$\displaystyle\\{S_{\alpha},\bar{S}_{\dot{\alpha}}\\}$ $\displaystyle=2{\rm
i}K_{{\alpha{\dot{\alpha}}}}~{},$ (6.102f) with all the other
(anti-)commutators vanishing. Finally, the algebra of $K^{A}$ and
$\mbox{\boldmath$\nabla$}_{B}$ is
$\displaystyle[K_{\alpha{\dot{\alpha}}},\mbox{\boldmath$\nabla$}_{\beta{\dot{\beta}}}]$
$\displaystyle=4\big{(}\varepsilon_{{\dot{\alpha}}{\dot{\beta}}}M_{\alpha\beta}+\varepsilon_{\alpha\beta}\bar{M}_{{\dot{\alpha}}{\dot{\beta}}}-\varepsilon_{\alpha\beta}\varepsilon_{{\dot{\alpha}}{\dot{\beta}}}\mathbb{D}\big{)}~{},$
(6.102g) $\displaystyle\\{S_{\alpha},\mbox{\boldmath$\nabla$}_{\beta}\\}$
$\displaystyle=\varepsilon_{\alpha\beta}\big{(}2\mathbb{D}-3\mathbb{Y}\big{)}-4M_{\alpha\beta}~{},$
(6.102h)
$\displaystyle\\{\bar{S}_{\dot{\alpha}},\bar{\mbox{\boldmath$\nabla$}}_{\dot{\beta}}\\}$
$\displaystyle=-\varepsilon_{{\dot{\alpha}}{\dot{\beta}}}\big{(}2\mathbb{D}+3\mathbb{Y})+4\bar{M}_{{\dot{\alpha}}{\dot{\beta}}}~{},$
(6.102i)
$\displaystyle[K_{\alpha{\dot{\alpha}}},\mbox{\boldmath$\nabla$}_{\beta}]$
$\displaystyle=-2{\rm i}\varepsilon_{\alpha\beta}\bar{S}_{{\dot{\alpha}}}\
,\qquad\qquad\qquad[K_{\alpha{\dot{\alpha}}},\bar{\mbox{\boldmath$\nabla$}}_{\dot{\beta}}]=2{\rm
i}\varepsilon_{{\dot{\alpha}}{\dot{\beta}}}S_{\alpha}~{},$ (6.102j)
$\displaystyle[S_{\alpha},\mbox{\boldmath$\nabla$}_{\beta{\dot{\beta}}}]$
$\displaystyle=2{\rm
i}\varepsilon_{\alpha\beta}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\
,\qquad\qquad\quad\qquad[\bar{S}_{\dot{\alpha}},\mbox{\boldmath$\nabla$}_{\beta{\dot{\beta}}}]=-2{\rm
i}\varepsilon_{{\dot{\alpha}}{\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\beta}\
,$ (6.102k)

where all other graded commutators vanish.

In this setting, the $\mathcal{N}=1$ conformal supergravity gauge group ${\cal
G}$ includes local $\Lambda$-transformations of the form

$\displaystyle\delta_{\Lambda}^{(\scriptsize\mbox{\boldmath$\mathcal{G}$})}\mbox{\boldmath$\nabla$}_{A}=\big{[}\Lambda,\mbox{\boldmath$\nabla$}_{A}\big{]}~{},\qquad\Lambda:=\xi^{B}\mbox{\boldmath$\nabla$}_{B}+\frac{1}{2}K^{bc}M_{bc}+{\rm
i}\rho\mathbb{Y}+\Sigma\mathbb{D}+\tau^{B}K_{B}~{}.~{}~{}~{}$ (6.103)

Here the gauge parameter $\Lambda$ incorporates several parameters describing
the general coordinate ($\xi^{B}$), local Lorentz ($K^{ab}$), chiral ($\rho$),
scaling ($\Sigma$), and special superconformal ($\tau^{B}$) transformations. A
tensor superfield $\Phi$ (with Lorentz indices suppressed) is said to be
${\cal G}$-covariant if it transforms under ${\cal G}$ according to the rule

$\displaystyle\delta_{\Lambda}^{(\mbox{\boldmath$\scriptstyle{{\cal
G}}$})}\Phi=\Lambda\Phi~{}.$ (6.104)

A ${\cal G}$-covariant superfield $\Phi$ is said to be primary with super-Weyl
weight $\Delta$ and $\mathsf{U}(1)_{R}$ charge $q$ if it satisfies

$\displaystyle
K_{A}\Phi=0~{},\qquad\mathbb{D}\Phi=\Delta\Phi~{},\qquad\mathbb{Y}\Phi=q\Phi~{}.$
(6.105)

We note that, due to the anticommutator (6.102f), the following (one way)
relation holds

$\displaystyle
S_{\alpha}\Phi=\bar{S}_{{\dot{\alpha}}}\Phi=0~{}\qquad\implies\qquad
K_{\alpha{\dot{\alpha}}}\Phi=0~{}.$ (6.106)

If we further specify that the primary superfield $\Phi$ is covariantly
chiral, then consistency of this constraint with e.g. the anti-commutator
(6.102h) fixes the charge-to-weight ratio:

$\displaystyle\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\Phi=0\qquad\implies\qquad
q=-\frac{2}{3}\Delta~{}.$ (6.107)

In conformal superspace, the torsion and curvature tensors in (6.101) are
subject to covariant constraints such that
$[\mbox{\boldmath$\nabla$}_{A},\mbox{\boldmath$\nabla$}_{B}\\}$ is expressed
solely in terms of the super-Weyl tensor
$\mbox{\boldmath$W$}_{\alpha\beta\gamma}$, its conjugate
$\mbox{\boldmath$\bar{W}$}_{{\dot{\alpha}}{\dot{\beta}}{\dot{\gamma}}}$ and
their covariant derivatives (cf. (6.4)). The latter is a primary and chiral
${\cal G}$-covariant superfield with weight 3/2 and charge -1,

$\displaystyle
K_{B}\mbox{\boldmath$W$}_{\alpha(3)}=0~{},\quad\bar{}\mbox{\boldmath$\nabla$}_{\dot{\beta}}\mbox{\boldmath$W$}_{\alpha(3)}=0~{},\quad{\mathbb{D}}\mbox{\boldmath$W$}_{\alpha(3)}=\frac{3}{2}\mbox{\boldmath$W$}_{\alpha(3)}~{},\quad\mathbb{Y}\mbox{\boldmath$W$}_{\alpha(3)}=-\mbox{\boldmath$W$}_{\alpha(3)}$
(6.108)

The solutions to the aforementioned constraints are given by [123]

$\displaystyle\\{\mbox{\boldmath$\nabla$}_{\alpha},\mbox{\boldmath$\nabla$}_{\beta}\\}$
$\displaystyle=$ $\displaystyle
0~{},\quad\\{\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}},\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\\}=0~{},\quad\\{\mbox{\boldmath$\nabla$}_{\alpha},\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\\}=-2{\rm
i}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\alpha}}}~{},$ (6.109a)
$\displaystyle\big{[}\mbox{\boldmath$\nabla$}_{\alpha},\mbox{\boldmath$\nabla$}_{\beta{\dot{\beta}}}\big{]}$
$\displaystyle=$ $\displaystyle{\rm
i}\varepsilon_{\alpha\beta}\Big{(}2\mbox{\boldmath$\bar{W}$}_{{\dot{\beta}}}{}^{{\dot{\gamma}}{\dot{\delta}}}\bar{M}_{{\dot{\gamma}}{\dot{\delta}}}-\frac{1}{2}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$\bar{W}$}_{{\dot{\alpha}}{\dot{\beta}}{\dot{\gamma}}}\bar{S}^{{\dot{\gamma}}}+\frac{1}{2}\mbox{\boldmath$\nabla$}^{\gamma{\dot{\alpha}}}\mbox{\boldmath$\bar{W}$}_{{\dot{\alpha}}{\dot{\beta}}}{}^{{\dot{\gamma}}}K_{\gamma{\dot{\gamma}}}\Big{)}~{},$
(6.109b)
$\displaystyle\big{[}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}},\mbox{\boldmath$\nabla$}_{\beta{\dot{\beta}}}\big{]}$
$\displaystyle=$ $\displaystyle-{\rm
i}\varepsilon_{{\dot{\alpha}}{\dot{\beta}}}\Big{(}2\mbox{\boldmath$W$}_{\beta}{}^{\gamma\delta}M_{\gamma\delta}+\frac{1}{2}\mbox{\boldmath$\nabla$}^{\alpha}\mbox{\boldmath$W$}_{\alpha\beta\gamma}S^{\gamma}+\frac{1}{2}\mbox{\boldmath$\nabla$}^{\alpha{\dot{\gamma}}}\mbox{\boldmath$W$}_{\alpha\beta}{}^{\gamma}K_{\gamma{\dot{\gamma}}}\Big{)}~{},$
(6.109c)
$\displaystyle\big{[}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\alpha}}},\mbox{\boldmath$\nabla$}_{\beta{\dot{\beta}}}\big{]}$
$\displaystyle=$
$\displaystyle\varepsilon_{{\dot{\alpha}}{\dot{\beta}}}\psi_{\alpha\beta}+\varepsilon_{\alpha\beta}\bar{\psi}_{{\dot{\alpha}}{\dot{\beta}}}~{}.$
(6.109d) In eq. (6.109d) we have defined the operators $\psi_{\alpha(2)}$ and
$\bar{\psi}_{{\dot{\alpha}}(2)}$ as follows $\displaystyle\psi_{\alpha\beta}$
$\displaystyle=$
$\displaystyle\mbox{\boldmath$W$}_{\alpha\beta}{}^{\gamma}\mbox{\boldmath$\nabla$}_{\gamma}+\mbox{\boldmath$\nabla$}^{\gamma}\mbox{\boldmath$W$}_{\alpha\beta}{}^{\delta}M_{\gamma\delta}-\frac{1}{8}\mbox{\boldmath$\nabla$}^{2}\mbox{\boldmath$W$}_{\alpha\beta\gamma}S^{\gamma}+\frac{{\rm
i}}{2}\mbox{\boldmath$\nabla$}^{\gamma{\dot{\gamma}}}\mbox{\boldmath$W$}_{\alpha\beta\gamma}\bar{S}_{{\dot{\gamma}}}$
(6.109e)
$\displaystyle+\frac{1}{4}\mbox{\boldmath$\nabla$}^{\gamma{\dot{\delta}}}\mbox{\boldmath$\nabla$}_{(\alpha}\mbox{\boldmath$W$}_{\beta)\gamma}{}^{\delta}K_{\delta{\dot{\delta}}}+\frac{1}{4}\mbox{\boldmath$\nabla$}^{\gamma}\mbox{\boldmath$W$}_{\alpha\beta\gamma}\big{(}2\mathbb{D}-3\mathbb{Y}\big{)}~{},$
$\displaystyle\bar{\psi}_{{\dot{\alpha}}{\dot{\beta}}}$ $\displaystyle=$
$\displaystyle-\mbox{\boldmath$\bar{W}$}_{{\dot{\alpha}}{\dot{\beta}}}{}^{{\dot{\gamma}}}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\gamma}}}-\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\gamma}}}\mbox{\boldmath$\bar{W}$}_{{\dot{\alpha}}{\dot{\beta}}}{}^{{\dot{\delta}}}\bar{M}_{{\dot{\gamma}}{\dot{\delta}}}+\frac{1}{8}\bar{\mbox{\boldmath$\nabla$}}^{2}\mbox{\boldmath$\bar{W}$}_{{\dot{\alpha}}{\dot{\beta}}{\dot{\gamma}}}\bar{S}^{{\dot{\gamma}}}+\frac{{\rm
i}}{2}\mbox{\boldmath$\nabla$}^{\gamma{\dot{\gamma}}}\mbox{\boldmath$\bar{W}$}_{{\dot{\alpha}}{\dot{\beta}}{\dot{\gamma}}}S_{\gamma}$
(6.109f)
$\displaystyle-\frac{1}{4}\mbox{\boldmath$\nabla$}^{\delta{\dot{\gamma}}}\bar{\mbox{\boldmath$\nabla$}}_{({\dot{\alpha}}}\mbox{\boldmath$\bar{W}$}_{{\dot{\beta}}){\dot{\gamma}}}{}^{{\dot{\delta}}}K_{\delta{\dot{\delta}}}-\frac{1}{4}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\gamma}}}\mbox{\boldmath$\bar{W}$}_{{\dot{\alpha}}{\dot{\beta}}{\dot{\gamma}}}\big{(}2\mathbb{D}+3\mathbb{Y}\big{)}~{}.$

Therefore, we see that if $\mathcal{M}^{4|4}$ is conformally-flat, then the
superconformal covariant derivatives obey the same algebra as the covariant
derivatives of $\mathbb{M}^{4|4}$,

$\displaystyle\mbox{\boldmath$W$}_{\alpha(3)}=0\qquad\implies\qquad\big{[}\mbox{\boldmath$\nabla$}_{A},\mbox{\boldmath$\nabla$}_{\beta{\dot{\beta}}}\big{]}=0~{}.$
(6.110)

We also find that $\mbox{\boldmath$W$}_{\alpha\beta\gamma}$ obeys the Bianchi
identity

$\displaystyle\mbox{\boldmath$B$}_{\alpha{\dot{\alpha}}}:={\rm
i}\mbox{\boldmath$\nabla$}^{\beta}{}_{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{\gamma}\mbox{\boldmath$W$}_{\alpha\beta\gamma}={\rm
i}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\beta}}}\bar{}\mbox{\boldmath$\nabla$}^{\dot{\gamma}}\mbox{\boldmath$\bar{W}$}_{{\dot{\alpha}}{\dot{\beta}}{\dot{\gamma}}}=\mbox{\boldmath$\bar{B}$}_{\alpha{\dot{\alpha}}}~{},$
(6.111)

where $\mbox{\boldmath$B$}_{\alpha{\dot{\alpha}}}$ is the super-Bach
tensor.191919It is instructive to check that (6.19) and (6.111) coincide upon
degauging to $\mathsf{U}(1)$ superspace.  It is a primary superfield with
weight 3,

$\displaystyle K_{B}\mbox{\boldmath$B$}_{\alpha{\dot{\alpha}}}$
$\displaystyle=$ $\displaystyle
0~{},\qquad{\mathbb{D}}\mbox{\boldmath$B$}_{\alpha{\dot{\alpha}}}=3\mbox{\boldmath$B$}_{\alpha{\dot{\alpha}}}~{},$
(6.112)

which obeys the conservation equation

$\displaystyle\mbox{\boldmath$\nabla$}^{\alpha}\mbox{\boldmath$B$}_{\alpha{\dot{\alpha}}}=0\quad$
$\displaystyle\Longleftrightarrow$
$\displaystyle\quad\bar{}\mbox{\boldmath$\nabla$}^{\dot{\alpha}}\mbox{\boldmath$B$}_{\alpha{\dot{\alpha}}}=0~{}.$
(6.113)

In appendix 6.A we review how the geometry of $\mathcal{N}=1$ conformal
superspace may be degauged to recover the $\mathsf{U}(1)$ superspace geometry
described in section 6.1.1. Additionally, in the context of conformal
superspace, we discuss the issues of (i) integration by parts; and (ii)
transitioning between integrations over the full superspace and its chiral
subspace.

Finally, we note that if we restrict our attention to bosonic backgrounds,
defined by

$\displaystyle\mbox{\boldmath$\nabla$}_{a}|=\nabla_{a}~{},\quad\mbox{\boldmath$W$}_{\alpha(3)}|=0~{},\quad\mbox{\boldmath$\nabla$}^{2}$
$\displaystyle\mbox{\boldmath$W$}_{\alpha(3)}|=0~{},\quad\mbox{\boldmath$\nabla$}^{\beta}\mbox{\boldmath$W$}_{\beta\alpha(2)}|=0~{},$
(6.114a)
$\displaystyle\mbox{\boldmath$\nabla$}_{\alpha}\mbox{\boldmath$W$}_{\beta(3)}|$
$\displaystyle=-W_{\alpha\beta(3)}~{},$ (6.114b)

then the resulting geometry describes conformal gravity (see section 2.4.3).
For such backgrounds, the only surviving component field of
$\mbox{\boldmath$B$}_{\alpha{\dot{\alpha}}}$ is the Bach tensor
$B_{\alpha(2){\dot{\alpha}}(2)}$

$\displaystyle-\frac{1}{2}\big{[}\mbox{\boldmath$\nabla$}_{(\alpha_{1}},\bar{\mbox{\boldmath$\nabla$}}_{({\dot{\alpha}}_{1}}\big{]}\mbox{\boldmath$B$}_{\alpha_{2}){\dot{\alpha}}_{2})}|=\nabla_{({\dot{\alpha}}_{1}}{}^{\beta_{1}}\nabla_{{\dot{\alpha}}_{2})}{}^{\beta_{2}}W_{\alpha(2)\beta(2)}=B_{\alpha(2){\dot{\alpha}}(2)}~{},$
(6.115)

where $W_{\alpha(4)}$ is the Weyl tensor. The gauge choice (6.114) means that
we are switching off the axial gauge field and all covariant fermionic fields
(such as the gravitino), which comes with a loss of local supersymmetry and
$\mathsf{U}(1)_{R}$ symmetry; only rigid versions remain.

####  6.4.2 SCHS models

We define SCHS prepotentials $H_{\alpha(m){\dot{\alpha}}(n)}$ in conformal
superspace by lifting the properties of the corresponding SCHS prepotentials
living in $\mathsf{U}(1)$ superspace. The latter were described in section
6.1.2, and were based on the earlier constructions [101, 102, 295]. In
particular, for all allowed values of $m$ and $n$, the SCHS prepotential
$H_{\alpha(m){\dot{\alpha}}(n)}$ is primary

$\displaystyle K_{A}H_{\alpha(m){\dot{\alpha}}(n)}=0~{},$ (6.116a) and has the
following super-Weyl weight and $\mathsf{U}(1)_{R}$ charge
$\displaystyle\mathbb{D}H_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=-\frac{1}{2}(m+n)H_{\alpha(m){\dot{\alpha}}(n)}~{},$ (6.116b)
$\displaystyle\mathbb{Y}H_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=\phantom{-}\frac{1}{3}(m-n)H_{\alpha(m){\dot{\alpha}}(n)}~{}.$
(6.116c)

The weight and charge of $H_{\alpha(m){\dot{\alpha}}(n)}$ is fixed by
requiring consistency of the algebra (6.102) with its corresponding gauge
transformations, which vary depending on $m$ and $n$:

  * •

For $1\leq m\neq n\geq 1$, $H_{\alpha(m){\dot{\alpha}}(n)}$ is defined modulo
gauge transformations

$\displaystyle\delta_{\Lambda,\Omega}H_{\alpha(m){\dot{\alpha}}(n)}=\mbox{\boldmath$\nabla$}_{\alpha}\Lambda_{\alpha(m-1){\dot{\alpha}}(n)}+\bar{}\mbox{\boldmath$\nabla$}_{{\dot{\alpha}}}\Omega_{\alpha(m){\dot{\alpha}}(n-1)}~{},$
(6.117a) for unconstrained primary complex gauge parameters
$\Lambda_{\alpha(m-1){\dot{\alpha}}(n)}$ and
$\Omega_{\alpha(m){\dot{\alpha}}(n-1)}$.

  * •

For $m=n=s\geq 1$, $H_{\alpha(s){\dot{\alpha}}(s)}$ is real and defined modulo
the gauge transformations

$\displaystyle\delta_{\Lambda}H_{\alpha(s){\dot{\alpha}}(s)}=\mbox{\boldmath$\nabla$}_{\alpha}\Lambda_{\alpha(s-1){\dot{\alpha}}(s)}-\bar{}\mbox{\boldmath$\nabla$}_{{\dot{\alpha}}}\bar{\Lambda}_{\alpha(s){\dot{\alpha}}(s-1)}~{},$
(6.117b)

for unconstrained primary complex gauge parameter
$\Lambda_{\alpha(s-1){\dot{\alpha}}(s)}$.

  * •

For $m>n=0$, $H_{\alpha(m)}$ is defined modulo the gauge transformations

$\displaystyle\delta_{\Lambda,\lambda}H_{\alpha(m)}=\mbox{\boldmath$\nabla$}_{\alpha}\Lambda_{\alpha(m-1)}+\lambda_{\alpha(m)}~{},\qquad\bar{}\mbox{\boldmath$\nabla$}_{{\dot{\alpha}}}\lambda_{\alpha(m)}=0~{}.$
(6.117c)

The complex gauge parameter $\Lambda_{\alpha(m-1)}$ is primary (but otherwise
unconstrained), whilst the complex gauge parameter $\lambda_{\alpha(m)}$ is
primary and covariantly chiral.

