Let us introduce the operator $\Pi^{\parallel}_{(n)}$ which is the orthogonal
complement of $\Pi^{\perp}_{(n)}$

$\Pi^{\parallel}_{(n)}:=\mathds{1}-\Pi^{\perp}_{(n)}~{}.$ (2.282)

By construction, the operators $\Pi^{\perp}_{(n)}$ and $\Pi^{\parallel}_{(n)}$
resolve the identity, $\mathds{1}=\Pi^{\parallel}_{(n)}+\Pi^{\perp}_{(n)}$,
and form an orthogonal set of projectors

$\displaystyle\Pi^{\perp}_{(n)}\Pi^{\perp}_{(n)}$
$\displaystyle=\Pi^{\perp}_{(n)}~{},\qquad\Pi^{\parallel}_{(n)}\Pi^{\parallel}_{(n)}=\Pi^{\parallel}_{(n)}~{},$
(2.283a) $\displaystyle\Pi^{\parallel}_{(n)}\Pi^{\perp}_{(n)}$
$\displaystyle=0~{},\qquad~{}~{}~{}\phantom{.}\Pi^{\perp}_{(n)}\Pi^{\parallel}_{(n)}=0~{}.$
(2.283b)

Moreover, it can be shown that $\Pi^{\parallel}_{(n)}$ projects a field
$\phi_{\alpha(n)}$ onto its longitudinal component. A rank-$n$ field
$\phi^{\parallel}_{\alpha(n)}$ is said to be longitudinal if there exists a
rank-$(n-2)$ field $\phi_{\alpha(n-2)}$ such that
$\phi^{\parallel}_{\alpha(n)}$ may be expressed as
$\phi^{\parallel}_{\alpha(n)}=\partial_{\alpha(2)}\phi_{\alpha(n-2)}$. Such
fields are sometimes referred to as being pure gauge. Therefore, we find that
[4]

$\displaystyle\phi^{\parallel}_{\alpha(n)}:=\Pi^{\parallel}_{(n)}\phi_{\alpha(n)}=\partial_{\alpha(2)}\phi_{\alpha(n-2)}~{},$
(2.284)

for some unconstrained field $\phi_{\alpha(n-2)}$.

Let $\phi^{\parallel}_{\alpha(n)}$ be some longitudinal field, for which we do
not assume to be in the image of $\Pi^{\parallel}_{(n)}$. Since
$\Pi_{(n)}^{\perp}$ commutes with $\partial_{\alpha(2)}$ and annihilates all
lower-rank fields (5.41), it follows that it also annihilates any rank-$n$
longitudinal field [4]333333This also implies that
$\widehat{\Pi}^{\perp}_{(n)}\psi_{\alpha(n)}=0$, since both
$\widehat{\Pi}^{\perp}_{(n)}$ and $\Pi^{\perp}_{(n)}$ are equal on
${V}_{(n)}$.

$\displaystyle\phi^{\parallel}_{\alpha(n)}=\partial_{\alpha(2)}\phi_{\alpha(n-2)}\qquad\implies\qquad\Pi^{\perp}_{(n)}\phi^{\parallel}_{\alpha(n)}=0~{}.$
(2.285)

As a consequence, given two integers $n,k$ satisfying $2\leq k\leq n$, it
immediately follows that $\Pi^{\parallel}_{(n)}$ acts as the identity operator
on the space of rank-$k$ longitudinal fields $\phi^{\parallel}_{\alpha(k)}$
[4],

$\displaystyle\phi^{\parallel}_{\alpha(k)}=\partial_{\alpha(2)}\phi_{\alpha(k-2)}\qquad\implies\qquad\Pi^{\parallel}_{(k+2s)}\phi^{\parallel}_{\alpha(k)}=\phi^{\parallel}_{\alpha(k)}~{},$
(2.286)

with $s$ a non-negative integer. These properties will be useful in the
subsequent section.

Using the fact that $\Pi^{\perp}_{(n)}$ and $\Pi^{\parallel}_{(n)}$ resolve
the identity, one can decompose an arbitrary field $\phi_{\alpha(n)}$ as
follows

$\phi_{\alpha(n)}=\phi^{\perp}_{\alpha(n)}+\partial_{\alpha(2)}\phi_{\alpha(n-2)}~{}.$
(2.287)

Here $\phi^{\perp}_{\alpha(n)}$ is transverse and $\phi_{\alpha(n-2)}$ is
unconstrained. Repeating this process iteratively, we obtain the following
decomposition [4]

$\displaystyle\phi_{\alpha(n)}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\lfloor
n/2\rfloor}\big{(}\partial_{\alpha(2)}\big{)}^{j}\phi^{\perp}_{\alpha(n-2j)}~{}.$
(2.288)

Here each of the fields $\phi^{\perp}_{\alpha(n-2j)}$ are transverse, except
of course $\phi^{\perp}$ and $\phi^{\perp}_{\alpha}$. Note that the spin
projection operators (2.269) will only select the component in the
decomposition (2.288) with the highest spin, as a consequence of the
properties (2.273) and (2.275). We note that, using (2.277), one may take the
decomposition (2.288) a step further and bisect each term into irreducible
components which are transverse and have positive or negative helicity [4],

$\displaystyle\phi_{\alpha(n)}=\sum_{j=0}^{\lfloor
n/2\rfloor}\big{(}\partial_{\alpha(2)}\big{)}^{j}\Big{(}\phi^{(+)}_{\alpha(n-2j)}+\phi^{(-)}_{\alpha(n-2j)}\Big{)}~{}.$
(2.289)

Making use of the spin projectors (2.269) and their corresponding properties,
one can construct operators which extract the component
$\phi_{\alpha(n-2j)}^{\perp}$ from the decomposition (2.288), where $1\leq
j\leq\lfloor n/2\rfloor$. In particular, we find that the spin
$\frac{1}{2}(n-2j)$ component may be extracted via [4]

$\displaystyle\phi_{\alpha(n)}\mapsto\phi^{\perp}_{\alpha(n-2j)}=\big{(}{S}_{(n-2j)}^{\perp}\phi\big{)}_{\alpha(n-2j)}\equiv{S}_{\alpha(n-2j)}^{\perp}(\phi)~{},$
(2.290)

where we have defined

$\displaystyle{S}_{\alpha(n-2j)}^{\perp}(\phi)$
$\displaystyle=\frac{(-1)^{j}}{2^{2j}}\binom{n}{j}\frac{1}{\Box^{j}}\Pi^{\perp}_{(n-2j)}\big{(}\partial^{\beta(2)}\big{)}^{j}\phi_{\alpha(n-2j)\beta(2j)}~{}.$
(2.291)

From this expression, it is clear that ${S}_{\alpha(n-2j)}^{\perp}(\phi)$ is
transverse,

$0=\partial^{\beta(2)}{S}_{\beta(2)\alpha(n-2j-2)}^{\perp}(\phi)~{}.$ (2.292)

Therefore it is appropriate to call ${S}_{(n-2j)}^{\perp}$ the transverse spin
$\frac{1}{2}(n-2j)$ extractor. It is not idempotent, since it is dimensionful
and reduces the rank of the field on which it acts.

####  2.3.4 Conformal higher-spin theory

In this section we explore applications for the spin projection operators in
modern conformal higher-spin theories in $\mathbb{M}^{3}$. For $n\geq 2$, a
real rank-$n$ spinor field $h_{\alpha(n)}$ is said to be a CHS field if it is
defined modulo gauge transformations of the form [117]343434For a detailed
discussion concerning the conformal nature of CHS fields in $\mathbb{M}^{3}$,
see [117].

$\delta_{\zeta}h_{\alpha(n)}=\partial_{\alpha(2)}\zeta_{\alpha(n-2)}~{},$
(2.293)

where $\zeta_{\alpha(n-2)}$ is a real unconstrained gauge parameter.

Associated with the gauge prepotentials $h_{\alpha(n)}$ are the real primary
descendents $C_{\alpha(n)}(h)$, which are known as the linearised higher-spin
Cotton tensors. The Cotton tensors $C_{\alpha(n)}(h)$ are uniquely defined by
the properties:

  1. 1.

$C_{\alpha(n)}(h)$ is transverse

$\partial^{\beta\gamma}C_{\beta\gamma\alpha(n-2)}(h)=0~{}.$ (2.294a)

  2. 2.

$C_{\alpha(n)}(h)$ is gauge-invariant

$\delta_{\zeta}C_{\alpha(n)}(h)=0~{}.$ (2.294b)

Modulo an overall normalisation factor, the linearised higher-spin bosonic and
fermionic Cotton tensors take the respective closed forms

$\displaystyle{C}_{\alpha(2s)}(h)$ $\displaystyle=$
$\displaystyle\frac{1}{2^{2s-1}}\sum_{j=0}^{s-1}2^{2j+1}\binom{s+j}{2j+1}\Box^{j}\partial_{\alpha(2)}^{s-j-1}\partial_{\alpha}{}^{\beta}\big{(}\partial^{\beta(2)}\big{)}^{s-j-1}h_{\alpha(2j+1)\beta(2s-2j-1)}~{},\hskip
39.83368pt$ (2.295a) $\displaystyle{C}_{\alpha(2s+1)}(h)$ $\displaystyle=$
$\displaystyle\frac{1}{2^{2s}}\sum_{j=0}^{s}2^{2j}\binom{s+j}{2j}\frac{(2s+1)}{(2j+1)}\Box^{j}\partial_{\alpha(2)}^{s-j}\big{(}\partial^{\beta(2)}\big{)}^{s-j}h_{\alpha(2j+1)\beta(2s-2j)}~{}.$
(2.295b)

The linearised higher-spin bosonic Cotton tensor (2.295a) was first derived by
Pope and Townsend [65] in 1989, while its fermionic counterpart (2.295b) was
only recently derived by Kuzenko [66] in 2016.

Since the Cotton tensors (2.295) are transverse by definition, it proves
natural to recast them in terms of the transverse spin projection operators
(2.3.3). Making use of the projectors $\widehat{\Pi}^{\perp}_{(n)}$ (2.3.3),
it can be shown that the higher-spin Cotton tensors (2.295) can be
reformulated in the simple form [67]

$\displaystyle{C}_{\alpha(2s)}(h)$ $\displaystyle=$
$\displaystyle\frac{1}{2s}\Box^{s-1}W\widehat{\Pi}^{\perp}_{(2s)}h_{\alpha(2s)}~{},$
(2.296a) $\displaystyle{C}_{\alpha(2s+1)}(h)$ $\displaystyle=$
$\displaystyle\Box^{s}\widehat{\Pi}^{\perp}_{(2s+1)}h_{\alpha(2s+1)}~{}.$
(2.296b)

Moreover, we can make use of the equivalent family of projectors
$\Pi^{\perp}_{(n)}$ to recast ${C}_{\alpha(n)}(h)$ (2.296) purely in terms of
the Casimir operators (2.233). Explicitly, they read [4]353535It can be shown
that the Cotton tensors are equivalent to those derived in [153, 154].

$\displaystyle C_{\alpha(2s)}(h)$ $\displaystyle=$
$\displaystyle\frac{W}{2^{2s-1}(2s-1)!}\prod_{j=1}^{s-1}\Big{(}W^{2}-(2j)^{2}\Box\Big{)}h_{\alpha(2s)}~{},$
(2.297a) $\displaystyle C_{\alpha(2s+1)}(h)$ $\displaystyle=$
$\displaystyle\frac{1}{2^{2s}(2s)!}\prod_{j=0}^{s-1}\Big{(}W^{2}-(2j+1)^{2}\Box\Big{)}h_{\alpha(2s+1)}~{}.$
(2.297b)

There are many advantages to expressing the Cotton tensors in terms of spin
projection operators. Firstly, in both (2.296) and (2.297), the properties of
transversality (2.294a) and gauge invariance (2.294b) are made manifest, as a
consequence of the projector properties (2.265b) and (2.285).

Using the gauge freedom (2.293), one may impose the transverse gauge condition

$\displaystyle h_{\alpha(n)}\equiv h^{\perp}_{\alpha(n)}~{},\qquad
0=\partial^{\beta(2)}h^{\perp}_{\beta(2)\alpha(n-2)}~{}.$ (2.298)

In this gauge, the Cotton tensors (2.296) become manifestly factorised into
products of (at most) second-order differential operators

$\displaystyle{C}_{\alpha(2s)}(h^{\perp})$ $\displaystyle=$
$\displaystyle\frac{1}{2s}\Box^{s-1}Wh^{\perp}_{\alpha(2s)}~{},$ (2.299a)
$\displaystyle{C}_{\alpha(2s+1)}(h^{\perp})$ $\displaystyle=$
$\displaystyle\Box^{s}h^{\perp}_{\alpha(2s+1)}~{},$ (2.299b)

on account of the projector property (2.265c). An interesting feature of the
new realisation (2.297) is that the Cotton tensors are manifestly factorised
in terms of (at most) second-order differential operators without having to
enter the transverse gauge.

By virtue of the above observations, it follows that the conformal higher-spin
action

$\displaystyle S_{\text{CHS}}^{(n)}[h]=\frac{\text{i}^{n}}{2^{\lceil
n/2\rceil+1}}\int\text{d}^{3}x\,h^{\alpha(n)}{C}_{\alpha(n)}(h)~{},$ (2.300)

is manifestly gauge invariant and factorised when ${C}_{\alpha(n)}(h)$ is
expressed as (2.297).

It is well known that in $\mathbb{M}^{3}$, the Cotton tensor
$C_{\alpha(n)}(h)$ vanishes if and only if the gauge field $h_{\alpha(n)}$ is
pure gauge

$C_{\alpha(n)}(h)=0\qquad\Longleftrightarrow\qquad
h_{\alpha(n)}=\partial_{\alpha(2)}\zeta_{\alpha(n-2)}~{}.$ (2.301)

This property, which is known as gauge completeness, was first proved in [153,
154].363636Note that a sketch of this proof which utilises the spin projection
operators (2.3.3) was given in [152].  Gauge completeness is useful in
determining whether a higher-spin theory in $\mathbb{M}^{3}$ is massless.
Specifically, if it can be shown that the Cotton tensor of the surviving gauge
field $h_{\alpha(n)}$ vanishes on the equations of motion, then
$h_{\alpha(n)}$ is massless since it propagates no physical degrees of
freedom.

####  2.3.5 Massless higher-spin theories

We wish to study free theories which, at the level of the equations of motion,
generate the massless conditions (2.252). These actions were first computed by
Fronsdal [27] and Fang and Fronsdal [28] in $\mathbb{M}^{4}$ for a massless
spin-$s$ and spin-$(s+\frac{1}{2})$ field, respectively. In this section we
review generalisations of these massless models to $\mathbb{M}^{3}$, which
were recently constructed by Kuzenko and Ogburn [122].

#####  2.3.5.1 Fronsdal action

Given an integer $s\geq 2$, the Fronsdal action is described by the real
bosonic fields $h_{\alpha(2s)}$ and $y_{\alpha(2s-4)}$ on $\mathbb{M}^{3}$,
which are defined modulo gauge transformations of the form

$\displaystyle\delta_{\zeta}h_{\alpha(2s)}$ $\displaystyle=$
$\displaystyle\partial_{\alpha(2)}\zeta_{\alpha(2s-2)}~{},$ (2.302a)
$\displaystyle\delta_{\zeta}y_{\alpha(2s-4)}$ $\displaystyle=$
$\displaystyle\frac{2s-2}{2s-1}\partial^{\beta(2)}\zeta_{\beta(2)\alpha(2s-4)}~{}.$
(2.302b)

Here the gauge parameter $\zeta_{\alpha(2s-2)}$ is real unconstrained. The
Fronsdal action is given by

$\displaystyle S^{\text{F}}_{(s)}[h,y]=$
$\displaystyle\frac{1}{2}\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3}x~{}\Big{\\{}h^{\alpha(2s)}\Box
h_{\alpha(2s)}-\frac{s}{2}\partial_{\beta(2)}h^{\beta(2)\alpha(2s-2)}\partial^{\gamma(2)}h_{\gamma(2)\alpha(2s-2)}$
(2.303)
$\displaystyle-\frac{1}{2}(2s-3)\Big{(}y^{\alpha(2s-4)}\partial^{\beta(2)}\partial^{\gamma(2)}h_{\beta(2)\gamma(2)\alpha(2s-4)}+\frac{2}{s}y^{\alpha(2s-4)}\Box
y_{\alpha(2s-4)}$
$\displaystyle+\frac{1}{4s(s-1)}(s-2)(2s-5)\partial_{\beta(2)}y^{\beta(2)\alpha(2s-6)}\partial^{\gamma(2)}y_{\gamma(2)\alpha(2s-6)}\Big{)}\Big{\\}}~{},$

which is invariant under the gauge transformations (2.302).

It is an instructive exercise to show that the action (2.303) does indeed
describe a massless spin-$s$ field. The equations of motion corresponding to
(2.303) are

$\displaystyle 0=$ $\displaystyle 2\Box
h_{\alpha(2s)}+s\partial_{\alpha(2)}\partial^{\beta(2)}h_{\beta(2)\alpha(2s-2)}-\frac{1}{2}(2s-3)\partial_{\alpha(2)}\partial_{\alpha(2)}y_{\alpha(2s-4)}~{},$
(2.304a) $\displaystyle 0=$ $\displaystyle 8(s-1)\Box
y_{\alpha(2s-4)}-(s-2)(2s-5)\partial_{\alpha(2)}\partial^{\beta(2)}y_{\beta(2)\alpha(2s-6)}$
(2.304b)
$\displaystyle+2s(s-1)\partial^{\beta(2)}\partial^{\gamma(2)}h_{\beta(2)\gamma(2)\alpha(2s-4)}~{}.$

The gauge transformation (2.302b) allows us to impose the gauge condition

$y_{\alpha(2s-4)}=0~{}.$ (2.305)

In accordance with (2.302b), the residual gauge freedom which preserves
(2.305) is described by the parameter $\zeta_{\alpha(2s-2)}$ constrained by

$\partial^{\beta(2)}\zeta_{\beta(2)\alpha(2s-4)}=0~{}.$ (2.306)

In the gauge (2.305), the equation of motion (2.304b) reduces to

$\partial^{\beta(2)}\partial^{\gamma(2)}h_{\beta(2)\gamma(2)\alpha(2s-4)}=0~{}.$
(2.307)

Using (2.302a), the divergence of $h_{\alpha(2s)}$ can be shown to transform
as

$\delta_{\zeta}\big{(}\partial^{\beta(2)}h_{\beta(2)\alpha(2s-2)}\big{)}=-\frac{2}{s}\Box\zeta_{\alpha(2s-2)}~{},$
(2.308)

where we have also used the fact that the gauge parameter is constrained by
(2.306). Taking into account (2.307), the gauge freedom (2.308) can be used to
gauge away $\partial^{\beta(2)}h_{\beta(2)\alpha(2s-2)}$

$\partial^{\beta(2)}h_{\beta(2)\alpha(2s-2)}=0~{}.$ (2.309)

The remaining residual gauge symmetry is described by the gauge parameter
satisfying (2.306), and the second-order equation

$\Box\zeta_{\alpha(2s-2)}=0~{}.$ (2.310)

Upon imposing the gauges (2.305) and (2.309), it follows that the equation of
motion (2.304a) reduces to

$\Box h_{\alpha(2s)}=0~{}.$ (2.311)

In accordance with (2.252), it follows from the above analysis that the
Fronsdal action (2.303) describes a massless spin-$s$ field on-shell.

#####  2.3.5.2 Fang-Fronsdal action

For $s\geq 2$, the Fang-Fronsdal model is described in terms of the real
fields $h_{\alpha(2s+1)}$, $y_{\alpha(2s-1)}$ and $y_{\alpha(2s-3)}$, which
are defined modulo gauge transformations of the form

$\displaystyle\delta_{\zeta}h_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle\partial_{\alpha(2)}\zeta_{\alpha(2s-1)}~{},$ (2.312a)
$\displaystyle\delta_{\zeta}y_{\alpha(2s-1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s+1}\partial^{\beta}{}_{(\alpha_{1}}\zeta_{\alpha_{2}\ldots\alpha_{2s-1})\beta}~{},$
(2.312b) $\displaystyle\delta_{\zeta}z_{\alpha(2s-3)}$ $\displaystyle=$
$\displaystyle\partial^{\beta(2)}\zeta_{\beta(2)\alpha(2s-3)}~{}.$ (2.312c)

Here, the gauge parameter $\zeta_{\alpha(2s-1)}$ is real and unconstrained.
The Fang-Fronsdal action is given by

$\displaystyle S^{\text{FF}}_{(s+\frac{1}{2})}[h,y,z]$
$\displaystyle=\frac{{\rm i}}{2}\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3}x~{}\Big{\\{}h^{\beta\alpha(2s)}\partial_{\beta}{}^{\gamma}h_{\gamma\alpha(2s)}+2(2s-1)y^{\alpha(2s-1)}\partial^{\beta(2)}h_{\beta(2)\alpha(2s-1)}$
$\displaystyle+4(2s-1)y^{\beta\alpha(2s-2)}\partial_{\beta}{}^{\gamma}y_{\gamma\alpha(2s-2)}+\frac{2}{s}(s-1)(2s+1)z^{\alpha(2s-3)}\partial^{\beta(2)}y_{\beta(2)\alpha(2s-3)}$
$\displaystyle-\frac{1}{s(2s-1)}(s-1)(2s-3)z^{\beta\alpha(2s-4)}\partial_{\beta}{}^{\gamma}z_{\gamma\alpha(2s-4)}\Big{\\}}~{},$
(2.313)

which is invariant under the gauge transformations (2.312).

