November, 2020

Revised version: June, 2021

Higher-spin gauge models with (1,1) supersymmetry in AdS3: Reduction to (1,0)
superspace

Daniel Hutchings, Jessica Hutomo and Sergei M. Kuzenko

Department of Physics M013, The University of Western Australia
35 Stirling Highway, Crawley W.A. 6009, Australia

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In three dimensions, there are two types of ${\cal N}=2$ anti-de Sitter (AdS)
supersymmetry, which are denoted (1,1) and (2,0). They are characterised by
different supercurrents and support different families of higher-spin gauge
models (massless and massive) which were constructed in Refs. [1] and [2] for
the (1,1) and (2,0) cases, respectively, using superspace techniques. It turns
out that the precise difference between the (1,1) and (2,0) higher-spin
supermultiplets can be pinned down by reducing these gauge theories to (1,0)
AdS superspace. The present paper is devoted to the $(1,1)\to(1,0)$ AdS
superspace reduction. In conjunction with the outcomes of the $(2,0)\to(1,0)$
AdS superspace reduction carried out in Ref. [3], we demonstrate that every
known higher-spin theory with (1,1) or (2,0) AdS supersymmetry decomposes into
a sum of two off-shell (1,0) supermultiplets which belong to four series of
inequivalent higher-spin gauge models. The latter are reduced to components.

###### Contents

  1. 1 Introduction
  2. 2 ${{\cal N}}=1$ supersymmetric field theory in AdS3
  3. 3 Massless higher-spin ${\cal N}=1$ supermultiplets
    1. 3.1 Massless half-integer superspin theories
      1. 3.1.1 Longitudinal formulation
      2. 3.1.2 Transverse formulation
    2. 3.2 Massless integer superspin theories
      1. 3.2.1 Longitudinal formulation
      2. 3.2.2 Transverse formulation
  4. 4 (1,1) $\rightarrow$ (1,0) AdS superspace reduction
    1. 4.1 (1,1) AdS superspace: Complex basis
    2. 4.2 (1,1) AdS superspace: Real basis
    3. 4.3 Reduction from (1,1) to ${\cal N}=1$ AdS superspace
    4. 4.4 The (1,1) AdS supersymmetric actions in AdS3|2
  5. 5 Massless half-integer superspin: Transverse formulation
    1. 5.1 Transverse formulation
    2. 5.2 Reduction of gauge prepotentials to $\text{AdS}^{3|2}$
    3. 5.3 Massless higher-spin ${\cal N}=1$ supermultiplets
    4. 5.4 Second supersymmetry transformations
  6. 6 Massless half-integer superspin: Longitudinal formulation
    1. 6.1 Longitudinal formulation
    2. 6.2 Reduction of gauge prepotentials to AdS3|2
    3. 6.3 Massless higher-spin ${\cal N}=1$ supermultiplets
    4. 6.4 Second supersymmetry transformations
  7. 7 Massless integer superspin: Longitudinal formulation
    1. 7.1 Longitudinal formulation
    2. 7.2 Reduction of gauge prepotentials to AdS3|2
    3. 7.3 Massless higher-spin ${\cal N}=1$ supermultiplets
    4. 7.4 Second supersymmetry transformations
  8. 8 Massless integer superspin: Transverse formulation
    1. 8.1 Transverse formulation
    2. 8.2 Reduction of gauge prepotentials to AdS3|2
    3. 8.3 Massless higher-spin ${\cal N}=1$ supermultiplets
    4. 8.4 Second supersymmetry transformations
  9. 9 Discussion
  10. A 3D notation and AdS identities
  11. B Component structure of ${\cal N}=1$ supersymmetric higher-spin actions in AdS3
    1. B.1 Fronsdal-type actions in AdS3
    2. B.2 Dual formulation of the Fronsdal-type action in Minkowski space
    3. B.3 Transverse formulation of the superspin-$(s+\frac{1}{2})$ multiplet
    4. B.4 Longitudinal formulation of the superspin-$(s+\frac{1}{2})$ multiplet
    5. B.5 Transverse formulation of the superspin-$s$ multiplet
    6. B.6 Longitudinal formulation of the superspin-$s$ multiplet

##  1 Introduction

To study ${\cal N}$-extended supersymmetric theories in $d$ dimensions, it is
advantageous to deal with a formulation that permits some amount of
supersymmetry to be realised manifestly. In general there exist two superspace
settings to achieve this. One of them makes use of the standard ${\cal
N}$-extended superspace (sometimes endowed with additional commuting variables
[4, 5, 6]) and provides a manifestly supersymmetric formulation. The other
employs a smaller $\widetilde{\cal N}$-extended superspace, with
$\widetilde{\cal N}<{\cal N}$, to keep manifest only $\widetilde{\cal N}$
supersymmetries. Both approaches have found numerous applications in the
literature. For instance, it is known that the geometric properties of general
${\cal N}=2$ supersymmetric nonlinear $\sigma$-models in Minkowski space
${\mathbb{M}}^{4}$ [7] are remarkably transparent if these theories are
realised in ${\cal N}=1$ superspace [8, 9]. One of the two supersymmetries is
manifest and off-shell in this setting, while the second supersymmetry is
hidden and on-shell (the commutator of the first and the second
supersymmetries closes only on-shell). On the other hand, in order to
construct general off-shell ${\cal N}=2$ supersymmetric nonlinear
$\sigma$-models, manifestly supersymmetric techniques are indispensable, and
there exist two powerful ${\cal N}=2$ superspace approaches: (i) the harmonic
superspace [5, 10]; and (ii) the projective superspace [6, 11, 12]. One of the
conceptual virtues of these manifestly supersymmetric formulations is the
possibility to generate ${\cal N}=2$ nonlinear $\sigma$-model actions (and
thus hyperkähler metrics) from Lagrangians of arbitrary functional
form.111This possibility was realised for the first time in an important paper
[13], where it was demonstrated that the most general interacting Lagrangian
of the $q^{+}$ hypermultiplet superfields can be treated as the hyperkähler
potential parametrising the complete set of hyperkähler metrics. Analogous
results exist for general ${\cal N}=2$ supersymmetric nonlinear
$\sigma$-models in four-dimensional anti-de Sitter space (AdS4), which were
originally formulated as off-shell theories in ${\cal N}=2$ AdS superspace
[14]. This approach makes ${\cal N}=2$ supersymmetry manifest, but the
hyperkähler geometry of the $\sigma$-model target space is hidden. Some time
later, the most general ${\cal N}=2$ supersymmetric $\sigma$-models in $\rm
AdS_{4}$ were constructed using a formulation in terms of ${\cal N}=1$
covariantly chiral superfields [15, 16]. One of the two supersymmetries is
hidden in this approach, but the geometric properties of the ${\cal N}=2$
supersymmetric $\sigma$-models in $\rm AdS_{4}$ become transparent.
Specifically, the target space must be a non-compact hyperkähler manifold
endowed with a Killing vector field which generates an $\mathsf{SO}(2)$ group
of rotations on the two-sphere of complex structures. The two $\sigma$-model
formulations developed in [14] and [15, 16] are related via the ${\cal
N}=2\to{\cal N}=1$ AdS4 superspace reduction worked out in [17].

Extended supersymmetric theories in AdS3 possess exceptionally many superspace
realisations. We recall that the (connected) AdS group in three dimensions is
not a simple group,

$\displaystyle\mathsf{SO}_{0}(2,2)\cong\Big{(}\mathsf{SL}(2,{\mathbb{R}})\times\mathsf{SL}(2,{\mathbb{R}})\Big{)}/{\mathbb{Z}}_{2}~{},$
(1.1)

and so are its supersymmetric extensions
${\mathsf{OSp}}(p|2;{\mathbb{R}})\times{\mathsf{OSp}}(q|2;{\mathbb{R}})$. This
means that there are several species of ${\cal N}$-extended AdS supersymmetry
[18], which are known as the $(p,q)$ AdS supersymmetry types, where the
integers $p\geq q\geq 0$ are such that ${\cal N}=p+q$. In principle, field
theories possessing the $(p,q)$ AdS supersymmetry may be realised in the so-
called $(p,q)$ AdS superspace [19], ${\rm AdS}^{(3|p,q)}$, which may be viewed
as a maximally supersymmetric solution of the $(p,q)$ AdS supergravity [18],
with anti-de Sitter space $\rm AdS_{3}$ being the bosonic body of the
superspace.222In order to realise field theories with the $(p,q)$ AdS
supersymmetry in ${\rm AdS}^{(3|p,q)}$, for $p+q\leq 4$ one can employ the
off-shell supergravity methods developed in [20, 21, 22]. More specifically,
within the supergravity framework of [19, 20] the superspace ${\rm
AdS}^{(3|p,q)}$ originates as a maximally symmetric supergeometry with
covariantly constant torsion and curvature generated by a symmetric real
torsion $S^{IJ}=S^{JI}$, with the structure-group indices $I,J$ taking values
from 1 to ${\cal N}$. It turns out that $S^{IJ}$ is nonsingular, and the
parameters $p$ and $q={\cal N}-p$ determine its signature. Since the isometry
group of ${\rm AdS}^{(3|p,q)}$ is
${\mathsf{OSp}}(p|2;{\mathbb{R}})\times{\mathsf{OSp}}(q|2;{\mathbb{R}})$ and
since $S^{IJ}$ is invariant under its compact subgroup
$\mathsf{SO}(p)\times{\mathsf{SO}}(q)$, the global realisation of the
superspace is

$\displaystyle{\rm
AdS}^{(3|p,q)}=\frac{{\mathsf{OSp}}(p|2;{\mathbb{R}})\times{\mathsf{OSp}}(q|2;{\mathbb{R}})}{{\mathsf{SL}}(2,{\mathbb{R}})\times{\mathsf{SO}}(p)\times{\mathsf{SO}}(q)}~{}.$
(1.2)

In fact, starting from the superspace geometry of ${\cal N}$-extended
conformal supergravity [20] and restricting the torsion to be covariantly
constant and Lorentz invariant, a general AdS superspace solution for ${\cal
N}\geq 4$ includes not only the torsion $S^{IJ}$, but also a completely
antisymmetric curvature $X^{IJKL}=X^{[IJKL]}$. It turns out that the latter
may be non-zero only if $S^{IJ}=S\delta^{IJ}$, which means $p={\cal N}$ and
$q=0$. Such solutions correspond to exotic AdS superspaces, ${\rm
AdS}^{3|2{\cal N}}_{S,X}$, for which the isometry group is, in general, a
subgroup of $\rm OSp({\cal N}|2;{\mathbb{R}})\times SL(2,{\mathbb{R}})$. The
superspace ${\rm AdS}^{3|2{\cal N}}_{S,X}$ is not conformally flat. All the
superspaces (1.2) are conformally flat, see [19] for the technical
details.333It should be pointed out that the superconformal flatness of AdS
superspaces in diverse dimensions, including the $d=3$ case, was studied in
Ref. [23] where the coset spaces (1.2) with $p+q=2$ were briefly discussed. It
should be mentioned that (1,0) AdS superspace (or simply ${\cal N}=1$ AdS
superspace, ${\rm AdS}^{(3|1,0)}\equiv\rm AdS^{3|2}$) was introduced in [24].

Consider a $(p,q)$ supersymmetric field theory formulated in ${\rm
AdS}^{(3|p,q)}$, with $p+q\geq 3$ and $p\geq q$. As argued in [25], it can
always be recast as a supersymmetric theory realised in $(2,0)$ AdS
superspace, with $(p+q-2)$ supersymmetries being hidden. In the case $p+q\geq
3$ and $p\geq q>0$, every supersymmetric field theory in ${\rm
AdS}^{(3|p,q)}$, can be reformulated as a theory in (1,1) AdS superspace. Such
reformulations were developed in [25] for general $(p,q)$ supersymmetric
nonlinear $\sigma$-models in AdS3, with $p+q\leq 4$. It is obvious that the
list of possible superspace embeddings can be continued. It is also clear that
every extended supersymmetric theory in AdS3, with $p+q>1$, may be realised in
${\cal N}=1$ AdS superspace.

This work is devoted to implementing the $(1,1)\to(1,0)$ AdS reduction of the
$(1,1)$ supersymmetric higher-spin models in AdS3 (both massless and massive),
which were constructed in [1] as an extension of the earlier results in ${\cal
N}=2$ Minkowski superspace [26].444The massless higher-spin theories proposed
in [1, 26] have natural counterparts in four dimensions [27, 28, 29], see also
[30] for a review. There are two main motivations for this project. Firstly,
the off-shell structure of the half-integer superspin multiplets with (2,0)
AdS supersymmetry [2] drastically differs from that of their (1,1)
counterparts [1].555It is pertinent to mention here that the (2,0) and (1,1)
AdS supersymmetries support different supercurrent multiplets [21]. Direct
comparison of these theories is difficult in a manifestly supersymmetric
setting, since they are formulated in different superspaces, ${\rm
AdS}^{(3|2,0)}$ and ${\rm AdS}^{(3|1,1)}$, respectively. However, both
families of higher-spin theories can be reformulated in the same (1,0) AdS
superspace, and then the precise difference between the (2,0) and (1,1)
higher-spin supermultiplets can be pinned down. The $(2,0)\to(1,0)$ reduction
of the (2,0) supersymmetric higher-spin theories in AdS3 [2] has already been
carried out in [3]. We will show that every massless higher-spin theory with
(1,1) or (2,0) AdS supersymmetry decomposes into a sum of two off-shell (1,0)
supermultiplets which belong to the four series of inequivalent higher-spin
gauge models constructed in [3].

The second motivation for this work is quantum mechanical. All known off-shell
higher-spin ${\cal N}=2$ supermultiplets in AdS3, with either (2,0) or (1,1)
AdS supersymmetry [2, 1], are reducible gauge theories (in the terminology of
the Batalin-Vilkovisky quantisation [31]), similar to the massless higher-spin
supermultiplets in AdS4 [29]. The Lagrangian quantisation of such theories
proves to be a nontrivial procedure. In four dimensions, the quantisation of
the theories proposed in [29] was achieved in [32]. On the other hand, all
off-shell higher-spin ${\cal N}=1$ supermultiplets in AdS3 studied in this
paper and [3], are irreducible gauge theories. They can be quantised using the
Faddeev-Popov procedure [33], as in the non-supersymmetric case, see e.g.
[34]. This opens the possibility to develop heat kernel techniques for higher-
spin theories in AdS3|2, by extending the four-dimensional results [30, 35,
36].

This paper is organised as follows. Section 2 provides a succinct review of
${{\cal N}}=1$ supersymmetric field theory in AdS3. In section 3, we review
the structure of the four series of off-shell massless higher-spin ${{\cal
N}}=1$ supersymmetric models in AdS3 constructed in [3, 37]. A formalism to
reduce every field theory with (1,1) AdS supersymmetry to ${{\cal N}}=1$ AdS
superspace is developed in section 4. This formalism is then applied
throughout sections 5$-$8 to carry out the (1,1) $\to$ (1,0) AdS superspace
reduction of massless higher-spin models with (1,1) AdS supersymmetry. These
are detailed in sections 5$-$6 for the half-integer $(s+\frac{1}{2})$
superspin case, and in sections 7$-$8 for the integer $(s)$ superspin case.
Our results are summarised and discussed in section 9, where we also analyse
the difference between the (2,0) and (1,1) higher-spin (massless and massive)
supermultiplets. Our spinor notation and conventions are collected in appendix
A. In appendix B, we study the component structure of the off-shell massless
higher-spin $\mathcal{N}=1$ supersymmetric models in AdS3 reviewed in section
3.

##  2 ${{\cal N}}=1$ supersymmetric field theory in AdS3

In this section we provide salient facts about the geometry of ${{\cal N}}=1$
AdS superspace, AdS3|2, and its isometries following [19]. We also recall
certain duality transformations in AdS3|2, following [3].

Let $z^{\cal M}=(x^{m},\theta^{\mu})$ be local coordinates parametrising
AdS3|2. The geometry of AdS3|2 is encoded in the set of covariant derivatives
of the form

$\nabla_{A}=(\nabla_{a},\nabla_{\alpha})=E_{A}{}^{M}{\partial}_{M}+\frac{1}{2}\omega_{A}{}^{bc}M_{bc}~{},$
(2.1)

where $E_{A}{}^{M}$ is the inverse vielbein and $\omega_{A}{}^{bc}$ the
Lorentz connection. The explicit relation between Lorentz generators with two
vector indices $(M_{ab})$, with one vector index $(M_{a})$ and with two spinor
indices $(M_{\alpha\beta})$ is described in appendix A. The covariant
derivatives obey the following algebra [24]

$\\{\nabla_{\alpha},\nabla_{\beta}\\}=2{\rm i}\nabla_{\alpha\beta}-4{\rm
i}|\mu|M_{\alpha\beta}~{},\\\ $ (2.2a) $\
[\nabla_{a},\nabla_{\beta}]=|\mu|(\gamma_{a})_{\beta}{}^{\gamma}\nabla_{\gamma}~{},\qquad\
[\nabla_{a},\nabla_{b}]=-4|\mu|^{2}M_{ab}~{}.$ (2.2b) In spinor notation, eqs.
(2.2b) take the form $\displaystyle\
[\nabla_{\alpha\beta},\nabla_{\gamma}]=-2|\mu|\varepsilon_{\gamma(\alpha}\nabla_{\beta)}~{},\qquad\
[\nabla_{\alpha\beta},\nabla_{\gamma\delta}]=4|\mu|^{2}\Big{(}\varepsilon_{\gamma(\alpha}M_{\beta)\delta}+\varepsilon_{\delta(\alpha}M_{\beta)\gamma}\Big{)}~{}.$
(2.2c)

Here $|\mu|>0$ is a constant parameter, which determines the curvature of
AdS3.

The parameter $|\mu|$ in (2.2) was denoted ${\cal S}$ in [19]. However, in
this paper we prefer to make use of the notation $|\mu|$ which is more
appropriate in the context of the $(1,1)\to(1,0)$ superspace reduction, which
will be considered later. It should be remarked that the geometry of ${\cal
N}=1$ AdS superspace can also be described by the graded commutation relations
which are obtained from (2.2) by replacing $|\mu|\to-|\mu|$. The two choices,
${\cal S}=|\mu|$ and ${\cal S}=-|\mu|$, correspond to the so-called $(1,0)$
and $(0,1)$ AdS superspaces [19], which are different realisations of ${\cal
N}=1$ AdS superspace. The $(1,0)$ and $(0,1)$ AdS superspaces are naturally
embedded in $(1,1)$ AdS superspace666 As follows from (4.19), in $(1,1)$ AdS
superspace two subsets of covariant derivatives,
$({\mbox{\boldmath${\nabla}$}}_{a},{\mbox{\boldmath${\nabla}$}}_{\alpha}^{{\underline{1}}})$
and
$({\mbox{\boldmath${\nabla}$}}_{a},{\mbox{\boldmath${\nabla}$}}_{\alpha}^{{\underline{2}}})$,
form closed algebras of the type (2.2), with the curvature parameter being
$|\mu|$ and $-|\mu|$, respectively. and are related to each other by a parity
transformation.

One may derive several useful identities:

$\displaystyle\nabla_{\alpha}\nabla_{\beta}$ $\displaystyle=$
$\displaystyle{\rm i}\nabla_{\alpha\beta}-2{\rm
i}|\mu|M_{\alpha\beta}+\frac{1}{2}\varepsilon_{\alpha\beta}\nabla^{2}~{},$
(2.3a) $\displaystyle\nabla^{\beta}\nabla_{\alpha}\nabla_{\beta}$
$\displaystyle=$ $\displaystyle 4{\rm
i}|\mu|\nabla_{\alpha}~{},\quad\\{\nabla^{2},\nabla_{\alpha}\\}=4{\rm
i}|\mu|\nabla_{\alpha}~{},$ (2.3b) $\displaystyle\nabla^{2}\nabla_{\alpha}$
$\displaystyle=$ $\displaystyle 2{\rm i}|\mu|\nabla_{\alpha}+2{\rm
i}\nabla_{\alpha\beta}\nabla^{\beta}-4{\rm
i}|\mu|\nabla^{\beta}M_{\alpha\beta}~{},$ (2.3c) $\displaystyle\qquad\
[\nabla_{\alpha}\nabla_{\beta},\nabla^{2}]$ $\displaystyle=$ $\displaystyle
0\quad\Longrightarrow\quad\ [\nabla_{\alpha\beta},\nabla^{2}]=0~{},$ (2.3d)
$\displaystyle\ [\nabla_{\alpha},\Box]$ $\displaystyle=$ $\displaystyle
2|\mu|\nabla_{\alpha\beta}\nabla^{\beta}+3|\mu|^{2}\nabla_{\alpha}~{}.$ (2.3e)

where we have denoted $\nabla^{2}=\nabla^{\alpha}\nabla_{\alpha}$ and
$\Box={\nabla}^{a}{\nabla}_{a}=-\frac{1}{2}{\nabla}^{\alpha\beta}{\nabla}_{\alpha\beta}$.
In particular, it follows from the algebra of the covariant derivatives that

$\displaystyle-\frac{1}{4}\nabla^{2}\nabla^{2}$ $\displaystyle=$
$\displaystyle\Box-2{\rm
i}|\mu|\nabla^{2}+2|\mu|\nabla^{\alpha\beta}M_{\alpha\beta}-2|\mu|^{2}M^{\alpha\beta}M_{\alpha\beta}~{}.$
(2.4)

This differential operator can be expressed in terms of the quadratic Casimir
operator of the ${\cal N}=1$ AdS supergroup [37]

$\mathbb{Q}=-\frac{1}{4}\nabla^{2}\nabla^{2}+{\rm
i}|\mu|\nabla^{2},\qquad[\mathbb{Q},\nabla_{A}]=0~{}.$ (2.5)

Given an arbitrary superfield $F$ and its complex conjugate $\bar{F}$, the
following relation holds

$\displaystyle\overline{\nabla_{\alpha}F}=-(-1)^{\epsilon(F)}\nabla_{\alpha}\bar{F}~{},$
(2.6)

where $\epsilon(F)$ denotes the Grassmann parity of $F$.