In $\mathcal{N}=1$ conformal superspace, the higher-spin super-Weyl tensor of
the prepotential $H_{\alpha(m){\dot{\alpha}}(n)}$ and its conjugate
$\bar{H}_{\alpha(n){\dot{\alpha}}(m)}$ is the minimal lift of (6.44),

$\displaystyle\mathfrak{W}_{\alpha(m+n+1)}(H)$
$\displaystyle=-\frac{1}{4}\bar{}\mbox{\boldmath$\nabla$}^{2}\mbox{\boldmath$\nabla$}_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\mbox{\boldmath$\nabla$}_{\alpha_{n}}{}^{{\dot{\beta}}_{n}}\mbox{\boldmath$\nabla$}_{\alpha_{n+1}}H_{\alpha_{n+2}\dots\alpha_{m+n+1}){\dot{\beta}}(n)}~{},$
(6.118a) $\displaystyle\mathfrak{W}_{\alpha(m+n+1)}(\bar{H})$
$\displaystyle=-\frac{1}{4}\bar{}\mbox{\boldmath$\nabla$}^{2}\mbox{\boldmath$\nabla$}_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\mbox{\boldmath$\nabla$}_{\alpha_{m}}{}^{{\dot{\beta}}_{m}}\mbox{\boldmath$\nabla$}_{\alpha_{m+1}}\bar{H}_{\alpha_{m+2}\dots\alpha_{m+n+1}){\dot{\beta}}(m)}~{}.$
(6.118b)

The former satisfies the following properties:

  1. (i)

$\mathfrak{W}_{\alpha(m+n+1)}(H)$ is a primary superfield,

$\displaystyle K_{A}\mathfrak{W}_{\alpha(m+n+1)}(H)$ $\displaystyle=0~{},$
(6.119a) with super-Weyl weight and $\mathsf{U}(1)_{R}$ charge given by
$\displaystyle\mathbb{D}\mathfrak{W}_{\alpha(m+n+1)}(H)$
$\displaystyle=\phantom{-}\frac{1}{2}\big{(}3+n-m\big{)}\mathfrak{W}_{\alpha(m+n+1)}(H)~{},$
(6.119b) $\displaystyle\mathbb{Y}\mathfrak{W}_{\alpha(m+n+1)}(H)$
$\displaystyle=-\frac{1}{3}\big{(}3+n-m\big{)}\mathfrak{W}_{\alpha(m+n+1)}(H)~{}.$
(6.119c)

  2. (ii)

$\mathfrak{W}_{\alpha(m+n+1)}(H)$ is covariantly chiral (which is consistent
with (6.107)),

$\displaystyle\bar{}\mbox{\boldmath$\nabla$}_{{\dot{\alpha}}}\mathfrak{W}_{\alpha(m+n+1)}(H)=0~{}.$
(6.120)

  3. (iii)

$\mathfrak{W}_{\alpha(m+n+1)}(H)$ is gauge invariant under (6.117) if
$\mathcal{M}^{4|4}$ is conformally-flat,

$\displaystyle\mbox{\boldmath$W$}_{\alpha(3)}=0\quad\implies\quad\delta_{\Lambda}\mathfrak{W}_{\alpha(m+n+1)}(H)$
$\displaystyle=0~{}.$ (6.121)

The descendent $\mathfrak{W}_{\alpha(m+n+1)}(\bar{H})$ satisfies the same
properties as $\mathfrak{W}_{\alpha(m+n+1)}(H)$, but with $m$ and $n$
interchanged in (6.119). In order to prove the validity of (6.119a), the
following identities are useful

$\displaystyle\big{[}S_{\beta},\big{(}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\alpha}}}\big{)}^{k}\big{]}$
$\displaystyle=\phantom{-}2k{\rm
i}\varepsilon_{\beta\alpha}\bar{}\mbox{\boldmath$\nabla$}_{{\dot{\alpha}}}\big{(}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\alpha}}}\big{)}^{k-1}~{},$
(6.122a)
$\displaystyle\big{[}\bar{S}_{{\dot{\beta}}},\big{(}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\alpha}}}\big{)}^{k}\big{]}$
$\displaystyle=-2k{\rm
i}\varepsilon_{{\dot{\beta}}{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\alpha}\big{(}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\alpha}}}\big{)}^{k-1}~{},$
(6.122b) $\displaystyle\big{[}S_{\alpha},\mbox{\boldmath$\nabla$}^{2}\big{]}$
$\displaystyle=-2\mbox{\boldmath$\nabla$}_{\alpha}\big{(}2\mathbb{D}-3\mathbb{Y}-4\big{)}+8\mbox{\boldmath$\nabla$}^{\beta}M_{\alpha\beta}~{},$
(6.122c)
$\displaystyle\big{[}\bar{S}_{{\dot{\alpha}}},\bar{}\mbox{\boldmath$\nabla$}^{2}\big{]}$
$\displaystyle=-2\bar{}\mbox{\boldmath$\nabla$}_{{\dot{\alpha}}}\big{(}2\mathbb{D}+3\mathbb{Y}-4\big{)}+8\bar{}\mbox{\boldmath$\nabla$}^{{\dot{\beta}}}\bar{M}_{{\dot{\alpha}}{\dot{\beta}}}~{}.$
(6.122d)

They hold for an arbitrary positive integer $k$ and on a generic background.
The property (6.121) is a simple consequence of (6.45) and (6.110).

Similarly, the minimal lifts of the higher-spin super-Bach tensors (6.47a) and
(6.47b) to conformal superspace are

$\displaystyle\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)$
$\displaystyle=-\frac{1}{4}\mbox{\boldmath$\nabla$}_{({\dot{\alpha}}_{1}}{}^{\beta_{1}}\cdots\mbox{\boldmath$\nabla$}_{{\dot{\alpha}}_{m})}{}^{\beta_{m}}\mbox{\boldmath$\nabla$}^{\gamma}\bar{}\mbox{\boldmath$\nabla$}^{2}\mbox{\boldmath$\nabla$}_{(\gamma}\mbox{\boldmath$\nabla$}_{\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\mbox{\boldmath$\nabla$}_{\alpha_{n}}{}^{{\dot{\beta}}_{n}}H_{\beta_{1}\dots\beta_{m}){\dot{\beta}}(n)}~{},$
(6.123a) $\displaystyle\widehat{\mathfrak{B}}_{\alpha(n){\dot{\alpha}}(m)}(H)$
$\displaystyle=\phantom{-}\frac{1}{4}\mbox{\boldmath$\nabla$}_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\cdots\mbox{\boldmath$\nabla$}_{\alpha_{n})}{}^{{\dot{\beta}}_{n}}\bar{}\mbox{\boldmath$\nabla$}^{{\dot{\gamma}}}\mbox{\boldmath$\nabla$}^{2}\bar{}\mbox{\boldmath$\nabla$}_{({\dot{\gamma}}}\mbox{\boldmath$\nabla$}_{{\dot{\alpha}}_{1}}{}^{\beta_{1}}\cdots\mbox{\boldmath$\nabla$}_{{\dot{\alpha}}_{m}}{}^{\beta_{m}}H_{\beta(m){\dot{\beta}}_{1}\dots\dot{\beta}_{n})}~{}.$
(6.123b)

They may also be expressed in terms of the higher-spin super-Weyl tensors, as
in (6.51). On a generic background, the descendent
$\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)$ is a primary superfield,

$\displaystyle K_{A}\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)=0~{},$
(6.124a) with Weyl weight and $\mathsf{U}(1)_{R}$ charge given by
$\displaystyle\mathbb{D}\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)$
$\displaystyle=\Big{(}2+\frac{1}{2}\big{(}m+n\big{)}\Big{)}\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)~{},$
(6.124b) $\displaystyle\mathbb{Y}\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)$
$\displaystyle=\frac{1}{3}\big{(}m-n\big{)}\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)~{}.$
(6.124c)

The other descendent $\widehat{\mathfrak{B}}_{\alpha(n){\dot{\alpha}}(m)}(H)$
carries the same superconformal properties. On any conformally-flat
background,

$\displaystyle\mbox{\boldmath$W$}_{\alpha(3)}=0~{},$ (6.125)

they possess the following additional properties:

  1. (i)

$\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)$ and
$\widehat{\mathfrak{B}}_{\alpha(n){\dot{\alpha}}(m)}(H)$ are equal to one
another

$\displaystyle\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)=\widehat{\mathfrak{B}}_{\alpha(n){\dot{\alpha}}(m)}(H)~{},$
(6.126)

which leads to the following reality relation

$\displaystyle\mathfrak{B}_{\alpha(m){\dot{\alpha}}(n)}(\bar{H})=\Big{(}\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)\Big{)}^{*}\equiv\overline{\mathfrak{B}}_{\alpha(m){\dot{\alpha}}(n)}(\bar{H})~{}.$
(6.127)

  2. (ii)

For $m\geq 1$ and $n\geq 1$ $\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)$ is
TLAL,

$\displaystyle\mbox{\boldmath$\nabla$}^{\beta}\mathfrak{B}_{\beta\alpha(n-1){\dot{\alpha}}(m)}(H)$
$\displaystyle=0\qquad\implies\qquad\mbox{\boldmath$\nabla$}^{2}\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)=0~{},$
(6.128a)
$\displaystyle\bar{}\mbox{\boldmath$\nabla$}^{{\dot{\beta}}}\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m-1){\dot{\beta}}}(H)$
$\displaystyle=0\qquad\implies\qquad\bar{}\mbox{\boldmath$\nabla$}^{2}\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)=0~{}.$
(6.128b)

For $m>n=0$ $\mathfrak{B}_{{\dot{\alpha}}(m)}(H)$ is ALTL, whilst for $n>m=0$
$\mathfrak{B}_{\alpha(n)}(H)$ is LTAL.

  3. (iii)

$\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)$ is invariant under the gauge
transformations (6.117)

$\displaystyle\delta_{\Lambda}\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)$
$\displaystyle=0~{}.$ (6.129)

The properties (6.121), (6.128) and (6.129) mean that on any conformally-flat
background (6.125), the minimal lift of the action (6.53) to conformal
superspace,202020Moving between the full superspace and its (anti-)chiral
subspace is non-trivial in conformal superspace, see appendix 6.A. However,
the two forms (6.130a) and (6.130b) of the SCHS action prove to be equivalent
(at least on conformally-flat backgrounds).

$\displaystyle\mathbb{S}_{\text{SCHS}}^{(m,n)}[H,\bar{H}]$
$\displaystyle=\frac{1}{2}{\rm i}^{m+n}\int{\rm d}^{4}x{\rm
d}^{2}\theta\,{\cal
E}\,\mathfrak{W}^{\alpha(m+n+1)}(H)\mathfrak{W}_{\alpha(m+n+1)}(\bar{H})+\text{c.c.}~{}$
(6.130a) $\displaystyle=\frac{1}{2}(-{\rm i})^{m+n+2}\int{\rm
d}^{4|4}z\,E\,\bar{H}^{\alpha(n){\dot{\alpha}}(m)}\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)+\text{c.c.}~{},$
(6.130b)

is gauge invariant
$\delta_{\Lambda}\mathbb{S}_{\text{SCHS}}^{(m,n)}[H,\bar{H}]=0$. Due to the
properties (6.116), (6.119) and (6.124), the same action is invariant under
the $\mathcal{N}=1$ conformal supergravity gauge group ${\cal G}$ on a generic
supergravity background.

The action (6.130) ceases to be gauge invariant on non-conformally-flat
backgrounds. In section 6.5 we will extend the gauge invariance of
$\mathbb{S}_{\text{SCHS}}^{(m,n)}[H,\bar{H}]$ to Bach-flat backgrounds for
some low values of $m$ and $n$.

####  6.4.3 Component reduction: SCHS model $\longrightarrow$ CHS models

In this section we discuss how the gauge freedom (6.117) may be partially
fixed to construct a Wess-Zumino gauge on the SCHS prepotential
$H_{\alpha(m){\dot{\alpha}}(n)}$. We then make use of this gauge to reduce the
corresponding SCHS action $\mathbb{S}_{\text{SCHS}}^{(m,n)}[H,\bar{H}]$ to
components, and explore the resulting CHS actions which they induce. Since the
relevant gauge freedom changes according to the values of $m$ and $n$, an
analysis should proceed on a case-by-case basis. Here, we focus only on two of
the three cases; $m=n=s\geq 1$ and $m>n=0$.

##### Case $m=n=s\geq 1$

For this case we restrict our attention to bosonic superspace backgrounds
(6.114) which are conformally-flat (6.125), i.e. those satisfying

$\displaystyle\mbox{\boldmath$\nabla$}_{a}|=\nabla_{a}~{},\qquad\mbox{\boldmath$W$}_{\alpha(3)}=0\quad\implies\quad
W_{\alpha(4)}=0~{}.$ (6.131)

This means that there is only a rigid supersymmetry present (which we will not
be concerned with).

The gauge freedom (6.117b) of the real prepotential
$H_{\alpha(s){\dot{\alpha}}(s)}$ allows one to impose a Wess-Zumino gauge such
that there are only three independent component fields:

$\displaystyle h_{\alpha(s+1){\dot{\alpha}}(s+1)}$
$\displaystyle:=\frac{1}{2}\big{[}\mbox{\boldmath$\nabla$}_{\alpha},\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\big{]}H_{\alpha(s){\dot{\alpha}}(s)}|=\bar{h}_{\alpha(s+1){\dot{\alpha}}(s+1)}~{},$
(6.132a) $\displaystyle\psi_{\alpha(s+1){\dot{\alpha}}(s)}$
$\displaystyle:=-\frac{1}{4}\mbox{\boldmath$\nabla$}_{\alpha}\bar{\mbox{\boldmath$\nabla$}}^{2}H_{\alpha(s){\dot{\alpha}}(s)}|~{},$
(6.132b) $\displaystyle\bar{\psi}_{\alpha(s){\dot{\alpha}}(s+1)}$
$\displaystyle:=-\frac{1}{4}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{2}H_{\alpha(s){\dot{\alpha}}(s)}|~{},$
(6.132c) $\displaystyle g_{\alpha(s){\dot{\alpha}}(s)}$
$\displaystyle:=\frac{1}{32}\\{\mbox{\boldmath$\nabla$}^{2},\bar{\mbox{\boldmath$\nabla$}}^{2}\\}H_{\alpha(s){\dot{\alpha}}(s)}|=\bar{g}_{\alpha(s){\dot{\alpha}}(s)}~{}.$
(6.132d)

Each of these fields are primary and have the following Weyl weights:

$\displaystyle K_{b}h_{\alpha(s+1){\dot{\alpha}}(s+1)}$
$\displaystyle=0~{},\qquad\mathbb{D}h_{\alpha(s+1){\dot{\alpha}}(s+1)}=\big{(}1-s\big{)}h_{\alpha(s+1){\dot{\alpha}}(s+1)}~{},$
(6.133a) $\displaystyle K_{b}\psi_{\alpha(s+1){\dot{\alpha}}(s)}$
$\displaystyle=0~{},\qquad~{}\phantom{..}\mathbb{D}\psi_{\alpha(s+1){\dot{\alpha}}(s)}=\big{(}\frac{3}{2}-s\big{)}\psi_{\alpha(s+1){\dot{\alpha}}(s)}~{},$
(6.133b) $\displaystyle K_{b}g_{\alpha(s){\dot{\alpha}}(s)}$
$\displaystyle=0~{},\qquad~{}~{}~{}~{}\phantom{..}\mathbb{D}g_{\alpha(s){\dot{\alpha}}(s)}=\big{(}2-s\big{)}g_{\alpha(s){\dot{\alpha}}(s)}~{}.$
(6.133c)

The residual gauge freedom is generated by the (primary) parameters

$\displaystyle\xi_{\alpha(s){\dot{\alpha}}(s)}$ $\displaystyle:=-{\rm
i}\Big{(}\mbox{\boldmath$\nabla$}_{\alpha}\Lambda_{\alpha(s-1){\dot{\alpha}}(s)}|+\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\bar{\Lambda}_{\alpha(s){\dot{\alpha}}(s-1)}|\Big{)}~{},$
(6.134a) $\displaystyle\zeta_{\alpha(s){\dot{\alpha}}(s-1)}$
$\displaystyle:=-\frac{{\rm
i}}{2}\mbox{\boldmath$\nabla$}^{2}\bar{\Lambda}_{\alpha(s){\dot{\alpha}}(s-1)}|~{},$
(6.134b) $\displaystyle\ell_{\alpha(s-1){\dot{\alpha}}(s-1)}$
$\displaystyle:=\frac{{\rm
i}s}{8(s+1)}\Big{(}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\beta}}}\mbox{\boldmath$\nabla$}^{2}\Lambda_{\alpha(s-1){\dot{\alpha}}(s-1){\dot{\beta}}}|+\mbox{\boldmath$\nabla$}^{\beta}\bar{\mbox{\boldmath$\nabla$}}^{2}\bar{\Lambda}_{\beta\alpha(s-1){\dot{\alpha}}(s-1)}|\Big{)}~{},$
(6.134c)

under which the fields (6.132) transform according to

$\displaystyle\delta_{\xi}h_{\alpha(s+1){\dot{\alpha}}(s+1)}$
$\displaystyle=\nabla_{\alpha{\dot{\alpha}}}\xi_{\alpha(s){\dot{\alpha}}(s)}~{},$
(6.135a) $\displaystyle\delta_{\zeta}\psi_{\alpha(s+1){\dot{\alpha}}(s)}$
$\displaystyle=\nabla_{\alpha{\dot{\alpha}}}\zeta_{\alpha(s){\dot{\alpha}}(s-1)}~{},$
(6.135b) $\displaystyle\delta_{\ell}g_{\alpha(s){\dot{\alpha}}(s)}$
$\displaystyle=\nabla_{\alpha{\dot{\alpha}}}\ell_{\alpha(s-1){\dot{\alpha}}(s-1)}~{}.$
(6.135c)

The properties (6.133) and (6.135) allows us to identify the component fields
$h_{\alpha(s+1){\dot{\alpha}}(s+1)}$, $\psi_{\alpha(s+1){\dot{\alpha}}(s)}$
and $g_{\alpha(s){\dot{\alpha}}(s)}$ as spin-$(s+1)$, spin-$(s+\frac{1}{2})$
and spin-$s$ conformal depth $t=1$ fields respectively (see section 4.4.1).

By making use of the above definitions, one can show that the only non-zero
component fields of the higher-spin super-Weyl tensor (6.118a) of
$H_{\alpha(s){\dot{\alpha}}(s)}$ are

$\displaystyle\mathfrak{W}_{\alpha(2s+1)}(H)|$
$\displaystyle=\mathcal{W}_{\alpha(2s+1)}(\psi)~{},\quad~{}\mbox{\boldmath$\nabla$}^{\beta}\mathfrak{W}_{\beta\alpha(2s)}(H)|=-2\frac{(2s+2)}{(2s+1)}\mathcal{W}_{\alpha(2s)}(g)~{},$
(6.136a)
$\displaystyle\mbox{\boldmath$\nabla$}_{\alpha}\mathfrak{W}_{\alpha(2s+1)}(H)|$
$\displaystyle={\rm
i}\mathcal{W}_{\alpha(2s+2)}(h)~{},\quad\mbox{\boldmath$\nabla$}^{2}\mathfrak{W}_{\alpha(2s+1)}(H)|=-4{\rm
i}\mathcal{W}_{\alpha(2s+1)}(\bar{\psi})~{}.$ (6.136b)

Using this data, one can readily reduce the action (6.130a) to components as
follows:

$\displaystyle\mathbb{S}_{\text{SCHS}}^{(s,s)}[H]$
$\displaystyle=(-1)^{s}\int{\rm
d}^{4}x\,e\,\Big{\\{}-\frac{1}{4}\mathcal{W}^{\alpha(2s+2)}(h)\mathcal{W}_{\alpha(2s+2)}(h)~{}$
$\displaystyle+{\rm
i}\mathcal{W}^{\alpha(2s+1)}(\bar{\psi})\mathcal{W}_{\alpha(2s+1)}(\psi)+\frac{(s+1)}{(2s+1)}\mathcal{W}^{\alpha(2s)}(g)\mathcal{W}_{\alpha(2s)}(g)\Big{\\}}+\text{c.c.}$
$\displaystyle=\frac{1}{2}S_{\text{CHS}}^{(s+1,s+1)}[h]+2S_{\text{CHS}}^{(s+1,s)}[\psi,\bar{\psi}]+\frac{(2s+2)}{(2s+1)}S_{\text{CHS}}^{(s,s)}[g]~{}.$
(6.137)

Here we have made use of the notation introduced in chapter 4.