Let us show that the Fang-Fronsdal model (2.3.5.2) describes a massless
spin-$(s+\frac{1}{2})$ field on-shell. The equations of motion corresponding
to the model (2.3.5.2) are

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{\alpha}{}^{\gamma}h_{\gamma\alpha(2s)}-(2s-1)\partial_{\alpha(2)}y_{\alpha(2s-1)}~{},$
(2.314a) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial^{\beta(2)}h_{\beta(2)\alpha(2s-1)}+4\partial_{\alpha}{}^{\beta}y_{\beta\alpha(2s-2)}-\frac{1}{s(2s-1)}(s-1)(2s+1)\partial_{\alpha(2)}z_{\alpha(2s-3)}~{},\hskip
18.49411pt$ (2.314b) $\displaystyle 0$ $\displaystyle=$
$\displaystyle(2s-1)(2s+1)\partial^{\beta(2)}y_{\beta(2)\alpha(2s-3)}-(2s-3)\partial_{\alpha}{}^{\beta}z_{\beta\alpha(2s-4)}~{}.$
(2.314c)

We can use the gauge transformation (2.312c) to impose the gauge

$z_{\alpha(2s-3)}=0~{}.$ (2.315)

In accordance with (2.312c), the residual gauge symmetry that preserves the
condition (2.315) is

$\partial^{\beta(2)}\zeta_{\beta(2)\alpha(2s-3)}=0~{}.$ (2.316)

In the gauge (2.315), the equation of motion (2.314c) reduces to

$\partial^{\beta(2)}y_{\beta(2)\alpha(2s-3)}=0~{}.$ (2.317)

Hence, due to (2.316) and the gauge transformation (2.312b), we are allowed to
impose the additional gauge

$y_{\alpha(2s-1)}=0~{},$ (2.318)

which is consistent with (2.317). The residual gauge freedom which preserves
(2.318) is given by

$W\zeta_{\alpha(2s-1)}~{}=0\quad\Longrightarrow\quad\Box\zeta_{\alpha(2s-1)}=0~{}.$
(2.319)

In the gauges (2.315) and (2.318), the EoMs (2.314a) and (2.314b) reduce to

$\displaystyle Wh_{\alpha(2s+1)}$ $\displaystyle=$ $\displaystyle 0~{},$
(2.320a) $\displaystyle\partial^{\beta(2)}h_{\beta(2)\alpha(2s-1)}$
$\displaystyle=$ $\displaystyle 0~{}.$ (2.320b)

The conditions (2.312a), (2.316) (2.317), (2.319) and (2.320) coincide with
the massless on-shell conditions (2.246) for the case $n=2s+1$, thus
confirming that the action (2.3.5.2) describes a massless
spin-$(s+\frac{1}{2})$ field.

#####  2.3.5.3 Dual formulation of the Fronsdal-type action in
$\mathbb{M}^{3}$

In this section we introduce a one-parameter family of dual formulations [2]
for the Fronsdal-type action [117]. This Fronsdal-type action was shown in
[117] to take the following form373737In the $n=2s$, the action (2.321)
coincides with the Fronsdal action (2.303).

$\displaystyle S_{(\frac{n}{2})}^{\rm F}[h,y]=\frac{{\rm i}^{n}}{2^{\lfloor
n/2\rfloor+1}}$ $\displaystyle\int{\rm
d}^{3}x\,\bigg{\\{}\frac{1}{2}\partial^{\beta\gamma}h^{\alpha(n)}\partial_{\beta\gamma}h_{\alpha(n)}-\frac{n}{4}\partial_{\beta\gamma}h^{\beta\gamma\alpha(n-2)}\partial^{\delta\lambda}h_{\delta\lambda\alpha(n-2)}~{}$
$\displaystyle+\frac{n-3}{n}\Big{(}\frac{n}{2}\partial^{\beta\gamma}y^{\alpha(n-4)}\partial^{\delta\lambda}h_{\beta\gamma\delta\lambda\alpha(n-4)}-\partial^{\beta\gamma}y^{\alpha(n-4)}\partial_{\beta\gamma}y_{\alpha(n-4)}~{}$
$\displaystyle-\frac{(n-4)(n-5)}{4(n-2)}\partial_{\beta\gamma}y^{\beta\gamma\alpha(n-6)}\partial^{\delta\lambda}y_{\delta\lambda\alpha(n-6)}\Big{)}\bigg{\\}}~{}.$
(2.321)

It is invariant under the gauge transformations

$\displaystyle\delta_{\zeta}h_{\alpha(n)}$ $\displaystyle=$
$\displaystyle\partial_{\alpha(2)}\zeta_{\alpha(n-2)}~{},$ (2.322a)
$\displaystyle\delta_{\zeta}y_{\alpha(n-4)}$ $\displaystyle=$
$\displaystyle\frac{n-2}{n-1}\partial^{\beta(2)}\zeta_{\beta(2)\alpha(n-4)}~{}.$
(2.322b)

Both fields $h_{\alpha(n)}$ and $y_{\alpha(n-4)}$ appear in (2.321) with
derivatives,

$\displaystyle S^{\rm F}_{(\frac{n}{2})}=\frac{{\rm i}^{n}}{2^{\lfloor
n/2\rfloor+1}}\int{\rm d}^{3}x\,{\cal
L}\big{(}\partial_{\beta\gamma}h_{\alpha(n)},\partial_{\beta\gamma}y_{\alpha(n-4)}\big{)}~{}.$
(2.323)

Therefore a duality transformation may be performed upon each of them. Here we
will only dualise $y_{\alpha(n-4)}$ and keep $h_{\alpha(n)}$ intact.383838The
gauge transformation law (2.322a) allows us to interpret $h_{\alpha(n)}$ as a
conformal spin-$\frac{n}{2}$ gauge field, while (2.322b) is compatible with
the interpretation of $y_{\alpha(n-4)}$ as a conformal compensator (following
the modern supergravity terminology). Performing a duality transformation on
$y_{\alpha(n-4)}$ is equivalent to the introduction of an alternative
conformal compensator.

Our dual formulation for (2.323) is obtained by introducing the following
first-order action

$\displaystyle S_{\text{first-order}}=\frac{{\rm i}^{n}}{2^{\lfloor
n/2\rfloor+1}}\int{\rm d}^{3}x\,\Big{\\{}{\cal
L}\big{(}\partial_{\beta\gamma}h_{\alpha(n)},{\cal
H}_{\beta\gamma;\,\alpha(n-4)}\big{)}$ $\displaystyle-(n-3){\cal
H}^{\beta\gamma;\,\alpha(n-4)}F_{\beta\gamma;\,\alpha(n-4)}\Big{\\}}~{},$
(2.324)

where the Lagrangian ${\cal L}(\partial_{\beta\gamma}h_{\alpha(n)},{\cal
H}_{\beta\gamma;\,\alpha(n-4)})$ explicitly reads

$\displaystyle{\cal L}\big{(}\partial_{\beta\gamma}h_{\alpha(n)},{\cal
H}_{\beta\gamma;\,\alpha(n-4)}\big{)}=\frac{1}{2}\partial^{\beta\gamma}h^{\alpha(n)}\partial_{\beta\gamma}h_{\alpha(n)}-\frac{n}{4}\partial_{\beta\gamma}h^{\beta\gamma\alpha(n-2)}\partial^{\delta\lambda}h_{\delta\lambda\alpha(n-2)}~{}$
$\displaystyle+\frac{n-3}{n}\Big{(}\frac{n}{2}{\cal
H}^{\beta\gamma;\,\alpha(n-4)}\partial^{\delta\lambda}h_{\beta\gamma\delta\lambda\alpha(n-4)}-{\cal
H}^{\beta\gamma;\,\alpha(n-4)}{\cal H}_{\beta\gamma;\,\alpha(n-4)}~{}$
$\displaystyle+A\,{\cal H}^{\beta\gamma;\,\delta\lambda\alpha(n-6)}{\cal
H}_{\delta\lambda;\,\beta\gamma\alpha(n-6)}+B\,{\cal
H}_{\beta\gamma;}\,^{\beta\gamma\alpha(n-6)}{\cal
H}^{\delta\lambda;}\,_{\delta\lambda\alpha(n-6)}\Big{)}~{}.$ (2.325)

Here the real coefficients $A$ and $B$ are constrained by

$\displaystyle A+B=-\frac{(n-4)(n-5)}{4(n-2)}~{}.$ (2.326)

The field ${\cal H}_{\beta\gamma;\,\alpha(n-4)}$ in (2.3.5.3) is
unconstrained, whilst the Lagrange multiplier $F_{\beta\gamma;\,\alpha(n-4)}$
is given by

$\displaystyle
F_{\beta\gamma;\,\alpha(n-4)}=\partial_{(\beta}{}^{\delta}\phi_{\gamma)\delta;\,\alpha(n-4)}\quad\implies\quad\partial^{\beta\gamma}F_{\beta\gamma;\,\alpha(n-4)}=0~{},$
(2.327)

for some unconstrained field $\phi_{\beta\gamma;\,\alpha(n-4)}$.

Varying the first-order action (2.3.5.3) with respect to
$\phi_{\beta\gamma;\,\alpha(n-4)}$ yields

$\displaystyle\partial_{(\delta}{}^{\beta}{\cal
H}_{\gamma)\beta;\,\alpha(n-4)}=0\quad\implies\quad{\cal
H}_{\beta\gamma;\,\alpha(n-4)}=\partial_{\beta\gamma}y_{\alpha(n-4)}~{},$
(2.328)

thus $S_{\text{first-order}}$ reduces to the original action (2.321). On the
other hand, we can integrate out ${\cal H}_{\beta\gamma;\,\alpha(n-4)}$ via
its equation motion,

$\displaystyle\frac{\partial}{\partial{\cal
H}^{\beta\gamma;\,\alpha(n-4)}}{\cal
L}\big{(}\partial_{\delta\lambda}h_{\alpha(n)},{\cal
H}_{\delta\lambda;\,\alpha(n-4)}\big{)}+F_{\beta\gamma;\,\alpha(n-4)}=0~{},$
(2.329)

which is equivalent to

$\displaystyle{\cal
H}_{\beta\gamma;\,\alpha(n-4)}-\frac{B}{2}\Big{(}\varepsilon_{\beta(\alpha_{1}}\varepsilon_{|\gamma|\alpha_{2}}{\cal
H}^{\delta\lambda;}\,_{\alpha_{3}\dots\alpha_{n-4})\delta\lambda}+\varepsilon_{\gamma(\alpha_{1}}\varepsilon_{|\beta|\alpha_{2}}{\cal
H}^{\delta\lambda;}\,_{\alpha_{3}\dots\alpha_{n-4})\delta\lambda}\Big{)}$
$\displaystyle\qquad\quad-A{\cal
H}_{(\alpha_{1}\alpha_{2};\,\alpha_{3}\dots\alpha_{n-4})\beta\gamma}=\frac{n}{4}\partial^{\delta\lambda}h_{\delta\lambda\beta\gamma\alpha(n-4)}-\frac{n}{2}F_{\beta\gamma;\,\alpha(n-4)}~{}.$
(2.330)

This equation allows us to express ${\cal H}_{\beta\gamma;\,\alpha(n-4)}$ in
terms of $h_{\alpha(n)}$ and $F_{\beta\gamma;\,\alpha(n-4)}$,

$\displaystyle{\cal
H}_{\beta\gamma;\alpha(n-4)}=\frac{n}{4(1-A)}\Big{\\{}\partial^{\delta\lambda}h_{\delta\lambda\beta\gamma\alpha(n-4)}-2F_{\beta\gamma;\alpha(n-4)}$
$\displaystyle+c_{1}(A)\Big{(}\varepsilon_{\beta(\alpha_{1}}F^{\delta}{}_{\alpha_{2};\,\alpha_{3}\dots\alpha_{n-4})\gamma\delta}+\varepsilon_{\gamma(\alpha_{1}}F^{\delta}{}_{\alpha_{2};\,\alpha_{3}\dots\alpha_{n-4})\beta\delta}\Big{)}$
$\displaystyle+c_{2}(A)\Big{(}\varepsilon_{\beta(\alpha_{1}}\varepsilon_{|\gamma|\alpha_{2}}F^{\delta\lambda;}\,_{\alpha_{3}\dots\alpha_{n-4})\delta\lambda}+\varepsilon_{\gamma(\alpha_{1}}\varepsilon_{|\beta|\alpha_{2}}F^{\delta\lambda;}\,_{\alpha_{3}\dots\alpha_{n-4})\delta\lambda}\Big{)}\Big{\\}}~{},~{}~{}~{}~{}~{}~{}$
(2.331)

where we have defined

$\displaystyle c_{1}(A)$ $\displaystyle=$
$\displaystyle\frac{2A(n-4)}{(n+2(A-2))}~{},$ (2.332a) $\displaystyle
c_{2}(A)$ $\displaystyle=$
$\displaystyle-\frac{(n-4)\Big{(}2A\,\big{(}2A+B(n-3)\big{)}+(n-4)(B-A)\big{(}n+2(A-2)\big{)}\Big{)}}{(n+2(A-2))\Big{(}(n-4)\big{(}n-5-B(n-3)\big{)}-2A\Big{)}}~{}.\hskip
28.45274pt$ (2.332b)

Substituting (2.331) into $S_{\text{first-order}}$ yields the following one-
parameter family of dual actions

$\displaystyle S_{\rm A}^{(\frac{n}{2})}[h,\phi]=\frac{{\rm i}^{n}}{2^{\lfloor
n/2\rfloor+1}}\int{\rm d}^{3}x\,{\cal L}_{\rm
dual}\Big{(}\partial_{\beta\gamma}h_{\alpha(n)},F_{\beta\gamma;\,\alpha(n-4)}\Big{)}~{}.$
(2.333)

The action (2.3.5.3) is invariant under the gauge $\lambda$-transformations
(2.322a), which act on $\mathcal{H}_{\beta\gamma;\,\alpha(n-4)}$ and
$\phi_{\beta\gamma;\alpha(n-4)}$ as follows:

$\displaystyle\delta_{\zeta}\mathcal{H}_{\beta\gamma;\,\alpha(n-4)}$
$\displaystyle=$
$\displaystyle\frac{n-2}{n-1}\partial_{\beta\gamma}\partial^{\delta\rho}\zeta_{\delta\rho\alpha(n-4)}~{},$
(2.334a) $\displaystyle\delta_{\zeta}\phi_{\beta\gamma;\alpha(n-4)}$
$\displaystyle=$
$\displaystyle-\frac{2}{n(n-1)}\Big{(}\big{(}2n-5-2B(n-2)\big{)}\partial_{(\beta}{}^{\delta}\zeta_{\gamma)\delta\alpha(n-4)}$
(2.334b)
$\displaystyle-\big{(}n-4-2B(n-2)\big{)}\partial_{(\alpha_{1}}{}^{\delta}\zeta_{\alpha_{2}\dots\alpha_{n-4})\beta\gamma\delta}\Big{)}~{}.$

The dual action (2.333) is also invariant under the following gauge
$\rho$-transformations

$\displaystyle\delta_{\rho}{\phi}_{\gamma\delta;\,\alpha(n-4)}$
$\displaystyle=$
$\displaystyle\partial_{\gamma\delta}\rho_{\alpha(n-4)}\quad\implies\quad\delta_{\rho}{F}_{\beta\gamma;\,\alpha(n-4)}=0~{},$
(2.335a) $\displaystyle\delta_{\rho}h_{\alpha(n-4)}$ $\displaystyle=$
$\displaystyle 0~{}.$ (2.335b)

####  2.3.6 New topologically massive theories

The so-called ‘new topologically massive’ (NTM) models were first formulated
in $\mathbb{M}^{3}$ for fields with integer spin by Bergshoeff, Kovacevic,
Rosseel, Townsend and Yin in [156]. They were recently extended to the case of
half-integer spin fields by Kuzenko and Ponds in [117].393939Note that the
spin-$3$ case was first treated in [150]. These NTM models are constructed
purely in terms of the gauge prepotentials $h_{\alpha(n)}$ and the associated
Cotton tensors ${C}_{\alpha(n)}(h)$.

Specifically, given an integer $n\geq 2$, the gauge-invariant NTM action for
the field $h_{\alpha(n)}$ given in [117] is

$\displaystyle S_{\text{NTM}}^{(n)}[h]=\frac{\text{i}^{n}}{2^{\lceil
n/2\rceil+1}}\frac{1}{m}\int\text{d}^{3}x\,h^{\alpha(n)}\big{(}W-\sigma
nm\big{)}{C}_{\alpha(n)}(h)~{},$ (2.336)

where $m$ is some positive mass parameter and $\sigma:=\pm 1$. Making use of
the representation (2.297) leads to a manifestly gauge invariant and
factorised form of the action (2.336) [4].

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

$\displaystyle 0=\big{(}W-\sigma nm\big{)}{C}_{\alpha(n)}(h)~{}.$ (2.337)

By analysing (2.337), it can be shown that on-shell, the action (2.336)
describes a propagating mode with pseudo-mass $m$, spin $\frac{n}{2}$ and
helicity $\frac{\sigma n}{2}$ (cf. eq (2.236a)).

Recently, a new variant of the NTM model for bosonic fields in
$\mathbb{M}^{3}$ was given in [155]. This model also does not require
auxiliary fields, but is of order $2s-1$ in derivatives, as compared to those
given in [156] which are of order $2s$.

Given an integer $s\geq 1$, the actions of [155] are given by

$\displaystyle\widetilde{S}_{\text{NTM}}^{(2s)}[h]=\int{\rm
d}^{3}x~{}h^{\alpha(2s)}\big{(}W-\sigma nm\big{)}F_{\alpha(2s)}(h)~{},$
(2.338)

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

$F_{\alpha(2s)}(h):=\Box^{s-1}{\Pi}^{\perp}_{(2s)}h_{\alpha(2s)}~{}.$ (2.339)

Due to the properties of ${\Pi}^{\perp}_{(2s)}$, the action (2.338) is
manifestly gauge invariant and factorised. The descendent $F_{\alpha(2s)}(h)$
may be obtained by stripping off a $W$ from ${C}_{\alpha(2s)}(h)$:

$\displaystyle C_{\alpha(2s)}(h)=\frac{1}{2s}WF_{\alpha(2s)}(h)~{}.$ (2.340)

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

The equation of motion obtained by varying (2.338) with respect to the field
$h_{\alpha(2s)}$ is

$\displaystyle 0=(W-\sigma nm)F_{\alpha(2s)}(h)~{}.$ (2.341)

By analysing (2.341), it can be shown that on-shell, the model (2.338) has the
same particle content as the NTM model (2.338).

## Appendix

##  Appendix 2.A Four-dimensional notation and identities

This appendix serves as a practical compendium on the $4d$ two-component
spinor notation which is employed in this thesis. We follow the notation and
conventions used in [105].404040We direct the interested reader to [105] for a
pedagogical treatment.

The double-covering group of $\mathsf{SO}_{0}(3,1)$ in four dimensions is the
spin group $\mathsf{SL}(2,\mathbb{C})$. The irreducible representations of
$\mathsf{SL}(2,\mathbb{C})$ are labelled by $(\frac{m}{2},\frac{n}{2})$ for
integers $m,n\geq 0$. These irreducible representations are naturally realised
on the space of complex rank-$(m,n)$ two-component spinor fields
$\Phi_{\alpha_{1}\ldots\alpha_{m}{\dot{\alpha}}_{1}\ldots{\dot{\alpha}}_{n}}$

$\Phi_{\alpha_{1}\ldots\alpha_{m}{\dot{\alpha}}_{1}\ldots{\dot{\alpha}}_{n}}=\Phi_{(\alpha_{1}\ldots\alpha_{m})({\dot{\alpha}}_{1}\ldots{\dot{\alpha}}_{n})}\equiv\Phi_{\alpha(m){\dot{\alpha}}(n)}~{},$
(2.342)

where the two-component spinor indices are independently symmetric in their
dotted and undotted indices, respectively. The spinor indices are always
denoted by Greek letters from the beginning of the alphabet, with $\alpha=1,2$
and ${\dot{\alpha}}=\dot{1},\dot{2}$. The fields
$\phi_{\alpha(m){\dot{\alpha}}(n)}$ will be called spinor fields of Lorentz
type $(\frac{m}{2},\frac{n}{2})$ in this thesis.

The $\mathsf{SL}(2,\mathbb{C})$ invariant tensors

$\displaystyle\varepsilon_{\alpha\beta}=\left(\begin{array}[]{cc}0{}&-1\\\
1{}&0\end{array}\right)~{},\qquad\varepsilon^{\alpha\beta}=\left(\begin{array}[]{cc}0{}&1\\\
-1{}&0\end{array}\right)~{},\qquad\varepsilon^{\alpha\gamma}\varepsilon_{\gamma\beta}=\delta^{\alpha}_{\beta}~{},$
(2.343e)
$\displaystyle\varepsilon_{{\dot{\alpha}}{\dot{\beta}}}=\left(\begin{array}[]{cc}0{}&-1\\\
1{}&0\end{array}\right)~{},\qquad\varepsilon^{{\dot{\alpha}}{\dot{\beta}}}=\left(\begin{array}[]{cc}0{}&1\\\
-1{}&0\end{array}\right)~{},\qquad\varepsilon^{{\dot{\alpha}}{\dot{\gamma}}}\varepsilon_{{\dot{\gamma}}{\dot{\beta}}}=\delta^{{\dot{\alpha}}}_{{\dot{\beta}}}~{},$
(2.343j)

are used to raise and lower the spinor indices:

$\displaystyle\Phi^{\alpha}=\varepsilon^{\alpha\beta}\Phi_{\beta}~{},\qquad\Phi_{\alpha}=\varepsilon_{\alpha\beta}\Phi^{\beta}~{},\qquad\Phi^{{\dot{\alpha}}}=\varepsilon^{{\dot{\alpha}}{\dot{\beta}}}\Phi_{{\dot{\beta}}}~{},\qquad\Phi_{{\dot{\alpha}}}=\varepsilon_{{\dot{\alpha}}{\dot{\beta}}}\Phi^{{\dot{\beta}}}~{}.$
(2.344)

Since the metric tensors $\varepsilon_{\alpha\beta}$ and
$\varepsilon_{{\dot{\alpha}}{\dot{\beta}}}$ are antisymmetric, it follows that
all spinor fields of Lorentz type $(\frac{m}{2},\frac{n}{2})$ are also
traceless in their spinor indices.