According to the general formalism of [30], the isometry transformations of
AdS3|2 are generated by the Killing supervector fields,

$\displaystyle\xi=\xi^{B}\nabla_{B}=\xi^{b}\nabla_{b}+\xi^{\beta}\nabla_{\beta}~{},$
(2.7)

which by definition solve the master equation [19]

$[\xi+\frac{1}{2}\zeta^{bc}M_{bc},\nabla_{A}]=0~{},$ (2.8)

for some Lorentz parameter $\zeta^{bc}=-\zeta^{cb}$. It can be shown that the
Killing equation (2.8) is equivalent to the set of relations [19] (see also
[37])

$\displaystyle\nabla_{(\alpha}\xi_{\beta\gamma)}$ $\displaystyle=$
$\displaystyle 0~{},\qquad\nabla_{\beta}\xi^{\beta\alpha}=-6{\rm
i}\xi^{\alpha}~{},$ (2.9a) $\displaystyle\nabla_{(\alpha}\zeta_{\beta\gamma)}$
$\displaystyle=$ $\displaystyle
0~{},\qquad\nabla_{\beta}\zeta^{\beta\alpha}=-12{\rm i}|\mu|\xi^{\alpha}~{},$
(2.9b) $\displaystyle\nabla_{\alpha}\xi_{\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\zeta_{\alpha\beta}+|\mu|\xi_{\alpha\beta}~{},$
(2.9c)

which imply

$\displaystyle{\nabla}_{a}\xi_{b}+{\nabla}_{b}\xi_{a}$ $\displaystyle=$
$\displaystyle 0~{},$ (2.10a)
$\displaystyle{\nabla}_{\alpha\beta}\xi^{\beta}+3|\mu|\xi_{\alpha}$
$\displaystyle=$ $\displaystyle 0~{},$ (2.10b) $\displaystyle\big{(}{\rm
i}{\nabla}^{2}+12|\mu|\big{)}\xi_{\alpha}$ $\displaystyle=$ $\displaystyle
0~{}.$ (2.10c)

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

$\displaystyle\delta_{\xi}U=\big{(}\xi^{a}\nabla_{a}+\xi^{\alpha}\nabla_{\alpha}+\frac{1}{2}\zeta^{ab}M_{ab}\big{)}U~{}.$
(2.11)

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

$\displaystyle S=\int{\rm d}^{3}x\,{\rm d}^{2}\theta E\,L~{},\qquad
E^{-1}={\rm Ber}(E_{A}{}^{M})~{}.$ (2.12)

In what follows, we make use of the notation ${\rm d}^{3|2}z:={\rm d}^{3}x{\rm
d}^{2}\theta$.

The component form of the action (2.12) is

$\displaystyle S=\frac{1}{4}\int{\rm d}^{3}x\,e\,({\rm
i}\nabla^{2}+8|\mu|)L\,\big{|}_{\,\theta=0}~{},\qquad e^{-1}={\rm
det}(e_{a}{}^{m})~{}.$ (2.13)

Here the component inverse vielbein is defined as
$e_{a}{}^{m}(x)=E_{a}{}^{m}|_{\theta=0}$. Making use of the ${\cal N}=1$ AdS
transformation law $\delta_{\xi}L=\xi L$ and the identities (2.9) and (2.10),
it can be shown that the action (2.12) is invariant under the isometry group
of AdS3|2.

To conclude this section, we review the duality transformation described in
[3]. Consider a field theory in AdS3|2 which is formulated in terms of a real
tensor superfield $V_{\alpha(n)}$ with the action

$\displaystyle S^{\parallel}[V_{\alpha(n)}]=\int{\rm d}^{3|2}z\,E\,{\cal
L}\big{(}{\rm i}^{n+1}\nabla_{\beta}V_{\alpha(n)}\big{)}~{}.$ (2.14)

We call $\nabla_{\beta}V_{\alpha(n)}$ a longitudinal superfield, by analogy
with a longitudinal vector field. The theory (2.14) has a dual formulation
that is obtained by introducing a first-order action

$\displaystyle S_{\text{first-order}}=\int{\rm d}^{3|2}z\,E\,\Big{\\{}{\cal
L}\big{(}\Sigma_{\beta;\,\alpha(n)}\big{)}+{\rm i}^{n+1}{\cal
W}^{\beta;\,\alpha(n)}\Sigma_{\beta;\,\alpha(n)}\Big{\\}}~{}.$ (2.15)

Here $\Sigma_{\beta;\alpha(n)}$ is unconstrained and the Lagrange multiplier
is

$\displaystyle{\cal W}_{\beta;\,\alpha(n)}={\rm
i}^{n+1}\Big{(}\nabla^{\gamma}\nabla_{\beta}-4{\rm
i}|\mu|\delta^{\gamma}_{\,\beta}\Big{)}{\Psi}_{\gamma;\,\alpha(n)}~{},\qquad\nabla^{\beta}{\cal
W}_{\beta;\,\alpha(n)}=0~{},$ (2.16)

for some unconstrained prepotential ${\Psi}_{\gamma;\,\alpha(n)}$. Varying
(2.15) with respect to ${\Psi}_{\gamma;\,\alpha(n)}$ gives

$\displaystyle\nabla^{\beta}\nabla_{\gamma}\Sigma_{\beta;\,\alpha(n)}-4{\rm
i}|\mu|\Sigma_{\gamma;\,\alpha(n)}=0\quad\implies\quad\Sigma_{\beta;\,\alpha(n)}={\rm
i}^{n+1}\nabla_{\beta}V_{\alpha(n)}~{},$ (2.17)

thus $S_{\text{first-order}}$ reduces to the original action (2.14). On the
other hand, integrating out $\Sigma_{\beta;\alpha(n)}$ from $S_{\text{first-
order}}$ leads to a dual action of the form

$\displaystyle S^{\perp}[{\Psi}_{\gamma;\,\alpha(n)}]=\int{\rm
d}^{3|2}z\,E\,{\cal L}_{\rm dual}\big{(}{\cal
W}_{\beta;\,\alpha(n)}\big{)}~{},$ (2.18)

which is invariant under gauge transformations

$\displaystyle\delta{\Psi}_{\gamma;\,\alpha(n)}={\rm
i}^{n+1}\nabla_{\gamma}\eta_{\alpha(n)}~{}.$ (2.19)

It is natural to call the gauge-invariant field strength ${\cal
W}_{\beta;\,\alpha(n)}$ a transverse superfield, due to constraint (2.16). The
dual formulations (2.14) and (2.18) are referred to as longitudinal and
transverse, respectively. In the $n=1$ case, the duality transformation
corresponds to the standard duality between the ${\cal N}=1$ scalar and vector
multiplets in three dimensions, see [38].

The above duality transformation naturally extends to Minkowski superspace
${\mathbb{M}}^{3|2}$ in the limit $|\mu|\to 0$. Therefore, given two dually
equivalent theories in $\rm AdS^{3|2}$, their flat superspace counterparts are
also dually equivalent. The opposite is not always true. For instance, the
flat superspace counterparts of the massless half-integer superspin models
(3.1.1) and (3.1.2) are dual. However, these models are not dual in AdS3. The
requirement of gauge invariance uniquely fixes the actions (3.1.1) and (3.1.2)
in $\rm AdS^{3|2}$, including the presence of certain $|\mu|$-dependent terms
which are incompatible with duality invariance.

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

According to [3], for each superspin value $\hat{s}\geq 1$, where $\hat{s}$ is
either integer $(\hat{s}=s)$ or half-integer $(\hat{s}=s+\frac{1}{2})$, there
exist two off-shell formulations for a massless ${\cal N}=1$
superspin-$\hat{s}$ multiplet in AdS3. In the flat superspace limit, these
models prove to related to each other via a superfield Legendre
transformation. However, when uplifted to ${\cal N}=1$ AdS superspace, it
becomes apparent that these duality relations no longer hold. In this section,
we review the explicit formulation of these respective theories, which were
derived in [3, 37].

As is known, any massless multiplet of superspin $\hat{s}>1$ has no
propagating degrees of freedom, and the notion of superspin is purely
kinematical. It is used by analogy with massive supermultiplets. The concept
of superspin is well defined in the massive case, and we follow the definition
used in [39] in the super-Poincaré case. Specifically, if $n$ is a positive
integer, an on-shell massive multiplet of superspin $n/2$ is described by a
real symmetric rank-$n$ spinor superfield, ${\cal
H}_{\alpha_{1}\cdots\alpha_{n}}=\bar{\cal H}_{\alpha_{1}\dots\alpha_{n}}={\cal
H}_{(\alpha_{1}\cdots\alpha_{n})}$, which obeys the differential conditions
[40]

$\displaystyle D^{\beta}{\cal H}_{\beta\alpha_{1}\cdots\alpha_{n-1}}$
$\displaystyle=$ $\displaystyle 0~{},$ (3.1a) $\displaystyle-\frac{{\rm
i}}{2}D^{2}{\cal H}_{\alpha_{1}\dots\alpha_{n}}$ $\displaystyle=$
$\displaystyle m\sigma{\cal H}_{\alpha_{1}\dots\alpha_{n}}~{},\qquad\sigma=\pm
1~{}.$ (3.1b)

Here, $m$ is a real constant of unit mass dimension, $D_{\alpha}$ is the
spinor covariant derivative of ${\cal N}=1$ Minkowski superspace, and
$D^{2}=D^{\alpha}D_{\alpha}$. The superfield ${\cal H}_{\alpha(n)}$ contains
two ordinary on-shell massive fields, which are

$\displaystyle\phi_{\alpha_{1}\dots\alpha_{n}}:={\cal
H}_{\alpha_{1}\dots\alpha_{n}}|_{\theta=0}~{},\qquad\phi_{\alpha_{1}\dots\alpha_{n+1}}:={\rm
i}^{n+1}D_{(\alpha_{1}}{\cal
H}_{\alpha_{2}\dots\alpha_{n+1})}|_{\theta=0}~{}.$ (3.2)

Their helicity values are $\frac{n}{2}\sigma$ and $\frac{n+1}{2}\sigma$,
respectively, see [39] for the technical details.

An off-shell massless superspin-$n/2$ gauge theory in AdS3 can be realised in
terms of two superfields, of which one is universally a superconformal gauge
prepotential $H_{\alpha(n)}$ and the other is a compensating multiplet. Here
$H_{\alpha(n)}:=H_{\alpha_{1}....\alpha_{n}}=H_{(\alpha_{1}...\alpha_{n})}$ is
a real symmetric rank-$n$ spinor which is defined modulo gauge transformations
of the form777In the flat superspace limit this gauge transformation reduces
to that introduced in [41, 39].

$\displaystyle\delta_{\zeta}H_{\alpha(n)}={\rm i}^{n}(-1)^{\lfloor
n/2\rfloor}\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{n})}~{},\ $ (3.3)

where the gauge parameter $\zeta_{\alpha(n-1)}$ is a real unconstrained
superfield, and ${\lfloor x\rfloor}$ stands for the floor function denoting
the integer part of a real number $x\geq 0$.

###  3.1 Massless half-integer superspin theories

There exist two off-shell formulations for the massless
superspin-$(s+\frac{1}{2})$ multiplet, which are referred to as longitudinal
and transverse.

####  3.1.1 Longitudinal formulation

The longitudinal formulation is realised in terms of the real unconstrained
dynamical variables

${\cal
V}^{\parallel}_{(s+\frac{1}{2})}=\big{\\{}H_{\alpha(2s+1)},L_{\alpha(2s-2)}\big{\\}}~{},$
(3.4)

which are defined modulo gauge transformations of the form

$\displaystyle\delta H_{\alpha(2s+1)}$ $\displaystyle=$ $\displaystyle{\rm
i}\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{2s+1})}~{},$ (3.5a)
$\displaystyle\delta L_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\frac{s}{2(2s+1)}\nabla^{\beta\gamma}\zeta_{\beta\gamma\alpha(2s-2)}~{},$
(3.5b)

where the real gauge parameter $\zeta_{\alpha(2s)}$ is unconstrained. The
unique gauge-invariant action formulated in terms of the superfields (3.4)
takes the following form

$\displaystyle{S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3|2}z\,E\,\bigg{\\{}-\frac{{\rm
i}}{2}H^{\alpha(2s+1)}{\mathbb{Q}}H_{\alpha(2s+1)}}$ $\displaystyle-\frac{{\rm
i}}{8}\nabla_{\beta}H^{\beta\alpha(2s)}\nabla^{2}\nabla^{\gamma}H_{\gamma\alpha(2s)}+\frac{{\rm
i}s}{4}{\nabla}_{\beta\gamma}H^{\beta\gamma\alpha(2s-1)}{\nabla}^{\rho\delta}H_{\rho\delta\alpha(2s-1)}$
$\displaystyle+(2s-1)L^{\alpha(2s-2)}\nabla^{\beta\gamma}\nabla^{\delta}H_{\beta\gamma\delta\alpha(2s-2)}$
$\displaystyle+2(2s-1)\Big{(}L^{\alpha(2s-2)}({\rm
i}\nabla^{2}-4|\mu|)L_{\alpha(2s-2)}-\frac{{\rm
i}}{s}(s-1)\nabla_{\beta}L^{\beta\alpha(2s-3)}\nabla^{\gamma}L_{\gamma\alpha(2s-3)}\Big{)}$
$\displaystyle+|\mu|\Big{(}s\,{\nabla}_{\beta}H^{\beta\alpha(2s)}{\nabla}^{\gamma}H_{\gamma\alpha(2s)}+\frac{1}{2}(2s+1)H^{\alpha(2s+1)}(\nabla^{2}-4{\rm
i}|\mu|)H_{\alpha(2s+1)}\Big{)}\bigg{\\}}~{},$ (3.6)

where $\mathbb{Q}$ is the quadratic Casimir operator
$\eqref{QuadraticCasimir}$. The action (3.1.1) was derived in [37]. In the
flat superspace limit, $|\mu|\rightarrow 0$, the action (3.1.1) coincides with
the model derived in [39].

####  3.1.2 Transverse formulation

The transverse formulation is constructed in terms of the real superfields

${\cal
V}^{\perp}_{(s+\frac{1}{2})}=\big{\\{}H_{\alpha(2s+1)},\Upsilon_{\beta;\alpha(2s-2)}\big{\\}}~{},$
(3.7)

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

$\Upsilon_{\beta;\alpha(2s-2)}=\Upsilon_{(\beta;\alpha_{1}...\alpha_{2s-2})}+\frac{1}{2s-1}\sum_{k=1}^{2s-2}\varepsilon_{\beta\alpha_{k}}\Upsilon^{\gamma;}{}_{\gamma\alpha_{1}...\hat{\alpha}_{k}...\alpha_{2s-2}}~{}.$
(3.8)

The dynamical variables (3.7) possess the following gauge freedom

$\displaystyle\delta H_{\alpha(2s+1)}$ $\displaystyle=$ $\displaystyle{\rm
i}\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{2s+1})}~{},$ (3.9a)
$\displaystyle\delta\Upsilon_{\beta;\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{{\rm
i}}{2s+1}\Big{(}\nabla^{\gamma}\zeta_{\gamma\beta\alpha(2s-2)}+(2s+1)\nabla_{\beta}\eta_{\alpha(2s-2)}\Big{)}~{},$
(3.9b)

where the gauge parameters $\zeta_{\alpha(2s)}$ and $\eta_{\alpha(2s-2)}$ are
real unconstrained. The unique quadratic action which is invariant under gauge
transformations (3.9) takes the form

$\displaystyle
S^{\perp}_{(s+\frac{1}{2})}[{H}_{\alpha(2s+1)},\Upsilon_{\beta;\,\alpha(2s-2)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3|2}z\,E\,\bigg{\\{}-\frac{{\rm
i}}{2}H^{\alpha(2s+1)}{\mathbb{Q}}H_{\alpha(2s+1)}$ $\displaystyle-\frac{{\rm
i}}{8}\nabla_{\beta}H^{\beta\alpha(2s)}\nabla^{2}\nabla^{\gamma}H_{\gamma\alpha(2s)}+\frac{{\rm
i}}{8}{\nabla}_{\beta\gamma}H^{\beta\gamma\alpha(2s-1)}{\nabla}^{\rho\delta}H_{\rho\delta\alpha(2s-1)}$
$\displaystyle-\frac{{\rm
i}}{4}(2s-1)\Omega^{\beta;\,\alpha(2s-2)}\nabla^{\gamma\delta}H_{\gamma\delta\beta\alpha(2s-2)}+{\rm
i}s(2s-1)|\mu|\,\Upsilon^{\beta;\,\alpha(2s-2)}\Omega_{\beta;\,\alpha(2s-2)}$
$\displaystyle-\frac{{\rm
i}}{8}(2s-1)\Big{(}\Omega^{\beta;\,\alpha(2s-2)}\Omega_{\beta;\,\alpha(2s-2)}-2(s-1)\Omega_{\beta;}\,^{\beta\alpha(2s-3)}\Omega^{\gamma;}\,_{\gamma\alpha(2s-3)}\Big{)}$
$\displaystyle+|\mu|\Big{(}H^{\alpha(2s+1)}\big{(}\nabla^{2}-4{\rm
i}|\mu|\big{)}H_{\alpha(2s+1)}+\frac{1}{2}\
{\nabla}_{\beta}H^{\beta\alpha(2s)}{\nabla}^{\gamma}H_{\gamma\alpha(2s)}\Big{)}\bigg{\\}}~{},$
(3.10)

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

$\displaystyle\Omega_{\beta;\,\alpha(2s-2)}=-{\rm
i}\big{(}\nabla^{\gamma}\nabla_{\beta}-4{\rm
i}|\mu|\delta_{\beta}\,^{\gamma}\big{)}{\Upsilon}_{\gamma;\,\alpha(2s-2)}~{},\qquad\nabla^{\beta}\Omega_{\beta;\,\alpha(2s-2)}=0~{}.$
(3.11)

The above theory was introduced in [3].

###  3.2 Massless integer superspin theories

Analogous to the half-integer case, there exist two off-shell formulations for
the massless superspin-$s$ multiplet, which are referred to as longitudinal
and transverse.

####  3.2.1 Longitudinal formulation

The longitudinal formulation is realised in terms of the real unconstrained
variables

${\cal
V}^{\parallel}_{(s)}=\big{\\{}H_{\alpha(2s)},V_{\alpha(2s-2)}\big{\\}}~{},$
(3.12)

which are defined modulo gauge transformations of the form

$\displaystyle\delta H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{2s})}~{},$
(3.13a) $\displaystyle\delta V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s}\nabla^{\beta}\zeta_{\beta\alpha(2s-2)}~{},$ (3.13b)

where the gauge parameter $\zeta_{\alpha(2s-1)}$ is real unconstrained. The
unique action which is invariant under gauge transformations (3.13) assumes
the form

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

Up to normalisation, action (3.2.1) coincides with the off-shell ${\cal N}=1$
supersymmetric action for the massless superspin-$s$ multiplet derived in
[37]. Taking the flat-superspace limit, action (3.2.1) reduces to the model
derived in [39].