From the decomposition (6.4.3) of the action
$\mathbb{S}_{\text{SCHS}}^{(s,s)}[H]$, it is clear that every gauge invariant
model of a superconformal higher-spin superfield gives rise to a multiplet of
gauge invariant models of conformal higher-spin fields. This fact may be
exploited to deduce non-trivial information regarding various properties of
the induced CHS actions, even if we do not know the explicit form of the SCHS
action. For example, in section 6.5.4 we are able to deduce (some of) the
spectrum of the conformal spin-3 model on a Bach-flat background simply by
supposing the existence of the SCHS action with $s=2$.

##### Case $m>n=0$

To begin this analysis, we restrict our attention to bosonic superspace
backgrounds (6.114) which may not necessarily be conformally-flat. Only, when
we reduce the action to components do we restrict to conformally-flat
backgrounds.

The gauge freedom (6.117c) of the prepotential $H_{\alpha(m)}$ allows one to
impose a Wess-Zumino gauge such that there are only four non-zero component
fields:

$\displaystyle h_{\alpha(m+1){\dot{\alpha}}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{[}\mbox{\boldmath$\nabla$}_{(\alpha_{1}},\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\big{]}H_{\alpha_{2}\dots\alpha_{m+1})}|~{},$
(6.138a) $\displaystyle\psi_{\alpha(m){\dot{\alpha}}}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{2}H_{\alpha(m)}|~{},$
(6.138b) $\displaystyle\varphi_{\alpha(m+1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\mbox{\boldmath$\nabla$}_{(\alpha_{1}}\bar{\mbox{\boldmath$\nabla$}}^{2}H_{\alpha_{2}\dots\alpha_{m+1})}|~{},$
(6.138c) $\displaystyle\rho_{\alpha(m)}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\mbox{\boldmath$\nabla$}^{\beta}\bar{\mbox{\boldmath$\nabla$}}^{2}\mbox{\boldmath$\nabla$}_{\beta}H_{\alpha(m)}|-\frac{{\rm
i}m}{4(m+2)}\nabla^{\beta{\dot{\beta}}}h_{\beta\alpha(m){\dot{\beta}}}~{}.$
(6.138d)

Here we have defined $\rho_{\alpha(m)}$ in such a way that it is annihilated
by $K_{a}$. Likewise, by a routine computation, one can show that each
component field is primary

$\displaystyle
K_{{\beta{\dot{\beta}}}}h_{\alpha(m+1){\dot{\alpha}}}=0~{},\quad
K_{{\beta{\dot{\beta}}}}\psi_{\alpha(m){\dot{\alpha}}}=0~{},\quad
K_{{\beta{\dot{\beta}}}}\varphi_{\alpha(m+1)}=0~{},\quad
K_{{\beta{\dot{\beta}}}}\rho_{\alpha(m)}=0~{}.$ (6.139)

Their Weyl weights may be easily deduced from (6.138). Underlying this gauge
fixing are the following constraints on the gauge parameters:

$\displaystyle\ell_{\alpha(m)}:=2{\rm i}\lambda_{\alpha(m)}|$ $\displaystyle=$
$\displaystyle-2{\rm
i}\mbox{\boldmath$\nabla$}_{(\alpha_{1}}\Lambda_{\alpha_{2}\dots\alpha_{m})}|~{},$
(6.140a) $\displaystyle\mu_{\alpha(m-1)}:=-\frac{{\rm
i}m}{m+1}\mbox{\boldmath$\nabla$}^{\beta}\lambda_{\beta\alpha(m-1)}|$
$\displaystyle=$ $\displaystyle\frac{{\rm
i}}{2}\mbox{\boldmath$\nabla$}^{2}\Lambda_{\alpha(m-1)}|~{},$ (6.140b)
$\displaystyle\mbox{\boldmath$\nabla$}_{(\alpha_{1}}\lambda_{\alpha_{2}\dots\alpha_{m+1})}|$
$\displaystyle=$ $\displaystyle 0~{},$ (6.140c)
$\displaystyle\mbox{\boldmath$\nabla$}^{2}\lambda_{\alpha(m)}|$
$\displaystyle=$ $\displaystyle 0~{},$ (6.140d)
$\displaystyle\big{[}\mbox{\boldmath$\nabla$}_{(\alpha_{1}},\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\big{]}\Lambda_{\alpha_{2}\dots\alpha_{m})}|$
$\displaystyle=$ $\displaystyle-2{\rm
i}\nabla_{(\alpha_{1}{\dot{\alpha}}}\Lambda_{\alpha_{2}\dots\alpha_{m})}|~{},$
(6.140e)
$\displaystyle\mbox{\boldmath$\nabla$}_{(\alpha_{1}}\bar{\mbox{\boldmath$\nabla$}}^{2}\Lambda_{\alpha_{2}\dots\alpha_{m})}|$
$\displaystyle=$ $\displaystyle-4{\rm
i}\nabla_{(\alpha_{1}{\dot{\alpha}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\Lambda_{\alpha_{2}\dots\alpha_{m})}|~{},$
(6.140f)
$\displaystyle\\{\mbox{\boldmath$\nabla$}^{2},\bar{\mbox{\boldmath$\nabla$}}^{2}\\}\Lambda_{\alpha(m-1)}|$
$\displaystyle=$ $\displaystyle-16\Box\Lambda_{\alpha(m-1)}|~{},$ (6.140g)
$\displaystyle\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{2}\Lambda_{\alpha(m-1)}|$
$\displaystyle=$
$\displaystyle\frac{2m}{m+1}\nabla^{\beta}{}_{{\dot{\alpha}}}\ell_{\beta\alpha(m-1)}~{}.$
(6.140h)

The residual transformations associated with this gauge are generated by the
fields $\ell_{\alpha(m)}$ and $\mu_{\alpha(m-1)}$. These transform
$h_{\alpha(m+1){\dot{\alpha}}}$ and $\psi_{\alpha(m){\dot{\alpha}}}$
independently of the background geometry

$\displaystyle\delta_{\ell}h_{\alpha(m+1){\dot{\alpha}}}=\nabla_{\alpha{\dot{\alpha}}}\ell_{\alpha(m)}~{},\qquad\delta_{\mu}\psi_{\alpha(m){\dot{\alpha}}}=\nabla_{\alpha{\dot{\alpha}}}\mu_{\alpha(m-1)}~{}.$
(6.141a) The two fields $\varphi_{\alpha(m+1)}$ and $\rho_{\alpha(m)}$ are
both conformal non-gauge fields (see appendix 4.C), which do not have their
own gauge freedom. However, the $\ell$-transformation also acts on
$\rho_{\alpha(m)}$ when the background Weyl tensor is non-vanishing
$\displaystyle\delta_{\ell}\rho_{\alpha(m)}={\rm
i}\frac{m(m-1)}{2(m+2)}W^{\beta(2)}{}_{\alpha(2)}\ell_{\alpha(m-2)\beta(2)}~{}.$
(6.141b)

From this transformation law, it follows that $\rho_{\alpha(m)}$ should play
an important role in ensuring gauge invariance of the model describing
$h_{\alpha(m+1){\dot{\alpha}}}$. Since (6.141b) is non-zero for $m\geq 2$, we
therefore expect that any gauge-invariant (non-supersymmetric) model
describing $h_{\alpha(m+1){\dot{\alpha}}}$ in a Bach-flat background should
couple to a lower-spin field.

Upon further restricting the geometry to be conformally flat, the action
(6.130) may be readily reduced to components. One finds

$\displaystyle(-1)^{m}\mathbb{S}^{(m,0)}_{\text{SCHS}}$ $\displaystyle=$
$\displaystyle-\frac{{\rm i}^{m+1}}{2}\int{\rm
d}^{4}x\,e\,\bigg{\\{}-\mathcal{W}^{\alpha(m+1)}(\psi)\mathcal{W}_{\alpha(m+1)}(\bar{\psi})-\frac{{\rm
i}}{2}\mathcal{W}^{\alpha(m+2)}(h)\mathcal{W}_{\alpha(m+2)}(\bar{h})$
$\displaystyle+\bar{\varphi}^{{\dot{\alpha}}(m+1)}\mathfrak{X}_{{\dot{\alpha}}(m+1)}(\varphi)+2{\rm
i}\frac{m+2}{m+1}\bar{\rho}^{{\dot{\alpha}}(m)}\mathfrak{X}_{{\dot{\alpha}}(m)}(\rho)\bigg{\\}}+\text{c.c.}$
$\displaystyle=$
$\displaystyle\frac{1}{2}S_{\text{CHS}}^{(m+1,1)}[h,\bar{h}]+S_{\text{CHS}}^{(m,1)}[\psi,\bar{\psi}]-S_{\text{NG}}^{(m+1,0)}[\varphi,\bar{\varphi}]+\frac{(m+2)}{(m+1)}S_{\text{NG}}^{(m,0)}[\rho,\bar{\rho}]~{},$

which has been presented in a manifestly gauge-invariant form and where we
have made use of the notation introduced in section 4.4.2 and appendix 4.C. We
would like to point out that upon fixing $m=2$ (and making the appropriate
rescalings) in (6.4.3), the relative coefficient between the pseudo-graviton
sector and the non-gauge sector (the $\rho_{\alpha(2)}$ field) does not agree
with that of (4.268a). This fact will play an important role in our analysis
of the superconformal pseudo-graviton multiplet in section 6.5.3.

This analysis also leads to non-trivial information concerning the
$H_{\alpha(3)}$ supermultiplet in a generic background. For instance, at the
component level, (6.141a) implies that $\psi_{\alpha(3){\dot{\alpha}}}$ is
identifiable with the conformal pseudo-graviton field and the remaining
component fields are inert under its gauge transformation. However, it was
shown in section 4.5.6 that $\psi_{\alpha(3){\dot{\alpha}}}$ must couple to
another field to ensure gauge invariance. Thus, $H_{\alpha(3)}$ must also
couple to a suitable supermultiplet containing this extra field. As we will
see in section 6.5.3, a lower-spin coupling is also required in the
superconformal pseudo-graviton multiplet. Therefore, it seems reasonable to
expect that similar couplings are necessary for all $H_{\alpha(m)}$, where
$m\geq 2$.

####  6.4.4 Generalised SCHS models

A specific feature of $H_{\alpha(m){\dot{\alpha}}(n)}$ with $m\geq 1$ and
$n\geq 1$ is the presence of only a single spinor derivative in its gauge
transformations (6.117a). In this section we generalise these SCHS multiplets
by increasing the number of spinor derivatives appearing in their gauge
transformations. The resulting prepotentials will be called generalised SCHS
multiplets.

A depth-$t$ SCHS multiplet $H^{(t)}_{\alpha(m){\dot{\alpha}}(n)}$, with $m\geq
1$ and $n\geq 1$, is defined modulo gauge transformations of the form

$\displaystyle\delta_{\Lambda}H^{(t)}_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=\big{[}\mbox{\boldmath$\nabla$}_{(\alpha_{1}},\bar{\mbox{\boldmath$\nabla$}}_{({\dot{\alpha}}_{1}}\big{]}\mbox{\boldmath$\nabla$}_{\alpha_{2}{\dot{\alpha}}_{2}}\cdots\mbox{\boldmath$\nabla$}_{\alpha_{t}{\dot{\alpha}}_{t}}\Lambda^{(t)}_{\alpha_{t+1}\dots\alpha_{m}){\dot{\alpha}}_{t+1}\dots\dot{\alpha}_{n})}$
$\displaystyle\equiv\big{[}\mbox{\boldmath$\nabla$}_{\alpha},\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\big{]}\big{(}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\alpha}}}\big{)}^{t-1}\Lambda^{(t)}_{\alpha(m-t){\dot{\alpha}}(n-t)}~{},$
(6.143)

where the gauge parameter $\Lambda^{(t)}_{\alpha(m-t){\dot{\alpha}}(n-t)}$ is
unconstrained and $1\leq t\leq\text{min}(m,n)$. If we require that both the
prepotential and its gauge parameter are primary,

$\displaystyle K_{B}H^{(t)}_{\alpha(m){\dot{\alpha}}(n)}=0~{},\qquad
K_{B}\Lambda^{(t)}_{\alpha(m-t){\dot{\alpha}}(n-t)}=0~{},$ (6.144a) then the
Weyl weight and $\mathsf{U}(1)_{R}$ charge carried by
$H^{(t)}_{\alpha(m){\dot{\alpha}}(n)}$ must take the values
$\displaystyle\mathbb{D}H^{(t)}_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=\big{(}t-\frac{1}{2}(m+n)\big{)}H^{(t)}_{\alpha(m){\dot{\alpha}}(n)}~{},$
(6.144b) $\displaystyle\mathbb{Y}H^{(t)}_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=\frac{1}{3}(m-n)H^{(t)}_{\alpha(m){\dot{\alpha}}(n)}~{}.$
(6.144c)

For $m=n=s$, this allows us to choose both
$H^{(t)}_{\alpha(s){\dot{\alpha}}(s)}$ and
$\Lambda^{(t)}_{\alpha(s-t){\dot{\alpha}}(s-t)}$ to be real,

$\displaystyle
H^{(t)}_{\alpha(s){\dot{\alpha}}(s)}=\bar{H}^{(t)}_{\alpha(s){\dot{\alpha}}(s)}~{},\qquad\Lambda^{(t)}_{\alpha(s-t){\dot{\alpha}}(s-t)}=\bar{\Lambda}^{(t)}_{\alpha(s-t){\dot{\alpha}}(s-t)}~{}.$
(6.145)

Unlike the non-supersymmetric case, minimal depth SCHS superfields (i.e. those
with $t=1$) do not correspond to the ordinary SCHS superfields described in
the previous section. Accordingly, in order to collectively discuss all types
of SCHS gauge multiplets, we will refer to ordinary SCHS multiplets
$H_{\alpha(m){\dot{\alpha}}(n)}$, defined modulo the gauge transformations
(6.117), as having depth $t=0$; $H_{\alpha(m){\dot{\alpha}}(n)}\equiv
H^{(0)}_{\alpha(m){\dot{\alpha}}(n)}$ .

In principle one can instead consider supermultiplets
$\hat{H}^{(t)}_{\alpha(m){\dot{\alpha}}(n)}$ with the gauge freedom

$\displaystyle\delta_{\hat{\Lambda}}\hat{H}^{(t)}_{\alpha(m){\dot{\alpha}}(n)}=\mbox{\boldmath$\nabla$}_{\alpha}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\big{(}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\alpha}}}\big{)}^{t-1}\hat{\Lambda}^{(t)}_{\alpha(m-t){\dot{\alpha}}(n-t)}~{}.$
(6.146)

In order for $\hat{H}^{(t)}_{\alpha(m){\dot{\alpha}}(n)}$ to be primary it
must have weight $\big{(}t-\frac{1}{2}(m+n)\big{)}$ and charge
$\frac{1}{3}\big{(}m-n+2t\big{)}$. In this case one may consistently define
$\hat{H}^{(t)}_{\alpha(m){\dot{\alpha}}(n)}$ to be longitudinal anti-linear,
$\mbox{\boldmath$\nabla$}_{\alpha}\hat{H}^{(t)}_{\alpha(m){\dot{\alpha}}(n)}=0$.
However, these supermultiplets possess several undesirable features. In
particular, it is not possible to impose a reality condition similar to
(6.145), nor does it seem possible to construct gauge and super-Weyl invariant
actions describing their dynamics. It is for this reason that we will focus
only on the supermultiplets $H^{(t)}_{\alpha(m){\dot{\alpha}}(n)}$.

From the prepotential $H^{(t)}_{\alpha(m){\dot{\alpha}}(n)}$ and its conjugate
one may construct the following higher-derivative descendants

$\displaystyle\mathfrak{W}_{\alpha(m+n-t+1){\dot{\alpha}}(t-1)}^{(t)}(H)$
$\displaystyle=\big{(}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\alpha}-\frac{\text{i}t}{n-t+1}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\beta}}}\big{)}\big{(}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\beta}}}\big{)}^{n-t}H^{(t)}_{\alpha(m){\dot{\alpha}}(t-1){\dot{\beta}}(n-t+1)}~{},$
(6.147a)
$\displaystyle\mathfrak{W}_{\alpha(m+n-t+1){\dot{\alpha}}(t-1)}^{(t)}(\bar{H})$
$\displaystyle=\big{(}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\alpha}-\frac{\text{i}t}{m-t+1}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\beta}}}\big{)}\big{(}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\beta}}}\big{)}^{m-t}\bar{H}^{(t)}_{\alpha(n){\dot{\alpha}}(t-1){\dot{\beta}}(m-t+1)}~{}.$
(6.147b)

They are the higher-depth analogues of the higher-spin super-Weyl tensors
(6.118). In particular, (6.147a) proves to be a primary superfield in a
generic background

$\displaystyle
K_{B}\mathfrak{W}_{\alpha(m+n-t+1){\dot{\alpha}}(t-1)}^{(t)}(H)=0~{},$
(6.148a) which carries the following super-Weyl weight and $\mathsf{U}(1)_{R}$
charge
$\displaystyle\mathbb{D}\mathfrak{W}_{\alpha(m+n-t+1){\dot{\alpha}}(t-1)}^{(t)}(H)$
$\displaystyle=\big{(}1-\frac{1}{2}(m-n)\big{)}\mathfrak{W}_{\alpha(m+n-t+1){\dot{\alpha}}(t-1)}^{(t)}(H)~{},$
(6.148b)
$\displaystyle\mathbb{Y}\mathfrak{W}_{\alpha(m+n-t+1){\dot{\alpha}}(t-1)}^{(t)}(H)$
$\displaystyle=\frac{1}{3}(m-n)\mathfrak{W}_{\alpha(m+n-t+1){\dot{\alpha}}(t-1)}^{(t)}(H)~{}.$
(6.148c)

The descendent (6.147b) possesses the same superconformal properties but with
$m$ interchanged with $n$. In addition, when restricted to conformally-flat
backgrounds,

$\displaystyle\mbox{\boldmath$W$}_{\alpha(3)}=0~{},$ (6.149)

one can show they are invariant under the higher-depth gauge transformations
(6.143),

$\displaystyle\delta_{\Lambda}\mathfrak{W}_{\alpha(m+n-t+1){\dot{\alpha}}(t-1)}^{(t)}(H)=0~{},\qquad\delta_{\Lambda}\mathfrak{W}_{\alpha(m+n-t+1){\dot{\alpha}}(t-1)}^{(t)}(\bar{H})=0~{}.$
(6.150)

Using the field strengths (6.147), one may construct the action functional

$\displaystyle\mathbb{S}_{\text{SCHS}}^{(m,n,t)}[H,\bar{H}]=\frac{1}{2}\text{i}^{m+n+2}\int\text{d}^{4|4}z\,E\,\mathfrak{W}^{\alpha(m+n-t+1){\dot{\alpha}}(t-1)}_{(t)}(H)\mathfrak{W}_{\alpha(m+n-t+1){\dot{\alpha}}(t-1)}^{(t)}(\bar{H})+\text{c.c.}~{},$
(6.151)

which is locally $\mathcal{N}=1$ superconformal (on any background) and gauge
invariant on conformally-flat backgrounds. The overall coefficient of
$\text{i}^{m+n+2}$ in (6.151) has been chosen because of the identity

$\displaystyle\text{i}^{m+n+1}\int\text{d}^{4|4}z\,E\,\mathfrak{W}^{\alpha(m+n-t+1){\dot{\alpha}}(t-1)}_{(t)}(H)\mathfrak{W}_{\alpha(m+n-t+1){\dot{\alpha}}(t-1)}^{(t)}(\bar{H})+\text{c.c.}~{}=~{}0~{},$
(6.152)

which holds on conformally-flat backgrounds up to an irrelevant total
derivative.

The action (6.151) may be recast into the alternative forms

$\displaystyle\mathbb{S}_{\text{SCHS}}^{(m,n,t)}[H,\bar{H}]$
$\displaystyle=\frac{1}{2}\text{i}^{m+n+2}\int\text{d}^{4|4}z\,E\,\bar{H}^{\alpha(n){\dot{\alpha}}(m)}_{(t)}\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}^{(t)}(H)+\text{c.c.}$
$\displaystyle=\frac{1}{2}\text{i}^{m+n+2}\int\text{d}^{4|4}z\,E\,\bar{H}^{\alpha(n){\dot{\alpha}}(m)}_{(t)}\widehat{\mathfrak{B}}_{\alpha(n){\dot{\alpha}}(m)}^{(t)}(H)+\text{c.c.}$
(6.153)

where we have made use of the definitions

$\displaystyle\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}^{(t)}(H)$
$\displaystyle=\phantom{..}\big{(}\mbox{\boldmath$\nabla$}^{\beta}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}-\frac{\text{i}t}{m-t+1}\mbox{\boldmath$\nabla$}_{{\dot{\alpha}}}{}^{\beta}\big{)}\big{(}\mbox{\boldmath$\nabla$}_{{\dot{\alpha}}}{}^{\beta}\big{)}^{m-t}\mathfrak{W}^{(t)}_{\beta(m-t+1)\alpha(n){\dot{\alpha}}(t-1)}(H)~{},$
(6.154a)
$\displaystyle\widehat{\mathfrak{B}}_{\alpha(n){\dot{\alpha}}(m)}^{(t)}(H)$
$\displaystyle=-\big{(}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\alpha}-\frac{\text{i}t}{n-t+1}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\beta}}}\big{)}\big{(}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\beta}}}\big{)}^{n-t}\overline{\mathfrak{W}}^{(t)}_{\alpha(t-1){\dot{\alpha}}(m){\dot{\beta}}(n-t+1)}(H)~{},$
(6.154b)

and where
$\overline{\mathfrak{W}}^{(t)}_{\alpha(t-1){\dot{\alpha}}(m+n-t+1)}(H)\equiv\big{(}\mathfrak{W}_{\alpha(m+n-t+1){\dot{\alpha}}(t-1)}^{(t)}(\bar{H})\big{)}^{*}$.