Let us introduce the $4d$ sigma matrices which are defined as
$\sigma_{a}\equiv(\sigma_{a})_{\alpha{\dot{\alpha}}}=(\mathds{1},\vec{\sigma})$,
where $\vec{\sigma}$ are the usual Pauli matrices. Explicitly, they read

$\displaystyle\sigma_{0}=\left(\begin{array}[]{cc}1{}&0\\\
0{}&1\end{array}\right)~{},\quad\sigma_{1}=\left(\begin{array}[]{cc}0{}&1\\\
1{}&0\end{array}\right)~{},\quad\sigma_{2}=\left(\begin{array}[]{cc}0{}&-{\rm
i}\\\ {\rm
i{}}&0\end{array}\right)~{},\quad\sigma_{3}=\left(\begin{array}[]{cc}1{}&0\\\
0{}&-1\end{array}\right)~{}.\quad$ (2.345i)

The sigma matrices satisfy the following useful properties

$\displaystyle(\sigma_{a}\tilde{\sigma}_{b}+\sigma_{b}\tilde{\sigma}_{a})_{\alpha}{}^{\beta}=-2\eta_{ab}\delta_{\alpha}{}^{\beta}~{},\qquad(\tilde{\sigma}_{a}{\sigma}_{b}+\tilde{\sigma}_{b}{\sigma}_{a})^{\dot{\alpha}}{}_{\dot{\beta}}=-2\eta_{ab}\delta^{\dot{\alpha}}{}_{\dot{\beta}}~{},$
(2.346a)
$\displaystyle\text{Tr}(\sigma_{a}\tilde{\sigma}_{b})=(\sigma_{a})_{\alpha{\dot{\alpha}}}(\tilde{\sigma}_{b})^{{\dot{\alpha}}\alpha}=-2\eta_{ab}~{},\qquad(\sigma^{a})_{\alpha{\dot{\alpha}}}(\tilde{\sigma}_{a})^{{\dot{\beta}}\beta}=-2\delta_{\alpha}{}^{\beta}\delta^{\dot{\beta}}{}_{\dot{\alpha}}~{},$
(2.346b)

where
$\tilde{\sigma}_{a}\equiv(\tilde{\sigma}_{a})^{{\dot{\alpha}}\alpha}:=\varepsilon^{\alpha\beta}\varepsilon^{{\dot{\alpha}}{\dot{\beta}}}(\sigma_{a})_{\beta{\dot{\beta}}}$.

A four-vector $V_{a}$ can be equivalently realised as a rank-$(1,1)$ spinor
field $V_{\alpha{\dot{\alpha}}}$ by

$\displaystyle
V_{\alpha{\dot{\alpha}}}=(\sigma^{a})_{\alpha{\dot{\alpha}}}V_{a}~{},\qquad
V_{a}=-\frac{1}{2}(\tilde{\sigma}_{a})^{{\dot{\alpha}}\alpha}V_{\alpha{\dot{\alpha}}}~{}.$
(2.347)

Let us introduce the antisymmetric double sigma matrices

$\displaystyle\big{(}\sigma^{ab}\big{)}_{\alpha}{}^{\beta}:=-\frac{1}{4}\big{(}\sigma^{a}\tilde{\sigma}^{b}-\sigma^{b}\tilde{\sigma}^{a}\big{)}_{\alpha}{}^{\beta}~{},\qquad\big{(}\tilde{\sigma}^{ab}\big{)}^{{\dot{\alpha}}}{}_{{\dot{\beta}}}:=-\frac{1}{4}\big{(}\tilde{\sigma}^{a}\sigma^{b}-\tilde{\sigma}^{b}\sigma^{a}\big{)}^{{\dot{\alpha}}}{}_{{\dot{\beta}}}~{}.$
(2.348)

These matrices may be used to relate a real antisymmetric tensor $F_{ab}$ to a
pair of symmetric two-spinors
$\bar{F}_{{\dot{\alpha}}(2)}=\big{(}F_{\alpha(2)}\big{)}^{*}$ via

$\displaystyle F_{ab}=$
$\displaystyle~{}(\sigma_{ab})^{\alpha\beta}F_{\alpha\beta}-(\tilde{\sigma}_{ab})^{{\dot{\alpha}}{\dot{\beta}}}\bar{F}_{{\dot{\alpha}}{\dot{\beta}}}~{},$
$\displaystyle F_{\alpha\beta}=\frac{1}{2}(\sigma^{ab})_{\alpha\beta}$
$\displaystyle
F_{ab}~{},\qquad\bar{F}_{{\dot{\alpha}}{\dot{\beta}}}=-\frac{1}{2}(\tilde{\sigma}^{ab})_{{\dot{\alpha}}{\dot{\beta}}}F_{ab}~{}.$
(2.349)

The Lorentz generators $M_{\alpha\beta}$ and
${\bar{M}}_{{\dot{\alpha}}{\dot{\beta}}}$ act on two-component spinors as
follows:

$\displaystyle
M_{\alpha\beta}\,\Phi_{\gamma}=\frac{1}{2}(\varepsilon_{\gamma\alpha}\Phi_{\beta}+\varepsilon_{\gamma\beta}\Phi_{\alpha})~{},\quad$
$\displaystyle\qquad M_{\alpha\beta}\,{\bar{\Phi}}_{{\dot{\gamma}}}=0~{},$
(2.350a)
$\displaystyle{\bar{M}}_{{\dot{\alpha}}{\dot{\beta}}}\,{\bar{\Phi}}_{{\dot{\gamma}}}=\frac{1}{2}(\varepsilon_{{\dot{\gamma}}{\dot{\alpha}}}{\bar{\Phi}}_{{\dot{\beta}}}+\varepsilon_{{\dot{\gamma}}{\dot{\beta}}}{\bar{\Phi}}_{{\dot{\alpha}}})~{},\quad$
$\displaystyle\qquad{\bar{M}}_{{\dot{\alpha}}{\dot{\beta}}}\,\Phi_{\gamma}=0~{}.$
(2.350b)

It can be shown that the following identities hold:

$\displaystyle
M_{\alpha_{1}}{}^{\beta}\Phi_{\beta\alpha_{2}...\alpha_{m}{\dot{\alpha}}(n)}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}(m+2)\Phi_{\alpha(m){\dot{\alpha}}(n)}~{},$ (2.351a)
$\displaystyle\bar{M}_{{\dot{\alpha}}_{1}}{}^{{\dot{\beta}}}\Phi_{\alpha(m){\dot{\beta}}{\dot{\alpha}}_{2}...{\dot{\alpha}}_{n}}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}(n+2)\Phi_{\alpha(m){\dot{\alpha}}(n)}~{},$ (2.351b)
$\displaystyle
M^{\beta\gamma}M_{\beta\gamma}\Phi_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}m(m+2)\Phi_{\alpha(m){\dot{\alpha}}(n)}~{},$
(2.351c)
$\displaystyle\bar{M}^{{\dot{\beta}}{\dot{\gamma}}}\bar{M}_{{\dot{\beta}}{\dot{\gamma}}}\Phi_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}n(n+2)\Phi_{\alpha(m){\dot{\alpha}}(n)}~{}.$
(2.351d)

In this thesis, there will always be an implicit symmetrisation over the
indices of any tensor field which share the same Greek letter

$U_{\alpha(m){\dot{\alpha}}(n)}V_{\alpha(p){\dot{\alpha}}(q)}=U_{(\alpha_{1}\ldots\alpha_{m}({\dot{\alpha}}_{1}\ldots{\dot{\alpha}}_{n}}V_{\alpha_{m+1}\ldots\alpha_{m+p}){\dot{\alpha}}_{n+1}\ldots{\dot{\alpha}}_{n+q})}~{}.$
(2.352)

The (brackets) parentheses correspond to (anti-)symmetrisation of tensor or
spinor indices, which encode a normalisation factor, for example

$\displaystyle V_{[a_{1}a_{2}\dots a_{n}]}:=\frac{1}{n!}\sum_{\pi\in
S_{n}}\mbox{sgn}(\pi)V_{a_{\pi(1)}\dots a_{\pi(n)}}~{},\quad
V_{(\alpha_{1}\dots\alpha_{n})}:=\frac{1}{n!}\sum_{\pi\in
S_{n}}V_{\alpha_{\pi(1)}\dots\alpha_{\pi(n)}}~{},~{}~{}~{}~{}~{}~{}$ (2.353)

with $S_{n}$ being the symmetric group of $n$ elements.

##  Appendix 2.B Three-dimensional notation and identities

We follow the notation and conventions adopted in [172]. The $\rm
SL(2,{\mathbb{R}})$ invariant tensors

$\displaystyle\varepsilon_{\alpha\beta}=\left(\begin{array}[]{cc}0{}&-1\\\
1{}&0\end{array}\right)~{},\qquad\varepsilon^{\alpha\beta}=\left(\begin{array}[]{cc}0{}&1\\\
-1{}&0\end{array}\right)~{},\qquad\varepsilon^{\alpha\gamma}\varepsilon_{\gamma\beta}=\delta^{\alpha}_{\beta}~{},$
(2.358)

are used to raise and lower spinor indices:

$\displaystyle\psi^{\alpha}=\varepsilon^{\alpha\beta}\psi_{\beta}~{},\qquad\psi_{\alpha}=\varepsilon_{\alpha\beta}\psi^{\beta}~{}.$
(2.359)

We make use of real Dirac gamma-matrices,
$\gamma_{a}:=\big{(}(\gamma_{a})_{\alpha}{}^{\beta}\big{)}$, which are defined
by

$(\gamma_{a})_{\alpha}{}^{\beta}:=\varepsilon^{\beta\gamma}(\gamma_{a})_{\alpha\gamma}=({\rm
i}\sigma_{2},\sigma_{3},\sigma_{1})~{}.$ (2.360)

They have the following properties:

$\displaystyle\gamma_{a}\gamma_{b}$ $\displaystyle=$
$\displaystyle\eta_{ab}{\mathbbm{1}}+\varepsilon_{abc}\gamma^{c}~{},$ (2.361a)
$\displaystyle(\gamma^{a})_{\alpha\beta}(\gamma_{a})^{\rho\sigma}$
$\displaystyle=$
$\displaystyle-(\delta_{\alpha}^{\rho}\delta_{\beta}^{\sigma}+\delta_{\alpha}^{\sigma}\delta_{\beta}^{\rho})~{},$
(2.361b)
$\displaystyle\varepsilon_{abc}(\gamma^{b})_{\alpha\beta}(\gamma^{c})_{\gamma\delta}$
$\displaystyle=$
$\displaystyle\varepsilon_{\gamma(\alpha}(\gamma_{a})_{\beta)\delta}+\varepsilon_{\delta(\alpha}(\gamma_{a})_{\beta)\gamma}~{},$
(2.361c) $\displaystyle\text{tr}[\gamma_{a}\gamma_{b}\gamma_{c}\gamma_{d}]$
$\displaystyle=$ $\displaystyle
2\eta_{ab}\eta_{cd}-2\eta_{ac}\eta_{db}+2\eta_{ad}\eta_{bc}~{}.$ (2.361d)

A three-vector $x_{a}$ can be equivalently realised as a symmetric second-rank
spinor $x_{\alpha\beta}$ defined as

$\displaystyle
x_{\alpha\beta}:=(\gamma^{a})_{\alpha\beta}x_{a}=x_{\beta\alpha}~{},\qquad
x_{a}=-\frac{1}{2}(\gamma_{a})^{\alpha\beta}x_{\alpha\beta}~{}.$ (2.362)

The relationships between Lorentz generators with two vector indices
($M_{ab}=-M_{ba}$), one vector index ($M_{a}$) and two spinor indices
($M_{\alpha\beta}=M_{\beta\alpha}$) are:

$\displaystyle
M_{ab}=-\varepsilon_{abc}M^{c}~{},\,\,\,M_{a}=\frac{1}{2}\varepsilon_{abc}M^{bc}~{},\,\,\,M_{\alpha\beta}=(\gamma^{a})_{\alpha\beta}M_{a}~{},\,\,\,M_{a}=-\frac{1}{2}(\gamma_{a})^{\alpha\beta}M_{\alpha\beta}~{}.~{}~{}~{}~{}~{}$
(2.363)

These generators act on a vector $V_{c}$ and a spinor $\Psi_{\gamma}$ by the
rules

$\displaystyle
M_{ab}V_{c}=2\eta_{c[a}V_{b]}~{},~{}~{}~{}~{}~{}~{}M_{\alpha\beta}\Psi_{\gamma}=\varepsilon_{\gamma(\alpha}\Psi_{\beta)}~{}.$
(2.364)

##  Chapter 3 Three-dimensional ${\cal N}$-extended Minkowski superspace

In this chapter we derive the transverse superspin projection operators in
three-dimensional ${\cal N}$-extended Minkowski superspace
$\mathbb{M}^{3|2\mathcal{N}}$, for $1\leq{\cal N}\leq 6$. These operators are
supersymmetric generalisations of the spin projectors in $\mathbb{M}^{3}$,
which were reviewed in section 2.3.3. The defining characteristic of a
superspin projection operator in $\mathbb{M}^{3|2\mathcal{N}}$ is that it
extracts out the irreducible component of a superfield with maximal superspin.
These superprojectors have numerous applications. Most importantly, we will
see that these operators are vital in the construction of linearised higher-
spin super-Cotton tensors.

These linearised super-Cotton tensors are the main ingredients in the off-
shell formulation of linearised ${\cal N}$-extended (higher-spin) conformal
supergravity. For $1\leq\mathcal{N}\leq 6$, the complete non-linear actions
for $\mathcal{N}$-extended conformal supergravity were derived in [157, 158]
using the off-shell formulation for $\mathcal{N}$-extended conformal
supergravity developed in [159].111The ${\cal N}=1$ and ${\cal N}=2$ conformal
supergravity theories were constructed for the first time by van Nieuwenhuizen
[160] and Roček and van Nieuwenhuizen [161], respectively. The off-shell
action for ${\cal N}=6$ conformal supergravity was independently derived by
Nishimura and Tanii [162]. On-shell formulations for ${\cal N}$-extended
conformal supergravity with ${\cal N}>2$ were given in [163, 164]. Since the
complete nonlinear actions are known, it is natural to ask the following
question: What is the point of constructing linearised conformal supergravity
actions? The answer is that the supergravity actions proposed in [157] are
realised using certain closed super three-forms, which are themselves
constructed in terms of the constrained geometry of $\mathcal{N}$-extended
conformal superspace [159]. However, it may be shown that these constraints
can be solved in terms of unconstrained gauge prepotentials $H$ (with indices
suppressed). Thus, the action for conformal supergravity, $S_{\text{CSG}}$,
may be reformulated in terms of $H$, $S_{\text{CSG}}=S_{\text{CSG}}[H]$. In
order to determine the structure of $S_{\text{CSG}}$, it is necessary to
construct a linearised conformal supergravity action. One of the aims of this
chapter is to construct these actions, and their higher-spin extensions.

This chapter is based on the publication [1] and is organised as follows. In
section 3.1 we review salient facts concerning massive UIRs of the three-
dimensional super-Poincaré algebra. In section 3.2 we discuss how these
massive UIRs can be realised on the space of real superfields in
$\mathbb{M}^{3|2\mathcal{N}}$. The construction of the superspin projection
operators in $\mathbb{M}^{3|2\mathcal{N}}$, for the cases $1\leq{\cal N}\leq
6$, is completed in section 3.3. The transverse superprojectors are then
utilised in section 3.4 to construct the linearised higher-spin super-Cotton
tensors, and subsequently linearised actions for ${\cal N}$-extended
superconformal gravity, in terms of unconstrained gauge prepotentials. In
section 3.5 we review massless (half-)integer superspin theories in
$\mathbb{M}^{3|2}$, which were first derived by Kuzenko and Tsulaia [123]. The
explicit computation of the linearised super-Cotton tensors enables the
derivation of new topologically massive actions. These models are investigated
in section 3.6. A summary of the results obtained is given in section 3.7. The
chapter is accompanied by a single technical appendix 3.A which contains
technical details concerning the ${\cal N}$-extended superconformal group.

###  3.1 Irreducible representations of the super-Poincaré algebra

The three-dimensional ${\cal N}$-extended super-Poincaré algebra is spanned by
the Poincaré generators $P_{\alpha\beta}$ and $J_{\alpha\beta}$ and the
supersymmetry generators $Q^{I}_{\alpha}$. Here the $R$-symmetry index takes
${\cal N}$ values, $I=\underline{1},\underline{2},\cdots,\underline{{\cal
N}}$.222Since the $R$-symmetry group is $\mathsf{SO}({\cal N})$, the
corresponding indices are raised and lowered using the Kronecker delta. Hence
we do not distinguish between upper and lower $\mathsf{SO}({\cal N})$ indices.
The generators of the ${\cal N}$-extended super-Poincaré algebra satisfy the
(anti-)commutation relations

$\displaystyle[P_{\alpha\beta},P_{\gamma\delta}]$ $\displaystyle=0~{},$
$\displaystyle\ [J_{\alpha\beta},P_{\gamma\delta}]$ $\displaystyle={\rm
i}\varepsilon_{\gamma(\alpha}P_{\beta)\delta}+{\rm
i}\varepsilon_{\delta(\alpha}P_{\beta)\gamma}~{},$ (3.1a) $\displaystyle\
[J_{\alpha\beta},J_{\gamma\delta}]$ $\displaystyle={\rm
i}\varepsilon_{\gamma(\alpha}J_{\beta)\delta}+{\rm
i}\varepsilon_{\delta(\alpha}J_{\beta)\gamma}~{},$ $\displaystyle\
[P_{\alpha\beta},Q^{I}_{\gamma}]$ $\displaystyle=0~{},$ (3.1b) $\displaystyle\
[J_{\alpha\beta},Q^{I}_{\gamma}]$ $\displaystyle={\rm
i}\varepsilon_{\gamma(\alpha}Q^{I}_{\beta)}~{},$
$\displaystyle\\{Q^{I}_{\alpha},Q^{J}_{\beta}\\}$
$\displaystyle=2\delta^{IJ}P_{\alpha\beta}~{}.$ (3.1c)

The ${\cal N}$-extended super-Poincaré algebra has two quadratic Casimir
operators, which we label $C_{i}$ for $i=1,2$. These operators take the
form[165]

$C_{1}:=-P^{a}P_{a}=\frac{1}{2}P^{\alpha\beta}P_{\alpha\beta}~{},\qquad
C_{2}:=2W-\frac{{\rm i}}{2}Q^{\alpha I}Q_{\alpha}^{I}~{},$ (3.2a)
$[C_{i},P_{\alpha\beta}]=[C_{i},J_{\alpha\beta}]=[C_{i},Q^{I}_{\alpha}]=0~{},$
(3.2b)

where $C_{2}$ is the supersymmetric extension of the Pauli-Lubanski scalar $W$
(2.234). Note that the Casimir operator $C_{2}$ is commonly referred to as the
superhelicity operator and is often denoted $Z\equiv C_{2}$.

The massive UIRs of the three-dimensional ${\cal N}$-extended super-Poincaré
algebra are classified by the quantum numbers mass $m$ and superhelicity
$\kappa$. Given a massive UIR, these quantum numbers are defined by the
eigenvalues of the Casimir operators (3.2a) [165]

$\displaystyle C_{1}=-m^{2}{\mathbbm{1}}~{},\qquad
C_{2}=4m\kappa{\mathbbm{1}}~{}.$ (3.3)

Note that the eigenvalue of $C_{2}$ in a massive UIR has been adjusted by a
factor of four, which is not common practice in the literature. This factor
was introduced in order for our definition of superhelicity to match that
given in [165]. We will denote by $\mathbb{D}(m,\kappa)$ a massive UIR
carrying strictly positive mass $m>0$ and superhelicity $\kappa$. In
principle, the superhelicity can be any real number [165]. However, we will
only be interested in particular values for $\kappa$ which ensure that the
UIRs of $\mathfrak{iso}(2,1)$ encoded in $\mathbb{D}(m,\kappa)$ carry
(half-)integer helicity. We recall that these were the types of massive UIRs
of $\mathfrak{iso}(2,1)$ we were interested in section 2.3.1.