####  3.2.2 Transverse formulation

The transverse formulation is constructed in terms of the real unconstrained
superfields

${\cal
V}^{\perp}_{(s)}=\big{\\{}H_{\alpha(2s)},\Psi_{\beta;\alpha(2s-2)}\big{\\}}~{},$
(3.15)

which have the following gauge freedom

$\displaystyle\delta H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{2s})}~{},$
(3.16a) $\displaystyle\delta\Psi_{\beta;\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\zeta_{\beta\alpha(2s-2)}+{\rm
i}\nabla_{\beta}\eta_{\alpha(2s-2)}~{},$ (3.16b)

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

$\displaystyle
S^{\perp}_{(s)}[H_{\alpha(2s)},{\Psi}_{\beta;\,\alpha(2s-2)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3|2}z\,E\,\bigg{\\{}\frac{1}{2}H^{\alpha(2s)}({\rm
i}\nabla^{2}+8s|\mu|)H_{\alpha(2s)}$ $\displaystyle-{\rm
i}s\nabla_{\beta}H^{\beta\alpha(2s-1)}\nabla^{\gamma}H_{\gamma\alpha(2s-1)}-(2s-1){\cal
W}^{\beta;\,\alpha(2s-2)}\nabla^{\gamma}H_{\gamma\beta\alpha(2s-2)}$
$\displaystyle-\frac{{\rm i}}{2}(2s-1)\Big{(}{\cal
W}^{\beta;\,\alpha(2s-2)}{\cal W}_{\beta;\,\alpha(2s-2)}+\frac{s-1}{s}{\cal
W}_{\beta;}\,^{\beta\alpha(2s-3)}{\cal
W}^{\gamma;}\,_{\gamma\alpha(2s-3)}\Big{)}$ $\displaystyle-2{\rm
i}(2s-1)|\mu|\Psi^{\beta;\,\alpha(2s-2)}{\cal
W}_{\beta;\,\alpha(2s-2)}\bigg{\\}}~{},$ (3.17)

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

$\displaystyle{\cal W}_{\beta;\,\alpha(2s-2)}=-{\rm
i}\big{(}\nabla^{\gamma}\nabla_{\beta}-4{\rm
i}|\mu|\delta_{\beta}{}^{\gamma}\big{)}{\Psi}_{\gamma;\,\alpha(2s-2)}~{},\qquad\nabla^{\beta}{\cal
W}_{\beta;\,\alpha(2s-2)}=0~{}.$ (3.18)

For $s>1$, the $\zeta$-gauge freedom (3.16b) can be used to impose the gauge
condition

$\displaystyle\Psi_{(\alpha_{1};\,\alpha_{2}\dots\alpha_{2s-1})}=0\quad\Longleftrightarrow\quad\Psi_{\beta;\,\alpha(2s-2)}=\sum_{k=1}^{2s-2}\varepsilon_{\beta\alpha_{k}}\varphi_{\alpha_{1}\dots\hat{\alpha}_{k}\dots\alpha_{2s-2}}~{},$
(3.19)

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

$\displaystyle\zeta_{\alpha(2s-1)}={\rm
i}\nabla_{(\alpha_{1}}\eta_{\alpha_{2}\dots\alpha_{2s-1})}~{},$ (3.20)

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

$\displaystyle\big{\\{}H_{\alpha(2s)},~{}\varphi_{\alpha(2s-3)}\big{\\}}~{},$
(3.21)

which are defined modulo gauge transformations of the form

$\displaystyle\delta H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle-\nabla_{(\alpha_{1}\alpha_{2}}\eta_{\alpha_{3}\dots\alpha_{2s})}~{},$
(3.22) $\displaystyle\delta\varphi_{\alpha(2s-3)}$ $\displaystyle=$
$\displaystyle{\rm i}\nabla^{\beta}\eta_{\beta\alpha(2s-3)}~{}.$ (3.23)

The corresponding gauge-invariant action assumes the following form

$\displaystyle
S^{\perp}_{(s)}[H_{\alpha(2s)},\varphi_{\alpha(2s-3)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3|2}z~{}E~{}\bigg{\\{}\frac{1}{2}H^{\alpha(2s)}({\rm
i}\nabla^{2}+8s|\mu|)H_{\alpha(2s)}~{}$ $\displaystyle-{\rm
i}s\nabla_{\beta}H^{\beta\alpha(2s-1)}\nabla^{\gamma}H_{\gamma\alpha(2s-1)}-2(s-1)\varphi^{\alpha(2s-3)}\nabla^{\beta}\nabla^{\gamma\delta}H_{\beta\gamma\delta\alpha(2s-3)}~{}$
$\displaystyle+\frac{1}{s}(s-1)\Big{(}8{\rm
i}(2s-1)|\mu|^{2}\varphi^{\alpha(2s-3)}\varphi_{\alpha(2s-3)}+2(2s-3)|\mu|\varphi^{\alpha(2s-3)}\nabla^{2}\varphi_{\alpha(2s-3)}$
$\displaystyle-2{\rm
i}\varphi^{\alpha(2s-3)}\mathbb{Q}\varphi_{\alpha(2s-3)}+\frac{{\rm
i}}{2(2s-1)}(2s-3)\nabla_{\beta}\varphi^{\beta\alpha(2s-4)}\nabla^{2}\nabla^{\gamma}\varphi_{\gamma\alpha(2s-4)}~{}$
$\displaystyle+\frac{4}{2s-1}(2s-3)(3s-2)|\mu|\nabla_{\beta}\varphi^{\beta\alpha(2s-4)}\nabla^{\gamma}\varphi_{\gamma\alpha(2s-4)}~{}$
$\displaystyle-\frac{{\rm
i}}{2s-1}(2s-3)(s-2)\nabla_{\beta\gamma}\varphi^{\beta\gamma\alpha(2s-5)}\nabla^{\rho\sigma}\varphi_{\rho\sigma\alpha(2s-5)}\Big{)}\bigg{\\}}~{}.$
(3.24)

The above theory was proposed in [3]. In the flat-superspace limit, the action
(3.2.2) coincides with the ${\cal N}=1$ model (B.25) presented in [1].

##  4 (1,1) $\rightarrow$ (1,0) AdS superspace reduction

In this section, we begin by reviewing the necessary aspects of $(1,1)$ AdS
superspace which are then utilised to develop a consistent reduction procedure
for field theories with (1,1) AdS supersymmetry to ${\cal N}=1$ AdS
superspace.

###  4.1 (1,1) AdS superspace: Complex basis

We summarise the key results concerning (1,1) AdS superspace [21, 1] and
superfield representations of the associated isometry group ${\rm
OSp(1|2;{\mathbb{R}})}\times{\rm OSp(1|2;{\mathbb{R}})}$, following the
presentation in [1]. The geometry of (1,1) AdS superspace, ${\rm
AdS}^{(3|1,1)}$, can be described using either a real or complex basis for the
spinor covariant derivatives. In this subsection we consider the formulation
in the complex basis.

The covariant derivatives of ${\rm AdS}^{(3|1,1)}$ have the form

$\displaystyle{{\cal D}}_{{\cal A}}=({\cal D}_{a},{\cal D}_{\alpha},\bar{{\cal
D}}^{\alpha})=E_{{\cal A}}+\Omega_{{{\cal A}}}~{},$ (4.1)

where $z^{{\cal M}}=(x^{m},\theta^{\mu},\bar{\theta}_{\mu})$ are local complex
superspace coordinates, while $E_{{\cal A}}$ and $\Omega_{{\cal A}}$ denote
the inverse supervielbein and Lorentz connection,

$\displaystyle E_{{\cal A}}=E_{{\cal A}}{}^{{\cal M}}\frac{\partial}{\partial
z^{{\cal M}}}~{},\qquad\Omega_{{\cal A}}=\frac{1}{2}\Omega_{{\cal
A}}{}^{bc}M_{bc}=\frac{1}{2}\Omega_{{\cal
A}}{}^{\beta\gamma}M_{\beta\gamma}~{}.$ (4.2)

The covariant derivatives satisfy the following algebra

$\displaystyle\qquad\\{{\cal D}_{\alpha},\bar{\cal D}_{\beta}\\}=-2\rm i{\cal
D}_{\alpha\beta}~{},$ (4.3a) $\displaystyle\qquad\\{{\cal D}_{\alpha},{\cal
D}_{\beta}\\}=-4\bar{\mu}\,M_{\alpha\beta}~{},\qquad\\{{\bar{\cal
D}}_{\alpha},{\bar{\cal D}}_{\beta}\\}=4\mu\,M_{\alpha\beta}~{},$ (4.3b)
$\displaystyle\qquad[{\cal D}_{\alpha\beta},{\cal D}_{\gamma}]=-2\rm
i\bar{\mu}\,\varepsilon_{\gamma(\alpha}\bar{\cal
D}_{\beta)}~{},\qquad\,\,[{\cal D}_{\alpha\beta},{\bar{\cal D}}_{\gamma}]=2\rm
i\mu\,\varepsilon_{\gamma(\alpha}{\cal D}_{\beta)}~{},$ (4.3c)
$\displaystyle\quad\,\,\,\,[{\cal D}_{\alpha\beta},{\cal
D}_{\gamma\delta}]=4\bar{\mu}\mu\Big{(}\varepsilon_{\gamma(\alpha}M_{\beta)\delta}+\varepsilon_{\delta(\alpha}M_{\beta)\gamma}\Big{)}~{},$
(4.3d)

with $\mu\neq 0$ being a complex parameter determining the constant curvature
of ${\rm AdS}^{(3|1,1)}$. The phase of $\mu=|\mu|{\rm e}^{{\rm i}\varphi}$ can
be given any fixed value by a re-definition ${\cal D}_{\alpha}\to{\rm e}^{{\rm
i}\rho}{\cal D}_{\alpha}$ and $\bar{\cal D}_{\alpha}\to{\rm e}^{-{\rm
i}\rho}\bar{\cal D}_{\alpha}$, with $\rho$ constant.

Let ${G}_{\alpha(n)}$ be a symmetric rank-$n$ spinor superfield. It is said to
be longitudinal linear if it obeys the following first-order constraint

$\displaystyle\bar{\cal
D}_{(\alpha_{1}}{G}_{\alpha_{2}\dots\alpha_{n+1})}=0~{},$ (4.4a) which implies
$\displaystyle\big{(}\bar{\cal D}^{2}+2n\mu\big{)}{G}_{\alpha(n)}$
$\displaystyle=$ $\displaystyle 0~{}.$ (4.4b)

In the scalar case, $n=0$, the constraint (4.4a) becomes the condition of
covariant chirality, $\bar{\cal D}_{\alpha}{G}=0$.

A symmetric rank-$n$ spinor superfield ${\Gamma}_{\alpha(n)}$ is called
transverse linear if it obeys the first-order constraint

$\displaystyle\bar{\cal
D}^{\beta}{\Gamma}_{\beta\alpha_{1}\dots\alpha_{n-1}}=0~{},\qquad n\neq 0~{},$
(4.5a) which implies $\displaystyle\big{(}\bar{\cal
D}^{2}-2(n+2)\mu\big{)}{\Gamma}_{\alpha(n)}=0~{}.$ (4.5b)

In the $n=0$ case, the constraint (4.5a) is not defined. However its corollary
(4.5b) is perfectly consistent,

$\displaystyle\big{(}\bar{\cal D}^{2}-4\mu\big{)}{\Gamma}=0~{},$ (4.6)

and defines a covariantly linear scalar superfield ${\Gamma}$.

The constraints (4.4a) and (4.5a) may be solved in terms of prepotentials
${\Psi}_{\alpha(n-1)}$ and ${\Phi}_{\alpha(n+1)}$ as follows:

$\displaystyle{G}_{\alpha(n)}$ $\displaystyle=$ $\displaystyle\bar{\cal
D}_{(\alpha_{1}}{\Psi}_{\alpha_{2}\dots\alpha_{n})}~{},$ (4.7a)
$\displaystyle{\Gamma}_{\alpha(n)}$ $\displaystyle=$ $\displaystyle\bar{\cal
D}^{\beta}{\Phi}_{(\beta\alpha_{1}\dots\alpha_{n})}~{}.$ (4.7b)

Provided the constraints (4.4a) and (4.5a) are the only conditions imposed on
${G}_{\alpha(n)}$ and ${\Gamma}_{\alpha(n)}$ respectively, the prepotentials
${\Psi}_{\alpha(n-1)}$ and ${\Phi}_{\alpha(n+1)}$ can be chosen to be
unconstrained complex. They are defined modulo gauge transformations of the
form

$\displaystyle\delta_{\zeta}{\Psi}_{\alpha(n-1)}$ $\displaystyle=$
$\displaystyle\bar{\cal
D}_{(\alpha_{1}}{\zeta}_{\alpha_{2}\dots\alpha_{n-1})}~{},$ (4.8a)
$\displaystyle\delta_{\xi}{\Phi}_{\alpha(n+1)}$ $\displaystyle=$
$\displaystyle\bar{\cal
D}^{\gamma}{\xi}_{(\gamma\alpha_{1}\dots\alpha_{n+1})}~{},$ (4.8b)

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

One can define projectors $P^{\perp}_{n}$ and $P^{||}_{n}$ on the spaces of
transverse linear and longitudinal linear superfields, respectively. The
projectors are

$\displaystyle P^{\perp}_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{4(n+1)\mu}(\bar{\cal D}^{2}+2n\mu)~{},$ (4.9a)
$\displaystyle P^{||}_{n}$ $\displaystyle=$
$\displaystyle-\frac{1}{4(n+1)\mu}(\bar{\cal D}^{2}-2(n+2)\mu)~{},$ (4.9b)

with the properties

$\displaystyle\big{(}P^{\perp}_{n}\big{)}^{2}=P^{\perp}_{n}~{},\quad\big{(}P^{||}_{n}\big{)}^{2}=P^{||}_{n}~{},\quad
P^{\perp}_{n}P^{||}_{n}=P^{||}_{n}P^{\perp}_{n}=0~{}.$ (4.10)

These projectors are three-dimensional (3D) cousins of those introduced by
Ivanov and Sorin [42] in the case of ${\cal N}=1$ AdS supersymmetry in four
dimensions.

Let ${V}_{\alpha(n)}$ ($n\neq 0$) be an arbitrary complex tensor superfield.
It can be represented as a sum of transverse linear and longitudinal linear
multiplets [1]

$\displaystyle{V}_{\alpha(n)}=$ $\displaystyle-$
$\displaystyle\frac{1}{2\mu(n+2)}{\bar{\cal D}}^{\gamma}{\bar{\cal
D}}_{(\gamma}{V}_{\alpha_{1}\dots\alpha_{n})}-\frac{1}{2\mu(n+1)}{\bar{\cal
D}}_{(\alpha_{1}}{\bar{\cal
D}}^{|\gamma|}{V}_{\alpha_{2}\dots\alpha_{n})\gamma}~{},~{}~{}~{}$ (4.11)

where indices placed between vertical bars (for example $|\gamma|$) are not
subject to symmetrisation. If we choose ${V}_{\alpha(n)}$ to be either
longitudinal linear (${G}_{\alpha(n)}$) or transverse linear
(${\Gamma}_{\alpha(n)}$), then the above identity produces the relations
(4.7a) and (4.7b) for some prepotentials ${\Psi}_{\alpha(n-1)}$ and
${\Phi}_{\alpha(n+1)}$, respectively.

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

$\displaystyle\Big{[}{\Lambda}+\frac{1}{2}l^{ab}M_{ab},{\cal D}_{{\cal
C}}\Big{]}=0~{},$ (4.12)

where

$\displaystyle\Lambda=\lambda^{{\cal A}}{\cal D}_{{\cal A}}=\lambda^{a}{\cal
D}_{a}+\lambda^{\alpha}{\cal D}_{\alpha}+\bar{\lambda}_{\alpha}\bar{\cal
D}^{\alpha}~{},\qquad\bar{\lambda}^{a}=\lambda^{a}~{},$ (4.13)

and $l^{ab}$ is some local Lorentz parameter. As shown in [21], this equation
implies that the parameters $\lambda^{\alpha}$ and $l^{ab}$ can be uniquely
determined in terms of the vector $\lambda^{a}$,

$\displaystyle\lambda_{\alpha}=\frac{{\rm i}}{6}\bar{\cal
D}^{\beta}\lambda_{\alpha\beta}~{},\qquad l_{\alpha\beta}=2{\cal
D}_{(\alpha}\lambda_{\beta)}~{},$ (4.14)

and the vector parameter obeys the equation

$\displaystyle{\cal
D}_{(\alpha}\lambda_{\beta\gamma)}=0\quad\Longleftrightarrow\quad\bar{\cal
D}_{(\alpha}\lambda_{\beta\gamma)}=0~{}.$ (4.15)

In comparison with the 3D ${\cal N}=2$ Minkowski superspace, the specific
feature of ${\rm AdS}^{(3|1,1)}$ is that any two of the three parameters
$\\{\lambda_{{\alpha\beta}},\lambda_{\alpha},l_{\alpha\beta}\\}$ can be
expressed in terms of the third parameter. In particular,

$\displaystyle\lambda_{\alpha\beta}=\frac{{\rm i}}{\mu}\bar{\cal
D}_{(\alpha}\lambda_{\beta)}~{},\qquad\lambda_{\alpha}=\frac{1}{12\bar{\mu}}{\cal
D}^{\beta}l_{\alpha\beta}~{}.$ (4.16)

From (4.14) and (4.16) we deduce

$\displaystyle\bar{\cal D}^{\alpha}\lambda_{\alpha}={\cal
D}_{\alpha}\lambda^{\alpha}=0~{}.$ (4.17)

The solution to these equations is given in [21].

###  4.2 (1,1) AdS superspace: Real basis

It proves beneficial to realise the (1,1) AdS covariant derivatives in a real
basis when performing the reduction procedure to ${\cal N}=1$ AdS superspace.
In accordance with [19], the algebra of covariant derivative (4.3) can be
converted to the real basis by (i) making a convenient choice $\mu=-{\rm
i}|\mu|$; and (ii) replacing the complex operators ${\cal
D}_{\alpha},\bar{\cal D}_{\alpha}$ with
${\mbox{\boldmath${\nabla}$}}^{I}_{\alpha}=({\mbox{\boldmath${\nabla}$}}^{{\underline{1}}}_{\alpha},{\mbox{\boldmath${\nabla}$}}^{{\underline{2}}}_{\alpha})$
defined by

$\displaystyle\qquad{\cal
D}_{\alpha}=\frac{1}{\sqrt{2}}({\mbox{\boldmath${\nabla}$}}^{{\underline{1}}}_{\alpha}-{\rm
i}{\mbox{\boldmath${\nabla}$}}^{{\underline{2}}}_{\alpha})~{},\qquad\bar{{\cal
D}}_{\alpha}=-\frac{1}{\sqrt{2}}({\mbox{\boldmath${\nabla}$}}^{{\underline{1}}}_{\alpha}+{\rm
i}{\mbox{\boldmath${\nabla}$}}^{{\underline{2}}}_{\alpha})~{}.$ (4.18)

Furthermore, we introduce the real coordinates $z^{{\cal
M}}=(x^{m},\theta^{\mu}_{I})$ which are used to parametrise $(1,1)$ AdS
superspace. Choosing to define ${\mbox{\boldmath${\nabla}$}_{a}}={\cal
D}_{a}$, it can be shown that the algebra of $(1,1)$ AdS covariant derivatives
assumes the following form in the real basis (4.18)

$\displaystyle\qquad\\{{\mbox{\boldmath${\nabla}$}}^{{\underline{1}}}_{\alpha},{\mbox{\boldmath${\nabla}$}}^{{\underline{2}}}_{\beta}\\}$
$\displaystyle=0~{},$ (4.19a)
$\displaystyle\qquad\\{{\mbox{\boldmath${\nabla}$}}^{{\underline{1}}}_{\alpha},{\mbox{\boldmath${\nabla}$}}^{{\underline{1}}}_{\beta}\\}$
$\displaystyle=2{\rm i}{\mbox{\boldmath${\nabla}$}}_{\alpha\beta}-4{\rm
i}|\mu|M_{\alpha\beta}~{},$
$\displaystyle\qquad\\{{\mbox{\boldmath${\nabla}$}}^{{\underline{2}}}_{\alpha},{\mbox{\boldmath${\nabla}$}}^{{\underline{2}}}_{\beta}\\}$
$\displaystyle=2{\rm i}{\mbox{\boldmath${\nabla}$}}_{\alpha\beta}+4{\rm
i}|\mu|M_{\alpha\beta}~{},$ (4.19b)
$\displaystyle\qquad[{\mbox{\boldmath${\nabla}$}}_{a},{\mbox{\boldmath${\nabla}$}}^{{\underline{1}}}_{\alpha}]$
$\displaystyle=|\mu|(\gamma_{a})_{\alpha}{}^{\beta}{\mbox{\boldmath${\nabla}$}}^{{\underline{1}}}_{\beta}~{},$
$\displaystyle\qquad[{\mbox{\boldmath${\nabla}$}}_{a},{\mbox{\boldmath${\nabla}$}}^{{\underline{2}}}_{\alpha}]$
$\displaystyle=-|\mu|(\gamma_{a})_{\alpha}{}^{\beta}{\mbox{\boldmath${\nabla}$}}^{{\underline{2}}}_{\beta}~{},$
(4.19c)
$\displaystyle\qquad[{\mbox{\boldmath${\nabla}$}}_{a},{\mbox{\boldmath${\nabla}$}}_{b}]$
$\displaystyle=-4|\mu|^{2}M_{ab}~{}.$ (4.19d)

It is apparent from (4.19) that the operators
${\mbox{\boldmath${\nabla}$}}_{a}$ and
${\mbox{\boldmath${\nabla}$}}_{\alpha}^{{\underline{1}}}$ possess the
following properties:

  1. 1.