The descendent (6.154a) is primary in a generic background,

$\displaystyle K_{B}\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}^{(t)}(H)=0~{},$
(6.155a) and carries the following weight and charge
$\displaystyle\mathbb{D}\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}^{(t)}(H)$
$\displaystyle=\big{(}2-t+\frac{1}{2}(m+n)\big{)}\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}^{(t)}(H)~{},$
(6.155b)
$\displaystyle\mathbb{Y}\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}^{(t)}(H)$
$\displaystyle=\frac{1}{3}(m-n)\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}^{(t)}(H)~{}.$
(6.155c)

The other descendent (6.154b) carries the same superconformal attributes as
(6.154a). On any conformally-flat background, they possess the additional
properties:

  1. (i)

$\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)$ and
$\widehat{\mathfrak{B}}_{\alpha(n){\dot{\alpha}}(m)}(H)$ are equal to one
another

$\displaystyle\mathfrak{B}^{(t)}_{\alpha(n){\dot{\alpha}}(m)}(H)=\widehat{\mathfrak{B}}^{(t)}_{\alpha(n){\dot{\alpha}}(m)}(H)~{},$
(6.156)

which leads to the following reality relation

$\displaystyle\mathfrak{B}^{(t)}_{\alpha(m){\dot{\alpha}}(n)}(\bar{H})=\Big{(}\mathfrak{B}^{(t)}_{\alpha(n){\dot{\alpha}}(m)}(H)\Big{)}^{*}\equiv\overline{\mathfrak{B}}^{(t)}_{\alpha(m){\dot{\alpha}}(n)}(\bar{H})~{}.$
(6.157)

  2. (ii)

$\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}^{(t)}$ is ‘partially conserved’,

$\displaystyle 0$
$\displaystyle=\big{[}\mbox{\boldmath$\nabla$}^{\beta},\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\beta}}}\big{]}\big{(}\mbox{\boldmath$\nabla$}^{\beta{\dot{\beta}}}\big{)}^{t-1}\mathfrak{B}_{\beta(t)\alpha(m-t){\dot{\beta}}(t){\dot{\alpha}}(n-t)}^{(t)}(H)~{}.$
(6.158)

  3. (iii)

$\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}^{(t)}(H)$ is invariant under the
gauge transformations (6.143)

$\displaystyle\delta_{\Lambda}\mathfrak{B}^{(t)}_{\alpha(n){\dot{\alpha}}(m)}(H)$
$\displaystyle=0~{}.$ (6.159)

On account of these properties, both (6.154a) and (6.154b) may be interpreted
as the higher-depth analogues of the higher-spin super-Bach tensors (6.123a)
and (6.123b).

Finally, we note that for the supermultiplets with depth $t>1$, the presence
of vector derivatives in the gauge transformations (6.143) prevents one from
constructing a Wess-Zumino gauge. Consequently, all possible component fields
are present in this case, some of which are depth $t-1$, $t$ and $t+1$ CHS
gauge fields (there are also conformal non-gauge fields present). However in
the special case $t=1$, when the depth assumes its minimal value, this is no
longer an obstruction and the multiplet shortens. For an analysis of the
corresponding Wess-Zumino gauge with $m$ and $n$ arbitrary, we refer the
reader to [7], where the component reduction of
$\mathbb{S}_{\text{SCHS}}^{(m,n,t)}[H,\bar{H}]$ with $t=1$ was also given. In
section 6.5.2 we give an in-depth analysis of the component content for the
case $m=n=t=1$.

###  6.5 Superconformal higher-spin models on Bachgrounds

At this point we extend our considerations to superspace backgrounds with non-
vanishing super-Weyl tensor, $\mbox{\boldmath$W$}_{\alpha\beta\gamma}\neq 0$,
and focus our attention on generic superspace backgrounds which solve the
equation of motion for $\mathcal{N}=1$ conformal supergravity (6.18). Such
backgrounds are said to be Bach-flat since they have vanishing super-Bach
tensor,

$\displaystyle\mbox{\boldmath$B$}_{\alpha{\dot{\alpha}}}=0~{},$ (6.160)

where $\mbox{\boldmath$B$}_{\alpha{\dot{\alpha}}}$ is defined in eq. (6.111).
In this case it turns out that the SCHS actions (6.130) and (6.151), which we
now denote by $\mathbb{S}_{\text{skeleton}}^{(m,n,t)}[H,\bar{H}]$, are no
longer gauge invariant. The ensuing situation is very similar to the situation
which occurs in the non-supersymmetric case, described in section 4.5, to
which we refer the reader for a thorough discussion.

In short, to restore gauge invariance it is necessary, at the very least, to
add primary non-minimal (i.e. vanishing in the conformally-flat limit
$\mbox{\boldmath$W$}_{\alpha\beta\gamma}=0$) counter-terms. Such terms are
quadratic in the prepotential $H_{\alpha(m){\dot{\alpha}}(n)}^{(t)}$ and take
the form

$\displaystyle\mathbb{S}_{\text{NM}}^{(m,n,t)}[H,\bar{H}]=\frac{1}{2}\text{i}^{m+n}\int{\rm
d}^{4|4}z\,E\,H_{(t)}^{\alpha(m){\dot{\alpha}}(n)}\mathfrak{J}^{(t)}_{\alpha(m){\dot{\alpha}}(n)}(\bar{H})+\text{c.c.}~{}$
(6.161)

The tensor $\mathfrak{J}^{(t)}_{\alpha(m){\dot{\alpha}}(n)}(\bar{H})$ is some
non-minimal primary descendant of $\bar{H}^{(t)}_{\alpha(n){\dot{\alpha}}(m)}$
which shares the same superconformal properties as the super-Bach tensor
$\mathfrak{B}^{(t)}_{\alpha(m){\dot{\alpha}}(n)}(\bar{H})$, see (6.124) and
(6.154). The would-be gauge invariant SCHS action then takes the form

$\displaystyle\mathbb{S}_{\text{SCHS}}^{(m,n,t)}[H,\bar{H}]=\mathbb{S}_{\text{skeleton}}^{(m,n,t)}[H,\bar{H}]+\mathbb{S}_{\text{NM}}^{(m,n,t)}[H,\bar{H}]~{},\qquad\delta_{\Lambda}\mathbb{S}_{\text{SCHS}}^{(m,n,t)}[H,\bar{H}]\approx
0~{}.$ (6.162)

Throughout this section the symbol $\approx$ represents equality modulo terms
involving the super-Bach tensor. In section 6.5.1 and 6.5.2 below we will see
two explicit examples of where this is valid, namely, the:212121Recall that we
regard ordinary SCHS multiplets $H_{\alpha(m){\dot{\alpha}}(n)}\equiv
H^{(0)}_{\alpha(m){\dot{\alpha}}(n)}$ as having depth $t=0$. (i) conformal
gravitino supermultiplet with $m=n+1=t+1=1$; and (ii) maximal depth conformal
graviton supermultiplet with $m=n=t=1$.222222It should also be possible to
express the gauge invariant action for the conformal supergravity multiplet,
with $m=n=t+1=1$, in the form (6.162). This is because the corresponding
action must be attainable by linearising the action for $\mathcal{N}=1$
conformal supergravity about a Bach-flat background.

However, we saw in the non-supersymmetric case that simply including non-
minimal terms was not sufficient to ensure gauge invariance of the CHS models
with higher spin. This must also be the case for SCHS models since, for
bosonic Bach-flat superspace backgrounds satisfying (6.114) and (6.160), a
gauge invariant SCHS action decomposes into a multiplet of gauge invariant CHS
actions at the component level. Indeed, in section 6.5.3 we will see this
explicitly for the pseudo-graviton supermultiplet with $m-2=n=t=0$. Finally,
in section 6.5.4, by supposing the existence of a gauge invariant SCHS model
with $m=n=t+2=2$, we deduce non-trivial information regarding the ingredients
necessary to construct a gauge invariant CHS model for the conformal spin-3
field.

####  6.5.1 The conformal gravitino supermultiplet

The gauge-invariant model describing the superconformal gravitino multiplet
was constructed in the earlier work [102] within the framework of the Grimm-
Wess-Zumino geometry [290, 291].232323See [118] for a review of the Grimm-
Wess-Zumino geometry. Below we review this model, but in the setting of
conformal superspace.

The conformal gravitino supermultiplet is described by the prepotential
$H_{\alpha}$, which is a primary superfield with Weyl weight and
$\mathsf{U}(1)_{R}$ charge given by

$\displaystyle
K_{B}H_{\alpha}=0~{},\qquad\mathbb{D}H_{\alpha}=-\frac{1}{2}H_{\alpha}~{},\qquad\mathbb{Y}H_{\alpha}=\frac{1}{3}H_{\alpha}~{}.$
(6.163)

It corresponds to the case $m-1=n=t=0$, and possesses the gauge freedom

$\displaystyle\delta_{\Lambda,\lambda}H_{\alpha}=\mbox{\boldmath$\nabla$}_{\alpha}\Lambda+\lambda_{\alpha}~{},\qquad\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\lambda_{\alpha}=0~{},$
(6.164)

where both $\Lambda$ and $\lambda_{\alpha}$ are complex primary superfields,
and $\lambda_{\alpha}$ is covariantly chiral. Under $\Lambda$ and $\lambda$
gauge transformations, the skeleton action

$\displaystyle\mathbb{S}^{(1,0,0)}_{\text{skeleton}}[H,\bar{H}]=\frac{1}{2}{\rm
i}\int{\rm d}^{4}x{\rm d}^{2}\theta\,{\cal
E}\,\mathfrak{W}^{\alpha(2)}(H)\mathfrak{W}_{\alpha(2)}(\bar{H})+{\rm
c.c.}~{},$ (6.165)

which is composed of the super-Weyl tensor for $H_{\alpha}$ and
$\bar{H}_{{\dot{\alpha}}}$

$\displaystyle\mathfrak{W}_{\alpha(2)}(H)=-\frac{1}{4}\bar{\mbox{\boldmath$\nabla$}}^{2}\mbox{\boldmath$\nabla$}_{(\alpha_{1}}H_{\alpha_{2})}~{},\qquad\mathfrak{W}_{\alpha(2)}(\bar{H})=-\frac{1}{4}\bar{\mbox{\boldmath$\nabla$}}^{2}\mbox{\boldmath$\nabla$}_{(\alpha_{1}}{}^{{\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\alpha_{2})}\bar{H}_{{\dot{\beta}}}~{},$
(6.166)

has the following variations

$\displaystyle\delta_{\Lambda}\mathbb{S}^{(1,0,0)}_{\text{skeleton}}$
$\displaystyle=-\frac{1}{4}\int\text{d}^{4|4}z\,E\,\Lambda\bigg{\\{}\mbox{\boldmath$\bar{W}$}^{{\dot{\beta}}(3)}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\mbox{\boldmath$\nabla$}^{2}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\bar{H}_{{\dot{\beta}}}-2\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\mbox{\boldmath$\bar{W}$}^{{\dot{\beta}}(3)}\mbox{\boldmath$\nabla$}^{2}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\bar{H}_{{\dot{\beta}}}\bigg{\\}}+\text{c.c.}~{},$
(6.167a) $\displaystyle\delta_{\lambda}\mathbb{S}^{(1,0,0)}_{\text{skeleton}}$
$\displaystyle=-\text{i}\int{\rm d}^{4}x{\rm d}^{2}\theta\,{\cal
E}\,\lambda^{\alpha}\mbox{\boldmath$W$}_{\alpha}{}^{\beta(2)}\mathfrak{W}_{\beta(2)}(\bar{H})+\text{c.c.}$
(6.167b)

To compensate for the non-zero variation of the skeleton, we need to introduce
a non-minimal primary action of the form

$\displaystyle\mathbb{S}_{\text{NM},i}^{(1,0,0)}[H,\bar{H}]=\frac{1}{2}\text{i}\int\text{d}^{4|4}z\,E\,H^{\alpha}\mathfrak{J}^{(i)}_{\alpha}(\bar{H})+\text{c.c.}~{},$
(6.168)

where $\mathfrak{J}_{\alpha}^{(i)}(\bar{H})$ is a primary spinor superfield of
Weyl weight $5/2$ and $\mathsf{U}(1)_{R}$ charge $-1/3$ that depends
explicitly on the super-Weyl tensor. It may be shown that there are two
structures which satisfy these requirements, and they take the form

$\displaystyle\mathfrak{J}^{(1)}_{\alpha}(\bar{H})$
$\displaystyle=2\mbox{\boldmath$\bar{W}$}^{{\dot{\beta}}(3)}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\beta}}}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\bar{H}_{{\dot{\beta}}}-\text{i}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\mbox{\boldmath$\bar{W}$}^{{\dot{\beta}}(3)}\mbox{\boldmath$\nabla$}_{\alpha}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\bar{H}_{{\dot{\beta}}}+2\mbox{\boldmath$\nabla$}_{\alpha{\dot{\beta}}}\mbox{\boldmath$\bar{W}$}^{{\dot{\beta}}(3)}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\bar{H}_{{\dot{\beta}}}~{},$
(6.169a) $\displaystyle\mathfrak{J}^{(2)}_{\alpha}(\bar{H})$
$\displaystyle=2\mbox{\boldmath$W$}_{\alpha}{}^{\beta(2)}\mbox{\boldmath$\nabla$}_{\beta}{}^{{\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\beta}}}-{\rm
i}\mbox{\boldmath$\nabla$}_{\beta}\mbox{\boldmath$W$}_{\alpha}{}^{\beta(2)}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\beta}}}-2\mbox{\boldmath$\nabla$}_{\beta}{}^{{\dot{\beta}}}\mbox{\boldmath$W$}_{\alpha}{}^{\beta(2)}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\beta}}}~{}.$
(6.169b)

However, the corresponding actions (6.168) are not independent of one another.
Modulo a total derivative one may show that they are related via242424See
appendix 6.A.2 for a discussion on the technical issue of integration by parts
(IBP) in conformal superspace. There, an IBP rule is proposed and all
instances of IBP in this subsection are justified.

$\displaystyle\text{i}\int\text{d}^{4|4}z\,E\,H^{\alpha}\mathfrak{J}^{(1)}_{\alpha}(\bar{H})+\text{c.c.}$
$\displaystyle=\text{i}\int\text{d}^{4|4}z\,E\,\bar{H}^{{\dot{\alpha}}}\bar{\mathfrak{J}}^{(1)}_{{\dot{\alpha}}}(H)+\text{c.c.}$
$\displaystyle=\text{i}\int\text{d}^{4|4}z\,E\,H^{\alpha}\bigg{\\{}\mathfrak{J}^{(2)}_{\alpha}(\bar{H})+2{\rm
i}\mbox{\boldmath$B$}_{\alpha{\dot{\alpha}}}\bar{H}^{{\dot{\alpha}}}\bigg{\\}}+\text{c.c.}$
(6.170)

Therefore, it suffices to consider only the first functional.

Using the identity (6.170), it can be shown that a generic (infinitesimal)
variation of the action (6.168) (i.e. with respect to both $H_{\alpha}$ and
$\bar{H}_{{\dot{\alpha}}}$), with $i=1$, takes the form

$\displaystyle\delta\mathbb{S}_{\text{NM},1}^{(1,0,0)}$
$\displaystyle=\frac{{\rm i}}{2}\int\text{d}^{4|4}z\,E\,\delta
H^{\alpha}\bigg{\\{}\mathfrak{J}^{(1)}_{\alpha}(\bar{H})+\mathfrak{J}^{(2)}_{\alpha}(\bar{H})+2{\rm
i}\mbox{\boldmath$B$}_{\alpha{\dot{\alpha}}}\bar{H}^{{\dot{\alpha}}}\bigg{\\}}+\text{c.c.}$
(6.171)

In addition to the aforementioned properties, the superfields
$\mathfrak{J}^{(i)}_{\alpha}(\bar{H})$ may be shown to satisfy the following
useful identities

$\displaystyle\mbox{\boldmath$\nabla$}^{\alpha}\mathfrak{J}^{(1)}_{\alpha}(\bar{H})$
$\displaystyle=\frac{1}{2}\mbox{\boldmath$\bar{W}$}^{{\dot{\beta}}(3)}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\mbox{\boldmath$\nabla$}^{2}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\bar{H}_{{\dot{\beta}}}-\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\mbox{\boldmath$\bar{W}$}^{{\dot{\beta}}(3)}\mbox{\boldmath$\nabla$}^{2}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\bar{H}_{{\dot{\beta}}}~{},\qquad$
(6.172a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\alpha}\mathfrak{J}^{(2)}_{\alpha}(\bar{H})$
$\displaystyle=-2{\rm
i}\mbox{\boldmath$B$}_{\alpha{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{\alpha}\bar{H}^{{\dot{\alpha}}}~{},$
(6.172b)
$\displaystyle\bar{\mbox{\boldmath$\nabla$}}^{2}\mathfrak{J}^{(1)}_{\alpha}(\bar{H})$
$\displaystyle=\phantom{-}2\text{i}\bar{\mbox{\boldmath$\nabla$}}^{2}\big{(}\mbox{\boldmath$B$}_{\alpha{\dot{\alpha}}}\bar{H}^{{\dot{\alpha}}}\big{)}~{},$
(6.172c)
$\displaystyle\bar{\mbox{\boldmath$\nabla$}}^{2}\mathfrak{J}^{(2)}_{\alpha}(\bar{H})$
$\displaystyle=-8\mbox{\boldmath$W$}_{\alpha}{}^{\beta(2)}\mathfrak{W}_{\beta(2)}(\bar{H})~{}.$
(6.172d)

The relations (6.172a) and (6.172b) are useful when computing the
$\Lambda$-gauge variation of (6.168). For technical reasons, the
$\lambda$-gauge variation is best done in the chiral subspace (see appendix
6.A.3 for a discussion on this), for which (6.172c) and (6.172d) are crucial.

By virtue of (6.171) and (6.172), one can see that the gauge variation of the
functional $\mathbb{S}_{\text{NM}}^{(1,0,0)}$ is given by

$\displaystyle\delta_{\Lambda,\lambda}\mathbb{S}_{\text{NM}}^{(1,0,0)}=-\delta_{\Lambda,\lambda}\mathbb{S}_{\text{Skeleton}}^{(1,0,0)}+\bigg{(}\frac{1}{2}\int{\rm
d}^{4}x{\rm d}^{2}\theta\,{\cal
E}\,\bar{\mbox{\boldmath$\nabla$}}^{2}\big{(}\lambda^{\alpha}\mbox{\boldmath$B$}_{\alpha{\dot{\alpha}}}\bar{H}^{{\dot{\alpha}}}\big{)}+\text{c.c.}\bigg{)}~{}.$
(6.173)

It follows that the action

$\displaystyle\mathbb{S}_{\text{SCHS}}^{(1,0,0)}[H,\bar{H}]$
$\displaystyle=\mathbb{S}^{(1,0,0)}_{\text{Skeleton}}[H,\bar{H}]+\mathbb{S}^{(1,0,0)}_{\text{NM},1}[H,\bar{H}]~{}$
(6.174a) $\displaystyle=\frac{{\rm i}}{2}\int{\rm d}^{4}x{\rm
d}^{2}\theta\,{\cal
E}\,\mathfrak{W}^{\alpha(2)}(H)\mathfrak{W}_{\alpha(2)}(\bar{H})+{\rm
i}\int\text{d}^{4|4}z\,E\,H^{\alpha}\bigg{\\{}\mbox{\boldmath$\bar{W}$}^{{\dot{\beta}}(3)}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\beta}}}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\bar{H}_{{\dot{\beta}}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}-\frac{\text{i}}{2}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\mbox{\boldmath$\bar{W}$}^{{\dot{\beta}}(3)}\mbox{\boldmath$\nabla$}_{\alpha}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\bar{H}_{{\dot{\beta}}}+\mbox{\boldmath$\nabla$}_{\alpha{\dot{\beta}}}\mbox{\boldmath$\bar{W}$}^{{\dot{\beta}}(3)}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\bar{H}_{{\dot{\beta}}}\bigg{\\}}+\text{c.c.}$
(6.174b)

has gauge variation that is strictly proportional to the super-Bach tensor,

$\displaystyle\delta_{\Lambda,\lambda}\mathbb{S}_{\text{SCHS}}^{(1,0,0)}=-2\int\text{d}^{4|4}z\,E\,\lambda^{\alpha}\mbox{\boldmath$B$}_{\alpha{\dot{\alpha}}}\bar{H}^{{\dot{\alpha}}}+\text{c.c.}~{},$
(6.175)

and is hence gauge invariant when restricted to a super-Bach-flat background

$\displaystyle\delta_{\Lambda,\lambda}\mathbb{S}_{\text{SCHS}}^{(1,0,0)}\approx
0~{}.$ (6.176)

One may check that upon using (6.170) and degauging to Grimm-Wess-Zumino
superspace, the above action coincides with the one given in [102] modulo a
term proportional to the super-Bach tensor.