###  3.2 Irreducible superfield representations

Three-dimensional ${\cal N}$-extended Minkowski superspace
$\mathbb{M}^{3|2{\cal N}}$ is parametrised by the real coordinates
$z^{A}=(x^{a},\theta^{\alpha}_{I})$[166].333We make use of the notations and
conventions adopted in [166]. The covariant derivatives
$D_{A}=(\partial_{\alpha\beta},D^{I}_{\alpha})$ obey the (anti-)commutation
relations

$\displaystyle[\partial_{\alpha\beta},\partial_{\gamma\delta}]=0~{},\qquad[\partial_{\alpha\beta},D^{I}_{\gamma}]=0~{},\qquad\\{D^{I}_{\alpha},D^{J}_{\beta}\\}=2{\rm
i}\,\delta^{IJ}\partial_{\alpha\beta}~{}.$ (3.4)

The relations (3.4) lead to the following useful identities

$\displaystyle D^{I}_{\alpha}D^{I}_{\beta}$ $\displaystyle=$
$\displaystyle{\rm
i}\partial_{\alpha\beta}+\frac{1}{2}\varepsilon_{\alpha\beta}(D^{I})^{2}~{},$
(3.5a) $\displaystyle D^{\alpha I}D^{I}_{\beta}D^{I}_{\alpha}$
$\displaystyle=$ $\displaystyle
0\quad\implies\quad[D^{I}_{\alpha}D^{I}_{\beta},D^{I}_{\gamma}D^{I}_{\delta}]=0~{},$
(3.5b) $\displaystyle D^{\beta I}D^{I}_{\beta}D^{I}_{\alpha}$ $\displaystyle=$
$\displaystyle-D^{I}_{\alpha}D^{\beta I}D^{I}_{\beta}=2{\rm
i}\partial_{\alpha\beta}D^{\beta I}~{},$ (3.5c) $\displaystyle D^{\beta
I}D^{I}_{\beta}D^{\gamma I}D^{I}_{\gamma}$ $\displaystyle=$
$\displaystyle-4\Box~{}.$ (3.5d)

It must be emphasised that the Einstein summation convention was not employed
over the $R$-symmetry indices in the above identities (3.5). This will be the
only instance where this occurs.

For superfield representations of the ${\cal N}$-extended super-Poincaré
group, the infinitesimal super-Poincaré transformation of a tensor superfield
$\Phi$ (with indices suppressed) is given by

$\displaystyle\delta\Phi={\rm
i}\Big{(}\frac{1}{2}b^{\alpha\beta}P_{\alpha\beta}+\frac{1}{2}\omega^{\alpha\beta}J_{\alpha\beta}+{\rm
i}\epsilon^{\alpha I}Q^{I}_{\alpha}\Big{)}\Phi~{},$ (3.6)

where the generators of the three-dimensional ${\cal N}$-extended super-
Poincaré group take the following form

$\displaystyle P_{\alpha\beta}$ $\displaystyle=$ $\displaystyle-{\rm
i}\partial_{\alpha\beta}~{},\qquad\partial_{\alpha\beta}=(\gamma^{a})_{\alpha\beta}\partial_{a}~{},$
(3.7a) $\displaystyle J_{\alpha\beta}$ $\displaystyle=$ $\displaystyle-{\rm
i}x^{\gamma}{}_{(\alpha}\partial_{\beta)\gamma}+{\rm
i}\theta^{I}_{(\alpha}\partial^{I}_{\beta)}-{\rm i}M_{\alpha\beta}~{},$ (3.7b)
$\displaystyle Q^{I}_{\alpha}$ $\displaystyle=$
$\displaystyle\partial^{I}_{\alpha}-{\rm i}\theta^{\beta
I}\partial_{\alpha\beta}~{},\qquad\partial^{I}_{\alpha}=\frac{\partial}{\partial\theta^{\alpha}_{I}}~{}.$
(3.7c)

For completeness, we also give the explicit form of the spinor covariant
derivative [166]

$\displaystyle D^{I}_{\alpha}=\partial^{I}_{\alpha}+{\rm i}\theta^{\beta
I}\partial_{\alpha\beta}~{}.$ (3.8)

The spinor covariant derivative $D^{I}_{\alpha}$ possesses unusual complex
conjugation properties which differ to those in four dimensions. In
particular, given an arbitrary superfield $\Phi$ (with indices suppressed),
its complex conjugate is defined via the rule

$(D^{I}_{\alpha}\Phi)^{*}=-(-1)^{\epsilon(\Phi)}D^{I}_{\alpha}\bar{\Phi}~{},$
(3.9)

where $(\Phi)^{*}:=\bar{\Phi}$ and $\epsilon(\Phi)$ denotes the Grassmann
parity of $\Phi$.

In the superfield representation, it follows from (3.7) that the superhelicity
operator can be written in the manifestly supersymmetric form

$\displaystyle Z=2\partial^{\alpha\beta}M_{\alpha\beta}+{\cal
N}\Delta~{},\qquad[Z,D^{I}_{\alpha}]=0~{},$ (3.10)

where the scalar operator $\Delta$ is defined by

$\displaystyle\Delta:=-\frac{{\rm i}}{2{\cal N}}D^{\beta I}D^{I}_{\beta}~{}.$
(3.11)

The operator (3.11) possesses the important property

$\big{[}\Delta\,,D^{\beta I}D^{I}_{\alpha}\big{]}=0~{}.$ (3.12)

The normalisation factor of $Z$, which differs to that given in [123] by a
factor of four, was chosen to ensure that the AdS3|2 analogue of $C_{2}$,
which is $\mathbb{C}_{2}$ (5.121b), reduces to $C_{2}$ in the flat superspace
limit.

Let $\mathbb{V}^{[{\cal N}]}_{(n)}$ denote the space of totally symmetric real
rank-$n$ superfields $\Phi_{\alpha(n)}=\bar{\Phi}_{\alpha(n)}$ on
$\mathbb{M}^{3|2{\cal N}}$. The two Casimir operators (3.2a) are related to
each other on $\mathbb{V}^{[{\cal N}]}_{(n)}$ as follows

$\displaystyle Z^{2}\Phi_{\alpha(n)}$ $\displaystyle=$ $\displaystyle
4n^{2}\Box\Phi_{\alpha(n)}+{\cal N}^{2}\Delta^{2}\Phi_{\alpha(n)}+4n{\cal
N}\Delta\partial_{\alpha}{}^{\beta}\Phi_{\beta\alpha(n-1)}$ (3.13)
$\displaystyle+4n(n-1)\partial_{\alpha(2)}\partial^{\beta(2)}\Phi_{\beta(2)\alpha(n-2)}~{}.$

For integers $n>1$, a superfield $\Phi^{\perp}_{\alpha(n)}$ on
$\mathbb{V}^{[{\cal N}]}_{(n)}$ is called transverse (or divergenceless) if it
obeys the constraint

$\displaystyle D^{\beta I}\Phi^{\perp}_{\beta\alpha(n-1)}=0~{}.$ (3.14)

The transverse condition (3.14) implies that $\Phi^{\perp}_{\alpha(n)}$ is
transverse in the usual sense, that is
$\partial^{\beta(2)}\Phi^{\perp}_{\beta(2)\alpha(n-2)}=0$. On the space of
transverse superfields $\Phi^{\perp}_{\alpha{(n)}}$, the scalar operator
$\Delta$ can be shown to possess the properties

$\displaystyle D^{\beta I}\Delta\Phi^{\perp}_{\beta\alpha(n-1)}$
$\displaystyle=$ $\displaystyle 0~{},$ (3.15a)
$\displaystyle\Delta^{2}\Phi^{\perp}_{\alpha(n)}$ $\displaystyle=$
$\displaystyle\Box\Phi^{\perp}_{\alpha(n)}~{}.$ (3.15b)

The relation (3.15a) states that the $\Delta$ operator preserves the
transverse nature of $\Phi^{\perp}_{\alpha(n)}$.

####  3.2.1 Massive superfields

For integer $n>0$, a superfield $\Phi_{\alpha(n)}$ on $\mathbb{V}^{[{\cal
N}]}_{(n)}$ is said to be an on-shell massive supermultiplet if it satisfies
the conditions444This definition generalises those given earlier in the ${\cal
N}=1$ and ${\cal N}=2$ cases [123, 122, 167].

$\displaystyle D^{\beta I}\Phi_{\beta\alpha(n-1)}$ $\displaystyle=$
$\displaystyle 0~{},$ (3.16a) $\displaystyle\Delta\Phi_{\alpha(n)}$
$\displaystyle=$ $\displaystyle m\sigma\Phi_{\alpha(n)}~{},\qquad\sigma=\pm
1~{},$ (3.16b)

where $m>0$ is a real mass. Making use of the condition (3.16a), it can be
shown that the first-order constraint (3.16b) can be expressed in the
equivalent form

$\big{(}Z-4m\kappa\big{)}\Phi_{\alpha(n)}=0~{},\qquad\kappa=\frac{\sigma}{2}\Big{(}n+\frac{{\cal
N}}{2}\Big{)}~{}.$ (3.17)

Any field $\Phi_{\alpha(n)}$ satisfying both constraints (3.16a) and (3.16b)
is an eigenvector of the Casimir operator $\Box$

$\big{(}\Box-m^{2}\big{)}\Phi_{\alpha(n)}=0~{}.$ (3.18)

In accordance with (3.3), it follows from (3.17) and (3.18) that the on-shell
superfield (3.16) furnishes the massive UIR $\mathbb{D}(m,\kappa)$. Such
massive supermultiplets are said to carry mass $m$, superhelicity
$\kappa=\frac{\sigma}{2}(n+\frac{{\cal N}}{2})$ and superspin
$|s|:=\frac{n}{2}$. Note that in the superfield representation we choose to
label the massive UIR by $\mathbb{D}(m:\sigma,|s|,{\cal N})$.555The notation
$\mathbb{D}(m:\sigma,|s|,{\cal N})$ is not conventional, since the UIR is not
labelled strictly in terms of mass $m$ and superhelicity $\kappa$. However,
the parameters $\sigma$, $|s|$ and ${\cal N}$ prove sufficient in determining
$\kappa$ exactly (cf. (3.17)).

Studying the ${\cal N}=0$ component structure of a massive superfield
$\Phi_{\alpha(n)}$, it follows from the constraints (3.16) that
$\Phi_{\alpha(n)}$ describes the independent component fields

$\displaystyle\phi^{I_{1}\dots I_{k}}_{\alpha_{1}\dots\alpha_{n+k}}(x):$
$\displaystyle=$ $\displaystyle{\rm
i}^{nk+\frac{1}{2}k(k+1)}D^{[I_{1}}_{\alpha_{1}}\dots
D^{I_{k}]}_{\alpha_{k}}\Phi_{\alpha_{k+1}\dots\alpha_{n+k}}\Big{|}_{\theta=0}~{},\qquad
0\leq k\leq{\cal N}~{}.$ (3.19)

Here we have made use of the $3d$ analogue of bar projection
$V|:=V(x,\theta)|_{\theta=0}$, in which all the Grassmann coordinates
$\theta_{a}^{I}$ are set to zero. Each of the component fields (3.19) are
completely symmetric in their spinor indices $\phi^{I_{1}\dots
I_{k}}_{\alpha_{1}\dots\alpha_{n+k}}(x)=\phi^{I_{1}\dots
I_{k}}_{(\alpha_{1}\dots\alpha_{n+k})}(x)$, and are also transverse

$\displaystyle\partial^{\beta\gamma}\phi^{I_{1}\dots
I_{k}}_{\beta\gamma\alpha(n+k-2)}(x)=0~{},\qquad n+k>1~{}.$ (3.20)

Recall that in order to elucidate the helicity content of $\phi^{I_{1}\dots
I_{k}}_{\alpha(n+k)}$, it is necessary to compute the eigenvalues of the
helicity operator $W$ (2.239) on $\phi^{I_{1}\dots I_{k}}_{\alpha(n+k)}$.
Given $\Phi^{I_{1}\ldots I_{K}}_{\alpha(n+k)}$ satisfies the massive on-shell
conditions (3.16), one can show that $\phi^{I_{1}\dots I_{k}}_{\alpha(n+k)}$
obeys

$\displaystyle\big{(}W-\sigma(n+k)m\big{)}\phi^{I_{1}\dots
I_{k}}_{\alpha(n+k)}(x)=0~{}.$ (3.21)

In accordance with the on-shell conditions (2.236) on $\mathbb{M}^{3}$, it
follows from (3.20) and (3.21) that the component field $\phi^{I_{1}\dots
I_{k}}_{\alpha(n+k)}(x)$ is a massive field carrying helicity
$\frac{\sigma}{2}(n+k)$. As expected, a massive supermultiplet describes a
multiplet of massive fields.

In place of the on-shell conditions (3.16), one may instead consider a
superfield $\Phi_{\alpha(n)}$ on $\mathbb{V}^{[{\cal N}]}_{(n)}$ which
satisfies the equations (3.16a) and (3.18)

$\displaystyle D^{\beta I}\Phi_{\beta\alpha(n-1)}$ $\displaystyle=$
$\displaystyle 0~{},$ (3.22a)
$\displaystyle\big{(}\Box-m^{2}\big{)}\Phi_{\alpha(n)}$ $\displaystyle=$
$\displaystyle 0~{}.$ (3.22b)

In this case, the relation (3.13) can be expressed in the form

$\displaystyle\big{(}Z+4m\kappa\big{)}\big{(}Z-4m\kappa\big{)}\Phi_{\alpha(n)}=0~{},\qquad\kappa=\frac{1}{2}\Big{(}n+\frac{{\cal
N}}{2}\Big{)}~{}.$ (3.23)

It follows that such a field $\Phi_{\alpha(n)}$ realises the reducible
representation

$\displaystyle\mathbb{D}\Big{(}m:-,\frac{n}{2},{\cal
N}\Big{)}\oplus\mathbb{D}\Big{(}m:+,\frac{n}{2},{\cal N}\Big{)}~{},$ (3.24)

in which both signs of the superhelicity $\pm\frac{1}{2}\big{(}n+\frac{{\cal
N}}{2}\big{)}$ are present.

####  3.2.2 Massless superfields

As is well known, any massless superfield $\Phi_{\alpha(n)}$ in
$\mathbb{M}^{3|2{\cal N}}$, with rank $n>2$, carries no propagating degrees of
freedom, and thus the notion of superspin is purely kinematical. Thus when
speaking of a massless higher-superspin model in $\mathbb{M}^{3|2{\cal N}}$,
we will refer to the kinematic structure of the field variables, their gauge
transformation laws and the gauge-invariant action which ensure that the
theory describes no propagating degrees of freedom on the equations of motion.
Let us elaborate on this point in the context of $\mathbb{M}^{3|2}$.

Given an integer $n\geq 1$, we say that a superfield $\Phi_{\alpha(n)}$ on
$\mathbb{V}^{[1]}_{(n)}$ is massless if it satisfies the constraints666Similar
to the story in $\mathbb{M}^{3}$, there exists an alternative set of
conditions which define a massless higher-spin superfield in
$\mathbb{M}^{3|2}$. These constraints were first given in [123], where they
were shown to describe the superfield content of the massless half-integer
superspin theory (3.5.2).

$\displaystyle D^{\beta}\Phi_{\beta\alpha(n-1)}$ $\displaystyle=$
$\displaystyle 0~{},$ (3.25a) $\displaystyle\Delta\Phi_{\alpha(n)}$
$\displaystyle=$ $\displaystyle 0~{}.$ (3.25b) It can be shown that system of
equations (3.25a) and (3.25b) are compatible with the gauge symmetry
$\delta_{\zeta}\Phi_{\alpha(n)}={\rm i}^{n}D_{\alpha}\zeta_{\alpha(n-1)}~{},$
(3.25c) given that the real gauge parameter $\zeta_{\alpha(n-1)}$ is also on-
shell $\displaystyle D^{\beta}\zeta_{\beta\alpha(n-2)}$ $\displaystyle=$
$\displaystyle 0~{},$ (3.25d) $\displaystyle\Delta\zeta_{\alpha(n-1)}$
$\displaystyle=$ $\displaystyle 0~{}.$ (3.25e)

Note that the massless constraint (3.25b) is obtained from the massive on-
shell condition (3.16b) by taking the massless limit $m\rightarrow 0$. The
conditions (3.25a) and (3.25b) lead to the massless Klein-Gordon equation
$\Box\Phi_{\alpha(n)}=0$. We will see in section 3.5 that the massless
superfields (3.25), for the case $n\geq 4$, describe the superfield content of
the massless integer superspin theory (3.144). We say that a massless
superfield $\Phi_{\alpha(n)}$ carries superspin $s:=\frac{n}{2}$.

In order to show that a superfield in $\mathbb{M}^{3|2}$ is massless, it
suffices to show that at the component level, the fields surviving upon
implementing an appropriate Wess-Zumino gauge do not carry any propagating
degrees of freedom. Let us show that this is indeed true for the massless
superfield (3.25).

A superfield $\Phi_{\alpha(n)}$ satisfying the conditions (3.25) can be shown
to describe two independent component fields, which we choose to define as

$A_{\alpha(n)}:=\Phi_{\alpha(n)}|~{},\qquad B_{\alpha(n+1)}:={\rm
i}^{n+1}D_{\alpha}\Phi_{\alpha(n)}|~{}.$ (3.26)

The gauge freedom (3.25c) allows us to implement the following Wess-Zumino
gauge

$\Phi_{\alpha(n)}|=0~{}.$ (3.27)

In this gauge, the component field $A_{\alpha(n)}$ vanishes, leaving
$B_{\alpha(n+1)}$ as the only remaining field. The residual gauge freedom is
characterised by the condition

$D_{\alpha}\zeta_{\alpha(n-1)}|=0~{}.$ (3.28)

It follows from (3.25c) and (3.28) that there is only one independent gauge
parameter at the component level, which we choose to define as

$\xi_{\alpha(n-1)}:=-(-1)^{n}\zeta_{\alpha(n-1)}|~{}.$ (3.29)

The sole surviving field $B_{\alpha(n+1)}$ can be shown to satisfy the
conditions

$\partial^{\beta(2)}B_{\beta(2)\alpha(n-1)}=0~{},\qquad
WB_{\alpha(n+1)}=0\quad\Longrightarrow\quad\Box B_{\alpha(n+1)}=0~{},$ (3.30)

as a consequence of the conditions (3.25a) and (3.25b). In accordance with
(3.25c) and (3.29), it can be shown that the gauge transformation associated
with $B_{\alpha(n+1)}$ is given by

$\delta_{\xi}B_{\alpha(n+1)}=\partial_{\alpha(2)}\xi_{\alpha(n-1)}~{},$ (3.31)

where the gauge parameter $\xi_{\alpha(n-1)}$ obeys the conditions

$\partial^{\beta(2)}\xi_{\beta(2)\alpha(n-3)}=0~{},\qquad
W\xi_{\alpha(n-1)}=0\quad\Longrightarrow\quad\Box\xi_{\alpha(n-1)}=0~{}.$
(3.32)

Here, we have made use of the results (3.25d), (3.25e), (3.28) and (3.29). It
follows from (3.30), (3.31) and (3.32) that $B_{\alpha(n+1)}$ describes a
massless field (cf. (2.246)), which was shown to be pure gauge in section
2.3.2.2. Hence a superfield satisfying the conditions (3.25) is indeed
massless, since it describes a single massless field at the component level
which propagates no physical degrees of freedom.

###  3.3 Superspin projection operators

In this section we compute the superspin projection operators on
$\mathbb{M}^{3|2{\cal N}}$ for $1\leq{\cal N}\leq 6$.777Note that there is
nothing preventing the computation of the superspin projection operators on
$\mathbb{M}^{3|2{\cal N}}$ for generic ${\cal N}$. However, they must computed
on a case by case basis for each ${\cal N}$. The rank-$n$ superspin projection
operator $\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ is defined by its action on
$\mathbb{V}^{[{\cal N}]}_{(n)}$ according to the rule

$\displaystyle\mbox{\boldmath$\mbox{\boldmath$\Pi$}$}^{\perp}_{(n)}:\mathbb{V}^{[{\cal
N}]}_{(n)}$ $\displaystyle\longrightarrow$ $\displaystyle\mathbb{V}^{[{\cal
N}]}_{(n)}~{},$ (3.33a) $\displaystyle\Phi_{\alpha(n)}$
$\displaystyle\longmapsto$
$\displaystyle\mbox{\boldmath$\Pi$}^{\perp}_{(n)}\Phi_{\alpha(n)}=:\mbox{\boldmath$\Pi$}^{\perp}_{\alpha(n)}(\phi)~{}.$
(3.33b)

For fixed integer $n\geq 1$, the differential operator
$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ satisfies the properties:

  1. 1.

Idempotence: The operator $\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ squares to
itself

$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}\mbox{\boldmath$\Pi$}^{\perp}_{(n)}=\mbox{\boldmath$\Pi$}^{\perp}_{(n)}~{}.$
(3.34a)

  2. 2.

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

$D^{\beta I}\mbox{\boldmath$\Pi$}^{\perp}_{\beta\alpha(n-1)}(\Phi)=0~{}.$
(3.34b)

  3. 3.

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

$D^{\beta
I}\Phi^{\perp}_{\beta\alpha(n-1)}=0\quad\Longrightarrow\quad\mbox{\boldmath$\Pi$}^{\perp}_{(n)}\Phi^{\perp}_{\alpha(n)}=\Phi^{\perp}_{\alpha(n)}~{}.$
(3.34c)

By construction, the superspin projection operator
$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ maps any superfield $\Phi_{\alpha(n)}$
on $\mathbb{V}^{[{\cal N}]}_{(n)}$, which is supplemented with the first-order
differential constraint (3.16b), to a massive field (3.16) which furnishes the
UIR $\mathbb{D}(m:\sigma,\frac{n}{2},{\cal N})$.