These operators form a closed algebra given by

$\\{{\mbox{\boldmath${\nabla}$}}_{\alpha}^{{\underline{1}}},{\mbox{\boldmath${\nabla}$}}^{{\underline{1}}}_{\beta}\\}=2{\rm
i}{\mbox{\boldmath${\nabla}$}}_{\alpha\beta}-4{\rm i}|\mu|M_{\alpha\beta}~{},$
(4.20a)
$\quad[{\mbox{\boldmath${\nabla}$}}_{a},{\mbox{\boldmath${\nabla}$}}_{\beta}^{{\underline{1}}}]=|\mu|(\gamma_{a})_{\beta}{}^{\gamma}{\mbox{\boldmath${\nabla}$}}^{{\underline{1}}}_{\gamma}~{},\quad[{\mbox{\boldmath${\nabla}$}}_{a},{\mbox{\boldmath${\nabla}$}}_{b}]=-4|\mu|^{2}M_{ab}~{}.$
(4.20b)

  2. 2.

Relations (4.20) coincide with the algebra of covariant derivatives of AdS3|2,
eq. (2.2).

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

We now wish to recast the fundamental properties of the Killing supervector
fields of (1,1) AdS superspace (4.13) in the real representation (4.18). The
isometries of (1,1) AdS superspace are generated by the (1,1) AdS Killing
supervector fields,

$\Lambda:=\lambda^{\cal A}\mbox{\boldmath$\nabla$}_{\cal
A}=\lambda^{a}\mbox{\boldmath$\nabla$}_{a}+\lambda^{\alpha}_{I}\mbox{\boldmath$\nabla$}^{I}_{\alpha}~{},\qquad
I={\underline{1}},{\underline{2}}~{},$ (4.21)

which are defined to satisfy the Killing equation

$[\Lambda+\frac{1}{2}l^{ab}M_{ab},\mbox{\boldmath$\nabla$}_{\cal A}]=0~{},$
(4.22)

for some real Lorentz parameter $l^{ab}=-l^{ba}$. It can be shown that
equation (4.22) is equivalent to the set of equations

$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha}\lambda^{\underline{1}}_{\beta}$
$\displaystyle=$
$\displaystyle\frac{1}{2}l_{\alpha\beta}+|\mu|\lambda_{\alpha\beta}~{},\qquad\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha}\lambda^{\underline{2}}_{\beta}=0~{},$
(4.23a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}}_{\alpha}\lambda^{\underline{2}}_{\beta}$
$\displaystyle=$
$\displaystyle\frac{1}{2}l_{\alpha\beta}-|\mu|\lambda_{\alpha\beta}~{},\qquad\mbox{\boldmath$\nabla$}^{\underline{2}}_{\alpha}\lambda^{\underline{1}}_{\beta}=0~{},$
(4.23b)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha}l_{\beta\gamma}$
$\displaystyle=$ $\displaystyle 8{\rm
i}|\mu|\varepsilon_{\alpha(\beta}\lambda^{\underline{1}}{}_{\gamma)}~{},\qquad\mbox{\boldmath$\nabla$}^{\underline{2}}_{\alpha}l_{\beta\gamma}=-8{\rm
i}|\mu|\varepsilon_{\alpha(\beta}\lambda^{\underline{2}}{}_{\gamma)}~{},$
(4.23c) $\displaystyle\mbox{\boldmath$\nabla$}^{I}_{\alpha}\lambda_{a}$
$\displaystyle=$ $\displaystyle 2{\rm
i}(\gamma_{a})_{\alpha\beta}\lambda^{I\beta}~{},$ (4.23d)

and

$\displaystyle\mbox{\boldmath$\nabla$}_{a}\lambda_{b}$ $\displaystyle=$
$\displaystyle l_{ab}~{},$ (4.24a)
$\displaystyle\mbox{\boldmath$\nabla$}_{a}\lambda^{\underline{1}}_{\alpha}$
$\displaystyle=$
$\displaystyle|\mu|(\gamma_{a})_{\alpha}{}^{\beta}\lambda^{\underline{1}}_{\beta}~{},\qquad\mbox{\boldmath$\nabla$}_{a}\lambda^{\underline{2}}_{\alpha}=-|\mu|(\gamma_{a})_{\alpha}{}^{\beta}\lambda^{\underline{2}}_{\beta}~{},$
(4.24b) $\displaystyle\mbox{\boldmath$\nabla$}_{a}l_{bc}$ $\displaystyle=$
$\displaystyle 8|\mu|^{2}\eta_{a[b}\lambda_{c]}~{}.$ (4.24c)

Equations (4.23) and (4.24) can be recast in the equivalent form

$\displaystyle\mbox{\boldmath$\nabla$}^{I}_{(\alpha}\lambda_{\beta\gamma)}$
$\displaystyle=$ $\displaystyle
0~{},\qquad\mbox{\boldmath$\nabla$}^{I}_{(\alpha}l_{\beta\gamma)}=0~{},$
(4.25a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{1}}_{(\alpha}\lambda^{\underline{1}}_{\beta)}$
$\displaystyle=$
$\displaystyle\frac{1}{2}l_{\alpha\beta}+|\mu|\lambda_{\alpha\beta}~{},\qquad\mbox{\boldmath$\nabla$}^{\underline{2}}_{(\alpha}\lambda^{\underline{2}}_{\beta)}=\frac{1}{2}l_{\alpha\beta}-|\mu|\lambda_{\alpha\beta}~{},$
(4.25b)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{1}}_{(\alpha}\lambda^{\underline{2}}_{\beta)}$
$\displaystyle=$
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}}_{(\alpha}\lambda^{\underline{1}}_{\beta)}=0,\qquad\mbox{\boldmath$\nabla$}^{\alpha(I}\lambda^{J)}_{\alpha}=0~{},$
(4.25c) $\displaystyle\lambda^{{\underline{1}}\alpha}$ $\displaystyle=$
$\displaystyle\frac{{\rm
i}}{6}\mbox{\boldmath$\nabla$}^{\underline{1}}_{\beta}\lambda^{\alpha\beta}=\frac{{\rm
i}}{12|\mu|}\mbox{\boldmath$\nabla$}^{\underline{1}}_{\beta}l^{\alpha\beta}~{},$
(4.25d) $\displaystyle\lambda^{{\underline{2}}\alpha}$ $\displaystyle=$
$\displaystyle\frac{{\rm
i}}{6}\mbox{\boldmath$\nabla$}^{\underline{2}}_{\beta}\lambda^{\alpha\beta}=-\frac{{\rm
i}}{12|\mu|}\mbox{\boldmath$\nabla$}^{\underline{2}}_{\beta}l^{\alpha\beta}~{}.$
(4.25e)

It follows from (4.24a) that the parameter $\lambda_{a}$ is a Killing vector
field

$\mbox{\boldmath$\nabla$}_{a}\lambda_{b}+\mbox{\boldmath$\nabla$}_{b}\lambda_{a}=0~{},$
(4.26)

and relations (4.24b) are Killing spinor equations.

###  4.3 Reduction from (1,1) to ${\cal N}=1$ AdS superspace

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

$U|:=U(x,\theta_{I})|_{\theta_{\underline{2}}=0}$ (4.27)

in a special coordinate system which will be described below. Given the
$(1,1)$ covariant derivative in the the real representation (4.18)

$\mbox{\boldmath$\nabla$}_{\cal
A}=(\mbox{\boldmath$\nabla$}_{a},\mbox{\boldmath$\nabla$}^{I}_{\alpha})=E_{{\cal
A}}{}^{{\cal M}}\frac{\partial}{\partial z^{{\cal
M}}}+\frac{1}{2}\Omega_{{\cal A}}{}^{bc}M_{bc}~{},$ (4.28)

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

$\mbox{\boldmath$\nabla$}_{\cal A}|=E_{{\cal A}}{}^{{\cal
M}}|\frac{\partial}{\partial z^{{\cal M}}}+\frac{1}{2}\Omega_{{\cal
A}}{}^{bc}|M_{bc}~{}.$ (4.29)

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$}^{\underline{1}}_{\alpha}|=\nabla_{\alpha}~{},$
(4.30)

where $\nabla_{A}=(\nabla_{a},\nabla_{\alpha})$ is the set of covariant
derivatives for AdS3|2, see eq. (2.1). 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$}^{\underline{1}}_{\alpha_{1}}...\mbox{\boldmath$\nabla$}^{\underline{1}}_{\alpha_{k}}U)|=\nabla_{\alpha_{1}}...\nabla_{\alpha_{k}}U|$.

We now consider the ${\cal N}=1$ projection of the $(1,1)$ AdS Killing
supervector (4.21)

$\Lambda|=\xi^{a}\nabla_{a}+\xi^{\alpha}\nabla_{\alpha}+\epsilon^{\alpha}\mbox{\boldmath$\nabla$}^{\underline{2}}_{\alpha}|~{},$
(4.31)

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

$\xi^{a}:=\lambda^{a}|~{},\qquad\xi^{\alpha}:=\lambda^{\alpha}_{\underline{1}}|~{},\qquad\epsilon^{\alpha}:=\lambda^{\alpha}_{\underline{2}}|~{}.$
(4.32)

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

$\zeta^{ab}:=l^{ab}|~{}.$ (4.33)

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. (2.8). Indeed, the relations (2.9) and
(2.10) automatically follow from the $(1,1)$ AdS Killing equations (4.25),
upon projection. The parameter $\epsilon_{\alpha}$, which generates the second
supersymmetry transformation, has the property

$\displaystyle\nabla_{a}\epsilon_{\alpha}$ $\displaystyle=$
$\displaystyle-|\mu|(\gamma_{a})_{\alpha}{}^{\beta}\epsilon_{\beta}~{}.$
(4.34)

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

$\delta_{\Lambda}U=\big{(}\Lambda+\frac{1}{2}l^{ab}M_{ab}\big{)}U~{},$ (4.35)

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

$\delta_{\Lambda}U|=\delta_{\xi}U|+\delta_{\epsilon}U|~{},$ (4.36)

where

$\displaystyle\delta_{\xi}U|$ $\displaystyle=$
$\displaystyle\big{(}\xi^{a}\nabla_{a}+\xi^{\alpha}\nabla_{\alpha}+\frac{1}{2}\zeta^{ab}M_{ab}\big{)}U|~{},$
(4.37a) $\displaystyle\delta_{\epsilon}U|$ $\displaystyle=$
$\displaystyle\epsilon^{\alpha}\big{(}\mbox{\boldmath$\nabla$}^{\underline{2}}_{\alpha}U\big{)}|~{}.$
(4.37b)

The first transformation (4.37a) coincides with the infinitesimal
transformation generated by a Killing supervector in AdS3|2, eq. (2.11). Thus,
$U|$ can be identified as a tensor superfield on ${\cal N}=1$ AdS superspace.
The other transformation (4.37b) corresponds to the second supersymmetry
transformation, which is generated by $\epsilon^{\alpha}$.

###  4.4 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 subsection, we explore the
necessary mathematical framework which will be employed to develop such a
reduction procedure. As presented in [20, 21, 22, 25], 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$}\,{\cal L}=\frac{1}{16}\int{\rm
d}^{3}x\,e\,({\cal D}^{2}-16\bar{\mu})(\bar{\cal D}^{2}-4\mu){\cal
L}\Big{|}_{\theta=0}$ $\displaystyle=\int{\rm
d}^{3}x\,e\,\Big{(}\frac{1}{16}{\cal D}^{\alpha}(\bar{{\cal D}}^{2}-6\mu){\cal
D}_{\alpha}-\frac{\mu}{4}{\cal D}^{2}-\frac{\bar{\mu}}{4}\bar{{\cal
D}}^{2}+4\mu\bar{\mu}\Big{)}{\cal L}\Big{|}_{\theta=0}~{},$ (4.38)

with $\mbox{\boldmath$E$}^{-1}={\rm Ber}(E_{{\cal A}}{}^{{\cal M}})$.

  2. 2.

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

$\int{\rm d}^{3}x\,{\rm d}^{2}\theta\,{\cal E}\,{\cal
L}_{c}=-\frac{1}{4}\int{\rm d}^{3}x\,e\,({\cal D}^{2}-16\bar{\mu}){\cal
L}_{c}\Big{|}_{\theta=0}~{},\qquad\bar{{\cal D}}_{\alpha}{\cal L}_{c}=0~{},$
(4.39)

where ${\cal E}$ is the chiral density. 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$}\,{\cal L}=\int{\rm d}^{3}x\,{\rm
d}^{2}\theta\,{\cal E}\,{\cal L}_{c}~{},\qquad{\cal
L}_{c}:=-\frac{1}{4}(\bar{\cal D}^{2}-4\mu){\cal L}~{}.$ (4.40)

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$}\,{\cal L}_{c}~{}.$ (4.41)

We will use the notation ${\rm d}^{3|4}z:={\rm d}^{3}x\,{\rm d}^{2}\theta{\rm
d}^{2}\bar{\theta}$ for 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 $L$

$S=\int{\rm d}^{3|2}z\,E\,L=\frac{1}{4}\int{\rm d}^{3}x\,e\,({\rm
i}\nabla^{2}+8|\mu|)L\,\big{|}_{\,\theta=0}~{}.$ (4.42)

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$}^{\underline{2}})^{2}+8{\rm
i}|\mu|\Big{\\}}{\cal L}\big{|}~{},$ (4.43)

By making use of the Killing equations (4.25) and (4.34), it can be shown that
the action (4.43) is invariant under $(1,1)$ AdS isometry transformations
given by equation (4.36).

In the remainder of this paper, we will carry out the $(1,1)\rightarrow(1,0)$
AdS superspace reduction of the massless higher-spin supermultiplets, and show
that the reduced actions coincide with those presented in section 3.

##  5 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 $s\geq 2$
[1]. 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.

###  5.1 Transverse formulation

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

${\cal
V}^{\perp}_{(s+\frac{1}{2})}=\big{\\{}\mathfrak{H}_{\alpha(2s)},\Gamma_{\alpha(2s-2)},\bar{\Gamma}_{\alpha(2s-2)}\big{\\}}~{},$
(5.1)

where $\mathfrak{H}_{\alpha(2s)}=\mathfrak{H}_{(\alpha_{1}...\alpha_{2s})}$ is
an unconstrained real superfield, and the complex superfield
$\Gamma_{\alpha(2s-2)}=\Gamma_{(\alpha_{1}...\alpha_{2s-2})}$ is transverse
linear (4.5a). The superfields $\mathfrak{H}_{\alpha(2s)}$ and
$\Gamma_{\alpha(2s-2)}$ are defined modulo gauge transformations of the form

$\displaystyle\delta_{\lambda}\mathfrak{H}_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle{\bar{\cal
D}}_{(\alpha_{1}}\lambda_{\alpha_{2}\dots\alpha_{2s})}-{{\cal
D}}_{(\alpha_{1}}\bar{\lambda}_{\alpha_{2}\dots\alpha_{2s})}\equiv
g_{\alpha(2s)}+\bar{g}_{\alpha(2s)}~{},$ (5.2a)
$\displaystyle\delta_{\lambda}\Gamma_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\bar{{\cal D}}^{\beta}\big{(}{{\cal
D}}^{2}+2(2s-1)\bar{\mu}\big{)}\bar{\lambda}_{\beta\alpha(2s-2)}\equiv\frac{s}{2s+1}\bar{{\cal
D}}^{\beta}{\cal D}^{\gamma}\bar{g}_{(\beta\gamma\alpha{(2s-2)})}~{},$ (5.2b)

where the gauge parameter $\lambda_{\alpha(2s-1)}$ is complex unconstrained.
By construction, the complex gauge parameter $g_{\alpha(2s)}:={\bar{\cal
D}}_{(\alpha_{1}}\lambda_{\alpha_{2}\dots\alpha_{2s})}$ is longitudinal linear
(4.4a). Note that gauge transformation (5.2a) indicates that the superfield
$\mathfrak{H}_{\alpha(2s)}$ corresponds to the ${\cal N}=1$ superconformal
gauge prepotential [26]. Up to normalisation, there exists a unique quadratic
action which is invariant under the gauge transformations (5.2). This action
takes the 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$}\,\bigg{\\{}2s(2s-1)\bar{\mu}\mu\mathfrak{H}^{\alpha(2s)}\mathfrak{H}_{\alpha(2s)}~{}$
$\displaystyle+\frac{1}{8}\mathfrak{H}^{\alpha(2s)}{\cal D}^{\beta}({\bar{\cal
D}}^{2}-6\mu){\cal
D}_{\beta}\mathfrak{H}_{\alpha(2s)}+\mathfrak{H}^{\alpha(2s)}\Big{(}{\cal
D}_{\alpha_{1}}{\bar{\cal
D}}_{\alpha_{2}}{\Gamma}_{\alpha_{3}\dots\alpha_{2s}}~{}-{\bar{\cal
D}}_{\alpha_{1}}{\cal
D}_{\alpha_{2}}\bar{\Gamma}_{\alpha_{3}\dots\alpha_{2s}}\Big{)}~{}$
$\displaystyle+\frac{2s-1}{s}\bar{\Gamma}^{\alpha(2s-2)}{\Gamma}_{\alpha(2s-2)}+\frac{2s+1}{2s}\Big{(}{\Gamma}^{\alpha(2s-2)}{\Gamma}_{\alpha(2s-2)}+\bar{\Gamma}^{\alpha(2s-2)}\bar{\Gamma}_{\alpha(2s-2)}\Big{)}\bigg{\\}}~{}.$
(5.3)

The above construction does not take into consideration the case where $s=1$,
since the transverse linear constraint (4.5a) is ill-defined for $n=0$.
However, corollary (4.5b) is consistent for $n=0$ and defines a covariantly
transverse linear scalar superfield (4.5a). So $\Gamma$ and $\Gamma$ can be
interpreted as compensators, which have the corresponding gauge
transformations

$\displaystyle\delta_{\lambda}\mathfrak{H}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\bar{{\cal D}}_{(\alpha}\lambda_{\beta)}-{\cal
D}_{(\alpha}\bar{\lambda}_{\beta)}~{},$ (5.4a)
$\displaystyle\delta_{\lambda}\Gamma$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\bar{{\cal D}}^{\beta}({\cal
D}^{2}+2\bar{\mu})\bar{\lambda}_{\beta}~{},$ (5.4b)

as a result of (5.2). It is simple to show that constraint (4.5b) is
consistent with the variation (5.4b). Choosing $s=1$ in (5.1) yields a gauge-
invariant action which can be identified as the linearised action for non-
minimal $(1,1)$ AdS supergravity [21].

