$\displaystyle\mbox{\boldmath$j$}^{\alpha{\dot{\alpha}}}=[\nabla^{\alpha},\bar{\nabla}^{{\dot{\alpha}}}]J|_{\theta^{\alpha}=\bar{\theta}_{\dot{\alpha}}=0}~{}.$
(4.16)

###  4.2 Conformal higher-spin (super)fields

In the previous section, we described the conformal higher-spin
(super)currents in general curved backgrounds. Now, by making use of the
method of supercurrent multiplets advocated in [99, 103, 55], we determine
their associated gauge (super)fields by requiring that the Noether couplings

$\displaystyle{\cal N}=0:$ $\displaystyle\qquad\mathcal{S}^{(m,n)}_{\rm
N.C.}=\int{\rm
d}^{4}x\,e\,h_{\alpha(m){\dot{\alpha}}(n)}j^{\alpha(m){\dot{\alpha}}(n)}~{},$
(4.17a) $\displaystyle{\cal N}>0:$
$\displaystyle\qquad\mathcal{S}^{(m,n)}_{\rm N.C.}=\int\text{d}^{4|4{\cal
N}}z\,E\,H_{\alpha(m){\dot{\alpha}}(n)}J^{\alpha(m){\dot{\alpha}}(n)}~{},$
(4.17b)

are locally (super)conformal and gauge-invariant.

####  4.2.1 Conformal higher-spin gauge models

We recall from the previous section that conformal currents are primary tensor
fields $j^{\alpha(m){\dot{\alpha}}(n)}$, with $m,n\geq 1$, subject to the
conservation equation (4.1). It then follows from (4.17a) that their duals,
$h_{\alpha(m){\dot{\alpha}}(n)}$, possess the gauge freedom [101, 102]

$\displaystyle\delta_{\ell}h_{\alpha(m){\dot{\alpha}}(n)}=\nabla_{{\alpha{\dot{\alpha}}}}\ell_{\alpha(m-1){\dot{\alpha}}(n-1)}~{},$
(4.18)

where $\ell_{\alpha(m-1){\dot{\alpha}}(n-1)}$ is, in general, complex
unconstrained. We note that the special cases $(m,n)=(1,1),~{}(2,1),~{}(2,2)$
and $(3,1)$ describe the Maxwell field, conformal gravitino, conformal
graviton and conformal hook field [153], respectively. Additionally, the
$m=n=s$ and $m=n+1=s+1$ cases describe a conformal spin-$s$ and
$s+\frac{1}{2}$ field, respectively [97]. Further, in former case, we are able
to choose $h_{\alpha(s){\dot{\alpha}}(s)}$ to be real.

Requiring that both $h_{\alpha(m){\dot{\alpha}}(n)}$ and
$\ell_{\alpha(m-1){\dot{\alpha}}(n-1)}$ are primary,

$\displaystyle
K^{{\beta{\dot{\beta}}}}h_{\alpha(m){\dot{\alpha}}(n)}=0~{},\qquad
K^{{\beta{\dot{\beta}}}}\ell_{\alpha(m-1){\dot{\alpha}}(n-1)}=0~{},$ (4.19a)
uniquely fixes the dimension of the gauge field
$\displaystyle\mathbb{D}h_{\alpha(m){\dot{\alpha}}(n)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{(}4-(m+n)\big{)}h_{\alpha(m){\dot{\alpha}}(n)}~{}.$
(4.19b)

From $h_{\alpha(m){\dot{\alpha}}(n)}$ and its conjugate
$\bar{h}_{\alpha(n){\dot{\alpha}}(m)}$, we construct the higher-derivative
descendants [101, 102]

$\displaystyle\hat{\mathfrak{C}}_{\alpha(m+n)}(h)=(\nabla_{\alpha}{}^{{\dot{\beta}}})^{n}h_{\alpha(m){\dot{\beta}}(n)}~{},$
(4.20a)
$\displaystyle\check{\mathfrak{C}}_{\alpha(m+n)}(\bar{h})=(\nabla_{\alpha}{}^{{\dot{\beta}}})^{m}\bar{h}_{\alpha(n){\dot{\beta}}(m)}~{}.$
(4.20b)

We note that when the gauge field is real $(m=n=s)$, the above descendants
coincide

$\displaystyle
m=n=s\quad\Longrightarrow\quad{\mathfrak{C}}_{\alpha(2s)}(h)\equiv\hat{\mathfrak{C}}_{\alpha(2s)}(h)=\check{\mathfrak{C}}_{\alpha(2s)}(h)~{}.$
(4.21)

Additionally, we point out that $\mathfrak{C}_{\alpha(2)}(h)$ is Maxwell’s
field strength and $\mathfrak{C}_{\alpha(4)}(h)$ is the linearised Weyl
tensor. The descendants (4.20) have the following dimensions:

$\displaystyle\hat{\mathfrak{C}}_{\alpha(m+n)}(h)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(n-m+4)\hat{\mathfrak{C}}_{\alpha(m+n)}(h)~{},$
(4.22a) $\displaystyle\check{\mathfrak{C}}_{\alpha(m+n)}(\bar{h})$
$\displaystyle=$
$\displaystyle\frac{1}{2}(m-n+4)\check{\mathfrak{C}}_{\alpha(m+n)}(\bar{h})~{}.$
(4.22b)

Further, it may be shown that they are primary,

$\displaystyle
K^{\beta{\dot{\beta}}}\hat{\mathfrak{C}}_{\alpha(m+n)}(h)=0~{},\quad
K^{\beta{\dot{\beta}}}\check{\mathfrak{C}}_{\alpha(m+n)}(\bar{h})=0~{}.$
(4.23)

These properties imply that the action [101, 102]

$\displaystyle{{\cal S}}^{(m,n)}[h,\bar{h}]={\rm i}^{m+n}\int{\rm
d}^{4}x\,e\,\hat{\mathfrak{C}}^{\alpha(m+n)}(h)\check{\mathfrak{C}}_{\alpha(m+n)}({\bar{h}})+\text{c.c.}$
(4.24)

is locally superconformal. The overall factor of ${\rm i}^{m+n}$ above has
been chosen due to the identity

$\displaystyle{\rm i}^{m+n+1}\int{\rm
d}^{4}x\,e\,\hat{\mathfrak{C}}^{\alpha(m+n)}(h)\check{\mathfrak{C}}_{\alpha(m+n)}({\bar{h}})+\text{c.c.}=0~{},$
(4.25)

which holds up to a total derivative for any conformally flat background.

We now restrict our attention to conformally flat spacetimes. In these
geometries it may be shown that the descendants (4.20) are gauge-invariant
field strengths

$\displaystyle\delta_{\ell}\hat{\mathfrak{C}}_{\alpha(m+n)}(h)=0~{},\quad\delta_{\ell}\check{\mathfrak{C}}_{\alpha(m+n)}({\bar{h}})=0~{}.$
(4.26)

As a result, it follows that the action (4.24) is also gauge-invariant.

####  4.2.2 Superconformal higher-spin gauge models

Having described the CHS fields $h_{\alpha(m){\dot{\alpha}}(n)}$ and computed
their free actions above, we now study their supersymmetric extensions by
utilising the Noether coupling (4.17b). We recall that the first family of
conformal supercurrents take the form $J^{\alpha(m){\dot{\alpha}}(n)}$,
$m,n\geq 1$, and are subject to the conservation equation (4.4). It then
follows that their duals, $H_{\alpha(m){\dot{\alpha}}(n)}$, are defined
modulo:

$\displaystyle\delta_{\zeta,\lambda}H_{\alpha(m){\dot{\alpha}}(n)}=\bar{\nabla}_{{\dot{\alpha}}i}\zeta_{\alpha(m){\dot{\alpha}}(n-1)}{}^{i}+\nabla_{\alpha}^{i}\lambda_{\alpha(m-1){\dot{\alpha}}(n)i}~{},$
(4.27a) where $\zeta_{\alpha(m){\dot{\alpha}}(n-1)}{}^{i}$ and
$\lambda_{\alpha(m-1){\dot{\alpha}}(n)i}$ are complex unconstrained. For
$m=n$, the flat analogue of the tranformation law (4.27a) first appeared in
[99]. Further, in the ${\cal N}=1$ case, $H_{{\alpha{\dot{\alpha}}}}$
describes the conformal supergravity multiplet [100] and the gauge
transformations (4.27a) were proposed in [101], see also [102].