For arbitrary supersymmetry type ${\cal N}$, the superspin projection operator
$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ acts on $\mathbb{V}^{[{\cal N}]}_{(n)}$
according to the rule

$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}\Phi_{\alpha(n)}:=\mbox{\boldmath$\Pi$}^{\beta_{1}}{}_{(\alpha_{1}}{}\cdots\mbox{\boldmath$\Pi$}^{\beta_{n}}{}_{\alpha_{n})}\Phi_{\beta(n)}~{},$
(3.35)

where $\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ is a differential operator
which should possess the general properties

$\displaystyle D^{\beta I}\mbox{\boldmath$\Pi$}^{\alpha}{}_{\beta}$
$\displaystyle=$ $\displaystyle 0~{},$ (3.36a)
$\displaystyle\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}D^{I}_{\beta}$
$\displaystyle=$ $\displaystyle 0~{},$ (3.36b)
$\displaystyle\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}\mbox{\boldmath$\Pi$}^{\gamma}{}_{\beta}$
$\displaystyle=$ $\displaystyle\mbox{\boldmath$\Pi$}^{\gamma}{}_{\alpha}~{},$
(3.36c)
$\displaystyle\big{[}\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha},\mbox{\boldmath$\Pi$}^{\delta}{}_{\gamma}\big{]}$
$\displaystyle=$ $\displaystyle 0~{}.$ (3.36d)

We propose the ansatz for the general structure of
$\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ to be

$\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}=D^{\beta
I}D^{I}_{\alpha}F(\Delta,\Box)~{},$ (3.37)

where $F(\Delta,\Box)$ is some function which needs to be computed
independently for each ${\cal N}$. The ansatz (3.37) was chosen on account
that it automatically satisfies property (3.36d), as a consequence of the
identities

$D^{\beta I}D^{I}_{\alpha}={\rm i}{\cal
N}\big{(}\partial_{\alpha}{}^{\beta}+\Delta\delta_{\alpha}{}^{\beta}\big{)}~{},$
(3.38)

and (3.12). We will see in the subsequent sections that the conditions (3.36)
suffice in determining $\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ uniquely for
each ${\cal N}$. Furthermore, each of these
$\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ can be shown to satisfy the
additional property

$\displaystyle D^{\beta
I}\Phi^{\perp}_{\beta}=0\quad\implies\quad\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}\Phi^{\perp}_{\beta}=\Phi^{\perp}_{\alpha}~{}.$
(3.39)

It follows from (3.36) and (3.39) that the operator
$\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ is a superspin projection operator
(3.34) on the space of spinor superfields $\mathbb{V}^{[{\cal N}]}_{(1)}$.
Given that $\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ is a superspin
projection operator, it follows that its higher-rank generalisation
$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ also satisfies the defining conditions
(3.34) of a superspin projection operator. These properties are immediately
inherited from $\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$. Thus it suffices to
only compute $\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ for each ${\cal N}$,
since these operators are the building blocks of the higher-rank
superprojectors via (3.35). The explicit form of the spinor superprojector for
$1\leq{\cal N}\leq 6$, and their higher-rank extensions, will be presented in
sections 3.3.1-3.3.6 for each supersymmetry type, respectively. Before giving
the explicit form of these superprojectors, it is worth taking a moment to
detail some of their properties which hold for generic ${\cal N}$.

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

The orthogonal complement $\mbox{\boldmath$\Pi$}^{\parallel}_{(n)}$ of
$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ is defined to act on $\mathbb{V}^{[{\cal
N}]}_{(n)}$ via the rule

$\mbox{\boldmath$\Pi$}^{\parallel}_{(n)}\Phi_{\alpha(n)}=\big{(}\mathds{1}-\mbox{\boldmath$\Pi$}^{\perp}_{(n)}\big{)}\Phi_{\alpha(n)}~{}.$
(3.40)

By construction, $\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ and
$\mbox{\boldmath$\Pi$}^{\parallel}_{(n)}$ are orthogonal projectors

$\mbox{\boldmath$\Pi$}^{\parallel}_{(n)}\mbox{\boldmath$\Pi$}^{\parallel}_{(n)}=\mbox{\boldmath$\Pi$}^{\parallel}_{(n)}~{},\qquad\mbox{\boldmath$\Pi$}^{\parallel}_{(n)}\mbox{\boldmath$\Pi$}^{\perp}_{(n)}=\mbox{\boldmath$\Pi$}^{\perp}_{(n)}\mbox{\boldmath$\Pi$}^{\parallel}_{(n)}=0~{}.$
(3.41)

Using the explicit form of the superprojectors given in the subsequent
sections 3.3.1-3.3.6, it can be shown that the action of the orthogonal
complement $\mbox{\boldmath$\Pi$}^{\parallel}_{(n)}$ on $\mathbb{V}^{[{\cal
N}]}_{(n)}$ takes the general form

$\mbox{\boldmath$\Pi$}^{\parallel}_{(n)}\Phi_{\alpha(n)}={\rm
i}^{n}D^{I}_{\alpha}\Phi^{I}_{\alpha(n-1)}~{},$ (3.42)

where $\Phi^{I}_{\alpha(n-1)}$ are unconstrained superfields. Any symmetric
rank-$n$ spinor superfield $\Phi^{\parallel
I}_{\alpha(n)}=D^{I}_{\alpha}\Phi_{\alpha(n-1)}$ is said to be longitudinal.
In accordance with (3.41) and (3.42), the operator
$\mbox{\boldmath$\Pi$}^{\parallel}_{(n)}$ projects onto the space of rank-$n$
longitudinal superfields, and thus is called the longitudinal superprojector.

Using the fact that $\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ and
$\mbox{\boldmath$\Pi$}^{\parallel}_{(n)}$ resolve the identity operator
(3.40), it follows from (3.42) that any unconstrained superfield
$\Phi_{\alpha(n)}$ on $\mathbb{V}^{[{\cal N}]}_{(n)}$ can be decomposed in the
following manner

$\Phi_{\alpha(n)}=\Phi^{\perp}_{\alpha(n)}+{\rm
i}^{n}D^{I}_{\alpha}\Phi^{I}_{\alpha(n-1)}~{}.$ (3.43)

Here $\Phi^{\perp}_{\alpha(n)}$ is irreducible, whilst
$\Phi^{I}_{\alpha(n-1)}$ is unconstrained.

The superprojector $\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ annihilates any
longitudinal superfield $\Phi^{\parallel I}_{\alpha(n)}$ in
$\mathbb{M}^{3|2{\cal N}}$

$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}\Phi^{\parallel I}_{\alpha(n)}=0~{},$
(3.44)

as a direct consequence of (3.36b). The property (3.44) ensures that the
$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ selects the pure superspin component
$\Phi^{\perp}_{\alpha(n)}$ from the decomposition (3.43).

It follows immediately from (3.40) and (3.44) that the longitudinal
superprojector acts like the identity operator on the space of longitudinal
superfields

$\mbox{\boldmath$\Pi$}^{\parallel}_{(n)}\Phi^{\parallel I}_{\alpha(n)}=0~{}.$
(3.45)

#####  3.3.0.2 Superhelicity projection operators

Let us consider a superfield $\Phi_{\alpha(n)}$ satisfying the mass-shell
equation (3.22b). The superspin projection operator
$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ maps $\Phi_{\alpha(n)}$ to the
superfield obeying the conditions

$\displaystyle D^{\beta
I}\mbox{\boldmath$\Pi$}^{\perp}_{\beta\alpha(n-1)}(\Phi)$ $\displaystyle=$
$\displaystyle 0~{},$ (3.46a)
$\displaystyle(\Box-m^{2})\mbox{\boldmath$\Pi$}^{\perp}_{\alpha(n)}(\Phi)$
$\displaystyle=$ $\displaystyle 0~{}.$ (3.46b)

In accordance with (3.22), it follows that the projected superfield
$\mbox{\boldmath$\Pi$}^{\perp}_{\alpha(n)}(\Phi)$ realises the reducible
representation $\mathbb{D}(m:-,\frac{n}{2},{\cal
N})\oplus\mathbb{D}(m:+,\frac{n}{2},{\cal N})$.

In order to extract the on-shell massive supermultiplet of definite
superhelicity from $\Phi_{\alpha(n)}$, we must split
$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ into the superhelicity projection
operators ${\mathbb{P}}^{(\pm)}_{(n)}$ according to

$\mbox{\boldmath$\Pi$}_{(n)}^{\perp}=\mathbb{P}^{(+)}_{(n)}+\mathbb{P}^{(-)}_{(n)}~{}.$
(3.47)

Each of the superhelicity projectors ${\mathbb{P}}^{(\pm)}_{(n)}$ should
satisfy (3.34a) and (3.34b). In order to project out the component carrying a
single value of superhelicity, the projected superfield
$\Phi^{(\pm)}_{\alpha(n)}:=\mathbb{P}^{(\pm)}_{(n)}\Phi_{\alpha(n)}$ must obey
the conditions

$\displaystyle D^{\beta I}\Phi^{(\pm)}_{\beta\alpha(n-1)}$ $\displaystyle=$
$\displaystyle 0~{},$ (3.48a) $\displaystyle(\Delta\mp
m)\Phi^{(\pm)}_{\alpha(n)}$ $\displaystyle=$ $\displaystyle 0~{}.$ (3.48b)

It follows that $\Phi^{(\pm)}_{\alpha(n)}$ furnishes the massive UIR
$\mathbb{D}(m:\pm,\frac{n}{2},{\cal N})$ (cf. (3.16)).

The operator projecting to the subspace (3.48) is given by

$\displaystyle{\mathbb{P}}^{(\pm)}_{(n)}:=\frac{1}{2}\Big{(}\mathbbm{1}\pm\frac{\Delta}{\sqrt{\Box}}\Big{)}\mbox{\boldmath$\Pi$}^{\perp}_{(n)}~{}.$
(3.49)

By construction, the operators ${\mathbb{P}}^{(+)}_{(n)}$ and
${\mathbb{P}}^{(-)}_{(n)}$ are orthogonal projectors on $\mathbb{V}^{[{\cal
N}]}_{(n)}$888The superhelicity projectors (3.49) are strictly not superspin
projection operators, since they do not satisfy the property (3.34c), but this
is just semantics.

$\displaystyle{\mathbb{P}}^{(+)}_{(n)}{\mathbb{P}}^{(-)}_{(n)}=0~{},\qquad{\mathbb{P}}^{(+)}_{(n)}{\mathbb{P}}^{(+)}_{(n)}={\mathbb{P}}^{(+)}_{(n)}~{},\qquad{\mathbb{P}}^{(-)}_{(n)}{\mathbb{P}}^{(-)}_{(n)}={\mathbb{P}}^{(-)}_{(n)}~{}.$
(3.50)

They also map a superfield $\Phi_{\alpha(n)}$ on $\mathbb{V}^{[{\cal
N}]}_{(n)}$ to a transverse superfield, since the operator $\Delta$ preserves
the transverse nature of the superprojector
$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$, by virtue of (3.15a). Moreover, they
satisfy the additional constraint

$\displaystyle\Delta\Phi^{(\pm)}_{\alpha(n)}=\pm\sqrt{\Box}\Phi^{(\pm)}_{\alpha(n)}~{}.$
(3.51)

If $\Phi^{(\pm)}_{\alpha(n)}$ is on the mass-shell (3.22b), then (3.51)
reduces to (3.48b). Thus, the superhelicity projectors
${\mathbb{P}}^{(\pm)}_{(n)}$ map a superfield $\Phi_{\alpha(n)}$ obeying
(3.22b) to a massive superfield which furnishes the UIR
$\mathbb{D}(m:\pm,\frac{n}{2},{\cal N})$ with definite superhelicity.

####  3.3.1 Superspin projection operators in $\mathbb{M}^{3|2}$

The superspin projection operator on the space of spinor superfields
$\mathbb{V}_{(1)}^{[1]}$ is given by

$\displaystyle\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}=-\frac{D^{2}}{4\Box}D^{\beta}D_{\alpha}=-\frac{{\rm
i}}{2}\frac{\Delta}{\Box}D^{\beta}D_{\alpha}~{}.$ (3.52)

Proving that the operator $\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ satisfies
the defining properties (3.36) and (3.39) is simple if one makes use of the
identities (3.12) and

$\Delta^{2}=\Box~{}.$ (3.53)

Note that in $\mathbb{M}^{3|2}$, the identity (3.53) is equivalent to (3.5d).
We can immediately build the higher-rank superprojector
$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ according to (3.35)

$\displaystyle\mbox{\boldmath$\Pi$}_{(n)}^{\perp}\Phi_{\alpha(n)}=\Big{(}-\frac{{\rm
i}\Delta}{2\Box}\Big{)}^{n}D^{\beta_{1}}D_{\alpha_{1}}\cdots
D^{\beta_{n}}D_{\alpha_{n}}\Phi_{\beta(n)}~{},$ (3.54)

where we have made use of the property (3.12). Note that the right hand side
of (3.54) is totally symmetric in its $\alpha$ indices, as a consequence of
the projector property (3.36d).

Studying $\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ for the particular cases $n=2s$
and $n=2s+1$, the superprojectors can be shown to take the simplified form

$\displaystyle\mbox{\boldmath$\Pi$}^{\perp}_{(2s)}\Phi_{\alpha(2s)}$
$\displaystyle=\Big{(}-\frac{1}{4\Box}\Big{)}^{s}D^{\beta_{1}}D_{\alpha_{1}}\cdots
D^{\beta_{2s}}D_{\alpha_{2s}}\Phi_{\beta(2s)}~{},$ $\displaystyle s$
$\displaystyle\geq 1~{},$ (3.55a)
$\displaystyle\mbox{\boldmath$\Pi$}^{\perp}_{(2s+1)}\Phi_{\alpha(2s+1)}$
$\displaystyle=-\frac{{\rm
i}\Delta}{2\Box}\Big{(}-\frac{1}{4\Box}\Big{)}^{s}D^{\beta_{1}}D_{\alpha_{1}}\cdots
D^{\beta_{2s+1}}D_{\alpha_{2s+1}}\Phi_{\beta(2s+1)}~{},$ $\displaystyle s$
$\displaystyle\geq 0~{},$ (3.55b)

as a consequence of the identity (3.53).

We now turn our attention to computing the explicit form of the longitudinal
superprojector $\mbox{\boldmath$\Pi$}^{\parallel}_{(n)}$, which is defined
according to (3.40). Given an arbitrary superfield $\Phi_{\alpha(n)}$ on
$\mathbb{V}^{[1]}_{(n)}$, it can be shown using (3.54) that the action of
$\mbox{\boldmath$\Pi$}^{\parallel}_{(n)}$ on $\Phi_{\alpha(n)}$ is given by

$\displaystyle\mbox{\boldmath$\Pi$}^{\parallel}_{(n)}\Phi_{\alpha(n)}$
$\displaystyle=$ $\displaystyle{\rm
i}^{n}D_{\alpha}\Upsilon_{\alpha(n-1)}~{},$ (3.56a) where the real superfield
$\Upsilon_{\alpha(n-1)}$ takes the explicit form
$\displaystyle\Upsilon_{\alpha(n-1)}$ $\displaystyle:=$ $\displaystyle-(-{\rm
i})^{n}\sum_{j=1}^{n}\frac{1}{(4\Box)^{j}}\binom{n}{j}(D^{2})^{j}D^{\beta_{n-1}}D_{(\alpha_{n-1}}\dots
D^{\beta_{n-j+1}}D_{\alpha_{n-j+1}}$ (3.56b) $\displaystyle\times
D^{\beta_{n}}\Phi_{\alpha_{1}\dots\alpha_{n-j})\beta_{n-j+1}\dots\beta_{n}}~{}.$

In order to compute (3.56), it proves useful to rewrite
$\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ in the form

$\displaystyle\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}=\delta_{\alpha}\,^{\beta}-\frac{D^{2}}{4\Box}D_{\alpha}D^{\beta}~{}.$
(3.57)

Since the superprojectors $\mbox{\boldmath$\Pi$}_{(n)}^{\perp}$ and
$\mbox{\boldmath$\Pi$}_{(n)}^{\parallel}$ resolve the unit operator (3.40) ,
it follows that any superfield on $\mathbb{V}^{[1]}_{(n)}$ can be decomposed
in the following manner

$\Phi_{\alpha(n)}=\big{(}\mbox{\boldmath$\Pi$}_{(n)}^{\perp}+\mbox{\boldmath$\Pi$}_{(n)}^{\parallel}\big{)}\Phi_{\alpha(n)}=\Phi^{\perp}_{\alpha(n)}+{\rm
i}^{n}D_{\alpha}\Phi_{\alpha(n-1)}~{},$ (3.58)

where $\Phi^{\perp}_{\alpha(n)}$ is transverse and $\Phi_{\alpha(n-1)}$ is
unconstrained. Repeating this procedure iteratively yields the decomposition
of $\Phi_{\alpha(n)}$ into irreducible components

$\Phi_{\alpha(n)}=\sum_{j=0}^{\lfloor
n/2\rfloor}\big{(}D_{\alpha(2)}\big{)}^{j}\Phi^{\perp}_{\alpha(n-2j)}+{\rm
i}^{n}\sum_{j=0}^{\lceil
n/2\rceil-1}(D_{\alpha(2)}\big{)}^{j}D_{\alpha}\Phi^{\perp}_{\alpha(n-2j-1)}~{}.$
(3.59)

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

####  3.3.2 Superspin projection operators in $\mathbb{M}^{3|4}$

In the case of ${\cal N}=2$ supersymmetry, it is often useful to work in a
complex basis for the spinor covariant derivatives. Such a basis was first
introduced in [166] by replacing the covariant derivatives
$D^{I}_{\alpha}=(D^{\underline{1}}_{\alpha},D^{\underline{2}}_{\alpha})$ with
$D_{\alpha}$ and $\bar{D}_{\alpha}$, which are defined by

$\displaystyle D_{\alpha}=\frac{1}{\sqrt{2}}(D^{\underline{1}}_{\alpha}-{\rm
i}D^{\underline{2}}_{\alpha})~{},\qquad\bar{D}_{\alpha}=-\frac{1}{\sqrt{2}}(D^{\underline{1}}_{\alpha}+{\rm
i}D^{\underline{2}}_{\alpha})~{}.$ (3.60)

It follows from (3.4) that $D_{\alpha}$ and $\bar{D}_{\alpha}$ satisfy the
anti-commutation relations999These anti-commutation relations are reminiscent
to those in $\mathbb{M}^{4|4}$ (2.120).

$\displaystyle\\{D_{\alpha},D_{\beta}\\}=0\,,\qquad\\{\bar{D}_{\alpha},\bar{D}_{\beta}\\}=0\,,\qquad\\{D_{\alpha},\bar{D}_{\beta}\\}=-2{\rm
i}\partial_{\alpha\beta}\,.$ (3.61)

In this section we will construct the superspin projection operators on
$\mathbb{M}^{3|4}$ in both the complex (3.61) and real (3.4) bases for the
spinor covariant derivatives.

#####  3.3.2.1 Superspin projection operators in the complex basis

Before diving into the construction of the superprojectors, it is worth
pausing to discuss the types of constrained superfields which will be
pertinent to our subsequent analysis. According to the terminology of [122],
for integers $n\geq 1$, a real symmetric rank-$n$ superfield
$T_{\alpha(n)}=\bar{T}_{\alpha(n)}$ constrained by

$\displaystyle\bar{D}^{\beta}T_{\beta\alpha(n-1)}=0\quad\Longleftrightarrow\quad
D^{\beta}T_{\beta\alpha(n-1)}=0~{},$ (3.62)

is said to be real transverse linear. It is linear since the above constraint
implies

$\displaystyle\bar{D}^{2}T_{\alpha(n)}=0\quad\Longleftrightarrow\quad
D^{2}T_{\alpha(n)}=0~{},$ (3.63)

as a consequence of (3.61).

Moreover, a complex superfield $\Phi_{\alpha(n)}$ is said to be simultaneously
transverse linear and transverse anti-linear if it obeys the constraints

$\displaystyle\bar{D}^{\beta}\Gamma_{\beta\alpha(n-1)}=0\quad$
$\displaystyle\implies$
$\displaystyle\quad\bar{D}^{2}\Gamma_{\alpha(n)}=0~{},$ (3.64a) $\displaystyle
D^{\beta}{\Gamma}_{\beta\alpha(n-1)}=0\quad$ $\displaystyle\implies$
$\displaystyle\quad D^{2}{\Gamma}_{\alpha(n)}=0~{}.$ (3.64b)

A complex superfield $G_{\alpha(n)}$ is said to be chiral if it satisfies the
condition

$\bar{D}_{\beta}G_{\alpha(n)}=0~{},$ (3.65)

or anti-chiral if it obeys the constraint

$D_{\beta}{G}_{\alpha(n)}=0~{}.$ (3.66)

In the case of ${\cal N}=1$ supersymmetry, $\Delta$ is an invertible operator
(cf. (3.53)). This is no longer true for ${\cal N}=2$, due to the relation

$\displaystyle\Box=\tilde{\Delta}^{2}+\frac{1}{16}\big{\\{}\bar{D}^{2},D^{2}\big{\\}}~{},$
(3.67)

where $\tilde{\Delta}$ denotes the ${\cal N}=2$ version of (3.11), which
written in the complex basis is

$\displaystyle\tilde{\Delta}$ $\displaystyle=$ $\displaystyle\frac{{\rm
i}}{2}D^{\beta}\bar{D}_{\beta}=\frac{{\rm i}}{2}\bar{D}^{\beta}D_{\beta}~{}.$
(3.68)

The identity (3.67) can be expressed in the alternative form

$\mathds{1}={\cal P}_{(0)}+{\cal P}_{(+)}+{\cal P}_{(-)}~{},$ (3.69)

where the differential operators ${\cal P}_{i}=\big{(}{\cal P}_{(0)},{\cal
P}_{(+)},{\cal P}_{(-)}\big{)}$

${\cal P}_{(0)}=\frac{1}{\Box}\tilde{\Delta}^{2}~{},\qquad{\cal
P}_{(+)}=\frac{1}{16\Box}\bar{D}^{2}D^{2}~{},\qquad{\cal
P}_{(-)}=\frac{1}{16\Box}D^{2}\bar{D}^{2}~{},$ (3.70)

are orthogonal projectors, ${\cal P}_{i}{\cal P}_{j}=\delta_{ij}{\cal P}_{i}$.
The operator ${\cal P}_{(0)}$ projects onto the space of real linear
superfields (3.63), while ${\cal P}_{(+)}$ and ${\cal P}_{(-)}$ project onto
the space of chiral (3.65) and anti-chiral superfields (3.66), respectively.
Note that the projector ${\cal P}_{(0)}$ maps a complex superfield to the
space of simultaneously linear and anti-linear superfields.