####  6.5.2 The maximal-depth conformal graviton supermultiplet

The maximal-depth conformal graviton supermultiplet is described by the
weightless real primary superfield252525In this section we drop all labels
referring to the depth since we deal only with $t=1$.
$H_{\alpha{\dot{\alpha}}}$ which is inert under $\mathsf{U}(1)_{R}$
transformations,

$\displaystyle
K_{B}H_{\alpha{\dot{\alpha}}}=0~{},\qquad\mathbb{D}H_{\alpha{\dot{\alpha}}}=0~{},\qquad\mathbb{Y}H_{\alpha{\dot{\alpha}}}=0~{},\qquad
H_{\alpha{\dot{\alpha}}}=\bar{H}_{\alpha{\dot{\alpha}}}~{},$ (6.177a) and
which is defined modulo the depth-1 gauge transformations
$\displaystyle\delta_{\Lambda}H_{\alpha{\dot{\alpha}}}=\big{[}\mbox{\boldmath$\nabla$}_{\alpha},\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\big{]}\Lambda~{}.$
(6.177b)

Here the real gauge parameter $\Lambda$ is primary and unconstrained. This
corresponds to the generalised supermultiplet of section 6.4.4 with $m=n=t=1$
and is called the maximal-depth conformal graviton supermultiplet because, as
will be shown shortly, it contains the maximal-depth conformal graviton at the
component level.

##### Gauge invariant action in Bach-flat background

The linearised super-Weyl tensor associated with $H_{\alpha{\dot{\alpha}}}$,
and its corresponding gauge variation under (6.177b), is given by

$\displaystyle\mathfrak{W}_{\alpha(2)}(H)$
$\displaystyle=\Big{(}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\beta}}}\mbox{\boldmath$\nabla$}_{(\alpha_{1}}-\text{i}\mbox{\boldmath$\nabla$}_{(\alpha_{1}}{}^{{\dot{\beta}}}\Big{)}H_{\alpha_{2}){\dot{\beta}}}~{},$
(6.178a) $\displaystyle\delta_{\Lambda}\mathfrak{W}_{\alpha(2)}(H)$
$\displaystyle=3\Big{(}2\mbox{\boldmath$W$}_{\alpha(2)}{}^{\beta}\mbox{\boldmath$\nabla$}_{\beta}\Lambda+\mbox{\boldmath$\nabla$}_{\beta}\mbox{\boldmath$W$}_{\alpha(2)}{}^{\beta}\Lambda\Big{)}~{}.$
(6.178b)

It follows that the skeleton action

$\displaystyle\mathbb{S}^{(1,1,1)}_{\text{skeleton}}[H]=\frac{1}{3}\int\text{d}^{4|4}z\,E\,\mathfrak{W}^{\alpha(2)}(H)\mathfrak{W}_{\alpha(2)}(H)+\text{c.c.}$
(6.179)

(here we have chosen a different overall normalisation as compared to (6.151))
has gauge variation proportional to the background super-Weyl tensor

$\displaystyle\delta_{\Lambda}S^{(1,1,1)}_{\text{skeleton}}[H]=2\int\text{d}^{4|4}z\,E\,\Lambda\bigg{\\{}3\mbox{\boldmath$\nabla$}_{\gamma}\mbox{\boldmath$W$}^{\gamma\alpha(2)}\mathfrak{W}_{\alpha(2)}(H)-2\mbox{\boldmath$W$}^{\gamma\alpha(2)}\mbox{\boldmath$\nabla$}_{\gamma}\mathfrak{W}_{\alpha(2)}(H)\bigg{\\}}+\text{c.c.}$
(6.180)

It is possible to restore gauge invariance to the skeleton by supplementing it
with the non-minimal primary action

$\displaystyle\mathbb{S}^{(1,1,1)}_{\text{NM}}[H]=\int\text{d}^{4|4}z\,E\,H^{\alpha{\dot{\alpha}}}\mbox{\boldmath$W$}_{\alpha}{}^{\beta(2)}\mbox{\boldmath$\nabla$}_{\beta}H_{\beta{\dot{\alpha}}}+\text{c.c.}$
(6.181)

The action which is gauge invariant in a Bach-flat background may then be
shown to be

$\displaystyle\mathbb{S}^{(1,1,1)}_{\text{SCHS}}[H]=\mathbb{S}^{(1,1,1)}_{\text{skeleton}}[H]-3\mathbb{S}^{(1,1,1)}_{\text{NM}}[H]~{},\qquad\delta_{\Lambda}\mathbb{S}^{(1,1,1)}_{\text{SCHS}}[H]\approx
0~{}.$ (6.182)

##### The component action

Let us study the component content of the model (6.182) when restricted to a
bosonic Bach-flat superspace background, i.e. one satisfying (6.114) and
(6.160). By making use of the gauge freedom (6.177b), one may choose a Wess-
Zumino gauge such that the only non-vanishing component fields contained
within $H_{\alpha{\dot{\alpha}}}$ are

$\displaystyle\psi_{\alpha(2){\dot{\alpha}}}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\nabla$}_{(\alpha_{1}}H_{\alpha_{2}){\dot{\alpha}}}|~{},$
(6.183a) $\displaystyle A_{{\alpha{\dot{\alpha}}}}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\mbox{\boldmath$\nabla$}^{2}H_{{\alpha{\dot{\alpha}}}}|~{},$
(6.183b) $\displaystyle h_{\alpha(2){\dot{\alpha}}(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{[}\mbox{\boldmath$\nabla$}_{(\alpha_{1}},\bar{\mbox{\boldmath$\nabla$}}_{({\dot{\alpha}}_{1}}\big{]}H_{\alpha_{2}){\dot{\alpha}}_{2})}|~{},$
(6.183c) $\displaystyle\chi_{\alpha(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{[}\mbox{\boldmath$\nabla$}_{(\alpha_{1}},\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\big{]}H_{\alpha_{2}){\dot{\alpha}}}|~{},$
(6.183d) $\displaystyle\varphi_{\alpha(2){\dot{\alpha}}}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\mbox{\boldmath$\nabla$}_{(\alpha_{1}}\bar{\mbox{\boldmath$\nabla$}}^{2}H_{\alpha_{2}){\dot{\alpha}}}|+\frac{3{\rm
i}}{2}\nabla_{(\alpha_{1}}{}^{{\dot{\beta}}}\bar{\psi}_{\alpha_{2}){\dot{\alpha}}{\dot{\beta}}}~{},$
(6.183e) $\displaystyle\phi_{\alpha}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{2}H_{{\alpha{\dot{\alpha}}}}|-\frac{{\rm
i}}{2}\nabla^{{\beta{\dot{\beta}}}}\psi_{\alpha\beta{\dot{\beta}}}~{},$
(6.183f) $\displaystyle V_{{\alpha{\dot{\alpha}}}}$ $\displaystyle=$
$\displaystyle\frac{1}{32}\\{\mbox{\boldmath$\nabla$}^{2},\bar{\mbox{\boldmath$\nabla$}}^{2}\\}H_{{\alpha{\dot{\alpha}}}}|+\frac{{\rm
i}}{4}\nabla^{\beta}{}_{{\dot{\alpha}}}\chi_{\alpha\beta}-\frac{{\rm
i}}{4}\nabla_{\alpha}{}^{{\dot{\beta}}}\bar{\chi}_{{\dot{\alpha}}{\dot{\beta}}}~{}.$
(6.183g)

Both $h_{\alpha(2){\dot{\alpha}}(2)}$ and $V_{\alpha{\dot{\alpha}}}$ are real
whilst all other component fields are complex. All the fields are primary, and
their Weyl weights may be deduced from (6.177). Associated with (6.183) are
the following gauge fixing conditions

$\displaystyle\big{[}\mbox{\boldmath$\nabla$}_{\alpha},\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\big{]}\Lambda|=0~{},\qquad\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{2}\Lambda|=3{\rm
i}\mbox{\boldmath$\nabla$}_{{\alpha{\dot{\alpha}}}}\mbox{\boldmath$\nabla$}^{\alpha}\Lambda|~{},\qquad\\{\mbox{\boldmath$\nabla$}^{2},\bar{\mbox{\boldmath$\nabla$}}^{2}\\}\Lambda|=-4\Box\Lambda|~{}.$
(6.184)

The residual gauge transformations are generated by the fields

$\displaystyle\lambda:=-\frac{1}{4}\mbox{\boldmath$\nabla$}^{2}\Lambda|~{},\qquad\epsilon_{\alpha}:=2{\rm
i}\mbox{\boldmath$\nabla$}_{\alpha}\Lambda|~{},\qquad\xi:=-2\Lambda|~{}.$
(6.185)

Consequently, the fields
$A_{\alpha{\dot{\alpha}}},\psi_{\alpha(2){\dot{\alpha}}}$ and
$h_{\alpha(2){\dot{\alpha}}(2)}$ have the gauge transformation laws

$\displaystyle\delta_{\lambda}A_{\alpha{\dot{\alpha}}}$ $\displaystyle=$
$\displaystyle\nabla_{{\alpha{\dot{\alpha}}}}\lambda~{},$ (6.186a)
$\displaystyle\delta_{\epsilon}\psi_{\alpha(2){\dot{\alpha}}}$
$\displaystyle=$
$\displaystyle\nabla_{\alpha{\dot{\alpha}}}\epsilon_{\alpha}~{},$ (6.186b)
$\displaystyle\delta_{\xi}h_{\alpha(2){\dot{\alpha}}(2)}$ $\displaystyle=$
$\displaystyle\nabla_{\alpha{\dot{\alpha}}}\nabla_{\alpha{\dot{\alpha}}}\xi~{}.$
(6.186c)

On the other hand, the three fields
$\chi_{\alpha(2)},\varphi_{\alpha(2){\dot{\alpha}}}$ and $\phi_{\alpha}$ are
type I conformal non-gauge fields (see appendix 4.C) and do not have any gauge
freedom. The field $V_{\alpha{\dot{\alpha}}}$ is auxiliary.

Using the above definitions, the action (6.182) may be readily reduced to
components:

$\displaystyle\mathbb{S}^{(1,1,1)}_{\text{SCHS}}[H]$ $\displaystyle=$
$\displaystyle-\int{\rm
d}^{4}x\,e\,\bigg{\\{}\frac{3}{4}\bigg{[}\mathcal{W}^{\alpha(3){\dot{\alpha}}}(h)\mathcal{W}_{\alpha(3){\dot{\alpha}}}(h)-h^{\alpha(2){\dot{\alpha}}(2)}W_{\alpha(2)}{}^{\beta(2)}h_{\beta(2){\dot{\alpha}}(2)}\bigg{]}$
$\displaystyle+$ $\displaystyle\frac{3{\rm
i}}{2}\bigg{[}\mathcal{W}^{\alpha(3)}(\psi)\mathcal{W}_{\alpha(3)}(\bar{\psi})-\psi^{\alpha(2){\dot{\alpha}}}\Big{(}W_{\alpha(2)}{}^{\beta(2)}\nabla_{\beta}{}^{{\dot{\beta}}}\bar{\psi}_{\beta{\dot{\alpha}}{\dot{\beta}}}-\nabla_{\beta}{}^{{\dot{\beta}}}W_{\alpha(2)}{}^{\beta(2)}\bar{\psi}_{\beta{\dot{\alpha}}{\dot{\beta}}}\Big{)}\bigg{]}$
$\displaystyle+$ $\displaystyle
2\mathcal{W}^{\alpha(2)}(A)\mathcal{W}_{\alpha(2)}(\bar{A})-2{\rm
i}\bar{\varphi}^{\alpha{\dot{\alpha}}(2)}\mathfrak{X}_{\alpha{\dot{\alpha}}(2)}(\varphi)+\frac{1}{4}\bar{\chi}^{{\dot{\alpha}}(2)}\mathfrak{X}_{{\dot{\alpha}}(2)}(\chi)-\frac{{\rm
i}}{2}\bar{\phi}^{{\dot{\alpha}}}\mathfrak{X}_{\dot{\alpha}}(\phi)$
$\displaystyle-$ $\displaystyle
2V^{{\alpha{\dot{\alpha}}}}V_{{\alpha{\dot{\alpha}}}}\bigg{\\}}+\text{c.c.}$
$\displaystyle\equiv$
$\displaystyle-\frac{3}{2}S^{(2,2,2)}_{\text{CHS}}[h]+3S^{(2,1,1)}_{\text{CHS}}[\psi,\bar{\psi}]+4S^{(1,1,1)}_{\text{CHS}}[A,\bar{A}]$
$\displaystyle-4S^{(2,1)}_{\text{NG}}[\varphi,\bar{\varphi}]+\frac{1}{2}S^{(2,0)}_{\text{NG}}[\chi,\bar{\chi}]+S^{(1,0)}_{\text{NG}}[\phi,\bar{\phi}]-4S_{\text{NG}}^{(2,2)}[V]~{}.$
(6.188)

This action has been expressed in a manifestly conformal and gauge invariant
(in a Bach-flat background) form by using the field strengths associated with
each field,

$\displaystyle\mathcal{W}_{\alpha(3)}(\psi)$
$\displaystyle=\nabla_{(\alpha_{1}}{}^{{\dot{\beta}}}\psi_{\alpha_{2}\alpha_{3}){\dot{\beta}}}~{},\qquad\qquad\quad\mathcal{W}_{\alpha(3){\dot{\alpha}}}(h)=\nabla_{(\alpha_{1}}{}^{{\dot{\beta}}}h_{\alpha_{2}\alpha_{3}){\dot{\alpha}}{\dot{\beta}}}~{},$
(6.189a) $\displaystyle\mathcal{W}_{\alpha(3)}(\bar{\psi})$
$\displaystyle=\nabla_{(\alpha_{1}}{}^{{\dot{\beta}}_{1}}\nabla_{\alpha_{2}}{}^{{\dot{\beta}}_{2}}\bar{\psi}_{\alpha_{3}){\dot{\beta}}(2)}~{},\qquad~{}~{}\mathfrak{X}_{{\dot{\alpha}}(2)}(\chi)=\nabla_{({\dot{\alpha}}_{1}}{}^{\beta_{1}}\nabla_{{\dot{\alpha}}_{2})}{}^{\beta_{2}}\chi_{\beta(2)}~{},$
(6.189b) $\displaystyle\mathcal{W}_{\alpha(2)}(A)$
$\displaystyle=\nabla_{(\alpha_{1}}{}^{{\dot{\beta}}}A_{\alpha_{2}){\dot{\beta}}}~{},\qquad\qquad\quad\phantom{..}~{}\mathfrak{X}_{\alpha{\dot{\alpha}}(2)}(\varphi)=\nabla_{({\dot{\alpha}}_{1}}{}^{\beta}\varphi_{\beta\alpha{\dot{\alpha}}_{2})}~{},$
(6.189c) $\displaystyle\mathcal{W}_{\alpha(2)}(\bar{A})$
$\displaystyle=\nabla_{(\alpha_{1}}{}^{{\dot{\beta}}}\bar{A}_{\alpha_{2}){\dot{\beta}}}~{},\qquad\qquad\qquad~{}\phantom{..}~{}\mathfrak{X}_{\dot{\alpha}}(\phi)=\nabla_{{\dot{\alpha}}}{}^{\beta}\phi_{\beta}~{},$
(6.189d)

along with the non-minimal counter terms necessary for gauge invariance.

The analysis above indicates that the component action decomposes into a
(diagonal) sum of gauge invariant CHS actions – denoted
$S^{(2,2,2)}_{\text{CHS}}[h],S^{(2,1,1)}_{\text{CHS}}[\psi,\bar{\psi}]$ and
$S^{(1,1,1)}_{\text{CHS}}[A,\bar{A}]$ – describing a maximal-depth conformal
graviton $h_{\alpha(2){\dot{\alpha}}(2)}$ (4.204), a conformal gravitino
$\psi_{\alpha(2){\dot{\alpha}}}$ (4.174) and a complex Maxwell field
$A_{\alpha{\dot{\alpha}}}$ respectively. In addition, there are also several
non-gauge fields $\chi_{\alpha(2)},\varphi_{\alpha(2){\dot{\alpha}}}$ and
$\phi_{\alpha}$ present, the latter of which describes a massless Weyl spinor.
Their corresponding actions may be found in eq. (4.322). The vector field
$V_{\alpha{\dot{\alpha}}}$ is auxiliary since it enters the action with no
derivatives.

The overall sign of the action (6.151) has been chosen so that the Maxwell
action

$\displaystyle
S^{(1,1,1)}_{\text{CHS}}[A,\bar{A}]=-\frac{1}{2}\int\text{d}^{4}x\,e\,F^{ab}(A)F_{ab}(\bar{A})~{},\qquad
F_{ab}(A):=\mathcal{D}_{a}A_{b}-\mathcal{D}_{b}A_{a}~{},$ (6.190)

in (6.5.2) comes with canonical sign. Finally, we would like to point out that
the action (6.182) may be recast into the form

$\displaystyle\mathbb{S}^{(1,1,1)}_{\text{SCHS}}[H]=\frac{3}{2}\int\text{d}^{4|4}zH^{\alpha{\dot{\alpha}}}$
$\displaystyle\bigg{\\{}\frac{3}{2}D^{\beta}\bar{D}^{2}D_{\beta}H_{\alpha{\dot{\alpha}}}-\frac{1}{2}\big{[}D_{\alpha},\bar{D}_{{\dot{\alpha}}}\big{]}\big{[}D_{\beta},\bar{D}_{{\dot{\beta}}}\big{]}H^{\beta{\dot{\beta}}}$
$\displaystyle~{}-4\partial_{\alpha{\dot{\alpha}}}\partial^{\beta{\dot{\beta}}}H_{\beta{\dot{\beta}}}-\frac{1}{4}\big{\\{}D^{2},\bar{D}^{2}\big{\\}}H_{\alpha{\dot{\alpha}}}\bigg{\\}}$
(6.191)

in Minkowski superspace and defines a superconformal field theory. The action
is invariant under the gauge transformation
$\delta_{\Lambda}H_{\alpha{\dot{\alpha}}}=\big{[}D_{\alpha},\bar{D}_{{\dot{\alpha}}}\big{]}\Lambda$.
There exists a model for linearised supergravity constructed in [320] with a
larger gauge freedom

$\displaystyle\delta
H_{\alpha{\dot{\alpha}}}=\big{[}D_{\alpha},\bar{D}_{{\dot{\alpha}}}\big{]}\Lambda+\lambda_{\alpha{\dot{\alpha}}}+\bar{\lambda}_{\alpha{\dot{\alpha}}}~{},\qquad\bar{D}_{\dot{\beta}}\lambda_{\alpha{\dot{\alpha}}}=0~{},$
(6.192)

than that which the action (6.191) possesses. However, the $\lambda$ gauge
symmetry proves to be incompatible with the superconformal invariance.

####  6.5.3 The conformal pseudo-graviton supermultiplet

Let us now examine a curved superspace extension of the theory described by
the prepotential $H_{\alpha(2)}$, with $m-2=n=t=0$. We recall that when
restricted to a bosonic background, the gauge freedom (6.117c) allows us to
adopt a Wess-Zumino gauge where the component of $H_{\alpha(2)}$ defined by
(6.138a) is identifiable with the conformal pseudo-graviton field examined in
section 4.5.6. In addition, the prepotential $H_{\alpha(2)}$ has the same
dimension and total number of spinor indices as the prepotential
$H_{\alpha{\dot{\alpha}}}$ of conformal supergravity [315, 297]. Hence we will
refer to this model as the superconformal pseudo-graviton multiplet.