When $m\geq 1,~{}n=0$, the supercurrents take the form $J^{\alpha(m)}$ and
satisfy eq. (4.6). It is clear that their corresponding prepotentials,
$H_{\alpha(m)}$, are characterised by the gauge transformation law

$\displaystyle\delta_{\zeta,\lambda}H_{\alpha(m)}=\bar{\nabla}_{ij}\zeta_{\alpha(m)}{}^{ij}+\nabla_{\alpha}^{i}\lambda_{\alpha(m-1)i}~{},$
(4.27b)

where $\zeta_{\alpha(m)}{}^{ij}$ and $\lambda_{\alpha(m-1)i}$ are complex
unconstrained. For ${\cal N}=1$, the prepotential $H_{\alpha}$ describes the
conformal gravitino [154] multiplet.

Finally, for $m=n=0$, we obtain the real scalar supercurrent $J$. It is
constrained by (4.8) and its dual, $H$, is real and defined modulo the gauge
transformations:

$\displaystyle\delta_{\zeta}H=\bar{\nabla}_{ij}\zeta^{ij}+\text{c.c.}$ (4.27c)

The gauge parameter $\zeta^{ij}$ is complex unconstrained. It should be noted
that, for ${\cal N}=1$ and ${\cal N}=2$, $H$ describes a vector multiplet, see
e.g. [39], and the conformal supergravity multiplet [152, 99, 123],
respectively.

##### The $H_{\alpha(m){\dot{\alpha}}(n)}$ prepotentials

First, we study the prepotentials $H_{\alpha(m){\dot{\alpha}}(n)}$, $m,n\geq
1$, which possess the gauge freedom (4.27a). Requiring that both this
multiplet and its corresponding gauge parameters,
$\zeta_{\alpha(m){\dot{\alpha}}(n-1)}{}^{i}$ and
$\lambda_{\alpha(m-1){\dot{\alpha}}(n)i}$, are primary,

$\displaystyle K^{B}H_{\alpha(m){\dot{\alpha}}(n)}=0~{},\quad
K^{B}\zeta_{\alpha(n){\dot{\alpha}}(n-1)}{}^{i}=0~{},\quad
K^{B}\lambda_{\alpha(m-1){\dot{\alpha}}(n)i}=0~{},$ (4.28)

uniquely fixes the dimension and $\mathsf{U}(1)_{R}$ charge of
$H_{\alpha(m){\dot{\alpha}}(n)}$

$\displaystyle\mathbb{D}H_{\alpha(m){\dot{\alpha}}(n)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(m+n+4{\cal N}-4)H_{\alpha(m){\dot{\alpha}}(n)}~{},$
(4.29a) $\displaystyle\mathbb{Y}H_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=$ $\displaystyle-\frac{{\cal N}(m-n)}{{\cal
N}-4}H_{\alpha(m){\dot{\alpha}}(n)}~{}.$ (4.29b)

This implies that for $m=n=s$, we are able to choose
$H_{\alpha(s){\dot{\alpha}}(s)}$ to be real, in which case (4.27a) reduces to

$\displaystyle\delta_{\zeta}H_{\alpha(s){\dot{\alpha}}(s)}=\bar{\nabla}_{{\dot{\alpha}}i}\zeta_{\alpha(s){\dot{\alpha}}(s-1)}{}^{i}+\text{c.c.}$
(4.30)

From the prepotential $H_{\alpha(m){\dot{\alpha}}(n)}$ and its conjugate
$\bar{H}_{\alpha(n){\dot{\alpha}}(m)}$, we construct the higher-derivative
chiral descendants222In the flat superspace limit, these chiral descendants
reduce to those introduced in [155, 38]. In the ${\cal N}=1$ case their
$\text{AdS}^{4|4}$ superspace extension was given in [101].

$\displaystyle\hat{\mathfrak{W}}_{\alpha(m+n+{{\cal N}})}(H)$ $\displaystyle=$
$\displaystyle-\frac{1}{2^{\cal N}{\cal N}!({\cal N}+1)!}\bar{\nabla}^{2{\cal
N}}(\nabla_{\alpha}{}^{{\dot{\beta}}})^{n}\nabla_{\alpha({\cal
N})}H_{\alpha(m){\dot{\beta}}(n)}~{},$ (4.31a)
$\displaystyle\check{\mathfrak{W}}_{\alpha(m+n+{\cal N})}(\bar{H})$
$\displaystyle=$ $\displaystyle-\frac{1}{2^{\cal N}{\cal N}!({\cal
N}+1)!}\bar{\nabla}^{2{\cal
N}}(\nabla_{\alpha}{}^{{\dot{\beta}}})^{m}\nabla_{\alpha({{\cal
N}})}\bar{H}_{\alpha(n){\dot{\beta}}(m)}~{}.$ (4.31b)

We note that when the gauge prepotential is real $(m=n=s)$, they coincide

$\displaystyle m=n=s\quad\Longrightarrow\quad{\mathfrak{W}}_{\alpha(2s+{{\cal
N}})}(H)\equiv\hat{\mathfrak{W}}_{\alpha(2s+{{\cal
N}})}(H)=\check{\mathfrak{W}}_{\alpha(2s+{{\cal N}})}(H)~{}.$ (4.32)

These chiral descendants have the following dimensions:

$\displaystyle\mathbb{D}\hat{\mathfrak{W}}_{\alpha(m+n+{\cal N})}(H)$
$\displaystyle=$ $\displaystyle\frac{1}{2}(n-m-{{\cal
N}}+4)\hat{\mathfrak{W}}_{\alpha(m+n+{{\cal N}})}(H)~{},$ (4.33a)
$\displaystyle\mathbb{D}\check{\mathfrak{W}}_{\alpha(m+n+{\cal N})}(\bar{H})$
$\displaystyle=$ $\displaystyle\frac{1}{2}(m-n-{{\cal
N}}+4)\check{\mathfrak{W}}_{\alpha(m+n+{{\cal N}})}(\bar{H})~{}.$ (4.33b)

Further, it may be shown that they are primary,

$\displaystyle K^{B}\hat{\mathfrak{W}}_{\alpha(m+n+{{\cal N}})}(H)=0~{},\quad
K^{B}\check{\mathfrak{W}}_{\alpha(m+n+{{\cal N}})}(\bar{H})=0~{}.$ (4.34)

These properties imply that the action

$\displaystyle{\cal S}^{(m,n)}[H,\bar{H}]={\rm i}^{m+n}\int{\rm d}^{4}x{\rm
d}^{2{{\cal N}}}\theta\,{\cal E}\,\hat{\mathfrak{W}}^{\alpha(m+n+{\cal
N})}(H)\check{\mathfrak{W}}_{\alpha(m+n+{\cal N})}(\bar{H})+\text{c.c.}$
(4.35)

is locally superconformal.333We point out that, in the ${\cal N}=1$ case,
action (4.35) was proposed in [101, 102]. The overall factor of ${\rm
i}^{m+n}$ in (4.35) has been chosen due to the identity

$\displaystyle{\rm i}^{m+n+1}\int{\rm d}^{4}x{\rm d}^{2{{\cal
N}}}\theta\,{\cal E}\,\hat{\mathfrak{W}}^{\alpha(m+n+{\cal
N})}(H)\check{\mathfrak{W}}_{\alpha(m+n+{\cal N})}(\bar{H})+\text{c.c.}=0~{},$
(4.36)

which holds up to a total derivative for any conformally flat background.

We restrict our attention to conformally flat superspaces. In these geometries
it may be shown that the chiral descendants (4.31) are gauge-invariant

$\displaystyle\delta_{\zeta,\lambda}\hat{\mathfrak{W}}_{\alpha(m+n+{{\cal
N}})}(H)=0~{},\quad\delta_{\zeta,\lambda}\check{\mathfrak{W}}_{\alpha(m+n+{{\cal
N}})}(\bar{H})=0~{}.$ (4.37)

As a result, the action (4.35) is also gauge-invariant.