Let us now introduce the ${\cal N}=2$ spinor superprojector

$\displaystyle\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}=\frac{{\rm
i}}{4\Box}{\tilde{\Delta}}\big{(}\bar{D}^{\beta}D_{\alpha}+D^{\beta}\bar{D}_{\alpha}\big{)}~{},$
(3.71)

which is written in terms of the covariant derivatives in the complex basis.
One may check that $\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ satisfies the
conditions (3.36) and (3.39), keeping in mind that $D^{I}_{\alpha}$ now
encodes $D_{\alpha}$ and $\bar{D}_{\alpha}$. Making use of the anti-
commutation relations (3.61), $\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ can
be rewritten in the form

$\displaystyle\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}=\frac{\tilde{\Delta}}{2\Box}\big{(}\partial_{\alpha}\,^{\beta}+\delta_{\alpha}\,^{\beta}\tilde{{\Delta}}\big{)}~{}.$
(3.72)

The higher-rank generalisation $\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ of (3.72)
is given by

$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}\Phi_{\alpha(n)}=\Big{(}\frac{\tilde{\Delta}}{2\Box}\Big{)}^{n}\big{(}\partial_{(\alpha_{1}}{}^{\beta_{1}}+\tilde{\Delta}\delta_{(\alpha_{1}}{}^{\beta_{1}}\big{)}\big{(}\partial_{\alpha_{2}}{}^{\beta_{2}}+\tilde{\Delta}\delta_{\alpha_{2}}{}^{\beta_{2}}\big{)}\cdots\big{(}\partial_{\alpha_{n})}{}^{\beta_{n}}+\tilde{\Delta}\delta_{\alpha_{n})}{}^{\beta_{n}}\big{)}\Phi_{\beta(n)}~{}.$
(3.73)

The superspin projection operator $\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ maps
any real symmetric rank-$n$ superfield
$\Phi_{\alpha(n)}=\bar{\Phi}_{\alpha(n)}$ to the subspace of real transverse
superfields (3.62). If we instead consider the space of complex superfields
$\Phi_{\alpha(n)}$, then $\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ projects onto
the space of simultaneously transverse linear and transverse anti-linear
superfields (3.64).

We now have the tools at our disposal to construct the longitudinal
superprojector (3.40) explicitly. For an arbitrary real symmetric rank-$n$
superfield $\Phi_{\alpha(n)}$, we find

$\displaystyle\mbox{\boldmath$\Pi$}^{\parallel}_{(n)}\Phi_{\alpha(n)}$
$\displaystyle=$
$\displaystyle\bar{D}_{\alpha}\Lambda_{\alpha(n-1)}-(-1)^{n}D_{\alpha}\bar{\Lambda}_{\alpha(n-1)}~{},$
(3.74a) where the complex superfield $\Lambda_{\alpha(n-1)}$ can be shown to
take the form $\displaystyle\Lambda_{\alpha(n-1)}$ $\displaystyle=$
$\displaystyle-\sum_{j=1}^{n}\bigg{(}\frac{{\rm
i}}{4\Box}\bigg{)}^{j}\binom{n}{j}D^{\beta_{n}}A_{(\alpha_{n-1}}^{\quad\beta_{n-1}}\dots
A_{\alpha_{n-j+1}}^{\quad\beta_{n-j+1}}\tilde{\Delta}^{j}\Phi_{\alpha_{1}\dots\alpha_{n-j})\beta_{n-j+1}\dots\beta_{n}}$
(3.74b)
$\displaystyle+\frac{1}{8\Box}D^{\beta}\bar{D}^{2}\Phi_{\beta\alpha(n-1)}~{}.$

Here the operator $A_{\alpha}{}^{\beta}$ is defined by

$\displaystyle
A_{\alpha}{}^{\beta}:=D_{\alpha}\bar{D}^{\beta}+\bar{D}_{\alpha}D^{\beta}~{},$
(3.75)

and satisfies the property

$\displaystyle[\tilde{\Delta},A_{\alpha}\,^{\beta}]=0~{},$ (3.76)

which is crucial to the analysis above.

The relation (3.74a) naturally leads to the gauge transformation law

$\displaystyle\delta_{\zeta}H_{\alpha(n)}=g_{\alpha(n)}+\bar{g}_{\alpha(n)}~{},\qquad
g_{\alpha(n)}=\bar{D}_{\alpha}\zeta_{\alpha(n-1)}~{},$ (3.77)

where $\zeta_{\alpha(n-1)}$ is complex unconstrained. The transformation law
(3.77) was postulated in [122] to describe the gauge freedom of a
superconformal gauge prepotential $H_{\alpha(n)}$.

#####  3.3.2.2 Superspin projection operators in the real basis

The ${\cal N}=2$ superspin projection operator (3.71) takes the following form
in the real basis for the spinor covariant derivatives (3.4)

$\displaystyle\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}=-\frac{{\rm
i}\Delta}{4\Box}D^{\beta I}D^{I}_{\alpha}~{}.$ (3.78)

It follows from (3.35) that the higher-rank superprojector
$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ takes the following form on the space of
real superfields $\Phi_{\alpha(n)}$

$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}\Phi_{\alpha(n)}=\Big{(}-\frac{{\rm
i}\Delta}{4\Box}\Big{)}^{n}D^{\beta_{1}}D_{\alpha_{1}}\cdots
D^{\beta_{n}}D_{\alpha_{n}}\Phi_{\beta(n)}~{}.$ (3.79)

Note that this expression can be simplified in the bosonic $n=2s$ and
fermionic cases $n=2s+1$, as a result of the identity (3.15b). In the
subsequent sections, the superspin projection operators, in the case of
$3\leq{\cal N}\leq 6$ supersymmetry, will be computed in terms of the spinor
covariant derivatives in the real basis (3.4).

####  3.3.3 Superspin projection operators in $\mathbb{M}^{3|6}$

The ${\cal N}=3$ superspin projection operator
$\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ takes the form

$\displaystyle\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ $\displaystyle=$
$\displaystyle-\frac{{\rm i}\Delta}{48\Box^{2}}D^{\beta
I}D_{\alpha}^{I}\big{(}9\Delta^{2}-\Box\big{)}~{}.$ (3.80)

To show that $\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ satisfies the
properties (3.36) and (3.39), it is beneficial to use the identities (3.12),
(3.15b) and

$\displaystyle\Delta^{4}$ $\displaystyle=$
$\displaystyle\frac{1}{9}\Box\big{(}10\Delta^{2}-\Box\big{)}~{}.$ (3.81)

The higher-rank generalisation $\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ of the
superprojector (3.80) is

$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}\Phi_{\alpha(n)}=\Big{(}-\frac{{\rm
i}\Delta}{48\Box^{2}}\Big{)}^{n}\big{(}9\Delta^{2}-\Box\big{)}^{n}D^{\beta_{1}I_{1}}D_{\alpha_{1}}^{I_{1}}\cdots
D^{\beta_{n}I_{n}}D_{\alpha_{n}}^{I_{n}}\Phi_{\beta(n)}~{}.$ (3.82)

Given an integer $n\geq 1$, the following identities can be derived in
$\mathbb{M}^{3|6}$

$\displaystyle(9\Delta^{2}-\Box)^{n}$ $\displaystyle=$
$\displaystyle(8\Box)^{n-1}(9\Delta^{2}-\Box)~{},$ (3.83a)
$\displaystyle\Delta^{2n}(9\Delta^{2}-\Box)$ $\displaystyle=$
$\displaystyle\Box^{n}(9\Delta^{2}-\Box)~{},$ (3.83b)

as a result of relation (3.81). Utilising (3.83a), the superspin projection
operator $\mbox{\boldmath$\Pi$}^{\perp}_{(n)}$ can be reduced to the simpler
form on $\mathbb{V}_{(n)}^{[3]}$

$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}\Phi_{\alpha(n)}=\frac{1}{8\Box}\Big{(}-\frac{{\rm
i}\Delta}{6\Box}\Big{)}^{n}\big{(}9\Delta^{2}-\Box\big{)}D^{\beta_{1}I_{1}}D_{\alpha_{1}}^{I_{1}}\cdots
D^{\beta_{n}I_{n}}D_{\alpha_{n}}^{I_{n}}\Phi_{\beta(n)}~{}.$ (3.84)

####  3.3.4 Superspin projection operators in $\mathbb{M}^{3|8}$

The ${\cal N}=4$ spinor superprojector
$\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ assumes the form

$\displaystyle\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ $\displaystyle=$
$\displaystyle-\frac{{\rm i}\Delta}{24\Box^{2}}D^{\beta
I}D_{\alpha}^{I}\big{(}4\Delta^{2}-\Box\big{)}~{}.$ (3.85)

It is possible to show that the operator
$\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ satisfies the defining properties
(3.36) and (3.39), given that one utilises the identities (3.12), (3.15b) and

$\displaystyle\Delta^{5}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Delta\Box\big{(}5\Delta^{2}-\Box\big{)}~{}.$ (3.86)

The higher-rank extension of the superprojector
$\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ can be shown to take the form

$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}\Phi_{\alpha(n)}=\frac{1}{3\Box}\Big{(}-\frac{{\rm
i}\Delta}{8\Box}\Big{)}^{n}(4\Delta^{2}-\Box)D^{\beta_{1}I_{1}}D_{\alpha_{1}}^{I_{1}}\cdots
D^{\beta_{n}I_{n}}D_{\alpha_{n}}^{I_{n}}\Phi_{\beta(n)}~{}.$ (3.87)

In order to arrive at this result, it is important to make use of the identity

$\Delta(4\Delta^{2}-\Box)^{n}=(3\Box)^{n-1}\Delta(4\Delta^{2}-\Box)~{},\qquad
n\geq 1~{},$ (3.88)

which can be derived from (3.86).

####  3.3.5 Superspin projection operators in $\mathbb{M}^{3|10}$

The ${\cal N}=5$ spinor superprojector
$\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ takes the form

$\displaystyle\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ $\displaystyle=$
$\displaystyle-\frac{{\rm i}\Delta}{3840\Box^{3}}D^{\beta
I}D_{\alpha}^{I}\big{(}25\Delta^{2}-\Box\big{)}\big{(}25\Delta^{2}-9\Box\big{)}~{}.$
(3.89)

The projector properties (3.36) and (3.39) can be proved using (3.12) and
(3.15b), in conjunction with the identity

$\displaystyle\Delta^{6}$ $\displaystyle=$
$\displaystyle\frac{1}{625}\Box\big{(}875\Delta^{4}-259\Box\Delta^{2}+9\Box^{2}\big{)}~{}.$
(3.90)

The higher-rank generalisation of (3.89) is given by

$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}\Phi_{\alpha(n)}=\Big{(}-\frac{{\rm
i}\Delta}{3840\Box^{3}}\Big{)}^{n}\big{(}25\Delta^{2}-\Box\big{)}^{n}\big{(}25\Delta^{2}-9\Box\big{)}^{n}D^{\beta_{1}I_{1}}D_{\alpha_{1}}^{I_{1}}\cdots
D^{\beta_{n}I_{n}}D_{\alpha_{n}}^{I_{n}}\Phi_{\beta(n)}~{}.$ (3.91)

This expression can be simplified if one studies the particular cases when $n$
is either even $n=2s$ or odd $n=2s+1$.

####  3.3.6 Superspin projection operators in $\mathbb{M}^{3|12}$

The ${\cal N}=6$ spinor superprojector
$\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}$ is given by

$\mbox{\boldmath$\Pi$}^{\beta}{}_{\alpha}=-\frac{{\rm
i}\Delta}{480\Box^{3}}D^{\beta
I}D_{\alpha}^{I}\big{(}9\Delta^{2}-\Box\big{)}\big{(}9\Delta^{2}-4\Box\big{)}~{}.$
(3.92)

The superprojector properties (3.36) and (3.39) can be proved using the
identities (3.12), (3.15b) and

$\displaystyle\Delta^{7}$ $\displaystyle=$
$\displaystyle\frac{1}{81}\Box\Delta\big{(}126\Delta^{4}-49\Box\Delta^{2}+4\Box^{2}\big{)}~{}.$
(3.93)

In accordance with (3.35), the higher-rank generalisation of (3.92) is

$\mbox{\boldmath$\Pi$}^{\perp}_{(n)}\Phi_{\alpha(n)}=\Big{(}-\frac{{\rm
i}\Delta}{480\Box^{3}}\Big{)}^{n}\big{(}9\Delta^{2}-\Box\big{)}^{n}\big{(}9\Delta^{2}-4\Box\big{)}^{n}D^{\beta_{1}I_{1}}D_{\alpha_{1}}^{I_{1}}\cdots
D^{\beta_{n}I_{n}}D_{\alpha_{n}}^{I_{n}}\Phi_{\beta(n)}~{}.$ (3.94)

It can be shown that the form of the superprojector (3.94) simplifies if one
analyses the cases $n=2s$ and $n=2s+1$ separately.

###  3.4 Superconformal higher-spin theory

The off-shell actions for ${\cal N}$-extended conformal supergravity theories
in $3d$ were formulated in [157, 158] for $1\leq{\cal N}\leq 6$ using the off-
shell formulation for ${\cal N}$-extended conformal supergravity developed in
[159]. Each action is generated by a closed super three-form which is
constructed in terms of the constrained geometry of ${\cal N}$-extended
conformal superspace. In this section we initiate a program to recast these
actions, and their higher-spin extensions, in terms of unconstrained gauge
prepotentials. In order to achieve this, we first need to compute the
linearised higher-spin super-Cotton tensors in $\mathbb{M}^{3|2{\cal N}}$, for
$1\leq{\cal N}\leq 6$, in terms of unconstrained gauge prepotentials. Before
doing this, it is worth commenting on the super-Cotton tensor in the context
of conformal supergravity.

The geometry of three-dimensional ${\cal N}$-extended conformal superspace
[159] is formulated in terms of a single curvature superfield, which is known
as the super-Cotton tensor ${\cal W}$ (with suppressed indices). The
functional structure of ${\cal W}$ is dependent on the choice of ${\cal N}$.
The ${\cal N}=1$ super-Cotton tensor [168] is a primary symmetric rank-3
spinor superfield ${\cal W}_{\alpha\beta\gamma}$ of dimension $\frac{5}{2}$,
which obeys the conformally invariant constraint [159]

$\displaystyle\nabla^{\alpha}{\cal W}_{\alpha\beta\gamma}=0~{}.$ (3.95)

In the ${\cal N}=2$ case, the super-Cotton tensor [169, 170] is a primary
symmetric rank-2 spinor superfield ${\cal W}_{\alpha\beta}$ of dimension 2,
which obeys the Bianchi identity [159]

$\displaystyle\nabla^{\alpha I}{\cal W}_{\alpha\beta}=0~{}.$ (3.96)

In the ${\cal N}=3$ case, the super-Cotton tensor is a primary spinor
superfield ${\cal W}_{\alpha}$ of dimension $\frac{3}{2}$ constrained by [159]

$\displaystyle\nabla^{\alpha I}{\cal W}_{\alpha}=0~{}.$ (3.97)

In the ${\cal N}=4$ case, the super-Cotton tensor is a primary scalar
superfield ${\cal W}$ of dimension 1 constrained by [159]

$\displaystyle\nabla^{\alpha I}\nabla_{\alpha}^{J}{\cal
W}=\frac{1}{4}\delta^{IJ}\nabla^{\alpha K}\nabla_{\alpha}^{K}{\cal W}~{}.$
(3.98)

For ${\cal N}>4$, the super-Cotton tensor [171, 172] is a completely
antisymmetric tensor ${\cal W}^{IJKL}$ of dimension 1 constrained by [159]

$\displaystyle\nabla_{\alpha}^{I}{\cal W}^{JKLP}=\nabla_{\alpha}^{[I}{\cal
W}^{JKLP]}-\frac{4}{{\cal N}-3}\nabla^{Q}_{\alpha}{\cal
W}^{Q[JKL}\delta^{P]I}\ .$ (3.99)

In the above relations, $\nabla^{I}_{\alpha}$ denotes the spinor covariant
derivative of ${\cal N}$-extended conformal superspace [159].

The super-Cotton tensor is encoded in the action for conformal supergravity
$S_{\rm CSG}$ by the rule [159, 158]

$\displaystyle{\cal W}\propto\frac{\delta S_{\rm CSG}[H]}{\delta H}~{},$
(3.100)

where $H$ denotes an unconstrained gauge prepotential for conformal
supergravity (with indices suppressed). Modulo pure gauge degrees of freedom,
the structure of the unconstrained conformal gauge prepotentials are as
follows: $H_{\alpha\beta\gamma}$ for ${\cal N}=1$ [41], $H_{\alpha\beta}$ for
${\cal N}=2$ [169, 170], $H_{\alpha}$ for ${\cal N}=3$ [159], and $H$ for
${\cal N}=4$ [159]. The above consideration implies that we need expressions
for linearised super-Cotton tensors in terms of the gauge prepotentials,
$W=W(H)$, in order to obtain linearised actions for conformal supergravity
actions for $1\leq{\cal N}\leq 4$.

As a generalisation of the earlier ${\cal N}=1$ [66, 123] and ${\cal N}=2$
[122] results, we propose a ${\cal N}$-extended superconformal gauge-invariant
action of the form

$\displaystyle{S}^{(n|{\cal N})}_{\text{SCHS}}[H_{\alpha(n)}]=\frac{{\rm
i}^{n}}{2}\int{\rm d}^{3|2{\cal
N}}z\,H^{\alpha(n)}{W}_{\alpha(n)}(H)~{},\qquad n>0~{},$ (3.101)

where the real unconstrained superfield $H_{\alpha(n)}$ is defined modulo
gauge transformations

$\displaystyle\delta_{\zeta}H_{\alpha(n)}$ $\displaystyle=$ $\displaystyle{\rm
i}^{n}D^{I}_{\alpha}\zeta^{I}_{\alpha(n-1)}~{}.$ (3.102)

Here the gauge parameter $\zeta_{\alpha(n-1)}$ is also real unconstrained. The
super-Cotton tensor $W_{\alpha(n)}(H)$ in (3.101) is a local descendant of
$H_{\alpha(n)}$. It is a totally symmetric real rank-$n$ superfield, which is
defined to obey the following conditions

  1. 1.

$W_{\alpha(n)}$ is gauge invariant

$\displaystyle\delta_{\zeta}{W}_{\alpha(n)}\big{(}H\big{)}=0~{}.$ (3.103)

  2. 2.

$W_{\alpha(n)}$ is transverse

$\displaystyle D^{\beta I}W_{\beta\alpha(n-1)}(H)=0~{}.$ (3.104)

  3. 3.

The field strength $W_{\alpha(n)}(H)$ is a primary superconformal multiplet
(all relevant technical details about the ${\cal N}$-extended superconformal
group are collected in Appendix 3.A).

The latter fact, in conjunction with (3.104), uniquely fixes the dimension
$d_{W_{\alpha(n)}}$ of ${W}_{\alpha(n)}(H)$. The dimension $d_{H_{\alpha(n)}}$
of ${H}_{\alpha(n)}$ is also fixed uniquely if we require $H_{\alpha(n)}$ and
the associated gauge parameter $\zeta_{\alpha(n-1)}$ in (3.102) to be
superconformal primary. The mass dimensions are:

$\displaystyle d_{H_{\alpha(n)}}=2-{\cal N}-\frac{n}{2}~{},\qquad
d_{W_{\alpha(n)}}=1+\frac{n}{2}~{}.$ (3.105)

For integers $n\geq 1$, the above conditions suffice in determining
$W_{\alpha(n)}(H)$, modulo an overall numerical factor, in the form

$\displaystyle W_{\alpha(n)}\big{(}H\big{)}=\Delta^{n+{\cal
N}-1}\mbox{\boldmath$\Pi$}^{\perp}_{[n]}H_{\alpha(n)}~{}.$ (3.106)

Here $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ is the superspin projection
operator, which was computed explicitly in section 3.3 for the supersymmetry
types $1\leq{\cal N}\leq 6$. Note that the super-Cotton tensor in the form
(3.106) is automatically invariant under the gauge transformations (3.102),
due to the superprojector property (3.44). Below, we will utilise the
superspin projection operators to compute closed form expressions for the
linearised super-Cotton tensors in terms of unconstrained gauge prepotentials.

####  3.4.1 Linearised higher-spin super-Cotton tensor in $\mathbb{M}^{3|2}$

Given an integer $n\geq 1$, the rank-$n$ super-Cotton tensor [66] (see also
[123, 117]) is

$\displaystyle W_{\alpha(n)}(H)=\Big{(}-\frac{{\rm
i}}{2}\Big{)}^{n}D^{\beta_{1}}D_{\alpha_{1}}\cdots
D^{\beta_{n}}D_{\alpha_{n}}H_{\beta(n)}~{}.$ (3.107)

The choice $n=1$ in (3.107) corresponds to the gauge-invariant field strength
of an Abelian vector multiplet [118]. The case $n=2$ corresponds to the super-
Cottino tensor [66], which is the gauge-invariant field strength of a
superconformal gravitino multiplet.101010Among the component fields of
$W_{\alpha\beta}$ is the so-called Cottino tensor
$C_{\alpha\beta\gamma}=C_{(\alpha\beta\gamma)}$, which is the gauge-invariant
field strength of a conformal gravitino [173, 174, 175].  Choosing $n=3$ in
(3.107) gives the linearised version of the ${\cal N}=1$ super-Cotton tensor
[168].