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

We wish to reduce the gauge prepotentials (5.1) to ${\cal N}=1$ AdS
superspace. We start by reducing the superconformal gauge multiplet
$\mathfrak{H}_{\alpha(2s)}$. Converting the longitudinal linear constraint of
$g_{\alpha(2s)}$ (4.4a) to the real representation (4.18) yields

$\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}g_{\alpha_{2}...\alpha_{2s+1})}={\rm
i}\mbox{\boldmath$\nabla$}^{{\underline{1}}}{}_{(\alpha_{1}}g_{\alpha_{2}...\alpha_{2s+1})}~{}.$
(5.5)

Performing a Taylor expansion of $g_{\alpha(2s)}(\theta^{I})$ about
$\theta^{\underline{2}}$, and using (5.5), we find the independent
$\theta^{\underline{2}}$ -components of $g_{\alpha(2s)}$ to be

$g_{\alpha(2s)}|~{},\qquad\mbox{\boldmath$\nabla$}^{\underline{2}\beta}g_{\beta\alpha(2s-1)}|~{}.$
(5.6)

The gauge transformation (5.2a) allows us to impose the gauge conditions

$\mathfrak{H}_{\alpha(2s)}|=0~{},\qquad\mbox{\boldmath$\nabla$}^{{\underline{2}}\beta}\mathfrak{H}_{\beta\alpha(2s-1)}|=0~{}.$
(5.7)

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$}^{{\underline{2}}}{}_{(\alpha_{1}}\mathfrak{H}_{\alpha_{2}...\alpha_{2s+1})}|~{},$
(5.8a) $\displaystyle H_{\alpha(2s)}:$ $\displaystyle=$
$\displaystyle\frac{{\rm
i}}{4}(\mbox{\boldmath$\nabla$}^{\underline{2}})^{2}\mathfrak{H}_{\alpha(2s)}|~{}.$
(5.8b)

The residual gauge freedom which preserves gauge conditions (5.7) are
described by the unconstrained real ${\cal N}=1$ superfields

$\displaystyle g_{\alpha(2s)}|$ $\displaystyle=$ $\displaystyle-\frac{{\rm
i}}{2}\zeta_{\alpha(2s)}~{},\qquad\bar{\zeta}_{\alpha(2s)}=\zeta_{\alpha(2s)}~{},$
(5.9a)
$\displaystyle\mbox{\boldmath$\nabla$}^{{\underline{2}}\beta}g_{\beta\alpha(2s-1)}|$
$\displaystyle=$
$\displaystyle\frac{2s+1}{2s}\zeta_{\alpha(2s-1)}~{},\qquad\bar{\zeta}_{\alpha(2s-1)}=\zeta_{\alpha(2s-1)}~{}.$
(5.9b)

From (5.9), we can readily determine the gauge transformations of (5.8)

$\displaystyle\delta H_{\alpha(2s+1)}$ $\displaystyle=$ $\displaystyle{\rm
i}\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{2s+1})}~{},$ (5.10a)
$\displaystyle\delta H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{2s})}~{}.$
(5.10b)

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 (4.5a), which takes the following form in the real representation
(4.18)

$\mbox{\boldmath$\nabla$}^{{\underline{2}}\beta}\Gamma_{\beta\alpha(2s-3)}={\rm
i}\mbox{\boldmath$\nabla$}^{{\underline{1}}\beta}\Gamma_{\beta\alpha(2s-3)}~{}.$
(5.11)

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

$\Gamma_{\alpha(2s-2)}|\
,\qquad\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\Gamma_{\alpha_{2}...\alpha_{2s-1})}|~{}.$
(5.12)

Utilising the gauge transformation of $\Gamma_{\alpha(2s-2)}$ (5.2b) and the
real representation (4.18), we find

$\displaystyle\delta\Gamma_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{{\rm
i}s}{2s+1}\Big{(}\mbox{\boldmath$\nabla$}^{{\underline{1}}\beta}\mbox{\boldmath$\nabla$}^{{\underline{2}}\gamma}-\mbox{\boldmath$\nabla$}^{\beta\gamma}\Big{)}\bar{g}_{\beta\gamma\alpha(2s-2)}~{},$
(5.13a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\delta\Gamma_{\alpha_{2}...\alpha_{2s-1})}$
$\displaystyle=$
$\displaystyle\frac{s}{2s+1}\Big{(}\mbox{\boldmath$\nabla$}^{{\underline{1}}\beta}\mbox{\boldmath$\nabla$}_{(\alpha_{1}}{}^{\gamma}\bar{g}_{\alpha_{2}...\alpha_{2s-1})\beta\gamma}-(4s+1)|\mu|\mbox{\boldmath$\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$}^{{\underline{2}}}{}_{(\alpha_{1}}\bar{g}_{\alpha_{2}...\alpha_{2s-1})\beta\gamma}\Big{)}~{}.$

From (5.13), we can immediately read off the gauge transformations of the
complex ${\cal N}=1$ superfields (5.12)

$\displaystyle\delta\Gamma_{\alpha(2s-2)}|$ $\displaystyle=$
$\displaystyle-\frac{{\rm
i}}{2}\nabla^{\beta}\zeta_{\beta\alpha(2s-2)}+\frac{s}{2(2s+1)}\nabla^{\beta\gamma}\zeta_{\beta\gamma\alpha(2s-2)}~{},$
(5.14a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\delta\Gamma_{\alpha_{2}...\alpha_{2s-1})}|$
$\displaystyle=$ $\displaystyle\frac{s}{2(2s+1)}\bigg{(}{\rm
i}\nabla^{\beta}\nabla_{(\alpha_{1}}{}^{\gamma}\zeta_{\alpha_{2}...\alpha_{2s-1})\beta\gamma}+{\rm
i}\nabla_{\beta}{}^{\gamma}\nabla^{\beta}\zeta_{\gamma\alpha(2s-1)}~{}$
$\displaystyle-$ $\displaystyle(4s+1){\rm
i}|\mu|\nabla^{\beta}\zeta_{\beta\alpha(2s-1)}+\frac{2s+1}{s}\Big{(}(2s+1){\rm
i}|\mu|\zeta_{\alpha(2s-1)}~{}$ $\displaystyle+$ $\displaystyle{\rm
i}\nabla_{(\alpha_{1}}{}^{\beta}\zeta_{\alpha_{2}...\alpha_{2s-1})\beta}+\frac{1}{2}\nabla^{2}\zeta_{\alpha(2s-1)}\Big{)}\bigg{)}~{}.$

Let us express the ${{\cal N}}=1$ superfields (5.12) in terms of their real
and imaginary parts,

$\displaystyle\Gamma_{\alpha(2s-2)}|$ $\displaystyle=$ $\displaystyle-
L_{\alpha(2s-2)}-{\rm i}sV_{\alpha(2s-2)}~{},$ (5.15a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\Gamma_{\alpha_{2}...\alpha_{2s-1})}|$
$\displaystyle=$ $\displaystyle\frac{1}{2}\Big{(}\Phi_{\alpha(2s-1)}+{\rm
i}\Omega_{\alpha(2s-1)}\Big{)}.$ (5.15b)

It then follows from gauge transformations (5.10) and (5.14) that we are in
fact dealing with two different gauge theories. The first model is formulated
in terms of the real unconstrained gauge superfields

${\cal
V}^{\parallel}_{(s+\frac{1}{2})}=\\{H_{\alpha(2s+1)},L_{\alpha(2s-2)},\Phi_{\alpha(2s-1)}\\}~{},$
(5.16)

which are defined modulo gauge transformations of the form

$\displaystyle\delta H_{\alpha(2s+1)}$ $\displaystyle=$ $\displaystyle{\rm
i}\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{2s+1})}~{},$ (5.17a)
$\displaystyle\delta L_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\frac{s}{2(2s+1)}\nabla^{\beta\gamma}\zeta_{\beta\gamma\alpha(2s-2)}~{},$
(5.17b) $\displaystyle\delta\Phi_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle\frac{{\rm
i}s}{2s+1}\Big{(}\nabla^{\beta}\nabla_{(\alpha_{1}}{}^{\gamma}\zeta_{\alpha_{2}...\alpha_{2s-1})\beta\gamma}-(4s+1)|\mu|\nabla^{\beta}\zeta_{\beta\alpha(2s-1)}~{}$
$\displaystyle-$
$\displaystyle\nabla^{\beta\gamma}\nabla_{\beta}\zeta_{\gamma\alpha(2s-1)}\Big{)}~{},$

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

${\cal
V}^{\parallel}_{(s)}=\\{H_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-1)}\\}~{},$
(5.18)

which possesses the gauge freedom

$\displaystyle\delta H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{2s})}~{},$
(5.19a) $\displaystyle\delta V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s}\nabla^{\beta}\zeta_{\beta\alpha(2s-2)}~{},$ (5.19b)
$\displaystyle\delta\Omega_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle(2s+1)|\mu|\zeta_{\alpha(2s-1)}+\nabla_{(\alpha_{1}}{}^{\beta}\zeta_{\alpha_{2}...\alpha_{2s-1})\beta}-\frac{{\rm
i}}{2}\nabla^{2}\zeta_{\alpha(2s-1)}~{},$ (5.19c)

where the parameter $\zeta_{\alpha(2s-1)}$ is real unconstrained. Applying the
superspace reduction procedure to the action (5.1) yields two decoupled ${\cal
N}=1$ supersymmetric theories, each formulated in terms of the gauge fields
(5.16) and (5.18) respectively

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

Explicit expressions for the decoupled supersymmetric actions are given in the
following subsection.

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

The first of the decoupled ${\cal N}=1$ supersymmetric actions, which is
realised by the dynamical variables (5.16), takes the following form

$\displaystyle
S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)},\Phi_{\alpha(2s-1)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\Big{(}-\frac{{\rm
i}}{8}\Big{)}\int{\rm
d}^{3|2}z~{}E~{}\bigg{\\{}2H^{\alpha(2s+1)}\mathbb{Q}H_{\alpha(2s+1)}~{}$
$\displaystyle-8(s-2)(2s+1)|\mu|^{2}H^{\alpha(2s+1)}H_{\alpha(2s+1)}+(2s+1){\rm
i}|\mu|H^{\alpha(2s+1)}\nabla^{2}H_{\alpha(2s+1)}~{}$
$\displaystyle-4(2s+1)|\mu|H^{\beta\alpha(2s)}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s)}-{\rm
i}H^{\beta\alpha(2s)}\nabla^{2}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s)}-4H^{\beta\gamma\alpha(2s-1)}\nabla_{\beta\gamma}\Phi_{\alpha(2s-1)}~{}$
$\displaystyle-8{\rm
i}H^{\beta\gamma\delta\alpha(2s-2)}\nabla_{\beta\gamma}\nabla_{\delta}L_{\alpha(2s-2)}-16{\rm
i}L^{\alpha(2s-2)}\nabla^{\beta}\Phi_{\beta\alpha(2s-2)}+\frac{2}{s}\Phi^{\alpha(2s-1)}\Phi_{\alpha(2s-1)}~{}$
$\displaystyle+\frac{8{\rm
i}}{s(2s-1)}\Big{(}2(s-1)(4s-1)L^{\beta\alpha(2s-3)}\nabla_{\beta}{}^{\gamma}L_{\gamma\alpha(2s-3)}~{}$
$\displaystyle-4s(4s^{2}-3s+1)|\mu|L^{\alpha(2s-2)}L_{\alpha(2s-2)}-(3s-1){\rm
i}L^{\alpha(2s-2)}\nabla^{2}L_{\alpha(2s-2)}\Big{)}\bigg{\\}}~{}.$ (5.21)

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

$\Phi_{\alpha(2s-1)}=-s\nabla^{\beta\gamma}H_{\beta\gamma\alpha(2s-1)}-4{\rm
i}s\nabla_{(\alpha_{1}}L_{\alpha_{2}...\alpha_{2s-1})}~{},$ (5.22)

we can eliminate the auxiliary field $\Phi_{\alpha(2s-1)}$ from the action
(5.3). It can be shown that the resulting action coincides with the off-shell
${\cal N}=1$ supersymmetric action for massless half-integer superspin in AdS3
(3.1.1).

The other ${\cal N}=1$ theory is formulated in terms of the gauge superfields
(5.18)

$\displaystyle
S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-1)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3|2}z~{}E~{}\bigg{\\{}2|\mu|H^{\alpha(2s)}H_{\alpha(2s)}$
$\displaystyle+\frac{{\rm
i}}{2}H^{\alpha(2s)}\nabla^{2}H_{\alpha(2s)}+2sH^{\beta\gamma\alpha(2s-2)}\nabla_{\beta\gamma}V_{\alpha(2s-2)}-H^{\beta\alpha(2s-1)}\nabla_{\beta}\Omega_{\alpha(2s-1)}~{}$
$\displaystyle+\frac{{\rm
i}}{2}\Omega^{\alpha(2s-1)}\Omega_{\alpha(2s-1)}+\frac{1}{2s-1}\Big{(}4s(s-1)^{2}V^{\beta\alpha(2s-3)}\nabla_{\beta}{}^{\gamma}V_{\gamma\alpha(2s-3)}~{}$
$\displaystyle+4s(2s^{3}-2s+1)|\mu|V^{\alpha(2s-2)}V_{\alpha(2s-2)}+{\rm
i}s(2s^{2}-2s+1)V^{\alpha(2s-2)}\nabla^{2}V_{\alpha(2s-2)}$
$\displaystyle+(2s-1)V^{\alpha(2s-2)}\nabla^{\beta}\Omega_{\beta\alpha(2s-2)}\Big{)}\bigg{\\}}~{}.$
(5.23)

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

$\Omega_{\alpha(2s-1)}=-{\rm i}\nabla^{\beta}H_{\beta\alpha(2s-1)}-{\rm
i}\nabla_{(\alpha_{1}}V_{\alpha_{2}...\alpha_{2s-1})}~{},$ (5.24)

we find that the resulting action coincides with the off-shell ${\cal N}=1$
action massless superspin-$s$ multiplet in AdS3 (3.2.1).

###  5.4 Second supersymmetry transformations

As discussed in subsection 4.4, by construction, the ${\cal N}=1$ reduced
actions (5.3) and (5.3) are invariant under the second supersymmetry
transformations (4.37b). For convenience, we recall the form of these
transformations

$\delta_{\epsilon}U|=\epsilon^{\alpha}(\mbox{\boldmath$\nabla$}^{\underline{2}}_{\alpha}U)|~{},$
(5.25)

where $U(x,\theta^{I})$ is a ${\cal N}=(1,1)$ superfield, with indices being
suppressed. The second supersymmetry tranformations act on the ${\cal N}=1$
fields (5.16) and (5.18) in the following fashion

$\displaystyle\delta_{\epsilon}\mathfrak{H}_{\alpha(2s)}|$ $\displaystyle=$
$\displaystyle-{\rm i}\epsilon^{\beta}H_{\beta\alpha(2s)}~{},$ (5.26a)
$\displaystyle\delta_{\epsilon}\mbox{\boldmath$\nabla$}^{{\underline{2}}\beta}\mathfrak{H}_{\beta\alpha(2s-1)}|$
$\displaystyle=$ $\displaystyle 2{\rm
i}\epsilon^{\beta}H_{\beta\alpha(2s-1)}~{},$ (5.26b)
$\displaystyle\delta_{\epsilon}H_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle-2\epsilon_{(\alpha_{1}}H_{\alpha_{2}...\alpha_{2s+1})}~{},$
(5.26c) $\displaystyle\delta_{\epsilon}H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle-\frac{{\rm
i}}{2}\Big{(}(2s+1)|\mu|\epsilon^{\beta}H_{\beta\alpha(2s)}-\epsilon^{\beta}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s)}\Big{)}~{},$
(5.26d) $\displaystyle\delta_{\epsilon}L_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\frac{{\rm
i}}{2}\epsilon^{\beta}\Omega_{\beta\alpha(2s-2)}+\frac{2s}{2s-1}(s-1)\epsilon_{(\alpha_{1}}\nabla^{\beta}V_{\alpha_{2}...\alpha_{2s-2})\beta}~{},~{}$
(5.26e) $\displaystyle\delta_{\epsilon}V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{{\rm
i}}{2s}\epsilon^{\beta}\Phi_{\beta\alpha(2s-2)}-\frac{2}{s(2s-1)}(s-1)\epsilon_{(\alpha_{1}}\nabla^{\beta}L_{\alpha_{2}...\alpha_{2s-2})\beta}~{},$
(5.26f) $\displaystyle\delta_{\epsilon}\Phi_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s-1}\Big{(}2s(2s-1)\epsilon^{\beta}\nabla_{\beta(\alpha_{1}}V_{\alpha_{2}...\alpha_{2s-1})}+{\rm
i}s\epsilon_{(\alpha_{1}}\nabla^{2}V_{\alpha_{2}...\alpha_{2s-1})}~{}$
$\displaystyle+$ $\displaystyle
4s(4s^{2}-3s+1)|\mu|\epsilon_{(\alpha_{1}}V_{\alpha_{2}...\alpha_{2s-1})}+(2s-1)\epsilon_{(\alpha_{1}}\nabla^{\beta}\Omega_{\alpha_{2}...\alpha_{2s-1})\beta}~{}$
$\displaystyle-$ $\displaystyle
4s(s-1)\epsilon_{(\alpha_{1}}\nabla_{\alpha_{2}}{}^{\beta}V_{\alpha_{3}...\alpha_{2s-1})\beta}\Big{)}~{},~{}$
$\displaystyle\delta_{\epsilon}\Omega_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2s-1}\Big{(}2(2s-1)\epsilon^{\beta}\nabla_{\beta(\alpha_{1}}L_{\alpha_{2}...\alpha_{2s-1})}+{\rm
i}\epsilon_{(\alpha_{1}}\nabla^{2}L_{\alpha_{2}...\alpha_{2s-1})}~{}$
$\displaystyle+$ $\displaystyle
4(4s^{2}-3s+1)|\mu|\epsilon_{(\alpha_{1}}L_{\alpha_{2}...\alpha_{2s-1})}+(2s-1)\epsilon_{(\alpha_{1}}\nabla^{\beta}\Phi_{\alpha_{2}...\alpha_{2s-1})\beta}~{}$
$\displaystyle-$ $\displaystyle
4(s-1)\epsilon_{(\alpha_{1}}\nabla_{\alpha_{2}}{}^{\beta}L_{\alpha_{3}...\alpha_{2s-1})\beta}\Big{)}~{}.~{}$

It is apparent from (5.26a) and (5.26b) that the second supersymmetry
transformation (5.25) breaks the WZ gauge conditions (5.7), which we recall
for convenience

$\mathfrak{H}_{\alpha(2s)}|=0~{},\qquad\mbox{\boldmath$\nabla$}^{{\underline{2}}\beta}\mathfrak{H}_{\beta\alpha(2s-1)}|=0~{}.$
(5.27)

In order to resolve this, it is necessary to supplement the variation (5.25)
with the $\epsilon$-dependent gauge transformations:

$\displaystyle\delta_{g(\epsilon)}\mathfrak{H}_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle g_{\alpha(2s)}(\epsilon)+\bar{g}_{\alpha(2s)}(\epsilon)~{},$
(5.28a) $\displaystyle\delta_{g(\epsilon)}\Gamma_{\alpha(2s-2)}$
$\displaystyle=$ $\displaystyle\frac{s}{2s+1}\bar{\cal D}^{\beta}{\cal
D}^{\gamma}\bar{g}_{\beta\gamma\alpha(2s-2)}(\epsilon)~{}.$ (5.28b)

The modified second supersymmetry transformations now take the following form

$\displaystyle\hat{\delta}_{\epsilon}\mathfrak{H}_{\alpha(2s)}$
$\displaystyle=$
$\displaystyle\delta_{\epsilon}\mathfrak{H}_{\alpha(2s)}+\delta_{g(\epsilon)}\mathfrak{H}_{\alpha(2s)}~{},$
(5.29a) $\displaystyle\hat{\delta}_{\epsilon}\Gamma_{\alpha(2s-2)}$
$\displaystyle=$
$\displaystyle\delta_{\epsilon}\Gamma_{\alpha(2s-2)}+\delta_{g(\epsilon)}\Gamma_{\alpha(2s-2)}~{}.$
(5.29b)

The sole purpose of introducing the $\epsilon$-dependent gauge transformations
(5.28) is to restore the original Wess-Zumino gauge (5.27). Fixing the form of
the ${{\cal N}}=1$ components of $g_{\alpha(2s)}(\epsilon)$ as

$\displaystyle g_{\alpha(2s)}(\epsilon)|$ $\displaystyle=$
$\displaystyle\frac{{\rm i}}{2}\epsilon^{\beta}H_{\beta\alpha(2s)}~{},$
(5.30a)
$\displaystyle\mbox{\boldmath$\nabla$}^{{\underline{2}}\beta}g_{\beta\alpha(2s-1)}(\epsilon)|$
$\displaystyle=$ $\displaystyle-{\rm
i}\epsilon^{\beta}H_{\beta\alpha(2s-1)}~{},$ (5.30b)

ensures that (5.29) takes us back to the original gauge (5.27). The
transformations (5.29) act on the ${{\cal N}}=1$ superfields by the rule

$\displaystyle\hat{\delta}_{\epsilon}H_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle-2\epsilon_{(\alpha_{1}}H_{\alpha_{2}...\alpha_{2s+1})}~{},$
(5.31a) $\displaystyle\hat{\delta}_{\epsilon}H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle-\frac{{\rm i}}{4(2s+1)}\Big{(}{\rm
i}\epsilon^{\beta}\nabla^{2}H_{\beta\alpha(2s)}+4s\epsilon_{(\alpha_{1}}\nabla^{\beta\gamma}H_{\alpha_{2}...\alpha_{2s})\beta\gamma}$
$\displaystyle+$ $\displaystyle
2(2s+1)(4s+1)|\mu|\epsilon^{\beta}H_{\beta\alpha(2s)}-2\epsilon^{\beta}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s)}\Big{)}~{},$
$\displaystyle\hat{\delta}_{\epsilon}L_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\frac{{\rm
i}}{2}\epsilon^{\beta}\Omega_{\beta\alpha(2s-2)}+\frac{2s}{2s-1}(s-1)\epsilon_{(\alpha_{1}}\nabla^{\beta}V_{\alpha_{2}...\alpha_{2s-2})\beta}$
$\displaystyle+$
$\displaystyle\frac{s}{2s+1}\epsilon^{\beta}\nabla^{\gamma}H_{\beta\gamma\alpha(2s-2)}~{},$
$\displaystyle\hat{\delta}_{\epsilon}V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{{\rm
i}}{2s}\epsilon^{\beta}\Phi_{\beta\alpha(2s-2)}-\frac{2}{s(2s-1)}(s-1)\epsilon_{(\alpha_{1}}\nabla^{\beta}L_{\alpha_{2}...\alpha_{2s-2})\beta}~{}$
$\displaystyle+$ $\displaystyle\frac{{\rm
i}}{2(2s+1)}\epsilon^{\beta}\nabla^{\gamma\delta}H_{\beta\gamma\delta\alpha(2s-2)}~{},$
$\displaystyle\hat{\delta}_{\epsilon}\Phi_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s-1}\Big{(}2s(2s-1)\epsilon^{\beta}\nabla_{\beta(\alpha_{1}}V_{\alpha_{2}...\alpha_{2s-1})}+{\rm
i}s\epsilon_{(\alpha_{1}}\nabla^{2}V_{\alpha_{2}...\alpha_{2s-1})}~{}$
$\displaystyle+$ $\displaystyle
4s(4s^{2}-3s+1)|\mu|\epsilon_{(\alpha_{1}}V_{\alpha_{2}...\alpha_{2s-1})}+(2s-1)\epsilon_{(\alpha_{1}}\nabla^{\beta}\Omega_{\alpha_{2}...\alpha_{2s-1})\beta}~{}$
$\displaystyle-$ $\displaystyle
4s(s-1)\epsilon_{(\alpha_{1}}\nabla_{\alpha_{2}}{}^{\beta}V_{\alpha_{3}...\alpha_{2s-1})\beta}\Big{)}-\frac{s}{2s+1}\epsilon^{\beta}\Big{(}4(s+1)|\mu|H_{\beta\alpha(2s-1)}~{}$
$\displaystyle+$ $\displaystyle
2\nabla_{(\alpha_{1}}{}^{\gamma}H_{\alpha_{2}...\alpha_{2s-1})\beta\gamma}-{\rm
i}\nabla^{2}H_{\beta\alpha(2s-1)}\Big{)}~{},$
$\displaystyle\hat{\delta}_{\epsilon}\Omega_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2s-1}\Big{(}2(2s-1)\epsilon^{\beta}\nabla_{\beta(\alpha_{1}}L_{\alpha_{2}...\alpha_{2s-1})}+{\rm
i}\epsilon_{(\alpha_{1}}\nabla^{2}L_{\alpha_{2}...\alpha_{2s-1})}~{}$
$\displaystyle+$ $\displaystyle
4(4s^{2}-3s+1)|\mu|\epsilon_{(\alpha_{1}}L_{\alpha_{2}...\alpha_{2s-1})}+(2s-1)\epsilon_{(\alpha_{1}}\nabla^{\beta}\Phi_{\alpha_{2}...\alpha_{2s-1})\beta}~{}$
$\displaystyle-$ $\displaystyle
4(s-1)\epsilon_{(\alpha_{1}}\nabla_{\alpha_{2}}{}^{\beta}L_{\alpha_{3}...\alpha_{2s-1})\beta}\Big{)}+\frac{s}{2s+1}\epsilon^{\beta}\Big{(}2(2s-1)|\mu|\nabla^{\gamma}H_{\beta\gamma\alpha(2s-1)}~{}$
$\displaystyle-$
$\displaystyle\nabla_{(\alpha_{1}}{}^{\gamma}\nabla^{\delta}H_{\alpha_{2}...\alpha_{2s-1})\beta\gamma\delta}+\nabla^{\gamma\delta}\nabla_{\gamma}H_{\beta\delta\alpha(2s-1)}\Big{)}~{}.$

##  6 Massless half-integer superspin: Longitudinal formulation

In this section, we develop the superspace reduction procedure for the
longitudinal formulation, following the prescription advocated in section 5.