##### The $H_{\alpha(m)}$ prepotentials

The gauge prepotentials $H_{\alpha(m)}$, $m\geq 1$ are defined modulo the
transformations (4.27b). Similar to the $n\neq 0$ case, requiring that both
the prepotentials and gauge parameters are superconformally primary,

$\displaystyle K^{B}H_{\alpha(m)}=0~{},\quad
K^{C}\zeta_{\alpha(n)}{}^{ij}=0~{},\quad K^{B}\lambda_{\alpha(m-1)i}=0~{},$
(4.38)

uniquely determines the dimension and $\mathsf{U}(1)_{R}$ charge of
$H_{\alpha(m)}$

$\displaystyle\mathbb{D}H_{\alpha(m)}=-\frac{1}{2}(m+4{\cal
N}-4)H_{\alpha(m)}~{},\quad\mathbb{Y}H_{\alpha(m)}=-\frac{{\cal N}m}{{\cal
N}-4}H_{\alpha(m)}~{}.$ (4.39)

Associated with the prepotential $H_{\alpha(m)}$ (and its conjugate
$\bar{H}_{{\dot{\alpha}}(m)}$) are the chiral descendants:

$\displaystyle\hat{\mathfrak{W}}_{\alpha(m+{{\cal N}})}(H)$ $\displaystyle=$
$\displaystyle-\frac{1}{2^{\cal N}{\cal N}!({\cal N}+1)!}\bar{\nabla}^{2{{\cal
N}}}\nabla_{\alpha({{\cal N}})}H_{\alpha(m)}~{},$ (4.40a)
$\displaystyle\check{\mathfrak{W}}_{\alpha(m+{{\cal N}})}(\bar{H})$
$\displaystyle=$ $\displaystyle-\frac{1}{2^{\cal N}{\cal N}!({\cal
N}+1)!}\bar{\nabla}^{2{{\cal
N}}}(\nabla_{\alpha}{}^{{\dot{\beta}}})^{m}\nabla_{\alpha({{\cal
N}})}\bar{H}_{{\dot{\beta}}(m)}~{}.$ (4.40b)

It may readily be shown that their dimensions are

$\displaystyle\mathbb{D}\hat{\mathfrak{W}}_{\alpha(m+{\cal N})}(H)$
$\displaystyle=$ $\displaystyle-\frac{1}{2}(m+{{\cal
N}}-2)\hat{\mathfrak{W}}_{\alpha(m+{{\cal N}})}(H)~{},$ (4.41a)
$\displaystyle\mathbb{D}\check{\mathfrak{W}}_{\alpha(m+{{\cal N}})}(\bar{H})$
$\displaystyle=$ $\displaystyle\frac{1}{2}(m-{{\cal
N}}+2)\check{\mathfrak{W}}_{\alpha(m+{{\cal N}})}(\bar{H})~{}.$ (4.41b)

and that they are primary

$\displaystyle K^{B}\hat{\mathfrak{W}}_{\alpha(m+{{\cal N}})}(H)=0~{},\quad
K^{B}\check{\mathfrak{W}}_{\alpha(m+{{\cal N}})}(\bar{H})=0~{}.$ (4.42)

As a result, the following action

$\displaystyle{\cal S}^{(m,0)}[H,\bar{H}]={\rm i}^{m}\int{\rm d}^{4}x{\rm
d}^{2{{\cal N}}}\theta\,{\cal E}\,\hat{\mathfrak{W}}^{\alpha(m+{{\cal
N}})}(H)\check{\mathfrak{W}}_{\alpha(m+{{\cal N}})}(\bar{H})+\text{c.c.}$
(4.43)

is locally superconformal. The overall coefficient of ${\rm i}^{m}$ has been
chosen since on conformally flat backgrounds

$\displaystyle{\rm i}^{m+1}\int{\rm d}^{4}x{\rm d}^{2{{\cal N}}}\theta\,{\cal
E}\,\hat{\mathfrak{W}}^{\alpha(m+{\cal
N})}(H)\check{\mathfrak{W}}_{\alpha(m+{\cal N})}(\bar{H})+\text{c.c.}=0~{},$
(4.44)

i.e. it is a total derivative.

In backgrounds with vanishing super-Weyl tensor, a routine calculation allows
us to show that (4.40) are gauge-invariant field strengths

$\displaystyle\delta_{\zeta,\lambda}\hat{\mathfrak{W}}_{\alpha(m+{\cal
N})}(H)=0~{},\quad\delta_{\zeta,\lambda}\check{\mathfrak{W}}_{\alpha(m+{\cal
N})}(\bar{H})=0~{}.$ (4.45)

Thus, the action (4.43) proves to be gauge-invariant.

##### The $H$ prepotential

The scalar gauge prepotential $H=\bar{H}$ possesses the gauge freedom (4.27c).
The requirement that both $H$ and $\zeta^{ij}$ are superconformally primary,

$\displaystyle K^{B}H=0~{},\quad K^{B}\zeta^{ij}=0~{},$ (4.46)

leads to

$\displaystyle\mathbb{D}H=-2({{\cal N}}-1)H~{},\quad\mathbb{Y}H=0~{}.$ (4.47)

From the prepotential $H$, we may construct the single chiral descendant444It
should be noted that for ${\cal N}=1$ and ${\cal N}=2$, the superfield (4.48)
coincides with the field strength of a $\mathsf{U}(1)$ vector multiplet and
the linearised super-Weyl tensor, respectively.

$\displaystyle\mathfrak{W}_{\alpha({{\cal N}})}(H)$ $\displaystyle=$
$\displaystyle-\frac{1}{2^{\cal N}{\cal N}!({\cal N}+1)!}\bar{\nabla}^{2{{\cal
N}}}\nabla_{\alpha({{\cal N}})}H~{}.$ (4.48)

It has the following superconformal properties

$\displaystyle\mathbb{D}\mathfrak{W}_{\alpha({{\cal
N}})}(H)=\frac{1}{2}(4-{\cal N})\mathfrak{W}_{\alpha({{\cal N}})}(H)~{},\quad
K^{B}\mathfrak{W}_{\alpha({{\cal N}})}(H)=0~{},$ (4.49)

which imply that the following action

$\displaystyle{\cal S}^{(0,0)}[H]=\int{\rm d}^{4}x{\rm d}^{2{{\cal
N}}}\theta\,{\cal E}\,\mathfrak{W}^{\alpha({{\cal
N}})}(H)\mathfrak{W}_{\alpha({{\cal N}})}(H)+\text{c.c.}$ (4.50)

is locally superconformal. We note that

$\displaystyle{\rm i}\int{\rm d}^{4}x{\rm d}^{2{{\cal N}}}\theta\,{\cal
E}\,\mathfrak{W}^{\alpha({{\cal N}})}(H)\mathfrak{W}_{\alpha({{\cal
N}})}(H)+\text{c.c.}=0~{},$ (4.51)

is a total derivative in backgrounds with vanishing super-Weyl tensor.

When the background geometry is conformally flat, it is easily shown that

$\displaystyle\delta_{\zeta}\mathfrak{W}_{\alpha({{\cal N}})}(H)=0~{}.$ (4.52)

Hence, (4.50) is gauge-invariant. Actually, in the ${\cal N}=1$ case,
$\mathfrak{W}_{\alpha}(H)$ proves to be gauge-invariant on arbitrary
backgrounds.

####  4.2.3 Superspace and component reduction

This subsection is devoted to a discussion of the component content of the
gauge superfields described above in conformally flat backgrounds.555In the
presence of conformal curvature, the gauge transformations of its constituent
fields may entangle to each other, see [2] for more details. To do this, we
will first reduce all ${\cal N}=N\geq 2$ prepotentials to ${\cal N}=1$
superspace and then complete the picture by performing a component reduction
of each ${\cal N}=1$ gauge multiplet.

We recall that a superconformal gauge multiplet
$H_{\alpha(m){\dot{\alpha}}(n)}$, $m,n\geq 0$, is defined modulo the
transformations (4.27). Utilising this freedom, we construct a Wess-Zumino
gauge on $H_{\alpha(m){\dot{\alpha}}(n)}$ such that the only non-vanishing
${{\cal N}}=N-1$ superfields in its multiplet are:

$\displaystyle\mbox{\boldmath$H$}_{\alpha(m+1){\dot{\alpha}}(n+1)}$
$\displaystyle=\frac{1}{2}[\nabla_{\alpha}^{\underline{N}},\bar{\nabla}_{{\dot{\alpha}}\underline{N}}]H_{\alpha(m){\dot{\alpha}}(n)}|_{\theta^{\alpha}_{\underline{N}}=\bar{\theta}_{\dot{\alpha}}^{\underline{N}}=0}~{},$
(4.53a) $\displaystyle\mbox{\boldmath$\Psi$}_{\alpha(m+1){\dot{\alpha}}(n)}$
$\displaystyle=-\frac{1}{4}\nabla_{\alpha}^{\underline{N}}(\bar{\nabla}_{\underline{N}})^{2}H_{\alpha(m){\dot{\alpha}}(n)}|_{\theta^{\alpha}_{\underline{N}}=\bar{\theta}_{\dot{\alpha}}^{\underline{N}}=0}~{},$
(4.53b) $\displaystyle\mbox{\boldmath$\Phi$}_{\alpha(m){\dot{\alpha}}(n+1)}$
$\displaystyle=-\frac{1}{4}\bar{\nabla}_{{\dot{\alpha}}\underline{N}}({\nabla}^{\underline{N}})^{2}H_{\alpha(m){\dot{\alpha}}(n)}|_{\theta^{\alpha}_{\underline{N}}=\bar{\theta}_{\dot{\alpha}}^{\underline{N}}=0}~{},$
(4.53c) $\displaystyle\mbox{\boldmath$G$}_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=\frac{1}{32}\\{(\nabla^{\underline{N}})^{2},(\bar{\nabla}_{\underline{N}})^{2}\\}H_{\alpha(m){\dot{\alpha}}(n)}|_{\theta^{\alpha}_{\underline{N}}=\bar{\theta}_{\dot{\alpha}}^{\underline{N}}=0}$
$\displaystyle\phantom{=}+\frac{1}{8(m+n+3)}[\mbox{\boldmath$\nabla$}^{\beta\underline{N}},\bar{\mbox{\boldmath$\nabla$}}^{{\dot{\beta}}}_{\underline{N}}]\mbox{\boldmath$H$}_{\alpha(m)\beta{\dot{\alpha}}(n){\dot{\beta}}}$
$\displaystyle\phantom{=}-\frac{{\rm
i}(m-n)}{4(m+n+3)}\mbox{\boldmath$\nabla$}^{{\beta{\dot{\beta}}}}\mbox{\boldmath$H$}_{\alpha(m)\beta{\dot{\alpha}}(n){\dot{\beta}}}~{}.$
(4.53d)

