath$\nabla$}^{\underline{1}})^{2},\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha}\\}$
$\displaystyle=$ $\displaystyle 4{\rm
i}|\mu|\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha}\qquad\
[\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha},\Box]=2|\mu|\mbox{\boldmath$\nabla$}_{\alpha\beta}\mbox{\boldmath$\nabla$}^{{\underline{1}}\beta}+3|\mu|^{2}\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha}~{},$
(6.202c)
$\displaystyle\mbox{\boldmath$($}\mbox{\boldmath$\nabla$}^{\underline{1}})^{2}\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha}$
$\displaystyle=$ $\displaystyle 2{\rm
i}|\mu|\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha}+2{\rm
i}\mbox{\boldmath$\nabla$}_{\alpha\beta}\mbox{\boldmath$\nabla$}^{{\underline{1}}\beta}-4{\rm
i}|\mu|\mbox{\boldmath$\nabla$}^{{\underline{1}}\beta}M_{\alpha\beta}~{},$
(6.202d)
$\displaystyle-\frac{1}{4}\mbox{\boldmath$($}\mbox{\boldmath$\nabla$}^{\underline{1}})^{2}\mbox{\boldmath$($}\mbox{\boldmath$\nabla$}^{\underline{1}})^{2}$
$\displaystyle=$ $\displaystyle\Box-2{\rm
i}|\mu|\mbox{\boldmath$($}\mbox{\boldmath$\nabla$}^{\underline{1}})^{2}+2|\mu|\mbox{\boldmath$\nabla$}^{\alpha\beta}M_{\alpha\beta}-2|\mu|^{2}M^{\alpha\beta}M_{\alpha\beta}~{},$
(6.202e)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}}_{\alpha}\mbox{\boldmath$\nabla$}^{\underline{2}}_{\beta}$
$\displaystyle=$ $\displaystyle{\rm
i}\mbox{\boldmath$\nabla$}_{\alpha\beta}+2{\rm
i}|\mu|M_{\alpha\beta}+\frac{1}{2}\varepsilon_{\alpha\beta}(\mbox{\boldmath$\nabla$}^{\underline{2}})^{2}~{},$
(6.202f)
$\displaystyle\mbox{\boldmath$\nabla$}^{{\underline{2}}\beta}\mbox{\boldmath$\nabla$}^{\underline{2}}_{\alpha}\mbox{\boldmath$\nabla$}^{\underline{2}}_{\beta}$
$\displaystyle=$ $\displaystyle-4{\rm