In accordance with (3.106), the super-Cotton tensor (3.107) can be recast in
terms of the superspin projection operator
$\mbox{\boldmath$\Pi$}_{[n]}^{\perp}$ (3.54) as follows

$\displaystyle
W_{\alpha(n)}(H)=\Delta^{n}\mbox{\boldmath$\Pi$}^{\perp}_{[n]}H_{\alpha(n)}~{}.$
(3.108)

In order to demonstrate that (3.108) is equivalent to (3.107), one simply
needs to recall the identity (3.53). This identity also allows us to simplify
$W_{\alpha(n)}$ when considering the cases of $n$ is even or odd.
Specifically, $W_{\alpha(2s)}$ and $W_{\alpha(2s+1)}$ take the following form

$\displaystyle W_{\alpha(2s)}(H)$
$\displaystyle=\Box^{s}\mbox{\boldmath$\Pi$}^{\perp}_{[2s]}H_{\alpha(2s)}~{},$
$\displaystyle s$ $\displaystyle\geq 1~{},$ (3.109a) $\displaystyle
W_{\alpha(2s+1)}(H)$
$\displaystyle=\Box^{s}\Delta\mbox{\boldmath$\Pi$}^{\perp}_{[2s+1]}H_{\alpha(2s+1)}~{},$
$\displaystyle s$ $\displaystyle\geq 0~{}.$ (3.109b)

####  3.4.2 Linearised higher-spin super-Cotton tensor in $\mathbb{M}^{3|4}$

In this section we derive a new representation for the linearised ${\cal N}=2$
rank-$n$ super-Cotton tensor $W_{\alpha(n)}$, with $n>1$. The real tensor
superfield $W_{\alpha(n)}$ was recently shown in [122] to take the following
form

$\displaystyle W_{\alpha(n)}(H)$ $\displaystyle:=$
$\displaystyle\frac{1}{2^{n-2}}\sum_{j=0}^{\lfloor\frac{n}{2}\rfloor}\Bigg{\\{}\binom{n}{2j}{\tilde{\Delta}}\Box^{j}\partial_{(\alpha_{1}}^{\,\,\,\beta_{1}}\cdots\partial_{\alpha_{n-2j}}^{\,\,\,\beta_{n-2j}}H_{\alpha_{n-2j+1}\ldots\alpha_{n})\beta(n-2j)}$
(3.110)
$\displaystyle+\binom{n}{2j+1}{\tilde{\Delta}}^{2}\Box^{j}\partial_{(\alpha_{1}}^{\,\,\,\beta_{1}}\cdots\partial_{\alpha_{n-2j-1}}^{\,\,\,\beta_{n-2j-1}}H_{\alpha_{n-2j}\ldots\alpha_{n})\beta(n-2j-1)}\Bigg{\\}}~{}.$

Note that $W_{\alpha(n)}(H)$ (3.110) is constructed in terms of the spinor
covariant derivatives in the complex basis. The super-Cotton tensor (3.110)
satisfies the following defining properties:

  1. 1.

$W_{\alpha(n)}(H)$ is invariant under the gauge transformations (3.77)

$\delta_{\zeta}W_{\alpha(n)}(H)=0~{}.$ (3.111)

  2. 2.

$W_{\alpha(n)}(H)$ is real transverse linear

$\displaystyle
D^{\beta}W_{\beta\alpha(n-1)}(H)=\bar{D}^{\beta}W_{\beta\alpha(n-1)}(H)=0~{}.$
(3.112)

The case $n=1$ corresponds to the super-Cottino tensor [66], which is the
gauge-invariant field strength of a superconformal gravitino multiplet. The
choice $n=2$ gives the linearised version of the ${\cal N}=2$ super-Cotton
tensor [169, 170].

One may show that $W_{\alpha(n)}(H)$ (3.110) can be recast in terms of the
superprojectors (3.73) in the complex basis as follows

$\displaystyle
W_{\alpha(n)}(H)={\tilde{\Delta}}^{n+1}\mbox{\boldmath$\Pi$}_{[n]}^{\perp}H_{\alpha(n)}~{}.$
(3.113)

Note that (3.113) is the analogue of (3.106) in the complex basis. The
properties of the operator $\tilde{\Delta}$

$\displaystyle\tilde{\Delta}^{2k}=\Box^{k-1}\tilde{\Delta}^{2}~{},\qquad\tilde{\Delta}^{2k+1}=\Box^{k}\tilde{\Delta}~{},\qquad
k=1,2,\dots~{},$ (3.114)

prove useful in showing that the super-Cotton tensors (3.110) and (3.113) are
equivalent.

A new representation of $W_{\alpha(n)}(H)$ (3.110) can be found using
$\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ (3.78) in the real basis. It follows
from (3.106) that $W_{\alpha(n)}(H)$ assumes the form

$\displaystyle
W_{\alpha(n)}\big{(}H\big{)}=\Delta^{n+1}\,\mbox{\boldmath$\Pi$}^{\perp}_{[n]}H_{\alpha(n)}~{}.$
(3.115)

Substituting the explicit form of the superprojector (3.79) into (3.115)
yields

$\displaystyle W_{\alpha(n)}(H)=\Big{(}-\frac{{\rm
i}}{4\Box}\Big{)}^{n}\Delta^{2n+1}D^{\beta_{1}I_{1}}D^{I_{1}}_{\alpha_{1}}\cdots
D^{\beta_{n}I_{n}}D^{I_{n}}_{\alpha_{n}}H_{\beta(n)}~{}.$ (3.116)

It can be shown that the above is equivalent to

$\displaystyle W_{\alpha(n)}(H)=\Big{(}-\frac{{\rm i}}{4}\Big{)}^{n}\Delta
D^{\beta_{1}I_{1}}D^{I_{1}}_{\alpha_{1}}\cdots
D^{\beta_{n}I_{n}}D^{I_{n}}_{\alpha_{n}}H_{\beta(n)}~{},$ (3.117)

where we have made use of the property (3.114). The new representation (3.117)
for the rank-$n$ super-Cotton tensor is clearly much simpler than the
expression (3.110) originally given in [122].

####  3.4.3 Linearised higher-spin super-Cotton tensor in $\mathbb{M}^{3|6}$

A linearised version $W_{\alpha}(H)$ of the ${\cal N}=3$ super-Cotton tensor
[159] has never been computed. Our goal in this section is to construct
$W_{\alpha}(H)$ and its higher-rank extension $W_{\alpha(n)}(H)$ using the
superspin projection operator (3.84).

In accordance with (3.105), the super-Cotton tensor $W_{\alpha(n)}(H)$ in
$\mathbb{M}^{3|6}$ is given by

$\displaystyle
W_{\alpha(n)}\big{(}H\big{)}=\Delta^{n+2}\,\mbox{\boldmath$\Pi$}^{\perp}_{[n]}H_{\alpha(n)}~{}.$
(3.118)

When $W_{\alpha(n)}(H)$ is represented in the form (3.118), both gauge
invariance (3.103) and transversality (3.104) are made manifest as a
consequence of the properties (3.34b), (3.36) and (3.39).

Making use of explicit form for the superprojector (3.84), the expression
(3.118) becomes

$\displaystyle W_{\alpha(n)}(H)=\frac{1}{8\Box}\Big{(}-\frac{{\rm
i}}{6\Box}\Big{)}^{n}\Delta^{2(n+1)}\big{(}9\Delta^{2}-\Box\big{)}D^{\beta_{1}I_{1}}D^{I_{1}}_{\alpha_{1}}\cdots
D^{\beta_{n}I_{n}}D^{I_{n}}_{\alpha_{n}}H_{\beta(n)}~{}.$ (3.119)

It appears that the super-Cotton tensor is not local, due to the presence of
inverse $\Box$. However, making use of the identity (3.83b), it follows that
the super-Cotton tensor (3.119) can be written in the manifestly local form

$\displaystyle W_{\alpha(n)}(H)=\frac{1}{8}\Big{(}-\frac{{\rm
i}}{6}\Big{)}^{n}\big{(}9\Delta^{2}-\Box\big{)}D^{\beta_{1}I_{1}}D^{I_{1}}_{\alpha_{1}}\cdots
D^{\beta_{n}I_{n}}D^{I_{n}}_{\alpha_{n}}H_{\beta(n)}~{}.$ (3.120)

In the $n=1$ case, the field strength $W_{\alpha}$ corresponds to the
linearised version of the ${\cal N}=3$ super-Cotton tensor. Thus, the field
strength (3.118) (or equivalently (3.120)) can be referred to as the rank-$n$
super-Cotton tensor.

#####  3.4.3.1 Superconformal gravitino multiplet

In accordance with the analysis of ${\cal N}=3$ supermultiplets of conserved
currents [176], the superconformal gravitino multiplet should be described by
a real scalar gauge prepotential $H$ of dimension $-1$, which is defined
modulo the gauge transformations

$\displaystyle\delta_{\zeta}H={\rm i}D^{\alpha
I}D^{J}_{\alpha}\zeta^{IJ}~{},\qquad\zeta^{IJ}=\zeta^{JI}~{},\quad\zeta^{II}=0~{}.$
(3.121)

Associated with $H$ is a primary descendant $W(H)$ of dimension $+1$, which
has the following properties:

  1. 1.

$W(H)$ is gauge invariant

$\displaystyle\delta_{\zeta}W(H)=0~{}.$ (3.122a)

  2. 2.

$W(H)$ obeys the constraint

$\displaystyle(D^{\alpha I}D^{J}_{\alpha}-2{\rm
i}\delta^{IJ}\Delta)W(H)=0~{}.$ (3.122b)

We normalise this super-Cottino tensor as

$\displaystyle W(H)=\frac{1}{8}(9\Delta^{2}-\Box)H~{}.$ (3.123)

It can be shown that the super-Cottino tensor (3.123) is consistent with the
conditions (3.122). Note that $W(H)$ naturally follows from the definition of
the ${\cal N}=3$ rank-$n$ super-Cotton tensor (3.120) in the case $n=0$, with
$\mbox{\boldmath$\Pi$}^{\perp}_{(0)}$ being identified as the scalar operator
$\mathcal{P}$

$\displaystyle{\cal P}=\frac{1}{8\Box}(9\Delta^{2}-\Box)~{}.$ (3.124)

It follows from (3.83a) that ${\cal P}$ is indeed a projector, ${\cal
P}^{2}={\cal P}$.

A linearised gauge-invariant action for the ${\cal N}=3$ superconformal
gravitino multiplet is fixed up to an overall constant. We propose the
following action

$\displaystyle S[H]=\frac{1}{2}\int{\rm d}^{3|6}z\,HW(H)~{},$ (3.125)

to describe the dynamics of the ${\cal N}=3$ superconformal gravitino
multiplet.

####  3.4.4 Linearised higher-spin super-Cotton tensor in $\mathbb{M}^{3|8}$

In accordance with (3.106), the super-Cotton tensor $W_{\alpha(n)}(H)$ in
$\mathbb{M}^{3|8}$ is fixed, modulo an overall factor, in the form

$\displaystyle
W_{\alpha(n)}\big{(}H\big{)}=\Delta^{n+3}\,\mbox{\boldmath$\Pi$}^{\perp}_{[n]}H_{\alpha(n)}~{},$
(3.126)

where $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ is the ${\cal N}=4$ superprojector
(3.87). Substituting $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ into (3.126) gives

$\displaystyle W_{\alpha(n)}(H)=\frac{1}{3\Box}\Big{(}-\frac{{\rm
i}}{8\Box}\Big{)}^{n}\Delta^{2n+3}(4\Delta^{2}-\Box)D^{\beta_{1}I_{1}}D_{\alpha_{1}}^{I_{1}}\cdots
D^{\beta_{n}I_{n}}D_{\alpha_{n}}^{I_{n}}\Phi_{\beta(n)}~{}.$ (3.127)

The super-Cotton tensor (3.127) can be simplified in such a way that the
resulting expression is manifestly local. Using the identity (3.15b), it
follows that the super-Cotton tensor (3.127) can be written in the manifestly
local form

$\displaystyle W_{\alpha(n)}(H)=\frac{1}{3}\Big{(}-\frac{{\rm
i}}{8}\Big{)}^{n}\Delta\big{(}4\Delta^{2}-\Box\big{)}D^{\beta_{1}I_{1}}D^{I_{1}}_{\alpha_{1}}\cdots
D^{\beta_{n}I_{n}}D^{I_{n}}_{\alpha_{n}}H_{\beta(n)}~{}.$ (3.128)

The superconformal field strength (3.126), or equivalently (3.128), can be
called the higher-rank ${\cal N}=4$ super-Cotton tensor.

#####  3.4.4.1 Linearised ${\cal N}=4$ conformal supergravity

In the ${\cal N}=4$ case, the linearised super-Cotton tensor proves to be a
real scalar superfield, which obeys the following condition [159]

$\displaystyle(D^{\alpha I}D^{J}_{\alpha}-2{\rm
i}\delta^{IJ}\Delta)W(H)=0~{}.$ (3.129)

In accordance with (3.105), both $W$ and $H$ are primary superfields of
dimension 1 and $-2$, respectively. The super-Cotton tensor is given by

$\displaystyle W(H)=\frac{\Delta}{3}(4\Delta^{2}-\Box)H~{}.$ (3.130)

It can be shown that (3.130) is a solution to (3.129). The super-Cotton tensor
$W(H)$ can be obtained from the rank-$n$ super-Cotton tensor (3.128) in the
case $n=0$. Note that in (3.128), the superprojector
$\mbox{\boldmath$\Pi$}_{(0)}$ is given by the scalar operator ${\cal P}$

$\displaystyle{\cal P}=\frac{\Delta^{2}}{3\Box^{2}}(4\Delta^{2}-\Box)~{}.$
(3.131)

As a direct consequence of (3.86) and (3.88), we deduce that the ${\cal P}$ a
projector, ${\cal P}^{2}={\cal P}$.

We define the linearised action for ${\cal N}=4$ conformal supergravity to be

$\displaystyle S[H]=\frac{1}{2}\int{\rm d}^{3|8}z\,HW(H)~{},$ (3.132)

which is invariant under the gauge transformations

$\displaystyle\delta_{\zeta}H={\rm i}D^{\alpha
I}D^{J}_{\alpha}\zeta^{IJ}~{},\qquad\zeta^{IJ}=\zeta^{JI}~{},\quad\zeta^{II}=0~{}.$
(3.133)

####  3.4.5 Linearised higher-spin super-Cotton tensor in $\mathbb{M}^{3|10}$

In accordance with (3.106), the higher-spin super-Cotton tensor
$W_{\alpha(n)}(H)$ assumes the following form in $\mathbb{M}^{3|10}$

$\displaystyle
W_{\alpha(n)}\big{(}H\big{)}=\Delta^{n+4}\,\mbox{\boldmath$\Pi$}^{\perp}_{[n]}H_{\alpha(n)}~{},$
(3.134)

where $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ is the superprojector (3.91). The
explicit form of (3.134) reads

$\displaystyle W_{\alpha(n)}(H)$ $\displaystyle=$
$\displaystyle\Big{(}-\frac{{\rm
i}}{3840\Box^{3}}\Big{)}^{n}\Delta^{2(n+2)}\big{(}25\Delta^{2}-\Box\big{)}^{n}\big{(}25\Delta^{2}-9\Box\big{)}^{n}$
$\displaystyle\times D^{\beta_{1}I_{1}}D^{I_{1}}_{\alpha_{1}}\cdots
D^{\beta_{n}I_{n}}D^{I_{n}}_{\alpha_{n}}H_{\beta(n)}~{}.$ (3.135)

It is possible to express (3.135) in a manifestly local form. First, equation
(3.90) leads to the following relation

$\displaystyle\Delta^{2n}\big{(}25\Delta^{2}-\Box\big{)}^{n}\big{(}25\Delta^{2}-9\Box\big{)}^{n}$
$\displaystyle=$ $\displaystyle
384^{n-1}\Box^{3(n-1)}\Delta^{2}\big{(}25\Delta^{2}-\Box\big{)}\big{(}25\Delta^{2}-9\Box\big{)}~{},\hskip
14.22636pt$ (3.136)

for $n\geq 1$. As a result, using (3.136) and (3.15b), one may show that
(3.135) reduces to

$\displaystyle W_{\alpha(n)}(H)=\frac{1}{384}\Big{(}-\frac{{\rm
i}}{10}\Big{)}^{n}\big{(}25\Delta^{2}-\Box\big{)}\big{(}25\Delta^{2}-9\Box\big{)}D^{\beta_{1}I_{1}}D^{I_{1}}_{\alpha_{1}}\cdots
D^{\beta_{n}I_{n}}D^{I_{n}}_{\alpha_{n}}H_{\beta(n)}~{}.\hskip 19.91684pt$
(3.137)

####  3.4.6 Linearised higher-spin super-Cotton tensor in $\mathbb{M}^{3|12}$

In accordance with (3.106), the super-Cotton tensor $W_{\alpha(n)}(H)$ takes
the following form

$\displaystyle
W_{\alpha(n)}\big{(}H\big{)}=\Delta^{n+5}\,\mbox{\boldmath$\Pi$}^{\perp}_{[n]}H_{\alpha(n)}~{},$
(3.138)

where $\mbox{\boldmath$\Pi$}^{\perp}_{[n]}$ is the higher-rank superprojector
(3.94). The field strength $W_{\alpha(n)}$ (3.138) takes the explicit form

$\displaystyle W_{\alpha(n)}(H)$ $\displaystyle=$
$\displaystyle\Big{(}-\frac{{\rm
i}}{480\Box^{3}}\Big{)}^{n}\Delta^{2n+5}\big{(}9\Delta^{2}-\Box\big{)}^{n}\big{(}9\Delta^{2}-4\Box\big{)}^{n}$
$\displaystyle\times D^{\beta_{1}I_{1}}D^{I_{1}}_{\alpha_{1}}\cdots
D^{\beta_{n}I_{n}}D^{I_{n}}_{\alpha_{n}}H_{\beta(n)}~{}.$ (3.139)

Again, we wish to express (3.139) in a manifestly local form. To do this, we
need to make use of the important relation

$\displaystyle\Delta^{2}\big{(}9\Delta^{2}-\Box\big{)}^{n}\big{(}9\Delta^{2}-4\Box\big{)}^{n}=40^{n-1}\Box^{2(n-1)}\Delta^{2}\big{(}9\Delta^{2}-\Box\big{)}\big{(}9\Delta^{2}-4\Box\big{)}~{},$
(3.140)

which follows from the identity (3.93). As a result, using (3.140) in
conjunction with (3.15b), one may show that $W_{\alpha(n)}(H)$ can be
expressed in the manifestly local form

$\displaystyle W_{\alpha(n)}(H)=\frac{1}{40}\Big{(}-\frac{{\rm
i}}{12}\Big{)}^{n}\Delta\big{(}9\Delta^{2}-\Box\big{)}\big{(}9\Delta^{2}-4\Box\big{)}D^{\beta_{1}I_{1}}D^{I_{1}}_{\alpha_{1}}\cdots
D^{\beta_{n}I_{n}}D^{I_{n}}_{\alpha_{n}}H_{\beta(n)}~{}.~{}~{}~{}$ (3.141)

Recall that in the context of conformal supergravity with ${\cal N}\geq 5$,
the super-Cotton tensors ${\cal W}_{IJKL}$ (3.99) possess only isospinor
indices. However, it is clearly evident that the proposed ${\cal N}=5$ (3.137)
and ${\cal N}=6$ (3.141) higher-spin Cotton tensors do not have the correct
tensorial form. We still call them linearised Cotton-tensors, in the sense
that they satisfy the defining conditions (3.4). It appears that the super-
Cotton tensors ${\cal W}_{IJKL}$ cannot be derived via superprojectors, thus
an alternative method is required for their computation.

###  3.5 Massless higher-spin theories in $\mathbb{M}^{3|2}$

In this section we review the (half-)integer massless theories in
$\mathbb{M}^{3|2}$, which were first studied by Kuzenko and Tsulaia in [123].
As discussed in section 3.2.2, an off-shell higher-spin gauge theory in
$\mathbb{M}^{3|2}$ is said to be massless if it does not propagate any
physical degrees of freedom on the equations of motion. These theories can be
constructed solely in terms of two superfields: a superconformal gauge
prepotential $H_{\alpha(n)}$, which was introduced in section 3.4;111111 Note
that when we refer to a massless theory in $\mathbb{M}^{3|2}$ as integer or
half-integer, we are referring to the index structure of $H_{\alpha(n)}$
possessing an even or odd number of indices respectively. and some
compensating multiplet. Let us begin by analysing the equations of motion for
the action describing a massless integer superspin multiplet.