We emphasise that these superfields have been defined such that they are
primary

$\displaystyle K^{B}\mbox{\boldmath$H$}_{\alpha(m+1){\dot{\alpha}}(n+1)}$
$\displaystyle=$ $\displaystyle 0~{},\qquad
K^{B}\mbox{\boldmath$\Psi$}_{\alpha(m+1){\dot{\alpha}}(n)}=0~{},$ (4.54a)
$\displaystyle K^{B}\mbox{\boldmath$\Phi$}_{\alpha(m){\dot{\alpha}}(n+1)}$
$\displaystyle=$ $\displaystyle 0~{},\qquad
K^{B}\mbox{\boldmath$G$}_{\alpha(m){\dot{\alpha}}(n)}=0~{}.$ (4.54b)

Further, owing to the residual gauge freedom associated with this Wess-Zumino
gauge, each prepotential in (4.53) is defined modulo the appropriate gauge
transformations (4.27).

This superspace reduction procedure allows us to systematically reduce any
superconformal higher-spin gauge multiplet to its constituent ${\cal N}=1$
superfields. We now complete our analysis by imposing an appropriate Wess-
Zumino gauge on some ${\cal N}=1$ gauge superfield
$H_{\alpha(m){\dot{\alpha}}(n)}$, $m,n\geq 0$, and then reduce it to
components. In such a gauge the only non-vanishing component fields are

$\displaystyle\mbox{\boldmath$h$}_{\alpha(m+1){\dot{\alpha}}(n+1)}$
$\displaystyle=\frac{1}{2}[\nabla_{\alpha},\bar{\nabla}_{{\dot{\alpha}}}]H_{\alpha(m){\dot{\alpha}}(n)}|_{\theta^{\alpha}=\bar{\theta}_{\dot{\alpha}}=0}~{},$
(4.55a) $\displaystyle\mbox{\boldmath$\psi$}_{\alpha(m+1){\dot{\alpha}}(n)}$
$\displaystyle=-\frac{1}{4}\nabla_{\alpha}\bar{\nabla}^{2}H_{\alpha(m){\dot{\alpha}}(n)}|_{\theta^{\alpha}=\bar{\theta}_{\dot{\alpha}}=0}~{},$
(4.55b) $\displaystyle\mbox{\boldmath$\phi$}_{\alpha(m){\dot{\alpha}}(n+1)}$
$\displaystyle=-\frac{1}{4}\bar{\nabla}_{{\dot{\alpha}}}{\nabla}^{2}H_{\alpha(m){\dot{\alpha}}(n)}|_{\theta^{\alpha}=\bar{\theta}_{\dot{\alpha}}=0}~{},$
(4.55c) $\displaystyle\mbox{\boldmath$g$}_{\alpha(m){\dot{\alpha}}(n)}$
$\displaystyle=\frac{1}{32}\\{\nabla^{2},\bar{\nabla}^{2}\\}H_{\alpha(m){\dot{\alpha}}(n)}|_{\theta^{\alpha}=\bar{\theta}_{\dot{\alpha}}=0}-\frac{{\rm
i}(m-n)}{4(m+n+3)}\mbox{\boldmath$\nabla$}^{{\beta{\dot{\beta}}}}\mbox{\boldmath$h$}_{\alpha(m)\beta{\dot{\alpha}}(n){\dot{\beta}}}~{}.$
(4.55d)

Each component field above has been defined such that it is primary

$\displaystyle
K^{\beta{\dot{\beta}}}\mbox{\boldmath$h$}_{\alpha(m+1){\dot{\alpha}}(n+1)}$
$\displaystyle=$ $\displaystyle 0~{},\qquad
K^{\beta{\dot{\beta}}}\mbox{\boldmath$\psi$}_{\alpha(m+1){\dot{\alpha}}(n)}=0~{},$
(4.56a) $\displaystyle
K^{\beta{\dot{\beta}}}\mbox{\boldmath$\phi$}_{\alpha(m){\dot{\alpha}}(n+1)}$
$\displaystyle=$ $\displaystyle 0~{},\qquad
K^{\beta{\dot{\beta}}}\mbox{\boldmath$g$}_{\alpha(m){\dot{\alpha}}(n)}=0~{}.$
(4.56b)

Additionally, due to the residual gauge freedom associated with this Wess-
Zumino gauge, each component field in (4.55) is defined modulo the gauge
transformations (4.18), provided that it possesses at least one undotted and
one dotted spinor index. The component fields (4.55) carrying only (un)dotted
spinor indices constitute conformal non-gauge fields [102]. These are known to
play a key role in the description of gauge-invariant models for CHS fields in
backgrounds with Weyl curvature, see e.g. [156].

###  4.3 Duality-invariant models for (S)CHS fields

Having identified the (S)CHS gauge multiplets and computed their corresponding
gauge-invariant free actions above, we now construct models for these
(super)fields with non-trivial dynamics. In particular, building on the
success of the so-called “ModMax” theory [157], see also [158], and its ${\cal
N}=1$ extension [159, 160], we formulate the theory of $\mathsf{U}(1)$ duality
invariant models for (S)CHS fields in conformally flat backgrounds.

####  4.3.1 Duality-invariant CHS models

Consider a dynamical system describing the propagation of a conformal spin-$s$
field $h_{\alpha(s){\dot{\alpha}}(s)}$, with $s\geq 1$, in a conformally flat
spacetime. Its action functional ${\cal
S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}}]$ is assumed to depend only on the
field strength $\mathfrak{C}_{\alpha(2s)}$666For notational simplicity we use
the shorthand $\mathfrak{C}_{\alpha(2s)}\equiv\mathfrak{C}_{\alpha(2s)}(h)$.
(and its conjugate $\bar{\mathfrak{C}}_{{\dot{\alpha}}(2s)}$), defined in
(4.20). It is important to note that the latter obeys the Bianchi identity

$(\nabla^{\beta}{}_{{\dot{\alpha}}})^{s}\mathfrak{C}_{\alpha(s)\beta(s)}=(\nabla_{\alpha}{}^{{\dot{\beta}}})^{s}\bar{\mathfrak{C}}_{{\dot{\alpha}}(s){\dot{\beta}}(s)}~{}.$
(4.57)

Now, we assume that ${\cal S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}}]$ is
extended to be a functional of an unconstrained field
$\mathfrak{C}_{\alpha(2s)}$ and its conjugate. We then introduce the dual
field

${\rm i}\mathfrak{M}_{\alpha(2s)}:=\frac{\delta{\cal
S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}}]}{\delta\mathfrak{C}^{\alpha(2s)}}~{},$
(4.58)

where we have defined

$\delta{\cal S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}}]=\int{\rm
d}^{4}x\,e\,\delta\mathfrak{C}^{\alpha(2s)}\frac{\delta{\cal
S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}}]}{\delta\mathfrak{C}^{\alpha(2s)}}+\text{c.c.}$
(4.59)

Varying ${\cal S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}}]$ with respect to
$h_{\alpha(s){\dot{\alpha}}(s)}$ yields the equation of motion

$(\nabla^{\beta}{}_{{\dot{\alpha}}})^{s}\mathfrak{M}_{\alpha(s)\beta(s)}=(\nabla_{\alpha}{}^{{\dot{\beta}}})^{s}\bar{\mathfrak{M}}_{{\dot{\alpha}}(s){\dot{\beta}}(s)}~{}.$
(4.60)

A crucial outcome of our analysis above is that the functional form of the
equation of motion (4.60) mirrors that of the Bianchi identity (4.57). As a
result, it is clear that their union is invariant under the
$\mathsf{SO}(2)\cong\mathsf{U}(1)$ duality transformations:

$\delta_{\lambda}\mathfrak{C}_{\alpha(2s)}=\lambda\mathfrak{M}_{\alpha(2s)}~{},\quad\delta_{\lambda}\mathfrak{M}_{\alpha(2s)}=-\lambda\mathfrak{C}_{\alpha(2s)}~{},$
(4.61)

where $\lambda$ is a constant, real parameter. One may then obtain two
equivalent expressions for the variation of ${\cal
S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}}]$ with respect to (4.61)

$\delta_{\lambda}{\cal S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}}]=\frac{{\rm
i}\lambda}{4}\int{\rm
d}^{4}x\,e\,\Big{\\{}\mathfrak{C}^{2}-\mathfrak{M}^{2}\Big{\\}}+\text{c.c.}=-\frac{{\rm
i}\lambda}{2}\int{\rm d}^{4}x\,e\,\mathfrak{M}^{2}+\text{c.c.}~{},$ (4.62)

as a generalisation of similar derivations in nonlinear electrodynamics [161,
162, 163].777In (4.62) we have employed the notational shorthand
$T^{2}=T^{\alpha(m)}T_{\alpha(m)}$ (similarly
$\bar{T}^{2}=\bar{T}_{{\dot{\alpha}}(m)}\bar{T}^{{\dot{\alpha}}(m)}$). This
implies the self-duality equation

$\text{Im}\int{\rm
d}^{4}x\,e\,\Big{\\{}\mathfrak{C}^{2}+\mathfrak{M}^{2}\Big{\\}}=0~{},$ (4.63)

which must hold for an unconstrained field $\mathfrak{C}_{\alpha(2s)}$ and its
conjugate. The simplest solution of (4.63) is the free CHS model (4.24).888For
the free CHS model (4.24), one can also consider scale transformations in
addition to the $\mathsf{U}(1)$ duality ones (4.61), which is similar to the
case of electrodynamics discussed, e.g., in [163]. The self-duality equation
for the CHS fields $h_{\alpha(m){\dot{\alpha}}(n)}$, with $m,n\geq 1$, is
derived in appendix 4.A.

In the $s=1$ case, the self-duality equation (4.63) was originally derived by
Bialynicki-Birula [164] and independently re-discovered by Gibbons and Rasheed
in 1995 [165]. Two years later, it was again re-derived by Gaillard and Zumino
[161] with the aid of their formalism developed in 1981 [166] but originally
applied only in the linear case.

If we allow for the action ${\cal S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}}]$
to depend on a dimension-$1$ conformal compensator $\Xi$, then the family of
$\mathsf{U}(1)$ duality-invariant theories is very large. For instance, the
following $\mathsf{U}(1)$ duality-invariant model

$\displaystyle{\cal
S}^{(s)}_{\text{BI}}[\mathfrak{C},\bar{\mathfrak{C}};\Xi]=-\int{\rm
d}^{4}x\,e\,\Xi^{4}\bigg{\\{}1-\bigg{(}1+(-1)^{s}\frac{\mathfrak{C}^{2}+\bar{\mathfrak{C}}^{2}}{\Xi^{4}}+\frac{(\mathfrak{C}^{2}-\mathfrak{C}^{2})^{2}}{4\Xi^{8}}\bigg{)}^{\frac{1}{2}}\bigg{\\}}$
(4.64)

is a higher-spin generalisation of Born-Infeld electrodynamics [167].999 The
latter is obtained from (4.64) for $s=1$ by making use of local dilatation
transformations to impose a gauge condition $\Xi^{2}=g^{-1}={\rm const}$.
Owing to the dependence of ${\cal
S}^{(s)}_{\text{BI}}[\mathfrak{C},\bar{\mathfrak{C}};\Xi]$ on $\Xi$, it is
clear that (4.64) is not conformal. Another important solution to (4.63) is
the following one-parameter duality-invariant extension of (4.64)

$\displaystyle{\cal
S}^{(s)}_{\text{BIgen}}[\mathfrak{C},\bar{\mathfrak{C}};\Xi]$
$\displaystyle=-\int{\rm
d}^{4}x\,e\,\Xi^{4}\bigg{\\{}1-\bigg{(}1+\frac{2}{\Xi^{4}}\bigg{[}\frac{(-1)^{s}}{2}\cosh\gamma(\mathfrak{C}^{2}+\bar{\mathfrak{C}}^{2})+\sinh\gamma(\mathfrak{C}^{2}\bar{\mathfrak{C}}^{2})^{\frac{1}{2}}\bigg{]}$
$\displaystyle\qquad\qquad\qquad\qquad+\frac{(\mathfrak{C}^{2}-\mathfrak{C}^{2})^{2}}{4\Xi^{8}}\bigg{)}^{\frac{1}{2}}\bigg{\\}}~{},\qquad\gamma\in\mathbb{R}~{}.$
(4.65)

For $s=1$ this model was introduced in [168].

##### Self-duality under Legendre transformation

In the case of nonlinear (super) electrodynamics, $\mathsf{U}(1)$ duality
invariance implies self-duality under Legendre transformations, see [163] for
a review. This remarkable property proves to extend to the higher-spin case,
as will be shown below.

We start by describing a Legendre transformation for a generic theory with
action ${\cal S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}}]$. For this we
introduce the parent action

$\displaystyle{\cal S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}},\mathfrak{C}^{\rm
D},\bar{\mathfrak{C}}^{\rm D}]={\cal
S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}}]+\int{\rm
d}^{4}x\,e\,\Big{(}\frac{{\rm i}}{2}\mathfrak{C}^{\alpha(2s)}\mathfrak{C}^{\rm
D}_{\alpha(2s)}+\text{c.c.}\Big{)}~{}.$ (4.66)

Here $\mathfrak{C}_{\alpha(2s)}$ is an unconstrained field and
$\mathfrak{C}^{\rm D}_{\alpha(2s)}$ takes the form

$\displaystyle\mathfrak{C}^{\rm
D}_{\alpha(2s)}=(\nabla_{\alpha}{}^{{\dot{\beta}}})^{s}h^{\rm
D}_{\alpha(s){\dot{\beta}}(s)}~{},$ (4.67)

where $h^{\rm D}_{\alpha(s){\dot{\alpha}}(s)}$ is a Lagrange multiplier field.
Indeed, upon varying (4.66) with respect to $h^{\rm
D}_{\alpha(s){\dot{\alpha}}(s)}$ one obtains the Bianchi identity (4.57), and
its general solution is given by eq. (4.20), for some real field
$h_{\alpha(s){\dot{\alpha}}(s)}$. As a result the second term in (4.66)
becomes a total derivative, and we end up with the original action ${\cal
S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}}]$. Alternatively, if we first vary
(4.66) with respect to $\mathfrak{C}^{\alpha(2s)}$, the equation of motion is

$\displaystyle\mathfrak{M}_{\alpha(2s)}=-\mathfrak{C}^{\rm
D}_{\alpha(2s)}~{},$ (4.68)

which we may solve to express $\mathfrak{C}_{\alpha(2s)}$ as a function of
$\mathfrak{C}^{\rm D}_{\alpha(2s)}$ and its conjugate. Inserting this solution
into (4.66), we obtain the dual model

$\displaystyle{\cal S}^{(s)}_{\rm D}[\mathfrak{C}^{\rm
D},\bar{\mathfrak{C}}^{\rm D}]:=\Big{[}{\cal
S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}}]+\int{\rm
d}^{4}x\,e\,\Big{(}\frac{{\rm i}}{2}\mathfrak{C}^{\alpha(2s)}\mathfrak{C}^{\rm
D}_{\alpha(2s)}+\text{c.c.}\Big{)}\Big{]}\Big{|}_{\mathfrak{C}=\mathfrak{C}(\mathfrak{C}^{\rm
D},\bar{\mathfrak{C}}^{\rm D})}~{}.$ (4.69)

Now, given an action ${\cal S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}}]$
satisfying (4.63), our aim is to show that it satisfies

$\displaystyle{\cal S}^{(s)}_{\rm D}[\mathfrak{C},\bar{\mathfrak{C}}]={\cal
S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}}]~{},$ (4.70)

which means that the corresponding Lagrangian is invariant under Legendre
transformations. A routine calculation allows one to show that the following
functional

$\displaystyle{\cal S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}}]+\int{\rm
d}^{4}x\,e\,\Big{(}\frac{{\rm
i}}{4}\mathfrak{C}^{\alpha(2s)}\mathfrak{M}_{\alpha(2s)}+\text{c.c.}\Big{)}$
(4.71)

is invariant under (4.61). The latter may be exponentiated to obtain the
finite $\mathsf{U}(1)$ duality transformations

$\displaystyle\mathfrak{C}^{\prime}_{\alpha(2s)}=\text{cos}\lambda\,\mathfrak{C}_{\alpha(2s)}+\text{sin}\lambda\,\mathfrak{M}_{\alpha(2s)}~{},\quad\mathfrak{M}^{\prime}_{\alpha(2s)}=-\text{sin}\lambda\,\mathfrak{C}_{\alpha(2s)}+\text{cos}\lambda\,\mathfrak{M}_{\alpha(2s)}~{}.$
(4.72)

Performing such a transformation with $\lambda=\frac{\pi}{2}$ on (4.71) yields

$\displaystyle{\cal S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}}]={\cal
S}^{(s)}[\mathfrak{C}^{\rm D},\bar{\mathfrak{C}}^{\rm D}]-\int{\rm
d}^{4}x\,e\,\Big{(}\frac{{\rm i}}{2}\mathfrak{C}^{\alpha(2s)}\mathfrak{C}^{\rm
D}_{\alpha(2s)}+\text{c.c.}\Big{)}~{}.$ (4.73)

Upon inserting this expression into (4.69), we obtain (4.70).