Conformal superspace is a powerful formalism for the construction of
superconformal invariants. However, its use also brings in certain technical
subtleties associated with integration by parts and with the transition
between integrals over the full superspace and the chiral
subspace.262626Unlike conformal superspace, such subtleties do not occur
within the $\mathsf{U}(1)$ superspace setting. However, $\mathsf{U}(1)$
superspace is much less powerful for constructing higher-derivative
superconformal invariants. It is fair to say that conformal superspace and
$\mathsf{U}(1)$ superspace are complementary. Without developing efficient
rules to perform these operations (and such rules are absent at the moment)
conformal superspace becomes impractical to do various field theoretic
calculations for higher-derivative models such as the ones under
consideration. In appendix 6.A we give a detailed discussion of these issues
and in some cases propose such a rule.

Fortunately, for the current model it can be shown that any contributions
arising from these subtleties are at least second order in the super-Weyl
tensor. In fact, it turns out that there is a shortcut which will allow us to
deduce the most important properties of the full gauge-invariant model, such
as the presence of a lower-spin superfield. It is for this reason that parts
of our subsequent analysis will be restricted to first order in
$\mbox{\boldmath$W$}_{\alpha(3)}$.

The superfield $H_{\alpha(2)}$ possesses the superconformal properties

$\displaystyle
K_{B}H_{\alpha(2)}=0~{},\qquad\mathbb{D}H_{\alpha(2)}=-H_{\alpha(2)}~{},\qquad\mathbb{Y}H_{\alpha(2)}=\frac{2}{3}H_{\alpha(2)}~{},$
(6.193)

and is defined modulo the gauge transformations

$\displaystyle\delta_{\Lambda,\lambda}H_{\alpha(2)}=\mbox{\boldmath$\nabla$}_{(\alpha_{1}}\Lambda_{\alpha_{2})}+\lambda_{\alpha(2)}~{},\qquad\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\lambda_{\alpha(2)}=0~{}.$
(6.194)

Just as in the previous cases, we begin with the skeleton action

$\displaystyle\mathbb{S}^{(2,0,0)}_{\text{skeleton}}[H,\bar{H}]=-\frac{1}{2}\int{\rm
d}^{4}x{\rm d}^{2}\theta\,{\cal
E}\,\mathfrak{W}^{\alpha(3)}(H)\mathfrak{W}_{\alpha(3)}(\bar{H})+{\rm c.c.}$
(6.195)

It has the following variations under the gauge transformations (6.194)

$\displaystyle\delta_{\Lambda}\mathbb{S}_{\text{skeleton}}^{(2,0,0)}$
$\displaystyle=\frac{1}{2}\int\text{d}^{4|4}z\,E\,\Lambda^{\alpha}\bigg{\\{}$
$\displaystyle~{}~{}~{}-{\rm
i}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\alpha}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}_{{\dot{\beta}}}{}^{{\dot{\alpha}}(2)}\mbox{\boldmath$\nabla$}^{2}\bar{\mbox{\boldmath$\nabla$}}_{({\dot{\alpha}}_{1}}\bar{H}_{{\dot{\alpha}}_{2}{\dot{\alpha}}_{3})}-{\rm
i}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\beta}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\bar{\mbox{\boldmath$W$}}_{{\dot{\beta}}}{}^{{\dot{\alpha}}(2)}\mbox{\boldmath$\nabla$}^{2}\bar{\mbox{\boldmath$\nabla$}}_{({\dot{\alpha}}_{1}}\bar{H}_{{\dot{\alpha}}_{2}{\dot{\alpha}}_{3})}$
$\displaystyle~{}~{}~{}-\frac{{\rm
i}}{2}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\alpha}}}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}(2){\dot{\beta}}}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\mbox{\boldmath$\nabla$}^{2}\bar{\mbox{\boldmath$\nabla$}}_{({\dot{\alpha}}_{1}}\bar{H}_{{\dot{\alpha}}_{2}{\dot{\alpha}}_{3})}+\frac{7{\rm
i}}{12}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}_{{\dot{\beta}}}{}^{{\dot{\alpha}}(2)}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\gamma}}}\mbox{\boldmath$\nabla$}^{2}\bar{\mbox{\boldmath$\nabla$}}_{\dot{\gamma}}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}+\frac{7{\rm
i}}{6}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}_{{\dot{\beta}}}{}^{{\dot{\alpha}}(2)}\bar{\mbox{\boldmath$\nabla$}}^{\dot{\gamma}}\bar{\mbox{\boldmath$\nabla$}}_{({\dot{\alpha}}_{1}}\bar{H}_{{\dot{\alpha}}_{2}){\dot{\gamma}}}-\frac{9{\rm
i}}{4}\mbox{\boldmath$\nabla$}^{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}_{{\dot{\beta}}}{}^{{\dot{\alpha}}(2)}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{2}\bar{\mbox{\boldmath$\nabla$}}_{({\dot{\alpha}}_{1}}\bar{H}_{{\dot{\alpha}}_{2}{\dot{\alpha}}_{3})}$
$\displaystyle~{}~{}~{}+\frac{{\rm
i}}{2}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\beta}}}\mbox{\boldmath$W$}^{{\dot{\alpha}}(3)}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\beta}}}\mbox{\boldmath$\nabla$}^{2}\bar{\mbox{\boldmath$\nabla$}}_{({\dot{\alpha}}_{1}}\bar{H}_{{\dot{\alpha}}_{2}{\dot{\alpha}}_{3})}-\frac{3{\rm
i}}{4}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}(2){\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\alpha}}}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\mbox{\boldmath$\nabla$}^{2}\bar{\mbox{\boldmath$\nabla$}}_{({\dot{\alpha}}_{1}}\bar{H}_{{\dot{\alpha}}_{2}{\dot{\alpha}}_{3})}$
$\displaystyle~{}~{}~{}-\frac{3{\rm
i}}{4}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}(2){\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\beta}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{2}\bar{\mbox{\boldmath$\nabla$}}_{({\dot{\alpha}}_{1}}\bar{H}_{{\dot{\alpha}}_{2}{\dot{\alpha}}_{3})}\bigg{\\}}+\text{c.c.}~{},$
(6.196a) $\displaystyle\delta_{\lambda}\mathbb{S}_{\text{skeleton}}^{(2,0,0)}$
$\displaystyle=-\int{\rm d}^{4}x{\rm d}^{2}\theta\,{\cal
E}\,\lambda^{\alpha\beta}\mbox{\boldmath$W$}^{\alpha(2)}{}_{\beta}\mathfrak{W}_{\alpha(3)}(\bar{H})+\text{c.c.}$
(6.196b)

Following the usual philosophy, in order to cancel the parts of the variation
that are linear in the super-Weyl tensor we must introduce primary non-minimal
corrections. In general, they will take the form

$\displaystyle\mathbb{S}_{\text{NM},i}^{(2,0,0)}[H,\bar{H}]=\frac{1}{2}\int\text{d}^{4|4}z\,E\,H^{\alpha(2)}\mathfrak{J}^{(i)}_{\alpha(2)}(\bar{H})+\text{c.c.}$
(6.197)

where $\mathfrak{J}_{\alpha(2)}^{(i)}(\bar{H})$ is a primary superfield of
Weyl weight $3$ and $\mathsf{U}(1)_{R}$ charge $-2/3$ that depends explicitly
on the super-Weyl tensor. It may be shown that to linear order in
$\mbox{\boldmath$W$}_{\alpha(3)}$, there are exactly two such structures, and
they are given by

$\displaystyle\mathfrak{J}^{(1)}_{\alpha(2)}(\bar{H})$
$\displaystyle=\mbox{\boldmath$\nabla$}^{\gamma{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{\beta{\dot{\alpha}}}\mbox{\boldmath$W$}_{\beta\alpha(2)}\mbox{\boldmath$\nabla$}_{\gamma}\bar{H}_{{\dot{\alpha}}(2)}+\frac{{\rm
i}}{4}\mbox{\boldmath$\nabla$}^{\beta{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\beta}\mbox{\boldmath$W$}_{\alpha(2)}{}^{\gamma}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\gamma}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}+\frac{{\rm
i}}{2}\mbox{\boldmath$\nabla$}^{\beta{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{\gamma}\mbox{\boldmath$W$}_{\gamma\alpha(2)}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\alpha}}(2)}+\frac{5{\rm
i}}{24}\mbox{\boldmath$\nabla$}^{\beta{\dot{\alpha}}}\mbox{\boldmath$W$}_{\beta\alpha(2)}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{2}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}-\frac{1}{3}\mbox{\boldmath$\nabla$}^{\beta{\dot{\alpha}}}\mbox{\boldmath$W$}_{\beta\alpha(2)}\mbox{\boldmath$\nabla$}^{\gamma{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\gamma}\bar{H}_{{\dot{\alpha}}(2)}+\frac{3}{2}\mbox{\boldmath$\nabla$}^{\beta{\dot{\alpha}}}\mbox{\boldmath$W$}_{\alpha(2)}{}^{\gamma}\mbox{\boldmath$\nabla$}_{\beta}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\gamma}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}-\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$W$}_{\alpha}{}^{\beta(2)}\mbox{\boldmath$\nabla$}_{\beta}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\alpha}}(2)}+\frac{7{\rm
i}}{12}\mbox{\boldmath$\nabla$}^{\beta}\mbox{\boldmath$W$}_{\beta\alpha(2)}\mbox{\boldmath$\nabla$}^{\gamma{\dot{\alpha}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\gamma}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}+\frac{{\rm
i}}{4}\mbox{\boldmath$\nabla$}^{\beta}\mbox{\boldmath$W$}_{\alpha(2)}{}^{\gamma}\mbox{\boldmath$\nabla$}_{\gamma}{}^{{\dot{\alpha}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\alpha}}(2)}+\frac{{\rm
i}}{2}\mbox{\boldmath$\nabla$}^{\beta}\mbox{\boldmath$W$}_{\alpha\beta}{}^{\gamma}\mbox{\boldmath$\nabla$}_{\gamma}{}^{{\dot{\alpha}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\alpha}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}+\frac{2}{3}\mbox{\boldmath$W$}_{\alpha(2)}{}^{\beta}\mbox{\boldmath$\nabla$}_{\beta}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{\gamma{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\gamma}\bar{H}_{{\dot{\alpha}}(2)}+\frac{5{\rm
i}}{24}\mbox{\boldmath$W$}_{\alpha(2)}{}^{\beta}\mbox{\boldmath$\nabla$}_{\beta}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{2}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}+\mbox{\boldmath$W$}_{\alpha}{}^{\beta(2)}\mbox{\boldmath$\nabla$}_{\beta}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\beta}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\alpha}\bar{H}_{{\dot{\alpha}}(2)}-\frac{2}{3}\mbox{\boldmath$W$}_{\alpha(2)}{}^{\beta}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}(2){\dot{\beta}}}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\alpha}}(2)}~{},$
(6.198a) $\displaystyle\mathfrak{J}^{(2)}_{\alpha(2)}(\bar{H})$
$\displaystyle=-\frac{2}{3}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}}{}_{{\dot{\beta}}}{}^{{\dot{\gamma}}}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\gamma}}}\bar{H}_{{\dot{\alpha}}(2)}+\frac{{\rm
i}}{3}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\beta}}}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}(2){\dot{\gamma}}}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\gamma}}}\mbox{\boldmath$\nabla$}_{\alpha}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}-\frac{2{\rm
i}}{3}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\gamma}}}\bar{\mbox{\boldmath$W$}}_{{\dot{\gamma}}}{}^{{\dot{\alpha}}(2)}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\alpha}\bar{H}_{{\dot{\alpha}}(2)}+\frac{4}{3}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\alpha}}}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}}{}_{{\dot{\beta}}}{}^{{\dot{\gamma}}}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\gamma}}}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}-\frac{2}{3}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}(2){\dot{\gamma}}}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\gamma}}}\bar{H}_{{\dot{\alpha}}(2)}-\frac{5{\rm
i}}{12}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}_{{\dot{\beta}}}{}^{{\dot{\alpha}}(2)}\bar{\mbox{\boldmath$\nabla$}}^{2}\mbox{\boldmath$\nabla$}_{\alpha}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}+\frac{1}{3}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}_{{\dot{\beta}}}{}^{{\dot{\alpha}}(2)}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\gamma}}}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\gamma}}}\bar{H}_{{\dot{\alpha}}(2)}-\frac{1}{3}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}(2){\dot{\gamma}}}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\gamma}}}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\beta}}}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}+\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\alpha}}}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}{\dot{\beta}}(2)}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\beta}}}\bar{H}_{{\dot{\alpha}}(2)}-\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}}{}_{{\dot{\beta}}}{}^{{\dot{\gamma}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\gamma}}}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}-\frac{{\rm
i}}{2}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}}{}_{{\dot{\beta}}}{}^{{\dot{\gamma}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\alpha}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\gamma}}}\bar{H}_{{\dot{\alpha}}(2)}+\frac{2}{3}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}(2){\dot{\gamma}}}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\gamma}}}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}+\frac{7{\rm
i}}{12}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}_{{\dot{\beta}}}{}^{{\dot{\alpha}}(2)}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\gamma}}}\mbox{\boldmath$\nabla$}_{\alpha}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\gamma}}}\bar{H}_{{\dot{\alpha}}(2)}-\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}{\dot{\beta}}(2)}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\beta}}}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}+\frac{{\rm
i}}{6}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}(2){\dot{\gamma}}}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\alpha}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\gamma}}}\bar{H}_{{\dot{\alpha}}(2)}-\frac{2}{3}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}(2){\dot{\beta}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\gamma}}}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\gamma}}}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\beta}}}\bar{H}_{{\dot{\alpha}}(2)}$
(6.198b) $\displaystyle~{}~{}~{}+\frac{5{\rm
i}}{24}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}(2){\dot{\beta}}}\bar{\mbox{\boldmath$\nabla$}}^{2}\mbox{\boldmath$\nabla$}_{\alpha}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\beta}}}\bar{H}_{{\dot{\alpha}}(2)}+\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}{\dot{\beta}}(2)}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\beta}}}\bar{H}_{{\dot{\alpha}}(2)}+\mathcal{O}(\mbox{\boldmath$W$}^{2})~{}.$

We remind the reader that we use the convention whereby all indices denoted by
indistinguishable characters are to be symmetrised over.

A few comments regarding the general structure of the primary superfields
(6.198) are in order. Firstly, the superfield (6.198a) is an exact primary
(i.e. it is primary to all orders in the super-Weyl tensor). There are terms
quadratic in $\mbox{\boldmath$W$}_{\alpha(3)}$ which ensure this property.
Their presence is new and did not appear in the first order structures for
previous models. Secondly, the superfield (6.198b) is primary only to first
order in the super-Weyl tensor. This does not present a problem since it may
be shown that (6.198b) and (6.198a) are not independent of one another and are
related via the identity

$\displaystyle\int\text{d}^{4|4}z\,E\,H^{\alpha(2)}\mathfrak{J}^{(1)}_{\alpha(2)}(\bar{H})+\text{c.c.}=\int\text{d}^{4|4}z\,E\,\bar{H}^{{\dot{\alpha}}(2)}\bar{\mathfrak{J}}^{(1)}_{{\dot{\alpha}}(2)}(H)+\text{c.c.}$
$\displaystyle=\int\text{d}^{4|4}z\,E\,H^{\alpha(2)}\bigg{\\{}\mathfrak{J}^{(2)}_{\alpha(2)}(\bar{H})+{\rm
i}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\bar{H}_{{\dot{\alpha}}(2)}+\frac{1}{6}\mbox{\boldmath$\nabla$}_{\alpha}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\bar{H}_{{\dot{\alpha}}(2)}+\frac{1}{3}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\alpha}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}+\frac{1}{6}\mbox{\boldmath$\nabla$}_{\alpha}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\bar{H}_{{\dot{\alpha}}(2)}+\frac{1}{2}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\alpha}\bar{H}_{{\dot{\alpha}}(2)}+\frac{2{\rm
i}}{3}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\alpha}}}\bar{H}_{{\dot{\alpha}}(2)}\bigg{\\}}+\text{c.c.}$
(6.199)

Hence, for the purpose of restoring gauge invariance it suffices to consider
only $\mathfrak{J}^{(1)}_{\alpha(2)}(\bar{H})$.

By making use of the identity (6.199), one can show that a generic
(infinitesimal) variation of the action (6.197), with $i=1$, takes the form

$\displaystyle\delta\mathbb{S}_{\text{NM},1}^{(2,0,0)}$
$\displaystyle=\frac{1}{2}\int\text{d}^{4|4}z\,E\,\delta
H^{\alpha(2)}\bigg{\\{}\mathfrak{J}^{(1)}_{\alpha(2)}(\bar{H})+\mathfrak{J}^{(2)}_{\alpha(2)}(\bar{H})+{\rm
i}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\bar{H}_{{\dot{\alpha}}(2)}+\frac{1}{6}\mbox{\boldmath$\nabla$}_{\alpha}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}+\frac{1}{3}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\alpha}\bar{H}_{{\dot{\alpha}}(2)}+\frac{1}{6}\mbox{\boldmath$\nabla$}_{\alpha}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\bar{H}_{{\dot{\alpha}}(2)}+\frac{1}{2}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\alpha}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}+\frac{2{\rm
i}}{3}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\alpha}}}\bar{H}_{{\dot{\alpha}}(2)}\bigg{\\}}+\text{c.c.}$
(6.200)

Furthermore, to first order it is possible to show that the superfields
(6.198) satisfy the following identities

$\displaystyle\mbox{\boldmath$\nabla$}^{\beta}\mathfrak{J}^{(1)}_{\alpha\beta}(\bar{H})$
$\displaystyle=-\mbox{\boldmath$\nabla$}^{\beta{\dot{\alpha}}}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\alpha}}(2)}-\frac{1}{4}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{2}\bar{H}_{{\dot{\alpha}}(2)}-\frac{1}{4}\mbox{\boldmath$\nabla$}^{\beta}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}-\frac{{\rm
i}}{2}\mbox{\boldmath$B$}^{\beta{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\alpha}}(2)}+\frac{7{\rm
i}}{12}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{\beta{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\alpha}}(2)}+\frac{1}{24}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{2}\bar{H}_{{\dot{\alpha}}(2)}~{},$
(6.201a)
$\displaystyle\mbox{\boldmath$\nabla$}^{\beta}\mathfrak{J}^{(2)}_{\alpha\beta}(\bar{H})$
$\displaystyle=\frac{1}{4}\bigg{\\{}{\rm
i}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}_{{\dot{\beta}}}{}^{{\dot{\alpha}}(2)}\mbox{\boldmath$\nabla$}^{2}\bar{\mbox{\boldmath$\nabla$}}_{({\dot{\alpha}}_{1}}\bar{H}_{{\dot{\alpha}}_{2}{\dot{\alpha}}_{3})}+{\rm
i}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\beta}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\bar{\mbox{\boldmath$W$}}_{{\dot{\beta}}}{}^{{\dot{\alpha}}(2)}\mbox{\boldmath$\nabla$}^{2}\bar{\mbox{\boldmath$\nabla$}}_{({\dot{\alpha}}_{1}}\bar{H}_{{\dot{\alpha}}_{2}{\dot{\alpha}}_{3})}$
$\displaystyle~{}~{}~{}+\frac{5{\rm
i}}{4}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}_{{\dot{\beta}}}{}^{{\dot{\alpha}}(2)}\bar{\mbox{\boldmath$\nabla$}}^{2}\mbox{\boldmath$\nabla$}^{2}\bar{H}_{{\dot{\alpha}}(2)}-2\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\gamma}}}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}(2){\dot{\beta}}}\mbox{\boldmath$\nabla$}^{\beta}{}_{{\dot{\gamma}}}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}+6\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}}{}_{{\dot{\beta}}}{}^{{\dot{\gamma}}}\mbox{\boldmath$\nabla$}^{\beta}{}_{{\dot{\gamma}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\alpha}}(2)}-\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}_{{\dot{\beta}}}{}^{{\dot{\alpha}}(2)}\mbox{\boldmath$\nabla$}^{\beta{\dot{\gamma}}}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\gamma}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}+\frac{9{\rm
i}}{4}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}_{{\dot{\beta}}}{}^{{\dot{\alpha}}(2)}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{2}\bar{\mbox{\boldmath$\nabla$}}_{({\dot{\alpha}}_{1}}\bar{H}_{{\dot{\alpha}}_{2}{\dot{\alpha}}_{3})}-\frac{{\rm
i}}{2}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}(3)}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\beta}}}\mbox{\boldmath$\nabla$}^{2}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}-\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}(2){\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\beta}}}\mbox{\boldmath$\nabla$}^{\beta{\dot{\gamma}}}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\gamma}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\alpha}}(2)}-4\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}{\dot{\beta}}(2)}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\beta}}}\mbox{\boldmath$\nabla$}^{\beta}{}_{{\dot{\beta}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}+\frac{1}{3}\mbox{\boldmath$\nabla$}_{(\alpha}\mbox{\boldmath$B$}_{\beta)}{}^{{\dot{\alpha}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{\beta}\bar{H}_{{\dot{\alpha}}(2)}+\frac{4{\rm
i}}{3}\mbox{\boldmath$B$}_{(\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{\beta}\mbox{\boldmath$\nabla$}_{\beta)}{}^{{\dot{\alpha}}}\bar{H}_{{\dot{\alpha}}(2)}\bigg{\\}}~{}.$
(6.201b)

Both relations (6.200) and (6.201) are fundamental in computing the gauge
variation of the superconformal action $\mathbb{S}_{\text{NM},1}^{(2,0,0)}$
and to first order it can be shown that the $\Lambda$ gauge variation is given
by

$\displaystyle\delta_{\Lambda}\mathbb{S}_{\text{NM,1}}^{(2,0,0)}$
$\displaystyle=-\frac{1}{4}\delta_{\Lambda}\mathbb{S}_{\text{skeleton}}^{(2,0,0)}+\int{\rm
d}^{4|4}z\,E\,\Lambda^{\alpha}\bigg{\\{}-\mbox{\boldmath$\nabla$}^{\beta{\dot{\alpha}}}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\alpha}}(2)}-\frac{1}{8}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{2}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}-\frac{1}{6}\mbox{\boldmath$\nabla$}^{\beta}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\alpha}}(2)}-\frac{15{\rm
i}}{16}\mbox{\boldmath$B$}^{\beta{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\alpha}}(2)}+\frac{47{\rm
i}}{48}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{\beta{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}-\frac{{\rm
i}}{12}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{\beta{\dot{\alpha}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\alpha}}(2)}-\frac{1}{6}\mbox{\boldmath$B$}_{\alpha}{}^{{\dot{\alpha}}}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\mbox{\boldmath$\nabla$}^{2}\bar{H}_{{\dot{\alpha}}(2)}\bigg{\\}}~{}.$
(6.202)

This implies that the superconformal action

$\displaystyle\mathbb{S}_{H\bar{H}}^{(2,0,0)}[H,\bar{H}]=\mathbb{S}^{(2,0,0)}_{\text{skeleton}}[H,\bar{H}]+4\mathbb{S}_{\text{NM,1}}^{(2,0,0)}[H,\bar{H}]$
(6.203)

is invariant under $\Lambda$-gauge transformations to leading order in the
super-Weyl tensor (when restricted to a Bach-flat background)

$\displaystyle\delta_{\Lambda,\lambda}\mathbb{S}_{H\bar{H}}^{(2,0,0)}\approx\mathcal{O}(\mbox{\boldmath$W$}^{2})~{}.$
(6.204)

It may also be checked that under a chiral gauge transformation, the action
(6.203) is proportional to terms quadratic in
$\mbox{\boldmath$W$}_{\alpha(3)}$ or linear in
$\mbox{\boldmath$B$}_{\alpha{\dot{\alpha}}}$.