###  6.1 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

$\displaystyle{\cal
V}^{\|}_{(s+\frac{1}{2})}=\big{\\{}\mathfrak{H}_{\alpha(2s)},G_{\alpha(2s-2)},\bar{G}_{\alpha(2s-2)}\big{\\}}~{}.$
(6.1)

Here, the real superfield $\mathfrak{H}_{\alpha(2s)}$ is identical to that of
(5.1), and the complex superfield ${G}_{\alpha(2s-2)}$ is longitudinal linear
(4.4a). Modulo an overall normalisation factor, the longitudinal formulation
is uniquely described by the action

$\displaystyle\mathbb{S}^{\|}_{(s+\frac{1}{2})}[\mathfrak{H}_{\alpha(2s)},G_{\alpha(2s-2)},\bar{G}_{\alpha(2s-2)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3|4}z\,\mbox{\boldmath$E$}~{}\bigg{\\{}\frac{1}{8}\mathfrak{H}^{\alpha(2s)}{\cal
D}^{\beta}(\bar{{\cal D}}^{2}-6\mu){\cal D}_{\beta}\mathfrak{H}_{\alpha(2s)}$
$\displaystyle+2s(s-1)\mu\bar{\mu}\mathfrak{H}^{\alpha(2s)}\mathfrak{H}_{\alpha(2s)}-\frac{1}{16}([{\cal
D}_{\beta},\bar{{\cal
D}}_{\gamma}]\mathfrak{H}^{\beta\gamma\alpha(2s-2)})[{\cal
D}^{\delta},\bar{{\cal D}}^{\rho}]\mathfrak{H}_{\delta\rho\alpha(2s-2)}$
$\displaystyle+\frac{s}{2}({\cal
D}_{\beta\gamma}\mathfrak{H}^{\beta\gamma\alpha(2s-2)}){\cal
D}^{\delta\rho}\mathfrak{H}_{\delta\rho\alpha(2s-2)}$
$\displaystyle+\frac{2s-1}{2s}\Big{(}{\rm i}{\cal
D}_{\beta\gamma}\mathfrak{H}^{\beta\gamma\alpha(2s-2)}\left(G_{\alpha(2s-2)}-\bar{G}_{\alpha(2s-2)}\right)+\frac{1}{s}\bar{G}^{\alpha(2s-2)}G_{\alpha(2s-2)}\Big{)}$
$\displaystyle-\frac{2s+1}{4s^{2}}\left(G^{\alpha(2s-2)}G_{\alpha(2s-2)}+\bar{G}^{\alpha(2s-2)}\bar{G}_{\alpha(2s-2)}\right)\bigg{\\}}~{},$
(6.2)

which is invariant under the gauge transformations

$\displaystyle\delta_{\lambda}\mathfrak{H}_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle{\bar{\cal
D}}_{(\alpha_{1}}\lambda_{\alpha_{2}\dots\alpha_{2s})}-{{\cal
D}}_{(\alpha_{1}}\bar{\lambda}_{\alpha_{2}\dots\alpha_{2s})}\equiv
g_{\alpha(2s)}+\bar{g}_{\alpha(2s)}~{},$ (6.3a)
$\displaystyle\delta_{\lambda}{G}_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\big{(}\bar{{\cal D}}^{2}-4s\mu\big{)}{\cal
D}^{\beta}\lambda_{\alpha(2s-2)\beta}+{\rm i}(s-1)\bar{{\cal
D}}_{(\alpha_{1}}{\cal
D}^{|\beta\gamma|}\lambda_{\alpha_{2}\dots\alpha_{2s-2})\beta\gamma}~{}$
(6.3b) $\displaystyle\equiv$ $\displaystyle\Big{(}\frac{s}{2s+1}{\cal
D}^{\beta}\bar{\cal D}^{\gamma}+{\rm i}s{\cal
D}^{\beta\gamma}\Big{)}g_{\beta\gamma\alpha(2s-2)}~{},$

where $\lambda_{\alpha(2s-1)}$ is complex unconstrained and $g_{\alpha(2s)}$
is longitudinal linear, as in (5.2a).

In the case where $s=1$, the compensator $G$ is covariantly chiral. If we
introduce the field redefinition $G=3\sigma$ in action (6.1), along with
choosing $s=1$, then the model can be shown to coincide with the linearised
action for minimal $(1,1)$ AdS supergravity [21]. In the case $s=1$, the gauge
transformations (6.3) yield

$\displaystyle\delta_{\lambda}\mathfrak{H}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\bar{{\cal D}}_{(\alpha}\lambda_{\beta)}-{\cal
D}_{(\alpha}\bar{\lambda}_{\beta)}~{},$ (6.4) $\displaystyle\delta_{\lambda}G$
$\displaystyle=$ $\displaystyle-\frac{1}{4}(\bar{{\cal
D}}^{2}-4\bar{\mu}){\cal D}^{\beta}\lambda_{\beta}~{}.$ (6.5)

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

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

The reduction of the superconformal gauge prepotential
$\mathfrak{H}_{\alpha(2s)}$ was addressed in section 5. We need only reduce
the compensator $G_{\alpha(2s-2)}$ to ${\cal N}=1$ AdS superspace. We start by
converting the longitudinal constraint of $G_{\alpha(2s-2)}$ (4.4a) to the
real basis (4.18)

$\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}G_{\alpha_{2}...\alpha_{2s-1})}={\rm
i}\mbox{\boldmath$\nabla$}^{{\underline{1}}}{}_{(\alpha_{1}}G_{\alpha_{2}...\alpha_{2s-1})}~{}.$
(6.6)

Performing a Taylor expansion of $G_{\alpha(2s-2)}(\theta^{I})$ about
$\theta^{\underline{2}}$, and using (6.6), we find the independent
$\theta^{\underline{2}}$ -components of $G_{\alpha(2s-2)}$ to be

$G_{\alpha(2s-2)}|~{},\qquad\mbox{\boldmath$\nabla$}^{{\underline{2}}\beta}G_{\beta\alpha(2s-3)}|~{}.$
(6.7)

Utilising the gauge transformations (6.3b) and the real representation (4.18),
we find

$\displaystyle\delta G_{\alpha(2s-2)}=\frac{{\rm
i}s}{2s+1}\Big{(}2s\mbox{\boldmath$\nabla$}^{\beta\gamma}-\mbox{\boldmath$\nabla$}^{{\underline{1}}\beta}\mbox{\boldmath$\nabla$}^{{\underline{2}}\gamma}\Big{)}g_{\beta\gamma\alpha(2s-2)}~{},$
(6.8a) $\displaystyle\mbox{\boldmath$\nabla$}^{{\underline{2}}\beta}\delta
G_{\beta\alpha(2s-3)}=\frac{s}{2s+1}\Big{(}2{\rm
i}s\mbox{\boldmath$\nabla$}^{\beta\gamma}\mbox{\boldmath$\nabla$}^{{\underline{2}}\delta}-\mbox{\boldmath$\nabla$}^{\beta\gamma}\mbox{\boldmath$\nabla$}^{{\underline{1}}\delta}\Big{)}g_{\beta\gamma\delta\alpha(2s-3)}~{}.$
(6.8b)

We can use the residual gauge freedom (5.9) to compute the gauge
transformations of the complex ${\cal N}=1$ superfields (6.7)

$\displaystyle\delta
G_{\alpha(2s-2)}|=\frac{s^{2}}{2s+1}\nabla^{\beta\gamma}\zeta_{\beta\gamma\alpha(2s-2)}-\frac{{\rm
i}}{2}\nabla^{\beta}\zeta_{\beta\alpha(2s-2)}~{},$ (6.9a)
$\displaystyle\mbox{\boldmath$\nabla$}^{{\underline{2}}\beta}\delta
G_{\beta\alpha(2s-3)}|={\rm
i}s\nabla^{\beta\gamma}\zeta_{\beta\gamma\alpha(2s-3)}+\frac{{\rm
i}s}{2(2s+1)}\nabla^{\beta\gamma}\nabla^{\delta}\zeta_{\beta\gamma\delta\alpha(2s-3)}~{}.$
(6.9b)

It is useful to separate the complex superfields (6.7) into real and imaginary
components

$\displaystyle G_{\alpha(2s-2)}|$ $\displaystyle=$
$\displaystyle-2sL_{\alpha(2s-2)}-{\rm i}sV_{\alpha(2s-2)}~{},$ (6.10a)
$\displaystyle\mbox{\boldmath$\nabla$}^{{\underline{2}}\beta}G_{\beta\alpha(2s-3)}|$
$\displaystyle=$ $\displaystyle\frac{1}{2}\Big{(}\Phi_{\alpha(2s-3)}+{\rm
i}\Omega_{\alpha(2s-3)}\Big{)}~{}.$ (6.10b)

It then becomes apparent from gauge transformations (5.10) and (6.9) that we
are in fact dealing with two different gauge theories. The first gauge theory
is formulated in terms of the dynamical variables

${\cal
V}^{\parallel}_{(s+\frac{1}{2})}=\big{\\{}H_{\alpha(2s+1)},L_{\alpha(2s-2)},\Phi_{\alpha(2s-3)}\big{\\}}~{},$
(6.11)

which are defined modulo gauge transfomations of the form

$\displaystyle\delta H_{\alpha(2s+1)}$ $\displaystyle=$ $\displaystyle{\rm
i}\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{2s+1)}}~{},$ (6.12a)
$\displaystyle\delta L_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\frac{s}{2(2s+1)}\nabla^{\beta\gamma}\zeta_{\beta\gamma\alpha(2s-2)}~{},$
(6.12b) $\displaystyle\delta\Phi_{\alpha(2s-3)}$ $\displaystyle=$
$\displaystyle\frac{{\rm
i}s}{2s+1}\nabla^{\beta\gamma}\nabla^{\delta}\zeta_{\beta\gamma\delta\alpha(2s-3)}~{}.$
(6.12c)

The other gauge model is described in terms of the superfields

${\cal
V}^{\parallel}_{(s)}=\big{\\{}H_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-3)}\big{\\}}~{},$
(6.13)

which possess the following gauge freedom

$\displaystyle\delta H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{2s})}~{},$
(6.14a) $\displaystyle\delta V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s}\nabla^{\beta}\zeta_{\beta\alpha(2s-2)}~{},$ (6.14b)
$\displaystyle\delta\Omega_{\alpha(2s-3)}$ $\displaystyle=$ $\displaystyle
2s\nabla^{\beta\gamma}\zeta_{\beta\gamma\alpha(2s-3)}~{}.$ (6.14c)

After carrying out the reduction to ${\cal N}=1$ AdS superspace, the action
(6.1) decouples into two ${\cal N}=1$ theories, which are formulated in terms
of superfields (6.11) and (6.13) respectively

$\displaystyle\mathbb{S}^{\parallel}_{(s+\frac{1}{2})}[\mathfrak{H}_{\alpha(2s)},G_{\alpha(2s-2)},\bar{G}_{\alpha(2s-2)}]=$
$\displaystyle
S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)},\Phi_{\alpha(2s-3)}]~{}$
(6.15)
$\displaystyle+S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-3)}]~{}.$

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

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

The first ${\cal N}=1$ gauge theory, described in terms of the dynamical
variables (6.11), takes the form

$\displaystyle
S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)},\Phi_{\alpha(2s-3)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\Big{(}-\frac{{\rm
i}}{8}\Big{)}\int{\rm
d}^{3|2}z\,E~{}\bigg{\\{}2H^{\alpha(2s+1)}\mathbb{Q}H_{\alpha(2s+1)}~{}$
$\displaystyle+(2s+1){\rm
i}|\mu|H^{\alpha(2s+1)}\nabla^{2}H_{\alpha(2s+1)}-4(2s+1)|\mu|H^{\beta\alpha(2s)}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s)}-{\rm
i}H^{\beta\alpha(2s)}\nabla^{2}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s)}~{}$
$\displaystyle-8(s-2)(2s+1)|\mu|^{2}H^{\alpha(2s+1)}H_{\alpha(2s+1)}-2s\nabla_{\beta\gamma}H^{\beta\gamma\alpha(2s-1)}\nabla^{\delta\rho}H_{\delta\rho\alpha(2s-1)}~{}$
$\displaystyle-\frac{4}{2s-1}\Big{(}16{\rm
i}|\mu|(2s^{3}-2s+1)L^{\alpha(2s-2)}L_{\alpha(2s-2)}+4(2s^{2}-1)L^{\alpha(2s-2)}\nabla^{2}L_{\alpha(2s-2)}~{}$
$\displaystyle+16{\rm
i}(s-1)^{2}L^{\beta\alpha(2s-3)}\nabla_{\beta}{}^{\gamma}L_{\gamma\alpha(2s-3)}+2{\rm
i}(2s-1)^{2}\nabla_{\beta\gamma}H^{\beta\gamma\delta\alpha(2s-2)}\nabla_{\delta}L_{\alpha(2s-2)}~{}$
$\displaystyle+\frac{4{\rm
i}}{s}(s-1)L^{\beta\alpha(2s-3)}\nabla_{\beta}\Phi_{\alpha(2s-3)}+\frac{1}{s}(s-1)\Phi^{\alpha(2s-3)}\Phi_{\alpha(2s-3)}\Big{)}\bigg{\\}}~{}.$
(6.16)

The action (6.3) is invariant under the gauge transformations (6.12). We can
eliminate the auxiliary field $\Phi_{\alpha(2s-3)}$ from (6.3) by using the
equation of motion

$\Phi_{\alpha(2s-3)}=-2{\rm i}\nabla^{\beta}L_{\beta\alpha(2s-3)}~{}.$ (6.17)

The resulting action, up to an overall factor, then coincides with model
(3.1.1).

The other ${\cal N}=1$ model which is constructed in terms of the dynamical
variables (6.13) assumes the form

$\displaystyle
S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-3)}]=-\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3|2}z~{}E~{}\bigg{\\{}(s-1)|\mu|H^{\alpha(2s)}H_{\alpha(2s)}~{}$
$\displaystyle-\frac{{\rm
i}}{4}H^{\alpha(2s)}\nabla^{2}H_{\alpha(2s)}+\frac{1}{2}H^{\beta\alpha(2s-1)}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s-1)}-(2s-1)H^{\alpha(2s)}\nabla_{(\alpha_{1}\alpha_{2}}V_{\alpha_{3}...\alpha_{2s})}~{}$
$\displaystyle-\frac{1}{4(2s-1)}\Big{(}2{\rm
i}sV^{\alpha(2s-2)}\nabla^{2}V_{\alpha(2s-2)}-4(s-1)(4s-1)V^{\beta\alpha(2s-3)}\nabla_{\beta}{}^{\gamma}V_{\gamma\alpha(2s-3)}~{}$
$\displaystyle+8s(4s^{2}-3s+1)|\mu|V^{\alpha(2s-2)}V_{\alpha(2s-2)}+8(s-1)V^{\beta\alpha(2s-3)}\nabla_{\beta}\Omega_{\alpha(2s-3)}~{}$
$\displaystyle-\frac{{\rm
i}}{s^{2}}(s-1)\Omega^{\alpha(2s-3)}\Omega_{\alpha(2s-3)}\Big{)}\bigg{\\}}~{},$
(6.18)

which is invariant under the gauge transformations (6.14). It is evident that
the superfield $\Omega_{\alpha(2s-3)}$ is auxiliary, so upon elimination via
its equation of motion

$\Omega_{\alpha(2s-3)}=-4{\rm i}s^{2}\nabla^{\beta}V_{\beta\alpha(2s-3)}~{},$
(6.19)

one obtains the action (3.2.1).

###  6.4 Second supersymmetry transformations

Let us note that the reduced actions (6.15)

$\displaystyle\mathbb{S}^{\parallel}_{(s+\frac{1}{2})}[\mathfrak{H}_{\alpha(2s)},G_{\alpha(2s-2)},\bar{G}_{\alpha(2s-2)}]=$
$\displaystyle
S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},L_{\alpha(2s-2)},\Phi_{\alpha(2s-3)}]~{}$
(6.20)
$\displaystyle+S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-3)}]~{},$

are invariant under second supersymmetry. More explicitly, the second
supersymmetry transformation acts on the dynamical superfields (6.11) and
(6.13) by the following rule

$\displaystyle\delta_{\epsilon}\mathfrak{H}_{\alpha(2s)}|$ $\displaystyle=$
$\displaystyle-{\rm i}\epsilon^{\beta}H_{\beta\alpha(2s)}~{},$ (6.21a)
$\displaystyle\delta_{\epsilon}\mbox{\boldmath$\nabla$}^{{\underline{2}}\beta}\mathfrak{H}_{\beta\alpha(2s-1)}|$
$\displaystyle=$ $\displaystyle 2{\rm
i}\epsilon^{\beta}H_{\beta\alpha(2s-1)}~{},$ (6.21b)
$\displaystyle\delta_{\epsilon}H_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle-2\epsilon_{(\alpha_{1}}H_{\alpha_{2}...\alpha_{2s+1})}~{},$
(6.21c) $\displaystyle\delta_{\epsilon}H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle-\frac{{\rm
i}}{2}\Big{(}(2s+1)|\mu|\epsilon^{\beta}H_{\beta\alpha(2s)}-\epsilon^{\beta}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s)}\Big{)}~{},$
(6.21d) $\displaystyle\delta_{\epsilon}L_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\epsilon^{\beta}\nabla_{(\beta}V_{\alpha(2s-2))}+\frac{{\rm
i}(s-1)}{2s(2s-1)}\epsilon_{(\alpha_{1}}\Omega_{\alpha_{2}...\alpha_{2s-2})}~{},$
(6.21e) $\displaystyle\delta_{\epsilon}V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle 2\epsilon^{\beta}\nabla_{(\beta}L_{\alpha(2s-2))}-\frac{{\rm
i}(s-1)}{s(2s-1)}\epsilon_{(\alpha_{1}}\Phi_{\alpha_{2}...\alpha_{2s-2})}~{},$
(6.21f) $\displaystyle\delta_{\epsilon}\Phi_{\alpha(2s-3)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s-1}\epsilon^{\beta}\Big{(}2s(2s-3)\nabla_{(\alpha_{1}}{}^{\gamma}V_{\alpha_{2}...\alpha_{2s-3})\beta\gamma}-{\rm
i}s\nabla^{2}V_{\beta\alpha(2s-3)}~{}$ (6.21g) $\displaystyle+$ $\displaystyle
4s^{2}\nabla_{\beta}{}^{\gamma}V_{\gamma\alpha(2s-3)}-4s(4s^{2}-5s+2)|\mu|V_{\beta\alpha(2s-3)}~{}$
$\displaystyle-$ $\displaystyle
2(s-1)\nabla_{(\beta}\Omega_{\alpha(2s-3))}\Big{)}~{},$
$\displaystyle\delta_{\epsilon}\Omega_{\alpha(2s-3)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s-1}\epsilon^{\beta}\Big{(}2{\rm
i}s\nabla^{2}L_{\beta\alpha(2s-3)}-4s(2s-3)\nabla_{(\alpha_{1}}{}^{\gamma}L_{\alpha_{2}...\alpha_{2s-3})\beta\gamma}~{}$
(6.21h) $\displaystyle-$ $\displaystyle
8s^{2}\nabla_{\beta}{}^{\gamma}L_{\gamma\alpha(2s-3)}+8s(4s^{2}-5s+2)|\mu|L_{\beta\alpha(2s-3)}~{}$
$\displaystyle+$ $\displaystyle
2(s-1)\nabla_{(\beta}\Phi_{\alpha(2s-3))}\Big{)}~{}.$

It is evident from the variations (6.21a) and (6.21b) that second
supersymmetry is not compatible with the imposed WZ gauge conditions (5.27).
Recall that the longitudinal and transverse formulations are both constructed
in terms of the real superfield $\mathfrak{H}_{\alpha(2s)}$, which is defined
modulo gauge transformations (5.2a). We used this gauge freedom to impose
(5.27) in both theories.