In the above analysis, we made use of the fact that the general solution of
the Bianchi identity (4.57) is given by (4.20). To justify this claim, it
suffices to work in Minkowski space. Let $\mathfrak{C}_{\alpha(2s)}$ be a
field subject to the equation (4.57), with $\nabla_{a}=\partial_{a}$.
Introduce its descendant defined by

$\displaystyle
h^{\perp}_{\alpha(s){\dot{\alpha}}(s)}:=(\partial^{\beta}{}_{{\dot{\alpha}}})^{s}\mathfrak{C}_{\alpha(s)\beta(s)}$
(4.74a) which is automatically transverse,
$\displaystyle\partial^{\beta{\dot{\beta}}}h^{\perp}_{\beta\alpha(s-1){\dot{\beta}}{\dot{\alpha}}(s-1)}$
$\displaystyle=$ $\displaystyle 0~{}.$ (4.74b)

The Bianchi identity (4.57) tells us that
$h^{\perp}_{\alpha(s){\dot{\alpha}}(s)}$ is real,
$\overline{h^{\perp}_{\alpha(s){\dot{\alpha}}(s)}}=h^{\perp}_{\alpha(s){\dot{\alpha}}(s)}$.
Now we can express $\mathfrak{C}_{\alpha(2s)}$ in terms of
$h^{\perp}_{\alpha(s){\dot{\alpha}}(s)}$,

$\displaystyle\mathfrak{C}_{\alpha(2s)}=\square^{-s}(\partial_{\alpha}{}^{{\dot{\beta}}})^{s}h^{\perp}_{\alpha(s){\dot{\beta}}(s)}=(\partial_{\alpha}{}^{{\dot{\beta}}})^{s}h_{\alpha(s){\dot{\beta}}(s)}~{}.$
(4.75)

In the final relation the real field $h_{\alpha(s){\dot{\alpha}}(s)}$ is not
assumed to be transverse. This field is related to
$\square^{-s}h^{\perp}_{\alpha(s){\dot{\alpha}}(s)}$ by a gauge transformation

$\displaystyle\delta_{\ell}h_{\alpha(s){\dot{\alpha}}(s)}=\partial_{{\alpha{\dot{\alpha}}}}\ell_{\alpha(s-1){\dot{\alpha}}(s-1)}~{},$
(4.76)

with a real gauge parameter $\ell_{\alpha(s-1){\dot{\alpha}}(s-1)}$. Our
consideration may be extended to the supersymmetric case considered in section
4.3.2.

##### Auxiliary variable formulation

As a generalisation of the Ivanov-Zupnik [169, 170, 171] approach, here we
will introduce a powerful formalism to generate duality-invariant models that
makes use of auxiliary variables.

Consider the following action functional

$\displaystyle{\cal S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}},\rho,\bar{\rho}]$
$\displaystyle=(-1)^{s}\int{\rm
d}^{4}x\,e\,\Big{\\{}2\rho\mathfrak{C}-\rho^{2}-\frac{1}{2}\mathfrak{C}^{2}\Big{\\}}+\text{c.c.}+\mathfrak{S}^{(s)}_{\text{int}}[\rho,\bar{\rho}]~{}.$
(4.77)

Here we have introduced the auxiliary variable $\rho_{\alpha(2s)}$, which is
an unconstrained primary dimension-2 field,

$\displaystyle
K^{\beta{\dot{\beta}}}\mathcal{\rho}_{\alpha(2s)}=0~{},\qquad{\mathbb{D}}\mathcal{\rho}_{\alpha(2s)}=2\mathcal{\rho}_{\alpha(2s)}~{}.$
(4.78)

The functional $\mathfrak{S}^{(s)}_{\text{int}}[\rho,\bar{\rho}]$, by
definition, contains cubic and higher powers of $\rho_{\alpha(2s)}$ and its
conjugate. The equation of motion for $\rho^{\alpha(2s)}$ is

$\rho_{\alpha(2s)}=\mathfrak{C}_{\alpha(2s)}+\frac{(-1)^{s}}{2}\frac{\delta\mathfrak{S}^{(s)}_{\text{int}}[\rho,\bar{\rho}]}{\delta\rho^{\alpha(2s)}}~{}.$
(4.79)

Equation (4.79) allows one to express $\rho_{\alpha(2s)}$ as a functional of
$\mathfrak{C}_{\alpha(2s)}$ and its conjugate. This means that (4.77) is
equivalent to a CHS theory with action

$\displaystyle{\cal S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}}]$
$\displaystyle=\frac{(-1)^{s}}{2}\int{\rm
d}^{4}x\,e\,\mathfrak{C}^{2}+\text{c.c.}+{{\cal
S}}^{(s)}_{\text{int}}[\mathfrak{C},\bar{\mathfrak{C}}]~{}.$ (4.80)

Thus, (4.77) and (4.80) provide two equivalent realisations of the same model.

The power of this formulation is most evident when the self-duality equation
(4.63) is applied. A routine computation reveals that this constraint is
equivalent to

$\text{Im}\int{\rm
d}^{4}x\,e\,\rho^{\alpha(2s)}\frac{\delta\mathfrak{S}^{(s)}_{\text{int}}[\rho,\bar{\rho}]}{\delta\rho^{\alpha(2s)}}=0~{}.$
(4.81)

Thus, self-duality of the action (4.77) is equivalent to the requirement that
$\mathcal{S}^{(s)}_{\text{int}}[\rho,\bar{\rho}]$ is invariant under rigid
$\mathsf{U}(1)$ phase transformations

$\mathfrak{S}^{(s)}_{\text{int}}[{\rm e}^{{\rm i}\varphi}\rho,{\rm e}^{-{\rm
i}\varphi}\bar{\rho}]=\mathfrak{S}^{(s)}_{\text{int}}[\rho,\bar{\rho}]~{},\quad\varphi\in\mathbb{R}~{}.$
(4.82)

As an example of a solution to (4.82), we can consider the model

$\displaystyle\mathfrak{S}^{(s)}_{\text{int}}[\rho,\bar{\rho};\Xi]=\int{\rm
d}^{4}x\,e\,\Xi^{4}{\mathfrak{F}}\Big{(}\frac{\mathcal{\rho}^{2}\bar{\mathcal{\rho}}^{2}}{\Xi^{8}}\Big{)}~{},$
(4.83)

where ${\mathfrak{F}}(x)$ is a real analytic function of a real variable.
However, such models are not conformal if the action depends on $\Xi$. Thus
condition of conformal invariance imposes additional nontrivial restrictions.

##### Conformal $\mathsf{U}(1)$ duality-invariant models

In the $s=1$ case, there is a unique conformal $\mathsf{U}(1)$ duality-
invariant extension of free electrodynamics proposed in [157] (see also
[158]). This was called “ModMax electrodynamics” in [157]. It turns out that
for $s>1$, families of conformal $\mathsf{U}(1)$ duality-invariant models
exist.