Due to the problems outlined in the beginning of this section, we have so far
restricted our analysis of gauge invariance to be of first order in the super-
Weyl tensor. However, in section 4.5.6 we constructed the non-supersymmetric
model for the conformal pseudo-graviton which was gauge invariant to all
orders (i.e. second order) in the background curvature. Actually, in section
4.5.6 we constructed a family (parametrised by $\Gamma$) of gauge invariant
models for the conformal hook, described by the action
$S^{(3,1)}_{\text{CHS},\Gamma}[h,\chi,\varphi]$, see eq. (4.275). It is
instructive to compare the model with $\Gamma=1$ to the supersymmetric model
considered in this section, and see what conclusions we can draw regarding the
second order completion of the latter. Comments regarding the other values of
$\Gamma$ will be given at the end of this section.

In the conformally-flat limit, we know from (4.277) with $\Gamma=1$ and
(6.4.3) that the non-supersymmetric model and the (bosonic sector of the)
supersymmetric skeleton (in the Wess-Zumino gauge) are given by the actions

$\displaystyle S_{\text{CHS}}^{(3,1)}$
$\displaystyle=\frac{1}{2}\int\text{d}^{4}x\,e\,\bigg{\\{}\mathcal{W}^{\alpha(4)}(h)\mathcal{W}_{\alpha(4)}(\bar{h})+\bar{\chi}^{{\dot{\alpha}}(2)}\mathfrak{X}_{{\dot{\alpha}}(2)}(\chi)\bigg{\\}}+\text{c.c.}~{},$
(6.205a) $\displaystyle\mathbb{S}_{{H\bar{H}}}^{(2,0,0)}\Big{|}_{\rm Bosonic}$
$\displaystyle=\frac{1}{4}\int\text{d}^{4}x\,e\,\bigg{\\{}\mathcal{W}^{\alpha(4)}(h)\mathcal{W}_{\alpha(4)}(\bar{h})-\frac{16}{3}\bar{\rho}^{{\dot{\alpha}}(2)}\mathfrak{X}_{{\dot{\alpha}}(2)}(\rho)\bigg{\\}}+\text{c.c.}~{}$
(6.205b)

From (6.205) one can see that the relative coefficients between the pseudo-
graviton $h_{\alpha(3){\dot{\alpha}}}$ and the non-gauge fields
$\chi_{\alpha(2)}$ and $\rho_{\alpha(2)}$ do not agree. The only way for these
models to be consistent with one another is if there is an additional lower-
spin superfield present in the supersymmetric pseudo-graviton model. By
construction, the model (6.203) is gauge invariant to first order in the
super-Weyl tensor, therefore the purpose of this lower-spin superfield must be
to ensure gauge invariance to second order.

It turns out that the only possible candidate for such a field is the chiral
non-gauge superfield $\Omega_{\alpha}$, which is characterised by the
superconformal properties

$\displaystyle
K_{B}\Omega_{\alpha}=0~{},\qquad\mathbb{D}\Omega_{\alpha}=\frac{1}{2}\Omega_{\alpha}~{},\qquad\mathbb{Y}\Omega_{\alpha}=-\frac{1}{3}\Omega_{\alpha}~{},\qquad\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\Omega_{\alpha}=0~{}.$
(6.206)

In appendix 6.B we discuss this model and its higher-rank extensions. In
particular, the superconformal kinetic action for $\Omega_{\alpha}$ is given
by

$\displaystyle\mathbb{S}_{\Omega\bar{\Omega}}[\Omega,\bar{\Omega}]=-\frac{{\rm
i}}{4}\int\text{d}^{4}x{\rm d}^{2}\theta\,{\cal
E}\,\Omega^{\alpha}\bar{\mbox{\boldmath$\nabla$}}^{2}\mbox{\boldmath$\nabla$}_{\alpha{\dot{\alpha}}}\bar{\Omega}^{{\dot{\alpha}}}~{},$
(6.207)

see appendix 6.B for more details.

In order to provide corrections to the overall gauge variation, it is clear
that $\Omega_{\alpha}$ must transform non-trivially and couple to the pseudo-
graviton $H_{\alpha(2)}$. The only transformation which preserves all of the
properties (6.206) is272727Given the chirality of $\Omega_{\alpha}$ and the
transformations (6.208), we can further conclude that the purpose of
$\Omega_{\alpha}$ is to ensure second-order invariance under chiral
($\lambda$) gauge transformations.

$\displaystyle\delta_{\lambda}\Omega_{\alpha}=\mbox{\boldmath$W$}_{\alpha}{}^{\beta(2)}\lambda_{\beta(2)}~{}.$
(6.208)

There is only a single possible quadratic non-minimal primary coupling between
$H$ and $\Omega$, and it takes the form

$\displaystyle\mathbb{S}_{H\bar{\Omega}}$ $\displaystyle[H,\Omega]={\rm
i}\int\text{d}^{4|4}z\,E\,H^{\alpha(2)}\mathfrak{J}_{\alpha(2)}(\bar{\Omega})+\text{c.c.}~{},$
(6.209a)
$\displaystyle\mathfrak{J}_{\alpha(2)}(\bar{\Omega})=\mbox{\boldmath$\nabla$}_{\beta{\dot{\beta}}}$
$\displaystyle\mbox{\boldmath$W$}_{\alpha(2)}{}^{\beta}\bar{\Omega}^{{\dot{\beta}}}-\mbox{\boldmath$W$}_{\alpha(2)}{}^{\beta}\mbox{\boldmath$\nabla$}_{\beta{\dot{\beta}}}\bar{\Omega}^{{\dot{\beta}}}+\frac{{\rm
i}}{2}\mbox{\boldmath$\nabla$}_{\beta}\mbox{\boldmath$W$}_{\alpha(2)}{}^{\beta}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\bar{\Omega}^{{\dot{\beta}}}~{}.$
(6.209b)

It follows that the total action for the pseudo-graviton multiplet is

$\displaystyle\mathbb{S}_{\text{SCHS}}^{(2,0,0)}=\mathbb{S}_{H\bar{H}}^{(2,0,0)}[H,\bar{H}]+\mu\mathbb{S}_{H\bar{\Omega}}[H,\Omega]+\mu\mathbb{S}_{\Omega\bar{\Omega}}[\Omega,\bar{\Omega}]+\mathcal{O}(\mbox{\boldmath$W$}^{2})$
(6.210)

for some constant $\mu\in\mathbb{R}$ and where
$\mathcal{O}(\mbox{\boldmath$W$}^{2})$ represent terms that are quadratic in
both $H$ and the super-Weyl tensor, and are primary. All possible structures
of this type may be shown to take the form (6.197) with

$\displaystyle\mathfrak{J}^{(3)}_{\alpha(2)}(\bar{H})$
$\displaystyle=\mbox{\boldmath$\nabla$}_{\beta}\mbox{\boldmath$W$}^{\beta}{}_{\alpha(2)}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}^{{\dot{\beta}}{\dot{\alpha}}(2)}\bar{H}_{{\dot{\alpha}}(2)}-2{\rm
i}\mbox{\boldmath$\nabla$}_{{\beta{\dot{\beta}}}}\mbox{\boldmath$W$}^{\beta}{}_{\alpha(2)}\bar{\mbox{\boldmath$W$}}^{{\dot{\beta}}{\dot{\alpha}}(2)}\bar{H}_{{\dot{\alpha}}(2)}$
$\displaystyle~{}~{}~{}+2{\rm
i}\mbox{\boldmath$W$}_{\alpha(2)}{}^{\beta}\mbox{\boldmath$\nabla$}_{{\beta{\dot{\beta}}}}\bar{\mbox{\boldmath$W$}}^{{\dot{\beta}}{\dot{\alpha}}(2)}\bar{H}_{{\dot{\alpha}}(2)}~{},$
(6.211a) $\displaystyle\mathfrak{J}^{(4)}_{\alpha(2)}(\bar{H})$
$\displaystyle=\mbox{\boldmath$\nabla$}_{\beta}\mbox{\boldmath$W$}^{\beta}{}_{\alpha(2)}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}(2){\dot{\beta}}}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\bar{H}_{{\dot{\alpha}}(2)}+\mbox{\boldmath$W$}_{\alpha(2)}{}^{\beta}\bar{\mbox{\boldmath$W$}}^{{\dot{\alpha}}(2){\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{\mbox{\boldmath$\nabla$}}_{\dot{\beta}}\bar{H}_{{\dot{\alpha}}(2)}~{},$
(6.211b) $\displaystyle\mathfrak{J}^{(5)}_{\alpha(2)}(\bar{H})$
$\displaystyle=\mbox{\boldmath$W$}_{\alpha(2)}{}^{\beta}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\bar{\mbox{\boldmath$W$}}^{{\dot{\beta}}{\dot{\alpha}}(2)}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\alpha}}(2)}-\mbox{\boldmath$W$}_{\alpha(2)}{}^{\beta}\bar{\mbox{\boldmath$W$}}^{{\dot{\beta}}{\dot{\alpha}}(2)}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\beta}}}\mbox{\boldmath$\nabla$}_{\beta}\bar{H}_{{\dot{\alpha}}(2)}~{},$
(6.211c)

or are expressible as linear combinations thereof. We note that the lower-spin
sector in (6.210) is $\Lambda$-gauge invariant in a Bach-flat background
whilst their relative coefficient of unity between
$\mathbb{S}_{H\bar{\Omega}}$ and $\mathbb{S}_{\Omega\bar{\Omega}}$ is fixed by
$\lambda$-invariance.

We can actually go a step further and deduce the value of $\mu$, but to do so
we must take a closer look at the component structure of
(6.210).282828According to our general procedure, $\mu$ would typically be
determined by second order $\lambda$-invariance.  For the component fields we
use the definitions (6.138), (6.140a) and (6.275). Using (6.205b) and the
results of appendix 6.B, one can show that in the conformally-flat limit the
bosonic sector of (6.210) is

$\displaystyle\mathbb{S}_{\text{SCHS}}^{(2,0,0)}\Big{|}_{\rm Bosonic}$
$\displaystyle=\frac{1}{4}\int\text{d}^{4}x\,e\,\bigg{\\{}\mathcal{W}^{\alpha(4)}(h)\mathcal{W}_{\alpha(4)}(\bar{h})-\frac{16}{3}\bar{\rho}^{{\dot{\alpha}}(2)}\mathfrak{X}_{{\dot{\alpha}}(2)}(\rho)$
$\displaystyle\phantom{SPACESPACE~{}~{}}-\mu\bigg{(}\bar{U}^{{\dot{\alpha}}(2)}\mathfrak{X}_{{\dot{\alpha}}(2)}(U)+\frac{1}{2}\bar{V}\Box
V\bigg{)}\bigg{\\}}+\text{c.c.}$ (6.212)

In a generic background we find that the fields in (6.205a) and (6.212) are
defined modulo the gauge transformations (see (4.263) and (6.141b) )

$\displaystyle\delta_{\ell}h_{\alpha(3){\dot{\alpha}}}=\nabla_{\alpha{\dot{\alpha}}}\ell_{\alpha(2)}~{},\quad\delta_{\ell}V=0~{},$
(6.213)
$\displaystyle\delta_{\ell}\chi_{\alpha(2)}=W_{\alpha(2)}{}^{\beta(2)}\ell_{\beta(2)}~{}$
$\displaystyle,\quad\delta_{\ell}\rho_{\alpha(2)}=\frac{{\rm
i}}{4}W_{\alpha(2)}{}^{\beta(2)}\ell_{\beta(2)}~{},\quad\delta_{\ell}U_{\alpha(2)}=\frac{{\rm
i}}{2}W_{\alpha(2)}{}^{\beta(2)}\ell_{\beta(2)}~{}.$

In (6.212) there are clearly two distinct non-gauge fields, $\rho_{\alpha(2)}$
and $U_{\alpha(2)}$, which are of the same tensor type as the non-gauge field
$\chi_{\alpha(2)}$ in the non-supersymmetric model and which have non-trivial
gauge transformations (and so play a role in ensuring gauge invariance).
However, if we make the following field redefinitions

$\displaystyle X_{\alpha(2)}=\frac{4{\rm
i}}{3}\big{(}\rho_{\alpha(2)}-2U_{\alpha(2)}\big{)}~{},\qquad
Y_{\alpha(2)}=\frac{4{\rm
i}}{3}\big{(}-2\rho_{\alpha(2)}+U_{\alpha(2)}\big{)}~{},$ (6.214)

then the fields $X_{\alpha(2)}$ and $Y_{\alpha(2)}$ transform in the manner

$\displaystyle\delta_{\ell}X_{\alpha(2)}=W_{\alpha(2)}{}^{\beta(2)}\ell_{\beta(2)}~{},\qquad\delta_{\ell}Y_{\alpha(2)}=0~{}.$
(6.215)

If in addition we choose $\mu=-16/3$ then the action (6.212) becomes

$\displaystyle\mathbb{S}_{\text{SCHS}}^{(2,0,0)}\Big{|}_{\rm Bosonic}=$
$\displaystyle\frac{1}{4}\int\text{d}^{4}x\,e\,\bigg{\\{}\mathcal{W}^{\alpha(4)}(h)\mathcal{W}_{\alpha(4)}(\bar{h})+\bar{X}^{{\dot{\alpha}}(2)}\mathfrak{X}_{{\dot{\alpha}}(2)}(X)$
$\displaystyle\phantom{SPACESPACE~{}~{}}-\bar{Y}^{{\dot{\alpha}}(2)}\mathfrak{X}_{{\dot{\alpha}}(2)}(Y)+\frac{8}{3}\bar{V}\Box
V\bigg{\\}}+\text{c.c.}~{}~{}~{}~{}~{}~{}$ (6.216)

The fields $Y_{\alpha(2)}$ and $V$ do not transform under $\ell_{\alpha(2)}$.
Consequently, they do not play a role in establishing gauge invariance and are
present only to ensure supersymmetry. Since the non-gauge fields
$\chi_{\alpha(2)}$ and $X_{\alpha(2)}$ share the same gauge transformations
and relative coefficient between their own kinetic sector and that of the
pseudo-graviton, we may identify them: $\chi_{\alpha(2)}\equiv X_{\alpha(2)}$.
Thus we conclude that the lower-spin sectors in the supersymmetric model
(6.210) have coefficients $\mu=-16/3$. This means that the coefficients of all
sectors in the pseudo-graviton multiplet, except for the second order ones
(6.211), have been determined.

##### Other superspace realisations of the conformal pseudo-graviton

In section 4.5.6 we derived a one-parameter family of gauge invariant models
for the conformal hook field, which was parameterised by $\Gamma$. By
analysing the resulting component structure, it was shown above that the model
(4.275) with $\Gamma=1$ (i.e. (4.268a)) could be embedded within a
supersymmetric gauge invariant action for a superfield $H_{\alpha(2)}$ coupled
to a chiral non-gauge superfield $\Omega_{\alpha}$. In this model, the latter
two transformed according to

$\displaystyle\delta_{\Lambda,\lambda}H_{\alpha(2)}=\mbox{\boldmath$\nabla$}_{(\alpha_{1}}$
$\displaystyle\Lambda_{\alpha_{2})}+\lambda_{\alpha(2)}~{},\qquad\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\lambda_{\alpha(2)}=0~{},$
(6.217a)
$\displaystyle\delta_{\lambda}\Omega_{\alpha}=\mbox{\boldmath$W$}_{\alpha}{}^{\beta(2)}\lambda_{\beta(2)}~{},$
(6.217b)

with $\Lambda_{\alpha}$ unconstrained and $\lambda_{\alpha(2)}$ covariantly
chiral.

For $\Gamma\neq 3$, it may be possible to embed (4.275) within a one-parameter
family of supersymmetric actions by introducing, in addition to
$H_{\alpha(2)}$ and $\Omega_{\alpha}$, the longitudinal linear non-gauge
superfield $\Omega_{\alpha(3){\dot{\alpha}}(2)}$ (see appendix 6.B for a
discussion of this type of non gauge field). The latter should be entangled
with $H_{\alpha(2)}$ through the transformations

$\displaystyle\delta_{\Lambda}\Omega_{\alpha(3){\dot{\alpha}}(2)}=\mbox{\boldmath$W$}_{\alpha(3)}\bar{\mbox{\boldmath$\nabla$}}_{({\dot{\alpha}}_{1}}\bar{\Lambda}_{{\dot{\alpha}}_{2})}~{}.$
(6.218)

Then, the component field of $\Omega_{\alpha(3){\dot{\alpha}}(2)}$ defined by
(6.283c) corresponds to $\varphi_{\alpha(4){\dot{\alpha}}(2)}$ and has the
correct gauge transformations (4.272) (in an appropriate Wess-Zumino gauge).

When $\Gamma=3$, the non-gauge field $\chi_{\alpha(2)}$ is no longer present
in the action (4.275). However, as explained previously, $\chi_{\alpha(2)}$
appears as a component field of the gauge superfield $H_{\alpha(2)}$. Hence,
at the component level, any gauge invariant model for $H_{\alpha(2)}$ will
necessarily contain a coupling between $h_{\alpha(3){\dot{\alpha}}}$ and
$\chi_{\alpha(2)}$. Therefore $H_{\alpha(2)}$ is not a suitable supermultiplet
in providing a supersymmetric embedding for the action (4.273).

A possible candidate for this role292929One can rule out the superfield
$H_{\alpha(3)}$ (which also contains $h_{\alpha(3){\dot{\alpha}}}$) as there
is no possible coupling to a non-gauge superfield which contains
$\varphi_{\alpha(4){\dot{\alpha}}(2)}$ at the component level and which
preserves all symmetries. is the spin-5/2 supermultiplet
$H_{\alpha(2){\dot{\alpha}}}$ coupled to a longitudinal non-gauge field
$\Omega_{\alpha(4){\dot{\alpha}}(2)}$ according to303030We would like to point
out that the gauge transformations (6.218) and (6.219b) preserve the gauge-
for-gauge symmetries
$\zeta_{\alpha}\rightarrow\zeta_{\alpha}+\mbox{\boldmath$\nabla$}_{\alpha}\sigma$
and
$\Lambda_{\alpha{\dot{\alpha}}}\rightarrow\Lambda_{\alpha{\dot{\alpha}}}+\mbox{\boldmath$\nabla$}_{\alpha}\bar{\eta}_{{\dot{\alpha}}}$
of (6.217a) and (6.219a), respectively.