####  3.5.1 Massless integer superspin action in $\mathbb{M}^{3|2}$

For integers $s\geq 2$, the model for the massless integer superspin multiplet
is realised in terms of the real unconstrained superfields [123]

${\cal V}_{\text{FO}}=\big{\\{}H_{\alpha(2s)},Y_{\alpha(2s-2)}\big{\\}}~{},$
(3.142)

which are defined modulo gauge transformations

$\displaystyle\delta_{\Lambda}H_{\alpha(2s)}$ $\displaystyle=$ $\displaystyle
D_{\alpha}\Lambda_{\alpha(2s-1)}~{},$ (3.143a)
$\displaystyle\delta_{\Lambda}Y_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2s}D^{\beta}\Lambda_{\beta\alpha(2s-2)}~{}.$ (3.143b)

Here the gauge parameter $\Lambda_{\alpha(2s-1)}$ is real unconstrained. The
massless integer superspin multiplet is described by the gauge-invariant
action [123]

$\displaystyle S_{(s)}^{\text{FO}}[H,Y]$
$\displaystyle=\bigg{(}-\frac{1}{2}\bigg{)}^{s}\frac{{\rm i}}{2}\int{\rm
d}^{3|2}z~{}\Big{\\{}H^{\alpha(2s)}D^{2}H_{\alpha(2s)}-D_{\beta}H^{\beta\alpha(2s-1)}D^{\gamma}H_{\gamma\alpha(2s-1)}$
(3.144)
$\displaystyle+(2s-1)\Big{(}Y^{\alpha(2s-2)}D^{2}Y_{\alpha(2s-2)}+2(s-1)D_{\beta}Y^{\beta\alpha(2s-3)}D^{\gamma}Y_{\gamma\alpha(2s-3)}\Big{)}$
$\displaystyle+2{\rm
i}(2s-1)Y^{\alpha(2s-2)}\partial^{\beta(2)}H_{\beta(2)\alpha(2s-2)}\Big{\\}}~{}.\hskip
14.22636pt$

Let us show that the model (3.144) describes no propagating degrees of freedom
by performing an on-shell analysis at the level of superfields. The equations
of motion corresponding to (3.144) are

$\displaystyle 0$ $\displaystyle=$ $\displaystyle 2{\rm
i}\partial_{\alpha}{}^{\beta}H_{\beta\alpha(2s-1)}+D^{2}H_{\alpha(2s)}-2{\rm
i}(2s-1)\partial_{\alpha(2)}Y_{\alpha(2s-2)}~{},$ (3.145a) $\displaystyle 0$
$\displaystyle=$ $\displaystyle{\rm
i}\partial^{\beta(2)}H_{\beta(2)\alpha(2s-2)}+sD^{2}Y_{\alpha(2s-2)}-2{\rm
i}(s-1)\partial_{\alpha}{}^{\beta}Y_{\beta\alpha(2s-3)}~{}.$ (3.145b)

The gauge freedom (3.143b) can be used to gauge away the superfield
$Y_{\alpha(2s-2)}$

$Y_{\alpha(2s-2)}=0~{}.$ (3.146)

The residual gauge freedom is described by the parameter
$\Lambda_{\alpha(2s-1)}$ constrained by

$D^{\beta}\Lambda_{\beta\alpha(2s-2)}=0\quad\Longrightarrow\quad\partial^{\beta\gamma}\Lambda_{\beta\gamma\alpha(2s-3)}=0~{},\quad{\rm
i}\partial_{\alpha}{}^{\beta}\Lambda_{\beta\alpha(2s-2)}=\frac{1}{2}D^{2}\Lambda_{\alpha(2s-1)}~{}.$
(3.147)

In the gauge (3.146), the equations of motion (3.145) reduce to

$\displaystyle 0$ $\displaystyle=$ $\displaystyle 2{\rm
i}\partial_{\alpha}{}^{\beta}H_{\beta\alpha(2s-1)}+D^{2}H_{\alpha(2s)}~{},$
(3.148a) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial^{\beta(2)}H_{\beta(2)\alpha(2s-2)}~{}.$ (3.148b)

According to (3.143a), in conjunction with the constraint (3.147), it can be
shown that $D^{\beta}H_{\beta\alpha(2s-1)}$ transforms as

$\delta_{\Lambda}(D^{\beta}H_{\beta\alpha(2s-1)})=D^{2}\Lambda_{\alpha(2s-1)}~{}.$
(3.149)

It follows from (3.149) that $D^{\beta}H_{\beta\alpha(2s-1)}$ can be
completely gauged away

$D^{\beta}H_{\beta\alpha(2s-1)}=0\quad\Longrightarrow\quad{\rm
i}\partial_{\alpha}{}^{\beta}H_{\beta\alpha(2s-1)}=\frac{1}{2}D^{2}H_{\alpha(2s)}~{}.$
(3.150)

The corresponding residual gauge freedom further constrains
$\Lambda_{\alpha(2s-1)}$ by

$D^{2}\Lambda_{\alpha(2s-1)}=0\quad\Longrightarrow\quad\Box\Lambda_{\alpha(2s-1)}=0~{}.$
(3.151)

In the gauges (3.146) and (3.150), the equation of motion (3.148a) reduces to

$0=D^{2}H_{\alpha(2s)}~{}.$ (3.152)

In summary, upon imposing the gauges (3.146) and (3.150), we are left with the
field $H_{\alpha(2s)}$ which satisfies the conditions

$D^{\beta}H_{\beta\alpha(2s-1)}=0~{},\qquad D^{2}H_{\alpha(2s)}=0~{}.$
(3.153a) The superfield $H_{\alpha(2s)}$ supports the gauge symmetry
$\delta_{\Lambda}H_{\alpha(2s)}=D_{\alpha}\Lambda_{\alpha(2s-1)}~{},$ (3.153b)
where the gauge parameter $\Lambda_{\alpha(2s-1)}$ is constrained via the
residual gauge symmetry $D^{\beta}\Lambda_{\beta\alpha(2s-2)}=0~{},\qquad
D^{2}\Lambda_{\alpha(2s-1)}=0~{}.$ (3.153c)

The conditions (3.153) coincide with those describing a massless superfield
(3.25) for $n=2s$. Thus, in accordance with the analysis of section 3.2.2, the
superfield $H_{\alpha(2s)}$ carries no physical propagating degrees of freedom
on-shell and hence the action (3.144) describes a massless superfield with
superspin $s$.

Alternatively, one could analyse the component structure of the massless
integer superspin multiplet in order to determine that it propagates no
physical degrees of freedom. The component analysis of the model describing
the massless integer superspin multiplet (3.144) was completed in [117]. Here
it was shown that the gauge freedom (3.143) can be used to impose the
following Wess-Zumino gauge

$H_{\alpha(2s)}|=0~{},\qquad D^{\beta}H_{\beta\alpha(2s-1)}|=0~{},\qquad
Y_{\alpha(2s-2)}|=0~{}.$ (3.154)

The residual gauge symmetry which preserves (3.154) are

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
D_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{2s})}|~{},$ (3.155a) $\displaystyle
0$ $\displaystyle=$ $\displaystyle D^{\beta}\zeta_{\beta\alpha(2s-2)}|~{},$
(3.155b) $\displaystyle D^{2}\zeta_{\alpha(2s-1)}|$ $\displaystyle=$
$\displaystyle-\frac{2{\rm
i}}{2s+1}(2s-1)\partial_{(\alpha_{1}}{}^{\beta}\zeta_{\beta\alpha_{2}...\alpha_{2s-1})}|~{}.$
(3.155c)

The relations (3.155) indicate that there is only one independent gauge
parameter, which we choose to define as

$\xi_{\alpha(2s-1)}:=\zeta_{\alpha(2s-1)}|~{}.$ (3.156)

Thus in the gauge (3.154), we are left with the remaining independent
component fields:

$\displaystyle h_{\alpha(2s+1)}$ $\displaystyle:=$ $\displaystyle-{\rm
i}D_{(\alpha_{1}}H_{\alpha_{2}...\alpha_{2s+1})}|~{},$ (3.157a) $\displaystyle
h_{\alpha(2s)}$ $\displaystyle:=$ $\displaystyle\frac{{\rm
i}}{4}D^{2}H_{\alpha(2s)}|~{},$ (3.157b) $\displaystyle y_{\alpha(2s-2)}$
$\displaystyle:=$ $\displaystyle\frac{{\rm i}}{4}D^{2}Y_{\alpha(2s-2)}|~{},$
(3.157c) $\displaystyle y_{\alpha(2s-1)}$ $\displaystyle:=$
$\displaystyle\frac{{\rm
i}}{2}D_{(\alpha_{1}}Y_{\alpha_{2}...\alpha_{2s-1})}|~{},$ (3.157d)
$\displaystyle z_{\alpha(2s-3)}$ $\displaystyle:=$ $\displaystyle-2{\rm
i}sD^{\beta}Y_{\beta\alpha(2s-3)}|~{}.$ (3.157e)

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

$\displaystyle\delta h_{\alpha(2s)}=0~{},\qquad\delta y_{\alpha(2s-2)}=0~{}.$
(3.158)

Applying the component reduction procedure to the action (3.144) yields the
following decoupled theories in $\mathbb{M}^{3}$

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

The bosonic sector is auxiliary, since the fields $h_{\alpha(2s)}$ and
$y_{\alpha(2s-2)}$ appear with no derivatives in the action (3.159). The
action $S^{\text{FF}}_{(s+\frac{1}{2})}\,[h,y,z]$ coincides with the Fang-
Fronsdal action (2.3.5.2), which was shown in section 2.3 to be a massless
theory. Thus, the massless integer superspin action (3.159) describes no
propagating degrees of freedom at the component level, as required. From this
point onwards, we will always determine whether a higher-spin gauge theory in
$3d$ (super)space is massless by analysing its component structure.

####  3.5.2 Massless half-integer superspin action in $\mathbb{M}^{3|2}$

For $s\geq 2$, the model for the massless half-integer superspin multiplet is
realised in terms of the real unconstrained superfields [123]

${\cal V}_{\text{SO}}=\big{\\{}H_{\alpha(2s+1)},Y_{\alpha(2s-2)}\big{\\}}~{}.$
(3.160)

These superfields are defined modulo gauge transformations

$\displaystyle\delta_{\Lambda}H_{\alpha(2s+1)}$ $\displaystyle=$
$\displaystyle{\rm i}D_{\alpha}\Lambda_{\alpha(2s)}~{},$ (3.161a)
$\displaystyle\delta_{\Lambda}Y_{\alpha(2s-2)}$ $\displaystyle=$
$\displaystyle\frac{s}{2s+1}\partial^{\beta(2)}\Lambda_{\beta(2)\alpha(2s-2)}~{},$
(3.161b)

where the gauge parameter $\Lambda_{\alpha(2s)}$ is real unconstrained.

The massless half-integer superspin multiplet is described by the gauge-
invariant action [123]

$\displaystyle S_{(s+\frac{1}{2})}^{\text{SO}}$
$\displaystyle[H,Y]=\bigg{(}-\frac{1}{2}\bigg{)}^{s}\int{\rm
d}^{3|2}z~{}\Big{\\{}-\frac{{\rm i}}{2}H^{\alpha(2s+1)}\Box
H_{\alpha(2s+1)}-\frac{{\rm
i}}{8}D_{\beta}H^{\beta\alpha(2s)}D^{2}D^{\gamma}H_{\gamma\alpha(2s)}$
$\displaystyle+\frac{{\rm
i}s}{4}\partial_{\beta\gamma}H^{\beta\gamma\alpha(2s-1)}\partial^{\delta\lambda}H_{\delta\lambda\alpha(2s-1)}+(2s-1)Y^{\alpha(2s-2)}\partial^{\beta\gamma}D^{\delta}H_{\beta\gamma\delta\alpha(2s-2)}$
$\displaystyle+2{\rm
i}(2s-1)\Big{(}Y^{\alpha(2s-2)}D^{2}Y_{\alpha(2s-2)}-\frac{1}{s}(s-1)D_{\beta}Y^{\beta\alpha(2s-3)}D^{\gamma}Y_{\gamma\alpha(2s-3)}\Big{)}\Big{\\}}~{}.$
(3.162)

As stated in the previous section, we will study the component structure of
the action (3.5.2) in order to determine that it does not describe any
physical degrees of freedom on-shell.121212The superfield analysis of the
massless half-integer superspin action (3.5.2) was completed in [123]. Note
that the component analysis for the massless half-integer superspin multiplet
in $\mathbb{M}^{3|2}$ was first completed in [117].

The gauge freedom (3.161) can be used to impose the following Wess-Zumino
gauge

$H_{\alpha(2s+1)}|=0~{},\qquad D^{\beta}H_{\beta\alpha(2s)}|=0~{}.$ (3.163)

The residual gauge freedom which preserves the conditions (3.163) is described
by

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
D_{(\alpha_{1}}\zeta_{\alpha_{2}...\alpha_{2s+1})}|~{},$ (3.164a)
$\displaystyle D^{2}\zeta_{\alpha(2s)}|$ $\displaystyle=$
$\displaystyle-\frac{2{\rm
i}s}{s+1}\partial^{\beta}{}_{(\alpha_{1}}\zeta_{\alpha_{2}\ldots\alpha_{2s})\beta}|~{}.$
(3.164b)

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

$\xi_{\alpha(2s)}:=\zeta_{\alpha(2s)}|~{},\qquad\lambda_{\alpha(2s-1)}:=-\frac{{\rm
i}s}{2s+1}D^{\beta}\zeta_{\beta\alpha(2s-1)}|~{}.$ (3.165)

Upon imposing the gauge (3.163), we are left with the following component
fields:

$\displaystyle h_{\alpha(2s+1)}$ $\displaystyle:=$ $\displaystyle\frac{{\rm
i}}{4}D^{2}H_{\alpha(2s+1)}|~{},$ (3.166a) $\displaystyle h_{\alpha(2s+2)}$
$\displaystyle:=$ $\displaystyle-
D_{(\alpha_{1}}H_{\alpha_{2}...\alpha_{2s+2})}|~{},$ (3.166b) $\displaystyle
y_{\alpha(2s-2)}$ $\displaystyle:=$ $\displaystyle-4Y_{\alpha(2s-2)}|~{},$
(3.166c) $\displaystyle y_{\alpha(2s-1)}$ $\displaystyle:=$
$\displaystyle(2s+1){\rm
i}D_{(\alpha_{1}}Y_{\alpha_{2}...\alpha_{2s-1})}|~{},$ (3.166d) $\displaystyle
z_{\alpha(2s-3)}$ $\displaystyle:=$ $\displaystyle 2{\rm
i}D^{\beta}Y_{\beta\alpha(2s-3)}|~{},$ (3.166e) $\displaystyle
F_{\alpha(2s-2)}$ $\displaystyle:=$ $\displaystyle\frac{{\rm
i}}{4}D^{2}Y_{\alpha(2s-2)}|~{}.$ (3.166f)

Performing a component reduction, we find that the higher-spin model (3.5.2)
decouples into a bosonic and a fermionic sector. The bosonic action is given
by

$\displaystyle S_{\text{bos}}[h,y,F]=\Big{(}-\frac{1}{2}\Big{)}^{s}\int{\rm
d}^{3}x\,\bigg{\\{}-\frac{1}{4}h^{\alpha(2s+2)}\Box
h_{\alpha(2s+2)}+\frac{1}{8}(s+1)\partial_{\beta\gamma}h^{\beta\gamma\alpha(2s)}\partial^{\delta\lambda}h_{\delta\lambda\alpha(2s)}~{}$
$\displaystyle+\frac{1}{8}(2s-1)y^{\alpha(2s-2)}\partial^{\beta\gamma}\partial^{\delta\lambda}h_{\beta\gamma\delta\lambda\alpha(2s-2)}+\frac{1}{16s}(2s-1)(s+1)y^{\alpha(2s-2)}\Box
y_{\alpha(2s-2)}~{}$
$\displaystyle+\frac{1}{s}(s-1)(2s-1)F^{\beta\alpha(2s-3)}\partial_{\beta}{}^{\gamma}y_{\gamma\alpha(2s-3)}+\frac{4}{s}(s+1)(2s-1)F^{\alpha(2s-2)}F_{\alpha(2s-2)}~{}\bigg{\\}}~{}.$
(3.167)

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

$F_{\alpha(2s-2)}=-\frac{1}{8(s+1)}(s-1)\partial_{(\alpha_{1}}{}^{\beta}y_{\alpha_{2}...\alpha_{2s-2})\beta}~{},$
(3.168)

we find that the resulting action coincides with the massless spin-$(s+1)$
action $S_{(s+1)}^{\text{F}}[h,y]$, which is given by (2.303) when setting
$s\rightarrow s+1$.

It can be shown that the fermionic action emerging from the reduction
procedure coincides with the Fang-Fronsdal type action
$S^{\text{FF}}_{(s+\frac{1}{2})}[h,y,z]$ (2.3.5.2). In summary, the massless
half-integer superspin action (3.5.2) decouples into the Fronsdal
$S_{(s+1)}^{\text{F}}[h,y]$ and Fang-Fronsdal
$S^{\text{FF}}_{(s+\frac{1}{2})}[h,y,z]$ actions at the component level

$\displaystyle
S^{\text{SO}}_{(s+\frac{1}{2})}[H,Y]=S_{(s+1)}^{\text{F}}[h,y]+S^{\text{FF}}_{(s+\frac{1}{2})}[h,y,z]~{}.$
(3.169)

It was shown in section 2.3 that such theories describe no propagating degrees
of freedom on-shell, hence the action (3.5.2) is indeed massless.

###  3.6 New topologically massive theories

New topological massive (NTM) theories were first proposed in
$\mathbb{M}^{3|2}$ by Kuzenko and Ponds in [117] as supersymmetric
generalisations of the NTM models (2.336) in $\mathbb{M}^{3}$. These theories
are generated by taking an appropriate deformation of the superconformal
action (3.101) such that the corresponding equations of motion coincide
identically with the massive on-shell conditions (3.16).

Given an integer $n\geq 1$, we propose that the NTM actions take the following
form in $\mathbb{M}^{3|2{\cal N}}$

$\displaystyle{S}^{(n|{\cal N})}_{\rm NTM}[H_{\alpha(n)}]\propto\frac{{\rm
i}^{n}}{2}\frac{1}{m}\int{\rm d}^{3|2{\cal
N}}z\,H^{\alpha(n)}\big{(}\Delta-m\sigma\big{)}{W}_{\alpha(n)}\big{(}H\big{)}~{},$
(3.170)

where $\sigma:=\pm 1$ and $m$ is an arbitrary mass parameter. Here, the
superfield $H_{\alpha(n)}$ is the superconformal gauge prepotential and
$W_{\alpha(n)}(H)$ is the higher-spin super-Cotton tensor (3.106). The action
(3.170) is invariant under the the gauge transformations (3.102).

The equation of motion obtained by varying (3.170) with respect to
$H_{\alpha(n)}$ is

$\big{(}\Delta-m\sigma\big{)}{W}_{\alpha(n)}(H)=0~{}.$ (3.171)

In accordance with ${W}_{\alpha(n)}(H)$ being transverse (6.59) by definition,
in conjunction with (3.171), it follows from (3.16) that the field strength
$W_{\alpha(n)}(H)$ describes a massive ${\cal N}$-extended supermultiplet
which carries mass $m$, spin $\frac{n}{2}$ and superhelicity
$\kappa=\frac{\sigma}{2}\big{(}n+\frac{{\cal N}}{2}\big{)}$.

It is worth taking a moment to comment on an alternative formulation for off-
shell theories describing massive supermultiplets in $\mathbb{M}^{3|2}$, known
as topologically massive theories. The corresponding actions of these theories
are built by appropriately coupling the off-shell superconformal action
(3.101) with the actions describing massless integer (3.144) and half-integer
(3.5.2) superspin multiplets. In topologically massive theories, the massless
actions are supplemented with a coupling constant which carries mass
dimension.

Topologically massive higher-spin theories were recently formulated in
$\mathbb{M}^{3}$ by Kuzenko and Ponds in [117]131313Topologically massive
higher-spin theories in AdS3 were also computed in [117]. as higher-spin
extensions of topologically massive gravity [120, 121]. Similarly, off-
shell141414The on-shell formulations for massive higher-spin ${\cal N}=1$
supermultiplets in ${\mathbb{M}}^{3}$ and AdS3 were developed in [178, 179] by
combining the massive bosonic and fermionic higher-spin actions described in
[180, 181]. The formulations given in [180, 181, 178, 179] are based on the
gauge-invariant approaches to the dynamics of massive higher-spin fields,
which were advocated by Zinoviev [92] and Metsaev [93]. topologically massive
higher-spin gauge theories were first constructed in $\mathbb{M}^{3|2}$ by
Kuzenko and Tsulaia [123] as extensions of ${\cal N}=1$ topologically massive
supergravity [173, 177]. These actions were shown to take the form151515These
topologically massive theories (3.172) were extended to $\mathbb{M}^{3|4}$ in
[122, 125].

$\displaystyle S^{\parallel}_{(s)}[H_{\alpha(2s)},V_{\alpha(2s-2)}|m]$
$\displaystyle=$
$\displaystyle{S}^{(2s|1)}_{\text{SCHS}}[H_{\alpha(2s)}]+m^{2s-1}S_{(s)}^{\text{FO}}[{\cal
V}_{\text{FO}}]~{},$ (3.172a) $\displaystyle
S^{\parallel}_{(s+\frac{1}{2})}[H_{\alpha(2s+1)},Y_{\alpha(2s-2)}|m]$
$\displaystyle=$
$\displaystyle{S}^{(2s+1|1)}_{\text{SCHS}}[H_{\alpha(2s+1)}]+m^{2s-1}S_{(s+\frac{1}{2})}^{\text{SO}}[{\cal
V}_{\text{SO}}]~{},$ (3.172b)

where $m$ is a coupling constant with mass dimension one. The superfield
analysis of the topologically massive theories (3.172) was conducted in [123],
where it was shown that on the equations of motion, the integer (3.172a) and
half-integer (3.172b) superspin theories each describe the massive
supermultiplet (3.16), with $n=2s$ and $n=2s+1$, respectively. The same
conclusion was arrived at via a component analysis in [117].

###  3.7 Summary of results

This chapter was dedicated to the construction of superspin projection