As a warm-up example, let us consider the following nonlinear conformal action

$\displaystyle{\cal
S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}}]=\frac{(-1)^{s}\alpha}{2}\int{\rm
d}^{4}x\,e\,\Big{\\{}\mathfrak{C}^{2}+\bar{\mathfrak{C}}^{2}\Big{\\}}+\beta\int{\rm
d}^{4}x\,e\,\sqrt{\mathfrak{C}^{2}\bar{\mathfrak{C}}^{2}}~{},\quad\alpha,\beta\in\mathbb{R}.$
(4.84)

Requiring this action to obey the self-duality equation (4.63), we obtain the
constraint

$\alpha^{2}-\beta^{2}=1\quad\Longrightarrow\quad\alpha=\text{cosh}\,\gamma~{},\quad\beta=\text{sinh}\,\gamma~{},\quad\gamma\in\mathbb{R}~{}.$
(4.85)

Thus, the nonlinear theory

$\displaystyle{\cal
S}^{(s)}[\mathfrak{C},\bar{\mathfrak{C}}]=\frac{(-1)^{s}\text{cosh}\,\gamma}{2}\int{\rm
d}^{4}x\,e\,\Big{\\{}\mathfrak{C}^{2}+\bar{\mathfrak{C}}^{2}\Big{\\}}+\text{sinh}\,\gamma\int{\rm
d}^{4}x\,e\,\sqrt{\mathfrak{C}^{2}\bar{\mathfrak{C}}^{2}}~{},\quad\gamma\in\mathbb{R},$
(4.86)

is a one-parameter conformal $\mathsf{U}(1)$ duality-invariant extension of
the free CHS action (4.20). In the $s=1$ case our model coincides with ModMax
electrodynamics. It should also be noted that this action may also be obtained
from (4.77) by choosing the interaction

$\displaystyle\mathfrak{S}_{\rm int}^{(s)}[\rho,\bar{\rho}]=\beta\int{\rm
d}^{4}x\,e\,\sqrt{\rho^{2}\bar{\rho}^{2}}~{},\qquad\beta\in\mathbb{R}~{}.$
(4.87)

Upon eliminating the auxiliary variables, it may be shown that the resulting
action coincides with (4.86) upon making the identification

$\displaystyle\cosh\gamma=\frac{1+(\beta/2)^{2}}{1-(\beta/2)^{2}}\quad\Longleftrightarrow\quad\sinh\gamma=\frac{\beta}{1-(\beta/2)^{2}}~{}.$
(4.88)

In order to construct more general models, it is advantageous to make use of
the auxiliary variable formulation described above. We introduce algebraic
invariants of the symmetric rank-$(2s)$ spinor $\rho_{\alpha(2s)}$, which has
the same algebraic properties as $\mathfrak{C}_{\alpha(2s)}$:

$\displaystyle\rho^{2}:=(-1)^{s}\rho_{\alpha(s)}{}^{\beta(s)}\rho_{\beta(s)}{}^{\alpha(s)}~{},\qquad\rho^{3}:=\rho_{\alpha(s)}{}^{\beta(s)}\rho_{\beta(s)}{}^{\gamma(s)}\rho_{\gamma(s)}{}^{\alpha(s)}~{},\qquad\dots$
(4.89)

If $s$ is odd, all invariants $\rho^{2n+1}$, with $n$ a non-negative integer,
vanish.

For simplicity, we restrict our analysis to the conformal graviton, $s=2$. In
this case the family of invariants (4.89) contains only two functionally
independent invariants [172], $\rho^{2}$ and $\rho^{3}$. In particular, one
may show that

$\displaystyle s=2:\qquad\rho^{4}=\frac{1}{2}(\rho^{2})^{2}~{}.$ (4.90)

Now we choose the self-interaction in (4.77) to be of the form

$\displaystyle\mathfrak{S}^{(2)}_{\text{int}}[\rho,\bar{\rho}]=\int{\rm
d}^{4}x\,e\,\Big{\\{}\beta\big{(}\rho^{2}\bar{\rho}^{2}\big{)}^{\frac{1}{2}}+\kappa\big{(}\rho^{3}\bar{\rho}^{3}\big{)}^{\frac{1}{3}}\Big{\\}}~{},$
(4.91)

where $\beta$ and $\kappa$ are real coupling constants. The resulting model is
clearly conformal and $\mathsf{U}(1)$ duality-invariant. For $\kappa\neq 0$,
elimination of the auxiliary variables $\rho_{\alpha(4)}$ and
$\bar{\rho}_{{\dot{\alpha}}(4)}$ does not result in a simple action like
(4.86). In particular, such an elimination, to quadratic order in the
couplings, yields the following self-dual model

$\displaystyle{\cal S}^{(2)}[\mathfrak{C},\bar{\mathfrak{C}}]$
$\displaystyle=$ $\displaystyle\int{\rm
d}^{4}x\,e\,\bigg{\\{}\frac{1}{2}\Big{(}1+\frac{1}{2}\beta^{2}\Big{)}(\mathfrak{C}^{2}+\bar{\mathfrak{C}}^{2})+\beta(\mathfrak{C}^{2}\bar{\mathfrak{C}}^{2})^{\frac{1}{2}}+\kappa(\mathfrak{C}^{3}\bar{\mathfrak{C}}^{3})^{\frac{1}{3}}$
(4.92)
$\displaystyle\quad+\frac{1}{2}\beta\kappa\frac{(\mathfrak{C}^{3})^{2}\bar{\mathfrak{C}}^{2}+(\bar{\mathfrak{C}^{3}})^{2}{\mathfrak{C}}^{2}}{(\mathfrak{C}^{3}\bar{\mathfrak{C}}^{3})^{\frac{2}{3}}(\mathfrak{C}^{2}\bar{\mathfrak{C}}^{2})^{\frac{1}{2}}}+\frac{1}{12}\kappa^{2}\frac{(\mathfrak{C}^{2})^{2}+(\bar{\mathfrak{C}}^{2})^{2}}{(\mathfrak{C}^{3}\bar{\mathfrak{C}}^{3})^{\frac{1}{3}}}$
$\displaystyle\quad-\frac{1}{24}\kappa^{2}\frac{(\mathfrak{C}^{3})^{2}(\bar{\mathfrak{C}}^{2})^{2}+(\bar{\mathfrak{C}^{3}})^{2}({\mathfrak{C}}^{2})^{2}}{(\mathfrak{C}^{3}\bar{\mathfrak{C}}^{3})^{\frac{4}{3}}}+\dots\bigg{\\}}~{}.$

The ellipsis in (4.92) denotes additional contributions to the full nonlinear
theory which are cubic or higher order in the coupling constants. We emphasise
that for the special case $\kappa=0$ the above action yields (4.86).

For $s>2$ the number of algebraic invariants of $\rho_{\alpha(2s)}$ grows, and
therefore one can define families of conformal $\mathsf{U}(1)$ duality-
invariant models.

####  4.3.2 Duality-invariant SCHS models

In this subsection we describe $\mathsf{U}(1)$ duality-invariant models for
SCHS multiplets. Such analyses require us to employ a supersymmetric
generalisation of the formalism described above. For the case of nonlinear
${\cal N}=1$ and ${\cal N}=2$ super electrodynamics, such a formalism was
developed in the rigid supersymmetric case in [173, 163] and extended to
supergravity in [174, 175]. Below, we will describe its extension to the case
of higher-spins.

To begin, we consider a dynamical system describing the propagation of a
superconformal gauge multiplet $H_{\alpha(s){\dot{\alpha}}(s)}$, $s\geq 0$, in
${\cal N}$-extended conformally flat superspace. We assume that its action
${\cal S}^{(s)}[\mathfrak{W},\bar{\mathfrak{W}}]$ depends only on the chiral
field strength $\mathfrak{W}_{\alpha(2s+{\cal N})}$101010As in the non-
supersymmetric case, we make use of the shorthand
$\mathfrak{W}_{\alpha(2s+{\cal N})}\equiv\mathfrak{W}_{\alpha(2s+{\cal
N})}(H)$. (and its conjugate $\bar{\mathfrak{W}}_{{\dot{\alpha}}(2s+{\cal
N})}$), defined in (4.31) ($s>0$) and (4.48) ($s=0$). It is important to note
that the field strengths obey the Bianchi identity

$(\nabla^{\beta}{}_{{\dot{\alpha}}})^{s}\nabla^{\beta({{\cal
N}})}\mathfrak{W}_{\alpha(s)\beta(s+{{\cal N}})}=(-1)^{\cal
N}(\nabla_{\alpha}{}^{{\dot{\beta}}})^{s}\bar{\nabla}^{{\dot{\beta}}({\cal
N})}\bar{\mathfrak{W}}_{{\dot{\alpha}}(s){\dot{\beta}}(s+{{\cal N}})}~{}.$
(4.93)

Here the real superfield
$(\nabla^{\beta}{}_{{\dot{\alpha}}})^{s}\nabla^{\beta({{\cal
N}})}\mathfrak{W}_{\alpha(s)\beta(s+{{\cal N}})}$ proves to be primary.