$\displaystyle\delta_{\Lambda,\Omega}H_{\alpha(2){\dot{\alpha}}}$
$\displaystyle=\mbox{\boldmath$\nabla$}_{\alpha}\Lambda_{\alpha{\dot{\alpha}}}+\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\zeta_{\alpha(2)}~{},$
(6.219a) $\displaystyle\delta_{\Lambda}\Omega_{\alpha(4){\dot{\alpha}}(2)}$
$\displaystyle=\mbox{\boldmath$W$}_{\alpha(3)}\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\bar{\Lambda}_{\alpha{\dot{\alpha}}}~{},$
(6.219b)

with unconstrained and complex gauge parameters
$\Lambda_{\alpha{\dot{\alpha}}}$ and $\zeta_{\alpha(2)}$. Here, it is the
component field (6.283e) of $\Omega_{\alpha(4){\dot{\alpha}}(2)}$ which may be
identified with the non-gauge field $\varphi_{\alpha(4){\dot{\alpha}}(2)}$.

####  6.5.4 The conformal spin-3 supermultiplet

Our model for the superconformal pseudo-graviton multiplet provides an example
of a universal pattern, that general CHS theories possess supersymmetric
embeddings. This gives the rationale to study SCHS theories as they may
uncover interesting features of non-supersymmetric CHS models. As an example,
let us assume the existence of a gauge-invariant model for the conformal
spin-3 supermultiplet in a Bach-flat background and see what can be deduced
regarding its non-supersymmetric counterpart. The former is described by the
real prepotential $H_{\alpha(2){\dot{\alpha}}(2)}$ which is a primary
superfield of weight $-2$ with the gauge freedom

$\displaystyle\delta_{\Lambda}H_{\alpha(2){\dot{\alpha}}(2)}=\bar{\mbox{\boldmath$\nabla$}}_{({\dot{\alpha}}_{1}}\Lambda_{\alpha(2){\dot{\alpha}}_{2})}-\mbox{\boldmath$\nabla$}_{(\alpha_{1}}\bar{\Lambda}_{\alpha_{2}){\dot{\alpha}}(2)}~{}.$
(6.220)

This transformation law defines a reducible gauge theory (in the terminology
of [303]) since the unconstrained gauge parameter
$\Lambda_{\alpha(2){\dot{\alpha}}}$ is defined modulo arbitrary local shifts

$\displaystyle\Lambda_{\alpha(2){\dot{\alpha}}}\to\Lambda^{\prime}_{\alpha(2){\dot{\alpha}}}=\Lambda_{\alpha(2){\dot{\alpha}}}+\bar{\mbox{\boldmath$\nabla$}}_{\dot{\alpha}}\sigma_{\alpha(2)}$
(6.221)

such that both parameters $\Lambda_{\alpha(2){\dot{\alpha}}}$ and
$\Lambda^{\prime}_{\alpha(2){\dot{\alpha}}}$ generate the same transformation
of the prepotential,
$\delta_{\Lambda^{\prime}}H_{\alpha(2){\dot{\alpha}}(2)}=\delta_{\Lambda}H_{\alpha(2){\dot{\alpha}}(2)}$.

A Wess-Zumino gauge may be chosen such that the only non-vanishing bosonic
component fields of $H_{\alpha(2){\dot{\alpha}}(2)}$ are

$\displaystyle h_{\alpha(3){\dot{\alpha}}(3)}$
$\displaystyle:=\frac{1}{2}\big{[}\mbox{\boldmath$\nabla$}_{(\alpha_{1}},\bar{\mbox{\boldmath$\nabla$}}_{({\dot{\alpha}}_{1}}\big{]}H_{\alpha_{2}\alpha_{3}){\dot{\alpha}}_{2}{\dot{\alpha}}_{3})}|~{},$
(6.222a) $\displaystyle h_{\alpha(2){\dot{\alpha}}(2)}$
$\displaystyle:=\frac{1}{32}\big{\\{}\mbox{\boldmath$\nabla$}^{2},\bar{\mbox{\boldmath$\nabla$}}^{2}\big{\\}}H_{\alpha(2){\dot{\alpha}}(2)}|~{}.$
(6.222b)

The residual gauge freedom is given by

$\displaystyle\delta_{\ell}h_{\alpha(3){\dot{\alpha}}(3)}$
$\displaystyle=\nabla_{(\alpha_{1}({\dot{\alpha}}_{1}}\ell_{\alpha_{2}\alpha_{3}){\dot{\alpha}}_{2}{\dot{\alpha}}_{3})}~{},$
(6.223a) $\displaystyle\delta_{\ell}h_{\alpha(2){\dot{\alpha}}(2)}$
$\displaystyle=\nabla_{(\alpha_{1}({\dot{\alpha}}_{1}}\ell_{\alpha_{2}){\dot{\alpha}}_{2})}+\bigg{[}\frac{{\rm
i}}{6}W_{\alpha(2)}{}^{\beta(2)}\ell_{\beta(2){\dot{\alpha}}(2)}+\text{c.c.}\bigg{]}~{}.$
(6.223b)

Eq. (6.223a) is the standard gauge transformation of the conformal spin-3
field. Similarly, the first term on the right of (6.223b) is the usual gauge
transformation of the conformal spin-2 field. However, the second term in
(6.223b) is a new feature. Its presence means that the spin-2 field also
varies under the spin-3 gauge transformation.

In principle, there is a possibility that the kinetic action for
$H_{\alpha(2){\dot{\alpha}}(2)}$ can be made gauge invariant without
introducing any extra supermultiplet. However, the analysis in this thesis
indicates that, most likely, $H_{\alpha(2){\dot{\alpha}}(2)}$ is to be
accompanied by a lower-spin supermultiplet in order to ensure gauge invariance
of the action on Bach-flat backgrounds. In addition, the results of [119, 120]
also indicate that for a consistent description of the conformal spin-3 field
in such backgrounds, it should be accompanied by a spin-1 gauge field which
gets shifted under the spin-3 gauge transformation. Thus our theory should
involve a lower-spin supermultiplet containing a gauge vector field. There are
only two options for such a gauge prepotential: (i) a real scalar $V$; and
(ii) a real vector $H_{\alpha{\dot{\alpha}}}$. Now we will consider both of
them in turn.

The gauge prepotential $V$ is a primary superfield with weight $0$ and
describes a vector multiplet. In addition to the standard gauge transformation

$\displaystyle\delta_{\lambda}V$ $\displaystyle=$
$\displaystyle\lambda+\bar{\lambda}~{},\qquad\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\lambda=0~{},$
(6.224a) there is a unique entanglement with the gauge parameter
$\Lambda_{\alpha(2){\dot{\alpha}}}$ of $H_{\alpha(2){\dot{\alpha}}(2)}$, given
by $\displaystyle\delta_{\Lambda}V$ $\displaystyle=$ $\displaystyle 4{\rm
i}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\alpha}}}\mbox{\boldmath$W$}^{\alpha(3)}\Lambda_{\alpha(2){\dot{\alpha}}}-\mbox{\boldmath$\nabla$}_{\alpha}\mbox{\boldmath$W$}^{\alpha(3)}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\Lambda_{\alpha(2){\dot{\alpha}}}-2{\rm
i}\mbox{\boldmath$W$}^{\alpha(3)}\mbox{\boldmath$\nabla$}_{\alpha}{}^{{\dot{\alpha}}}\Lambda_{\alpha(2){\dot{\alpha}}}$
(6.224b)
$\displaystyle-\frac{1}{2}\mbox{\boldmath$W$}^{\alpha(3)}\mbox{\boldmath$\nabla$}_{\alpha}\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\alpha}}}\Lambda_{\alpha(2){\dot{\alpha}}}+\text{c.c.}~{}$

The local shift (6.221) leaves the variation
$\delta_{\Lambda}V+\delta_{\lambda}V$ invariant provided the parameter
$\lambda$ is also shifted in the following way

$\displaystyle\lambda\rightarrow\lambda^{\prime}=\lambda-\bar{\mbox{\boldmath$\nabla$}}^{2}\Big{(}\mbox{\boldmath$\nabla$}_{\alpha}\mbox{\boldmath$W$}^{\alpha(3)}\sigma_{\alpha(2)}+\frac{1}{2}\mbox{\boldmath$W$}^{\alpha(3)}\mbox{\boldmath$\nabla$}_{\alpha}\sigma_{\alpha(2)}\Big{)}~{}.$
(6.225)

We choose a Wess-Zumino gauge such that the only bosonic fields of $V$ are

$\displaystyle
h_{\alpha{\dot{\alpha}}}=\frac{1}{2}\big{[}\mbox{\boldmath$\nabla$}_{\alpha},\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\big{]}V|~{},\qquad
D:=\frac{1}{32}\big{\\{}\mbox{\boldmath$\nabla$}^{2},\bar{\mbox{\boldmath$\nabla$}}^{2}\big{\\}}V|~{}.$
(6.226)

The auxiliary field $D$ is irrelevant to our discussion. For the complete
gauge transformation of $h_{\alpha{\dot{\alpha}}}$ we obtain

$\displaystyle\delta_{\ell}h_{\alpha{\dot{\alpha}}}=\nabla_{{\alpha{\dot{\alpha}}}}\ell+\frac{2}{3}\bigg{[}W_{\alpha}{}^{\beta(3)}\nabla_{\beta}{}^{{\dot{\beta}}}\ell_{\beta(2){\dot{\alpha}}{\dot{\beta}}}-3\nabla_{\beta}{}^{{\dot{\beta}}}W_{\alpha}{}^{\beta(3)}\ell_{\beta(2){\dot{\alpha}}{\dot{\beta}}}+\text{c.c.}\bigg{]}~{}.$
(6.227)

The first term on the right of (6.227) is the standard gauge transformation of
the spin-1 field. The second term in (6.227) tells us that the spin-1 field
also varies under the spin-3 gauge transformation. The complete variation
(6.227) is equivalent to the gauge transformation law postulated by Grigoriev
and Tseytlin [119].313131Indeed, the overall coefficient of (6.224b) was
chosen so that this is the case.

Next, let us consider the case where a coupling between
$H_{\alpha(2){\dot{\alpha}}(2)}$ and a superconformal spin-2 multiplet
$H_{{\alpha{\dot{\alpha}}}}$ is switched on. In addition to the standard law
(6.117b), the gauge transformation of $H_{\alpha{\dot{\alpha}}}$ may be
entangled with the spin-3 gauge parameter as follows

$\displaystyle\delta_{\Lambda}H_{\alpha{\dot{\alpha}}}=\bar{\mbox{\boldmath$\nabla$}}_{{\dot{\alpha}}}\Lambda_{\alpha}-\mbox{\boldmath$\nabla$}_{\alpha}\bar{\Lambda}_{{\dot{\alpha}}}+\mbox{\boldmath$W$}_{\alpha}{}^{\beta(2)}\Lambda_{\beta(2){\dot{\alpha}}}-\bar{\mbox{\boldmath$W$}}_{{\dot{\alpha}}}{}^{{\dot{\beta}}(2)}\bar{\Lambda}_{\alpha{\dot{\beta}}(2)}~{}.$
(6.228)

The local shift (6.221) does not have any effect on
$\delta_{\Lambda}H_{\alpha{\dot{\alpha}}}$ provided the parameter
$\Lambda_{\alpha}$ also gets shifted as follows

$\displaystyle\Lambda_{\alpha}\to\Lambda^{\prime}_{\alpha}=\Lambda_{\alpha}+\mbox{\boldmath$W$}_{\alpha}{}^{\beta(2)}\sigma_{\beta(2)}~{}.$
(6.229)

A Wess-Zumino gauge may also be constructed for the field
$H_{\alpha{\dot{\alpha}}}$ such that its only non-vanishing bosonic component
fields are

$\displaystyle\tilde{h}_{\alpha(2){\dot{\alpha}}(2)}$
$\displaystyle:=\frac{1}{2}\big{[}\mbox{\boldmath$\nabla$}_{(\alpha_{1}},\bar{\mbox{\boldmath$\nabla$}}_{({\dot{\alpha}}_{1}}\big{]}H_{\alpha_{2}){\dot{\alpha}}_{2})}|~{},$
(6.230a) $\displaystyle\tilde{h}_{\alpha{\dot{\alpha}}}$
$\displaystyle:=\frac{1}{32}\big{\\{}\mbox{\boldmath$\nabla$}^{2},\bar{\mbox{\boldmath$\nabla$}}^{2}\big{\\}}H_{\alpha{\dot{\alpha}}}|~{},$
(6.230b)

with gauge transformation laws

$\displaystyle\delta_{\ell,\tilde{\ell}}\tilde{h}_{\alpha(2){\dot{\alpha}}(2)}$
$\displaystyle=\nabla_{(\alpha_{1}({\dot{\alpha}}_{1}}\tilde{\ell}_{\alpha_{2}){\dot{\alpha}}_{2})}+\bigg{[}\frac{{\rm
i}}{2}W_{\alpha(2)}{}^{\beta(2)}\ell_{\beta(2){\dot{\alpha}}(2)}+\text{c.c.}\bigg{]}~{},$
(6.231a)
$\displaystyle\delta_{\ell,\tilde{\ell}}\tilde{h}_{\alpha{\dot{\alpha}}}$
$\displaystyle=\nabla_{\alpha{\dot{\alpha}}}\tilde{\ell}-\frac{1}{24}\bigg{[}W_{\alpha}{}^{\beta(3)}\nabla_{\beta}{}^{{\dot{\beta}}}\ell_{\beta(2){\dot{\beta}}{\dot{\alpha}}}-3\nabla_{\beta}{}^{{\dot{\beta}}}W_{\alpha}{}^{\beta(3)}\ell_{\beta(2){\dot{\beta}}{\dot{\alpha}}}+{\rm
c.c.}\bigg{]}$ (6.231b)

These transformations are analogous to (6.223b) and (6.227).

It is important to note that $h_{\alpha(2){\dot{\alpha}}(2)}$ and
$\tilde{h}_{\alpha(2){\dot{\alpha}}(2)}$, defined by (6.222b) and (6.230a),
describe two independent conformal spin-2 fields,323232This is reminiscent of
the situation in section 6.5.3 where there were two non-gauge fields,
$\rho_{\alpha(2)}$ and $U_{\alpha(2)}$, of the same tensor type. both of which
possess their own gauge transformations and are shifted under spin-3 gauge
transformations. However, upon performing the field redefinitions

$\displaystyle{\mbox{\boldmath$h$}}_{\alpha(2){\dot{\alpha}}(2)}=h_{\alpha(2){\dot{\alpha}}(2)}-\frac{1}{3}\tilde{h}_{\alpha(2){\dot{\alpha}}(2)}~{},$
$\displaystyle\tilde{\mbox{\boldmath$h$}}_{\alpha(2){\dot{\alpha}}(2)}=h_{\alpha(2){\dot{\alpha}}(2)}+\frac{1}{3}\tilde{h}_{\alpha(2){\dot{\alpha}}(2)}~{},$
(6.232a)
$\displaystyle\lambda_{\alpha{\dot{\alpha}}}=\ell_{\alpha{\dot{\alpha}}}-\frac{1}{3}\tilde{\ell}_{\alpha{\dot{\alpha}}}~{},$
$\displaystyle\tilde{\lambda}_{\alpha{\dot{\alpha}}}=\ell_{\alpha{\dot{\alpha}}}+\frac{1}{3}\tilde{\ell}_{\alpha{\dot{\alpha}}}~{},$
(6.232b) the fields ${\mbox{\boldmath$h$}}_{\alpha(2){\dot{\alpha}}(2)}$ and
$\tilde{\mbox{\boldmath$h$}}_{\alpha(2){\dot{\alpha}}(2)}$ transform according
to
$\displaystyle\delta_{\lambda}{\mbox{\boldmath$h$}}_{\alpha(2){\dot{\alpha}}(2)}$
$\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}({\dot{\alpha}}_{1}}\lambda_{\alpha_{2}){\dot{\alpha}}_{2})}~{},$
(6.232c)
$\displaystyle\delta_{\tilde{\lambda},\ell}\tilde{\mbox{\boldmath$h$}}_{\alpha(2){\dot{\alpha}}(2)}$
$\displaystyle=$
$\displaystyle\nabla_{(\alpha_{1}({\dot{\alpha}}_{1}}\tilde{\lambda}_{\alpha_{2}){\dot{\alpha}}_{2})}+\bigg{[}\frac{{\rm
i}}{3}W_{\alpha(2)}{}^{\beta(2)}\ell_{\beta(2){\dot{\alpha}}(2)}+\text{c.c.}\bigg{]}.$
(6.232d)

In particular, we see that
${\mbox{\boldmath$h$}}_{\alpha(2){\dot{\alpha}}(2)}$ transforms independently
of the spin-3 gauge parameter $\ell_{\alpha(2){\dot{\alpha}}(2)}$ and so it
must decouple from the other fields in the action. It is present only to
ensure supersymmetry. As it is not possible to eliminate the shift
transformation present in (6.232d),
$\tilde{\mbox{\boldmath$h$}}_{\alpha(2){\dot{\alpha}}(2)}$ and the spin-3
field $h_{\alpha(3){\dot{\alpha}}(3)}$ cannot be decoupled.

The need for a vector field, which accompanies the spin-3 field and transforms
according to (6.227), was proposed in [119] and detailed calculations were
carried out in [120]. However, the coupling between the spin-3 and spin-2
fields, described by (6.232d), was not considered. Existence of the gauge
invariant supersymmetric model dictates that any gauge invariant model of the
non-supersymmetric conformal spin-3 field must necessarily possess this
coupling. Similar considerations regarding the fermionic component fields of
$H_{\alpha(2){\dot{\alpha}}(2)}$, $H_{{\alpha{\dot{\alpha}}}}$ and $V$ also
allows us to conclude that a gauge invariant model of the conformal spin-$5/2$
field must necessarily include a coupling between the spin-$5/2$ field, the
conformal gravitino and/or the conformal Weyl spinor.

The analyses in [119, 120] were based on the use of the interacting bosonic
CHS theory developed in [41] (see also [40, 43, 42, 44] ). This theory
involves a single instance of each conformal spin-$s$ field
$h_{\alpha(s){\dot{\alpha}}(s)}$ (with $s=1,2\dots$). There also exists a
superconformal scenario briefly discussed in [102], which involves a single
instance of each gauge supermultiplet $H_{\alpha(s){\dot{\alpha}}(s)}$ (with
$s=0,1\dots$). At the component level, such a superconformal theory contains
two copies of each conformal spin-$s$ gauge field
$h_{\alpha(s){\dot{\alpha}}(s)}$. This is exactly the situation in our
analysis above.

###  6.6 Summary of results

This chapter was dedicated to the construction of SCHS models in curved $4d$
$\mathcal{N}=1$ superspace and applications thereof. We began in section 6.1
by describing the general features of the mixed symmetry SCHS field
$H_{\alpha(m){\dot{\alpha}}(n)}$, and its higher-spin super-Bach tensor
$\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)$, on a generic supergravity
background. The definition of $\mathfrak{B}_{\alpha(n){\dot{\alpha}}(m)}(H)$
is such that it has the characteristic features of a covariantly conserved
conformal supercurrent if superspace is conformally-flat, leading to the gauge
and super-Weyl invariant SCHS action (6.38).

In section 6.2 we reviewed the corresponding SCHS models in $\mathcal{N}=1$
Minkowski superspace $\mathbb{M}^{4|4}$. In section 6.3 we discussed on-shell
partially-massless $\mathcal{N}=1$ superfields in AdS4|4 and elaborated on
their gauge freedom for the first time [9], see eq. (6.67). We then
constructed gauge invariant actions (6.84) for the SCHS prepotential
$H_{\alpha(m){\dot{\alpha}}(n)}$ on AdS4|4 [9]. We found that the
corresponding SCHS kinetic operator factorises into products of minimal second
order operators associated with partially-massless supermultiplets of all
(super)depths, see e.g. (6.88). To facilitate this derivation, we made use of
the TLAL (i.e. simultaneously transverse linear and transverse anti-linear)
superprojectors which were constructed in [9] (the latter are not strictly
part of this thesis).

In section 6.4 we provided a review of $\mathcal{N}=1$ conformal superspace.
This was used to derive closed-form models for the SCHS field