Using a similar approach as in subsection 5.4, the WZ gauge conditions (5.27)
in the longitudinal formulation can be restored provided we accompany the
second supersymmetry transformations with the following $\epsilon$-dependent
gauge transformations:

$\displaystyle\delta_{g(\epsilon)}\mathfrak{H}_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle g_{\alpha(2s)}(\epsilon)+\bar{g}_{\alpha(2s)}(\epsilon)~{},$
(6.22a) $\displaystyle\delta_{g(\epsilon)}G_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{s}{2s+1}{\cal D}^{\beta}\bar{\cal D}^{\gamma}+{\rm
i}s{\cal D}^{\beta\gamma}\bigg{)}g_{\beta\gamma\alpha(2s-2)}(\epsilon)~{},$
(6.22b)

where

$\displaystyle g_{\alpha(2s)}(\epsilon)|$ $\displaystyle=$
$\displaystyle\frac{{\rm i}}{2}\epsilon^{\beta}H_{\beta\alpha(2s)}~{},$
(6.23a)
$\displaystyle\mbox{\boldmath$\nabla$}^{{\underline{2}}\beta}g_{\beta\alpha(2s-1)}(\epsilon)|$
$\displaystyle=$ $\displaystyle-{\rm
i}\epsilon^{\beta}H_{\beta\alpha(2s-1)}~{}.$ (6.23b)

The modified second supersymmetry transformations now read

$\displaystyle\hat{\delta}_{\epsilon}\mathfrak{H}_{\alpha(2s)}$
$\displaystyle=$
$\displaystyle\delta_{\epsilon}\mathfrak{H}_{\alpha(2s)}+\delta_{g(\epsilon)}\mathfrak{H}_{\alpha(2s)}~{},$
(6.24a) $\displaystyle\hat{\delta}_{\epsilon}G_{\alpha(2s-2)}$
$\displaystyle=$
$\displaystyle\delta_{\epsilon}G_{\alpha(2s-2)}+\delta_{g(\epsilon)}G_{\alpha(2s-2)}~{}.$
(6.24b)

The transformations (6.24) act on the ${\cal N}=1$ superfields (6.11) and
(6.13) as follows

$\displaystyle\hat{\delta}_{\epsilon}H_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle-2\epsilon_{(\alpha_{1}}H_{\alpha_{2}...\alpha_{2s+1})}~{},$
(6.25a) $\displaystyle\hat{\delta}_{\epsilon}H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle-\frac{{\rm i}}{4(2s+1)}\Big{(}{\rm
i}\epsilon^{\beta}\nabla^{2}H_{\beta\alpha(2s)}+4s\epsilon_{(\alpha_{1}}\nabla^{\beta\gamma}H_{\alpha_{2}...\alpha_{2s})\beta\gamma}$
$\displaystyle+$ $\displaystyle
2(2s+1)(4s+1)|\mu|\epsilon^{\beta}H_{\beta\alpha(2s)}-2\epsilon^{\beta}\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s)}\Big{)}~{},$
$\displaystyle\hat{\delta}_{\epsilon}L_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\epsilon^{\beta}\nabla_{(\beta}V_{\alpha(2s-2))}+\frac{{\rm
i}(s-1)}{2s(2s-1)}\epsilon_{(\alpha_{1}}\Omega_{\alpha_{2}...\alpha_{2s-2})}$
$\displaystyle-$
$\displaystyle\frac{1}{2(2s+1)}\epsilon^{\beta}\nabla^{\gamma}H_{\beta\gamma\alpha(2s-2)}~{},$
$\displaystyle\hat{\delta}_{\epsilon}V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle 2\epsilon^{\beta}\nabla_{(\beta}L_{\alpha(2s-2))}-\frac{{\rm
i}(s-1)}{s(2s-1)}\epsilon_{(\alpha_{1}}\Phi_{\alpha_{2}...\alpha_{2s-2})}~{}$
$\displaystyle-$ $\displaystyle\frac{{\rm
i}s}{2s+1}\epsilon^{\beta}\nabla^{\gamma\delta}H_{\beta\gamma\delta\alpha(2s-2)}~{},$
$\displaystyle\hat{\delta}_{\epsilon}\Phi_{\alpha(2s-3)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s-1}\epsilon^{\beta}\Big{(}2s(2s-3)\nabla_{(\alpha_{1}}{}^{\gamma}V_{\alpha_{2}...\alpha_{2s-3})\beta\gamma}-{\rm
i}s\nabla^{2}V_{\beta\alpha(2s-3)}~{}$ $\displaystyle+$ $\displaystyle
4s^{2}\nabla_{\beta}{}^{\gamma}V_{\gamma\alpha(2s-3)}-4s(4s^{2}-5s+2)|\mu|V_{\beta\alpha(2s-3)}~{}$
$\displaystyle-$ $\displaystyle
2(s-1)\nabla_{(\beta}\Omega_{\alpha(2s-3))}\Big{)}+\frac{4s^{2}}{2s+1}\epsilon^{\beta}\nabla^{\gamma\delta}H_{\beta\gamma\delta\alpha(2s-3)}~{},$
$\displaystyle\hat{\delta}_{\epsilon}\Omega_{\alpha(2s-3)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s-1}\epsilon^{\beta}\Big{(}2{\rm
i}s\nabla^{2}L_{\beta\alpha(2s-3)}-4s(2s-3)\nabla_{(\alpha_{1}}{}^{\gamma}L_{\alpha_{2}...\alpha_{2s-3})\beta\gamma}~{}$
$\displaystyle-$ $\displaystyle
8s^{2}\nabla_{\beta}{}^{\gamma}L_{\gamma\alpha(2s-3)}+8s(4s^{2}-5s+2)|\mu|L_{\beta\alpha(2s-3)}~{}$
$\displaystyle+$ $\displaystyle
2(s-1)\nabla_{(\beta}\Phi_{\alpha(2s-3))}\Big{)}+\frac{s}{2s+1}\epsilon^{\beta}\nabla^{\gamma\delta}\nabla^{\lambda}H_{\beta\gamma\delta\lambda\alpha(2s-3)}~{}.$

##  7 Massless integer superspin: Longitudinal formulation

In [1], 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 ${\cal N}=1$ AdS superspace.

###  7.1 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

$\displaystyle{\cal
V}^{\|}_{(s)}=\\{U_{\alpha(2s-2)},G_{\alpha(2s)},\bar{G}_{\alpha(2s)}\\}~{}.$
(7.1)

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

$\displaystyle\delta_{L}U_{\alpha(2s-2)}$ $\displaystyle=$ $\displaystyle{\cal
D}^{\beta}L_{\beta\alpha(2s-2)}-\bar{{\cal
D}}^{\beta}\bar{L}_{\beta\alpha(2s-2)}\equiv\bar{\gamma}_{\alpha(2s-2)}+{\gamma}_{\alpha(2s-2)}~{},$
(7.2a) $\displaystyle\delta_{L}G_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\bar{{\cal D}}_{(\alpha_{1}}\big{(}{\cal
D}^{2}-2(2s+1)\bar{\mu}\big{)}L_{\alpha_{2}\dots\alpha_{2s})}\equiv\bar{{\cal
D}}_{(\alpha_{1}}{\cal
D}_{\alpha_{2}}\bar{\gamma}_{\alpha_{3}\dots\alpha_{2s})}~{}.$ (7.2b)

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

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$}\,\bigg{\\{}\frac{1}{8}U^{\alpha(2s-2)}{\cal
D}^{\gamma}({\bar{\cal D}}^{2}-6\mu){\cal D}_{\gamma}U_{\alpha(2s-2)}$
$\displaystyle+\frac{s}{2s+1}U^{\alpha(2s-2)}\Big{(}{\cal D}^{\beta}{\bar{\cal
D}}^{\gamma}G_{\beta\gamma\alpha(2s-2)}-{\bar{\cal D}}^{\beta}{{\cal
D}}^{\gamma}\bar{G}_{\beta\gamma\alpha(2s-2)}\Big{)}+\frac{s}{2s-1}\bar{G}^{\alpha(2s)}G_{\alpha(2s)}$
$\displaystyle+\frac{s}{2(2s+1)}\Big{(}G^{\alpha(2s)}G_{\alpha(2s)}+\bar{G}^{\alpha(2s)}\bar{G}_{\alpha(2s)}\Big{)}+2s(s+1)\mu\bar{\mu}U^{\alpha(2s-2)}U_{\alpha(2s-2)}\bigg{\\}}~{},$
(7.3)

which is invariant under the gauge transformations (7.2).

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

We wish to reduce the gauge superfields (7.1) to ${\cal N}=1$ AdS superspace.
We start by reducing the superfield $U_{\alpha(2s-2)}$. Converting the
transverse linear constraint of $\gamma_{\alpha(2s-2)}$ (4.5a) to the real
basis (4.18) gives

$\mbox{\boldmath$\nabla$}^{{\underline{2}}\beta}\gamma_{\beta\alpha(2s-3)}={\rm
i}\mbox{\boldmath$\nabla$}^{{\underline{1}}\beta}\gamma_{\beta\alpha(2s-3)}~{}.$
(7.4)

Taking a Taylor expansion of $\gamma_{\alpha(2s-2)}(\theta^{I})$ about
$\theta^{\underline{2}}$, then using (7.4), we find the independent
$\theta^{\underline{2}}$ -components of $\gamma_{\alpha(2s-2)}$ to be

$\gamma_{\alpha(2s-2)}|,\qquad\qquad\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\gamma_{\alpha_{2}...\alpha_{2s-1})}|~{}.$
(7.5)

The gauge transformation (7.2a) allows us to choose the gauge conditions

$U_{\alpha(2s-2)}|=0~{}.$ (7.6)

It must be noted that the gauge condition (7.6) is less restrictive than those
proposed in the analogous reduction procedure performed in Minkowski
superspace, as given in appendix B of [1]. This was done in order to ensure
that one of the decoupled ${\cal N}=1$ actions coincides with (3.2.2). It can
be shown that if one chooses to impose the same gauge conditions as detailed
in [1], i.e

$U_{\alpha(2s-2)}|=0~{},\qquad\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}U_{\alpha_{2}...\alpha_{2s-1})}|=0~{},$
(7.7)

then one will obtain a ${\cal N}=1$ action which would instead coincide with
(3.2.2).

The residual gauge symmetry which preserves the gauge condition (7.6) is
described by the real unconstrained ${\cal N}=1$ superfield

$\gamma_{\alpha(2s-2)}|=-\frac{{\rm
i}}{2}\zeta_{\alpha(2s-2)}~{},\qquad\bar{\zeta}_{\alpha(2s-2)}=\zeta_{\alpha(2s-2)}~{}.$
(7.8)

We now wish to reduce the superfield $\bar{G}_{\alpha(2s)}$ to ${\cal N}=1$
AdS superspace. Converting the longitudinal linear constraint of
$\bar{G}_{\alpha(2s)}$ (4.4a) to the real basis yields

$\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\bar{G}_{\alpha_{2}...\alpha_{2s+1})}=-{\rm
i}\mbox{\boldmath$\nabla$}^{\underline{1}}{}_{(\alpha_{1}}\bar{G}_{\alpha_{1}...\alpha_{2s+1})}~{}.$
(7.9)

Upon reduction to ${\cal N}=1$ AdS superspace, it follows from constraint
(7.9) that $\bar{G}_{\alpha(2s)}$ has two independent $\theta^{\underline{2}}$
-components:

$\bar{G}_{\alpha(2s)}|~{},\qquad\mbox{\boldmath$\nabla$}^{{\underline{2}}\beta}\bar{G}_{\beta\alpha(2s-1)}|~{}.$
(7.10)

Making use of the gauge transformations (7.2b) and the real representation
(4.18), we find

$\displaystyle\delta\bar{G}_{\alpha(2s)}$ $\displaystyle=$ $\displaystyle{\rm
i}\Big{(}\mbox{\boldmath$\nabla$}^{\underline{1}}{}_{(\alpha_{1}}\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{\alpha_{2}}+\mbox{\boldmath$\nabla$}_{(\alpha_{1}\alpha_{2}}\Big{)}\gamma_{\alpha_{3}...{\alpha_{2s})}}~{},$
(7.11a)
$\displaystyle\mbox{\boldmath$\nabla$}^{{\underline{2}}\beta}\delta\bar{G}_{\beta\alpha(2s-1)}$
$\displaystyle=$ $\displaystyle\frac{{\rm
i}}{2s}\Big{(}2s\mbox{\boldmath$\nabla$}_{(\alpha_{1}}{}^{\beta}\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{\alpha_{2}}\gamma_{\alpha_{3}...\alpha_{2s-1}\beta)}+(4s^{2}-1)|\mu|\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\gamma_{\alpha_{2}...\alpha_{2s-1})}~{}$
$\displaystyle+$ $\displaystyle 2{\rm
i}(s+1)\mbox{\boldmath$\nabla$}_{\beta(\alpha_{1}}\mbox{\boldmath$\nabla$}^{{\underline{1}}\beta}\gamma_{\alpha_{2}...\alpha_{2s-1})}+6{\rm
i}s(2s+1)|\mu|\mbox{\boldmath$\nabla$}^{\underline{1}}{}_{(\alpha_{1}}\gamma_{\alpha_{2}...\alpha_{2s-1})}~{}$
$\displaystyle+$ $\displaystyle\frac{{\rm
i}}{2}(2s+1)(\mbox{\boldmath$\nabla$}^{\underline{1}})^{2}\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\gamma_{\alpha_{2}...\alpha_{2s-1})}-2{\rm
i}(s-1)\mbox{\boldmath$\nabla$}_{(\alpha_{1}}{}^{\beta}\mbox{\boldmath$\nabla$}^{\underline{1}}{}_{\alpha_{2}}\gamma_{\alpha_{3}...\alpha_{2s-1})\beta}\Big{)}~{}.$

Using the residual gauge freedom (7.8), we can directly compute the gauge
transformations of the ${\cal N}=1$ superfields (7.10) from (7.11)

$\displaystyle\delta\bar{G}_{\alpha(2s)}|$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}\nabla_{(\alpha_{1}\alpha_{2}}\zeta_{\alpha_{3}...\alpha_{2s})}+2{\rm
i}\nabla_{(\alpha_{1}}\big{(}\tau_{\alpha_{2}...\alpha_{2s})}+{\rm
i}\tilde{\tau}_{\alpha_{2}\dots\alpha_{2s})}\big{)}\Big{)}~{},$ (7.12a)
$\displaystyle\mbox{\boldmath$\nabla$}^{{\underline{2}}\beta}\delta\bar{G}_{\beta\alpha(2s-1)}|$
$\displaystyle=$ $\displaystyle\frac{{\rm
i}}{2s}\Big{(}2s(2s+1)|\mu|\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{2s-1})}+(s+1)\nabla_{\beta(\alpha_{1}}\nabla^{\beta}\zeta_{\alpha_{2}...\alpha_{2s-1})}~{}$
$\displaystyle+$ $\displaystyle(4s^{2}-1)|\mu|\big{(}\tau_{\alpha(2s-1)}+{\rm
i}\tilde{\tau}_{\alpha(2s-1)}\big{)}+\frac{{\rm
i}}{2}(2s+1)\nabla^{2}\big{(}\tau_{\alpha(2s-1)}+{\rm
i}\tilde{\tau}_{\alpha(2s-1)}\big{)}~{}$ $\displaystyle+$
$\displaystyle(2s-1)\nabla_{(\alpha_{1}}{}^{\beta}\big{(}\tau_{\alpha_{2}...\alpha_{2s-1})\beta}+{\rm
i}\tilde{\tau}_{\alpha_{2}...\alpha_{2s-1})\beta}\big{)}-(s-1)\nabla_{(\alpha_{1}}{}^{\beta}\nabla_{\alpha_{2}}\zeta_{\alpha_{3}...\alpha_{2s-1})\beta}\Big{)}~{},$

where we have defined

$\mbox{\boldmath$\nabla$}^{\underline{2}}{}_{(\alpha_{1}}\gamma_{\alpha_{2}...\alpha_{2s-1})}|=\tau_{\alpha{(2s-1)}}+{\rm
i}\tilde{\tau}_{\alpha{(2s-1)}}~{}.$ (7.13)

It is useful to decompose the complex ${\cal N}=1$ superfields (7.10) into
real and imaginary parts

$\displaystyle\bar{G}_{\alpha(2s)}|$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\big{(}H_{\alpha(2s)}-2{\rm
i}\tilde{H}_{\alpha(2s)}\big{)}~{},$ (7.14a)
$\displaystyle\mbox{\boldmath$\nabla$}^{{\underline{2}}\beta}\bar{G}_{\beta\alpha(2s-1)}|$
$\displaystyle=$ $\displaystyle\Phi_{\alpha(2s-1)}-2{\rm
i}\Omega_{\alpha(2s-1)}~{}.$ (7.14b)

The gauge transformations of these four real ${\cal N}=1$ superfields
immediately follow from gauge transformations (7.12)

$\displaystyle\delta H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\rho_{\alpha_{2}...\alpha_{2s})}~{},$
(7.15a) $\displaystyle\delta\tilde{H}_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\tau_{\alpha_{2}...\alpha_{2s})}~{},$
(7.15b) $\displaystyle\delta\Phi_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{8s}\Big{(}2(2s-1)\nabla_{(\alpha_{1}}{}^{\beta}\rho_{\alpha_{2}...\alpha_{2s-1})\beta}+2(4s^{2}-1)|\mu|\rho_{\alpha(2s-1)}$
$\displaystyle+(2s+1){\rm i}\nabla^{2}\rho_{\alpha(2s-1)}-8{\rm
i}(4s^{2}-1)|\mu|\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{2s-1})}\Big{)}~{},$
$\displaystyle\delta\Omega_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{8s}\Big{(}2(2s-1)\nabla_{(\alpha_{1}}{}^{\beta}\tau_{\alpha_{2}...\alpha_{2s-1})\beta}+2(4s^{2}-1)|\mu|\tau_{\alpha(2s-1)}~{}$
$\displaystyle+{\rm i}(2s+1)\nabla^{2}\tau_{\alpha(2s-1)}\Big{)}~{},$

where we have introduced the gauge parameter redefinition

$\rho_{\alpha(2s-1)}:=2\tilde{\tau}_{\alpha(2s-1)}+{\rm
i}\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{2s-1})}~{}.$ (7.16)

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

$\displaystyle{\cal U}_{\beta;\alpha(2s-2)}:={\rm
i}\mbox{\boldmath$\nabla$}^{{\underline{2}}}_{\beta}U_{\alpha(2s-2)}|~{},$
(7.17a) $\displaystyle V_{\alpha(2s-2)}:=-\frac{{\rm
i}}{8s}(\mbox{\boldmath$\nabla$}^{\underline{2}})^{2}U_{\alpha(2s-2)}|~{}.$
(7.17b)

Utilising the gauge transformations (7.2a), in conjunction with the residual
gauge symmetry (7.8) and defintion (7.13), we find that the superfields (7.17)
possess the following gauge freedom

$\displaystyle\delta{\cal U}_{\beta;\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\rho_{\beta\alpha(2s-2)}+{\rm
i}\nabla_{\beta}\zeta_{\alpha(2s-2)}~{},$ (7.18a) $\displaystyle\delta
V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s}\nabla^{\beta}\tau_{\beta\alpha(2s-2)}~{}.$ (7.18b)

It then becomes apparent from gauge transformations (7.15) and (7.18) that we
are dealing with two different gauge theories. The first of these gauge
theories is formulated in terms of the dynamical variables

${\cal V}^{\perp}_{(s)}=\big{\\{}H_{\alpha(2s)},{\cal
U}_{\beta;\alpha(2s-2)},\Phi_{\alpha(2s-1)}\big{\\}}~{},$ (7.19)

which are defined modulo gauge transformations

$\displaystyle\delta H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\rho_{\alpha_{2}...\alpha_{2s})}~{},$
(7.20a) $\displaystyle\delta{\cal U}_{\beta;\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle-\rho_{\beta\alpha(2s-2)}+{\rm
i}\nabla_{\beta}\zeta_{\alpha(2s-2)}~{},$ (7.20b)
$\displaystyle\delta\Phi_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{8s}\Big{(}2(2s-1)\nabla_{(\alpha_{1}}{}^{\beta}\rho_{\alpha_{2}...\alpha_{2s-1})\beta}+2(4s^{2}-1)|\mu|\rho_{\alpha(2s-1)}$
$\displaystyle+(2s+1){\rm i}\nabla^{2}\rho_{\alpha(2s-1)}-8{\rm
i}(4s^{2}-1)|\mu|\nabla_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{2s-1})}\Big{)}~{}.$

The other gauge model is described by the superfields

${\cal
V}^{\parallel}_{(s)}=\big{\\{}\tilde{H}_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-1)}\big{\\}}~{},$
(7.21)

which possesses the following gauge freedom

$\displaystyle\delta\tilde{H}_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}}\tau_{\alpha_{2}...\alpha_{2s})}~{},$
(7.22a) $\displaystyle\delta V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s}\nabla^{\beta}\tau_{\beta\alpha(2s-2)}~{},$ (7.22b)
$\displaystyle\delta\Omega_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{8s}\Big{(}2(2s-1)\nabla_{(\alpha_{1}}{}^{\beta}\tau_{\alpha_{2}...\alpha_{2s-1})\beta}+2(4s^{2}-1)|\mu|\tau_{\alpha(2s-1)}~{}$
$\displaystyle+{\rm i}(2s+1)\nabla^{2}\tau_{\alpha(2s-1)}\Big{)}~{}.$

Applying the reduction prescription to the action (7.1), we obtain two
decoupled ${\cal N}=1$ supersymmetric actions, which are formulated in terms
of superfields (7.19) and (7.21) respectively.

$\displaystyle\mathbb{S}^{\parallel}_{(s)}[U_{\alpha(2s-2)},G_{\alpha(2s)},\bar{G}_{\alpha(2s)}]$
$\displaystyle=$ $\displaystyle S^{\perp}_{(s)}[H_{\alpha(2s)},{\cal
U}_{\beta;\alpha(2s-2)},\Phi_{\alpha(2s-1)}]~{}$ (7.23) $\displaystyle+$
$\displaystyle
S^{\parallel}_{(s)}[\tilde{H}_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-1)}]~{}.$

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 [1]. This
formulation proves to be a generalised version of the longitudinal action
(7.3), for 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 [1]. 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. (7.23).

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

The first ${\cal N}=1$ supersymmetric action which is described by the
dynamical variables (7.19) takes the form

$\displaystyle S^{\perp}_{(s)}[H_{\alpha(2s)},{\cal
U}_{\beta;\alpha(2s-2)},\Phi_{\alpha(2s-1)}]=\bigg{(}-\frac{1}{2}\bigg{)}^{s}\frac{1}{4}\int{\rm
d}^{3|2}z\,E\,\bigg{\\{}-{\rm i}\,{\cal U}^{\beta;\,\alpha(2s-2)}\Box{\cal
U}_{\beta;\,\alpha(2s-2)}$ $\displaystyle-\frac{1}{2}{\cal
U}^{\beta;\,\alpha(2s-2)}{\nabla}^{2}{\nabla}_{\beta}{}^{\gamma}{\cal
U}_{\gamma;\,\alpha(2s-2)}+\frac{1}{2}(2s-3)|\mu|{\cal
U}^{\beta;\,\alpha(2s-2)}{\nabla}^{2}{\cal U}_{\beta;\,\alpha(2s-2)}$
$\displaystyle+2(s-1)|\mu|{\cal
U}_{\beta;}{}^{\beta;\,\alpha(2s-3)}{\nabla}^{2}{\cal
U}^{\gamma;}{}_{\gamma\alpha(2s-3)}+{\rm i}(4s^{2}-1)|\mu|^{2}{\cal
U}^{\beta;\,\alpha(2s-2)}{\cal U}_{\beta;\,\alpha(2s-2)}$ $\displaystyle-8{\rm
i}(s-1)|\mu|^{2}{\cal U}_{\beta;}{}^{\beta;\,\alpha(2s-3)}{\cal
U}^{\gamma;}{}_{\gamma\alpha(2s-3)}$ $\displaystyle-\frac{2s}{2s+1}{\cal
U}^{\beta;\,\alpha(2s-2)}\Big{(}-2{\nabla}^{\gamma\delta}{\nabla}_{\gamma}H_{\delta\beta\alpha(2s-2)}+{\nabla}^{\gamma\delta}{\nabla}_{\beta}H_{\gamma\delta\alpha(2s-2)}$
$\displaystyle+{\nabla}^{2}\Phi_{\beta\alpha(2s-2)}+2{\rm
i}\,{\nabla}_{\beta}{}^{\gamma}\Phi_{\alpha(2s-2)\gamma}$
$\displaystyle-4s|\mu|{\nabla}^{\gamma}H_{\gamma\beta\alpha(2s-2)}+2{\rm
i}(2s+1)|\mu|\Phi_{\beta\alpha(2s-2)}\Big{)}$
$\displaystyle-\frac{s}{(2s+1)^{2}(2s-1)}\Big{(}-{\rm
i}(3s+1)H^{\alpha(2s)}{\nabla}^{2}H_{\alpha(2s)}$
$\displaystyle+2s(4s+1)H^{\alpha(2s-1)\beta}{\nabla}_{\beta}{}^{\gamma}H_{\alpha(2s-1)\gamma}+16s^{2}H^{\alpha(2s)}{\nabla}_{(\alpha_{1}}\Phi_{\alpha_{2}\dots\alpha_{2s})}$
$\displaystyle-8{\rm
i}s\Phi^{\alpha(2s-1)}\Phi_{\alpha(2s-1)}-4s(4s^{2}+3s+1)|\mu|H^{\alpha(2s)}H_{\alpha(2s)}\Big{)}\bigg{\\}}~{}.$
(7.24)

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

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

we arrive at the resulting action

$\displaystyle S^{\perp}_{(s)}[H_{\alpha(2s)},{\cal
U}_{\beta;\alpha(2s-2)}]=\bigg{(}-\frac{1}{2}\bigg{)}^{s}\frac{1}{4}\int{\rm
d}^{3|2}z\,E\,\bigg{\\{}-\frac{{\rm
i}s(s-1)}{2s-1}H^{\alpha(2s)}{\nabla}^{2}H_{\alpha(2s)}$
$\displaystyle-\frac{2s^{2}}{2s-1}H^{\alpha(2s-1)\beta}{\nabla}_{\beta}{}^{\gamma}H_{\alpha(2s-1)\gamma}-\frac{4s^{2}(s-1)}{2s-1}|\mu|H^{\alpha(2s)}H_{\alpha(2s)}$
$\displaystyle-2s\,{\cal
U}^{\delta\alpha(2s-2)}{\nabla}^{\beta\gamma}\Big{(}{\nabla}_{(\delta}H_{\alpha_{1}\dots\alpha_{2s-2})\beta\gamma}-2{\nabla}_{\beta}H_{\gamma\delta\alpha(2s-2)}\Big{)}$
$\displaystyle+\frac{4s(s-1)}{2s-1}{\cal
U}_{\lambda;}{}^{\lambda\alpha(2s-3)}{\nabla}^{\beta\gamma}{\nabla}^{\delta}H_{\beta\gamma\delta\alpha(2s-3)}-8s^{2}|\mu|\,{\cal
U}^{\beta;\,\alpha(2s-2)}{\nabla}^{\gamma}H_{\gamma\beta\alpha(2s-2)}$
$\displaystyle+{\cal U}^{\alpha(2s-1)}\bigg{(}-2{\rm i}s\Box{\cal
U}_{\alpha(2s-1)}-s{\nabla}^{2}{\nabla}^{\beta}{}_{(\alpha_{1}}{\cal
U}_{\alpha_{2}\dots\alpha_{2s-1})\beta}$ $\displaystyle-{\rm
i}(s-1){\nabla}_{(\alpha_{1}\alpha_{2}}{\nabla}^{\beta\gamma}{\cal
U}_{\alpha_{3}\dots\alpha_{2s-1})\beta\gamma}+2{\rm
i}\frac{(s-1)(2s-3)}{2s-1}{\nabla}_{(\alpha_{1}\alpha_{2}}{\nabla}_{\alpha_{3}}{}^{\beta}{\cal
U}^{\gamma;}{}_{\alpha_{4}\dots\alpha_{2s-1})\beta\gamma}$
$\displaystyle+\frac{(s-1)(2s+1)}{2s-1}{\nabla}^{2}{\nabla}_{(\alpha_{1}\alpha_{2}}{\cal
U}^{\beta;}{}_{\alpha_{3}\dots\alpha_{2s-1})\beta}-s(2s+1)|\mu|{\nabla}^{2}{\cal
U}_{\alpha(2s-1)}$ $\displaystyle-4{\rm
i}\,s(2s-1)|\mu|{\nabla}^{\beta}{}_{(\alpha_{1}}{\cal
U}_{\alpha_{2}\dots\alpha_{2s-1})\beta}+2{\rm
i}(s-1)(2s+1)|\mu|{\nabla}_{(\alpha_{1}\alpha_{2}}{\cal
U}^{\beta;}{}_{\alpha_{3}\dots\alpha_{2s-1})\beta}$ $\displaystyle-2{\rm
i}s(2s-1)(2s+1)|\mu|^{2}{\cal U}_{\alpha(2s-1)}\bigg{)}$ $\displaystyle+{\cal
U}_{\lambda;}{}^{\lambda\alpha(2s-3)}\bigg{(}2{\rm
i}\frac{(s-1)(6s-5)}{(2s-1)^{2}}\Box{\cal U}^{\beta;}{}_{\beta\alpha(2s-3)}$
$\displaystyle-\frac{(s-1)(2s-3)}{(2s-1)^{2}}\Big{(}2{\rm
i}(s-2){\nabla}^{\beta\gamma}{\nabla}_{(\alpha_{1}\alpha_{2}}{\cal
U}^{\delta;}{}_{\alpha_{3}\dots\alpha_{2s-3})\beta\gamma\delta}$
$\displaystyle+{\nabla}^{2}{\nabla}^{\beta}{}_{(\alpha_{1}}{\cal
U}^{\gamma;}{}_{\alpha_{2}\dots\alpha_{2s-3})\beta\gamma}\Big{)}+\frac{(s-1)(2s+1)}{(2s-1)}|\mu|{\nabla}^{2}{\cal
U}^{\beta;}{}_{\beta\alpha(2s-3)}$ $\displaystyle-2{\rm
i}\frac{s-1}{2s-1}(4s^{2}+16s-17)|\mu|^{2}\,{\cal
U}^{\beta;}{}_{\beta\alpha(2s-3)}\bigg{)}\bigg{\\}}~{}.$ (7.26)

Here we have made use of the notation ${\cal U}_{\alpha(2s-1)}={\cal
U}_{(\alpha_{1};\alpha_{2}...\alpha_{2s-1})}$. It is a tedious but
straightforward exercise to verify that the above ${{\cal N}}=1$ action
coincides with the transverse formulation for massless superspin-$s$ multiplet
given by (3.2.2). To prove this, one first needs to introduce the ${{\cal
N}}=1$ field strength ${\cal W}_{\beta;\,\alpha(2s-2)}$ associated with the
prepotential ${\cal U}_{\beta;\alpha(2s-2)}$,

$\displaystyle{\cal W}_{\beta;\,\alpha(2s-2)}=-{\rm
i}({\nabla}^{\gamma}{\nabla}_{\beta}-4{\rm
i}|\mu|\delta_{\beta}{}^{\gamma})\,{\cal
U}_{\gamma;\,\alpha(2s-2)}~{},\qquad\nabla^{\beta}{\cal
W}_{\beta;\,\alpha(2s-2)}=0~{}.$ (7.27)

The action (7.3) can then be written more compactly as

$\displaystyle S^{\perp}_{(s)}[H_{\alpha(2s)},{\cal
U}_{\beta;\alpha(2s-2)}]=\Big{(}-\frac{1}{2}\Big{)}^{s}\bigg{(}\frac{2s}{2s-1}\bigg{)}\int{\rm
d}^{3|2}z\,E\,\bigg{\\{}\frac{1}{2}H^{\alpha(2s)}({\rm
i}\nabla^{2}+8s|\mu|)H_{\alpha(2s)}$ $\displaystyle-{\rm
i}s\nabla_{\beta}H^{\beta\alpha(2s-1)}\nabla^{\gamma}H_{\gamma\alpha(2s-1)}-(2s-1){\cal
W}^{\beta;\,\alpha(2s-2)}\nabla^{\gamma}H_{\gamma\beta\alpha(2s-2)}$
$\displaystyle-\frac{{\rm i}}{2}(2s-1)\Big{(}{\cal
W}^{\beta;\,\alpha(2s-2)}{\cal W}_{\beta;\,\alpha(2s-2)}+\frac{s-1}{s}{\cal
W}_{\beta;}\,^{\beta\alpha(2s-3)}{\cal
W}^{\gamma;}\,_{\gamma\alpha(2s-3)}\Big{)}$ $\displaystyle-2{\rm
i}(2s-1)|\mu|\,{\cal U}^{\beta;\,\alpha(2s-2)}{\cal
W}_{\beta;\,\alpha(2s-2)}\bigg{\\}}~{},$ (7.28)

which is exactly (3.2.2) modulo an overall normalisation factor.

The other decoupled ${\cal N}=1$ action which is formulated in terms of the
dynamical variables (7.21) assumes the form

$\displaystyle
S^{\parallel}_{(s)}[\tilde{H}_{\alpha(2s)},V_{\alpha(2s-2)},\Omega_{\alpha(2s-1)}]=\bigg{(}-\frac{1}{2}\bigg{)}^{s}\int{\rm
d}^{3|2}z\,E\,\bigg{\\{}2s^{2}V^{\alpha(2s-2)}\big{(}{\rm
i}\nabla^{2}+4|\mu|\big{)}V_{\alpha(2s-2)}~{}$
$\displaystyle-\frac{4s^{2}}{2s+1}V^{\alpha(2s-2)}\big{(}{\nabla}^{\beta\gamma}\tilde{H}_{\beta\gamma\alpha(2s-2)}+2{\nabla}^{\beta}\Omega_{\beta\alpha(2s-2)}\big{)}$
$\displaystyle+\frac{s}{(2s+1)^{2}(2s-1)}\Big{(}(2s+1)^{2}\tilde{H}^{\alpha(2s)}({\rm
i}{\nabla}^{2}+4|\mu|)\tilde{H}_{\alpha(2s)}$ $\displaystyle-4{\rm
i}s(s+1){\nabla}_{\beta}\tilde{H}^{\beta\alpha(2s-1)}{\nabla}^{\gamma}\tilde{H}_{\gamma\alpha(2s-1)}-8s\tilde{H}^{\alpha(2s)}{\nabla}_{(\alpha_{1}}\Omega_{\alpha_{2}\dots\alpha_{2s})}$
$\displaystyle+16{\rm
i}s^{2}\Omega^{\alpha(2s-1)}\Omega_{\alpha(2s-1)}\Big{)}\bigg{\\}}~{}.$ (7.29)

It is clear from the action (7.3) that the superfield $\Omega_{\alpha(2s-1)}$
is auxiliary. Integrating it out by using its equation of motion,

$\Omega_{\alpha(2s-1)}=\frac{{\rm
i}}{4s}\Big{(}(4s^{2}-1)\nabla_{(\alpha_{1}}V_{\alpha_{2}...\alpha_{2s-1})}-\nabla^{\beta}\tilde{H}_{\beta\alpha(2s-1)}\Big{)}~{},$
(7.30)

we find that the resulting action, modulo a normalisation factor, coincides
with (3.2.1).

###  7.4 Second supersymmetry transformations

We begin by computing the second supersymmetry transformations (5.25) of the
superfields (7.19) and (7.21)

$\displaystyle\delta_{\epsilon}U_{\alpha(2s-2)}|$ $\displaystyle=$
$\displaystyle-{\rm i}\epsilon^{\beta}{\cal U}_{\beta;\alpha(2s-2)}~{},$
(7.31a) $\displaystyle\delta_{\epsilon}{\cal U}_{\beta;\alpha(2s-2)}$
$\displaystyle=$ $\displaystyle 4s\epsilon_{\beta}V_{\alpha(2s-2)}~{},$
(7.31b) $\displaystyle\delta_{\epsilon}V_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{{\rm i}}{4s}\Big{(}(2s-1)|\mu|\epsilon^{\beta}{\cal
U}_{\beta;\alpha(2s-2)}-\epsilon^{\beta}\nabla_{\beta}{}^{\gamma}{\cal
U}_{\gamma;\alpha(2s-2)}~{}$ $\displaystyle+$ $\displaystyle
4(s-1)|\mu|\epsilon_{(\alpha_{1}}{\cal
U}^{\beta;}{}_{\alpha_{2}...\alpha_{2s-2})\beta}\Big{)}~{},$
$\displaystyle\delta_{\epsilon}H_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle-2\epsilon^{\beta}\nabla_{(\beta}\tilde{H}_{\alpha(2s))}-\frac{8{\rm
i}s}{2s+1}\epsilon_{(\alpha_{1}}\Omega_{\alpha_{2}...\alpha_{2s})}~{},$
(7.31d) $\displaystyle\delta_{\epsilon}\tilde{H}_{\alpha(2s)}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\epsilon^{\beta}\nabla_{(\beta}H_{\alpha(2s))}+\frac{2{\rm
i}s}{2s+1}\epsilon_{(\alpha_{1}}\Phi_{\alpha_{2}...\alpha_{2s})}~{},$ (7.31e)
$\displaystyle\delta_{\epsilon}\Phi_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2(2s+1)}\epsilon^{\beta}\Big{(}2(4s^{2}+3s+1)|\mu|\tilde{H}_{\beta\alpha(2s-1)}+{\rm
i}\nabla^{2}\tilde{H}_{\beta\alpha(2s-1)}~{}$ $\displaystyle-$ $\displaystyle
2(4s+3)\nabla_{\beta}{}^{\gamma}\tilde{H}_{\gamma\alpha(2s-1)}-2(2s-1)\nabla_{(\alpha_{1}}{}^{\gamma}\tilde{H}_{\alpha_{2}...\alpha_{2s-1})\beta\gamma}~{}$
$\displaystyle+$ $\displaystyle
8s\nabla_{(\beta}\Omega_{\alpha(2s-1))}\Big{)}~{},$
$\displaystyle\delta_{\epsilon}\Omega_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{8(2s+1)}\epsilon^{\beta}\Big{(}4(4s^{2}+3s+1)|\mu|H_{\beta\alpha(2s-1)}+{\rm
i}\nabla^{2}H_{\beta\alpha(2s-1)}~{}$ $\displaystyle-$ $\displaystyle
2(4s+3)\nabla_{\beta}{}^{\gamma}H_{\gamma\alpha(2s-1)}-2(2s-1)\nabla_{(\alpha_{1}}{}^{\gamma}H_{\alpha_{2}...\alpha_{2s-1})\beta\gamma}~{}$
$\displaystyle+$ $\displaystyle
8s\nabla_{(\beta}\Phi_{(\alpha(2s-1))}\Big{)}~{}.$

It is clear from the variation (7.31a) that the second supersymmetry
transformation (5.25) breaks the WZ gauge (7.6). To restore this gauge
condition, it is necessary to supplement the second supersymmetry
transformation with the $\epsilon$-dependent gauge transformations:

$\displaystyle\delta_{\gamma(\epsilon)}U_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\gamma_{\alpha(2s-2)}(\epsilon)+\bar{\gamma}_{\alpha(2s-2)}(\epsilon)~{},$
(7.32a) $\displaystyle\delta_{\gamma(\epsilon)}G_{\alpha(2s)}$
$\displaystyle=$ $\displaystyle\bar{{\cal D}}_{(\alpha_{1}}{\cal