We assume that ${\cal S}^{(s)}[\mathfrak{W},\bar{\mathfrak{W}}]$ is
consistently defined as a functional of a general chiral superfield
$\mathfrak{W}_{\alpha(2s+{\cal N})}$ and its conjugate. This allows us to
introduce the dual superfield

${\rm i}\mathfrak{M}_{\alpha(2s+{{\cal N}})}:=2\frac{\delta{\cal
S}^{(s)}[\mathfrak{W},\bar{\mathfrak{W}}]}{\delta\mathfrak{W}^{\alpha(2s+{{\cal
N}})}}~{},$ (4.94)

where the variational derivative is defined by

$\delta{\cal S}^{(s)}[\mathfrak{W},\bar{\mathfrak{W}}]=\int{\rm d}^{4}x{\rm
d}^{2{{\cal N}}}\theta\,\mathcal{E}\,\delta\mathfrak{W}^{\alpha(2s+{{\cal
N}})}\frac{\delta{\cal
S}^{(s)}[\mathfrak{W},\bar{\mathfrak{W}}]}{\delta\mathfrak{W}^{\alpha(2s+{{\cal
N}})}}+\text{c.c.}$ (4.95)

It follows that $\mathfrak{M}_{\alpha(2s+{\cal N})}$ is a primary chiral
superfield,

$\displaystyle K^{B}\mathfrak{M}_{\alpha(2s+{\cal
N})}=0~{},\qquad\bar{\nabla}^{\dot{\beta}}_{i}\mathfrak{M}_{\alpha(2s+{\cal
N})}=0~{},\qquad\mathbb{D}\mathfrak{M}_{\alpha(2s+{\cal N})}=\frac{4-{\cal
N}}{2}\mathfrak{M}_{\alpha(2s+{\cal N})}~{}.$ (4.96)

Varying the action with respect to the prepotential
$H_{\alpha(s){\dot{\alpha}}(s)}$ yields the equation of motion

$(\nabla^{\beta}{}_{{\dot{\alpha}}})^{s}\nabla^{\beta({{\cal
N}})}\mathfrak{M}_{\alpha(s)\beta(s+{{\cal N}})}=(-1)^{\cal
N}(\nabla_{\alpha}{}^{{\dot{\beta}}})^{s}\bar{\nabla}^{{\dot{\beta}}({\cal
N})}\bar{\mathfrak{M}}_{{\dot{\alpha}}(s){\dot{\beta}}(s+{{\cal N}})}~{}.$
(4.97)

We emphasise that the real superfield
$(\nabla^{\beta}{}_{{\dot{\alpha}}})^{s}\nabla^{\beta({{\cal
N}})}\mathfrak{M}_{\alpha(s)\beta(s+{{\cal N}})}$ is primary.

The analysis above indicates that the functional form of (4.93) coincides with
that of (4.97). It then follows that their union is invariant under the
$\mathsf{U}(1)$ duality rotations:

$\delta_{\lambda}\mathfrak{W}_{\alpha(2s+{\cal
N})}=\lambda\mathfrak{M}_{\alpha(2s+{\cal
N})}~{},\quad\delta_{\lambda}\mathfrak{M}_{\alpha(2s+{\cal
N})}=-\lambda\mathfrak{W}_{\alpha(2s+{\cal N})}~{},$ (4.98)

where $\lambda$ is a constant, real parameter. One may then obtain two
equivalent expressions for the variation of ${\cal
S}^{(s)}[\mathfrak{W},\bar{\mathfrak{W}}]$ with respect to (4.98)

$\displaystyle\delta_{\lambda}{\cal S}^{(s)}[\mathfrak{W},\bar{\mathfrak{W}}]$
$\displaystyle=$ $\displaystyle\frac{{\rm i}\lambda}{4}\int{\rm d}^{4}x\,{\rm
d}^{2{{\cal N}}}\theta\,{\cal
E}\,\Big{\\{}\mathfrak{W}^{2}-\mathfrak{M}^{2}\Big{\\}}+\text{c.c.}$ (4.99)
$\displaystyle=$ $\displaystyle-\frac{{\rm i}\lambda}{2}\int{\rm d}^{4}x\,{\rm
d}^{2{\cal N}}\theta\,{\cal E}\,\mathfrak{M}^{2}+\text{c.c.}~{},$

as a generalisation of similar derivations in nonlinear ${\cal N}=1$ and
${\cal N}=2$ supersymmetric electrodynamics [173, 163, 174, 175]. This implies
the self-duality equation

$\text{Im}\int{\rm d}^{4}x{\rm d}^{2{{\cal
N}}}\theta\,\mathcal{E}\,\Big{\\{}\mathfrak{W}^{2}+\mathfrak{M}^{2}\Big{\\}}=0~{},$
(4.100)

which must hold for a general chiral superfield $\mathfrak{W}_{\alpha(2s+{\cal
N})}$ and its conjugate. Every solution ${\cal
S}^{(s)}[\mathfrak{W},\bar{\mathfrak{W}}]$ of the self-duality equation
describes a $\mathsf{U}(1)$ duality-invariant theory. For the $s=0$ case,
equation (4.100) was originally derived in the ${\cal N}=1$ (${\cal N}=2$)
case in [173] ([163]) in Minkowski superspace and extended to supergravity in
[174] ([175]). The simplest solutions of the self-duality equation (4.100) are
the free SCHS models (4.35) ($s>0$) and (4.50) ($s=0$).

The above results allow one to prove, in complete analogy with the non-
supersymmetric analysis conducted in section 4.3.1, that $\mathsf{U}(1)$
duality-invariant theories are self-dual under Legendre transformations.

##### Auxiliary variable formulation

As a generalisation of the auxiliary variable formalism sketched in section
4.3.1, here we will develop a reformulation of the supersymmetric
$\mathsf{U}(1)$ duality-invariant theories introduced above. In the $s=0$
case, it will reduce to the auxiliary superfield approach for $\mathsf{U}(1)$
duality-invariant supersymmetric electrodynamics introduced in [176, 177].

Consider the action functional

$\displaystyle{\cal S}^{(s)}[\mathfrak{W},\bar{\mathfrak{W}},\eta,\bar{\eta}]$
$\displaystyle=(-1)^{s}\int{\rm d}^{4}x{\rm d}^{2{{\cal
N}}}\theta\,\mathcal{E}\,\Big{\\{}\eta\mathfrak{W}-\frac{1}{2}\eta^{2}-\frac{1}{4}\mathfrak{W}^{2}\Big{\\}}+\text{c.c.}+\mathfrak{S}^{(s)}_{\text{int}}[\eta,\bar{\eta}]~{}.$
(4.101)

Here we have introduced the new multiplet $\eta_{\alpha(2s+{\cal N})}$, which
is required to be primary and covariantly chiral,

$\displaystyle K^{B}\eta_{\alpha(2s+{{\cal
N}})}=0~{},\qquad\bar{\nabla}^{\dot{\beta}}_{i}\eta_{\alpha(2s+{{\cal
N}})}=0~{},\qquad{\mathbb{D}}\eta_{\alpha(2s+{{\cal N}})}=\frac{4-{\cal
N}}{2}\eta_{\alpha(2s+{{\cal N}})}~{}.$ (4.102)

By definition, the functional
$\mathfrak{S}^{(s)}_{\text{int}}[\eta,\bar{\eta}]$ contains cubic and higher
powers of $\eta_{\alpha(2s+{\cal N})}$ and its conjugate.

The equation of motion for $\eta^{\alpha(2s+{{\cal N}})}$ is

$\eta_{\alpha(2s+{{\cal N}})}=\mathfrak{W}_{\alpha(2s+{{\cal
N}})}+(-1)^{s}\frac{\delta\mathfrak{S}^{(s)}_{\text{int}}[\eta,\bar{\eta}]}{\delta\eta^{\alpha(2s+{{\cal
N}})}}~{}.$ (4.103)

Employing perturbation theory, equation (4.103) allows one to express
$\eta_{\alpha(2s+{{\cal N}})}$ as a functional of
$\mathfrak{W}_{\alpha(2s+{{\cal N}})}$ and its conjugate. This means that
(4.101) is dual to a SCHS theory with action

$\displaystyle{\cal S}^{(s)}[\mathfrak{W},\bar{\mathfrak{W}}]$
$\displaystyle=\frac{(-1)^{s}}{4}\int{\rm d}^{4}x{\rm d}^{2{\cal
N}}\theta\,\mathcal{E}\,\mathfrak{W}^{2}+\text{c.c.}+{{\cal
S}}^{(s)}_{\text{int}}[\mathfrak{W},\bar{\mathfrak{W}}]~{}.$ (4.104)

Thus, (4.101) and (4.104) provide two equivalent realisations of the same
model.

The power of this formulation is most evident when the self duality equation
(4.100) is applied. A routine computation reveals that this constraint is
equivalent to

$\text{Im}\int{\rm d}^{4}x{\rm d}^{2{{\cal
N}}}\theta\,\mathcal{E}\,\eta^{\alpha(2s+{{\cal
N}})}\frac{\delta\mathfrak{S}^{(s)}_{\text{int}}[\eta,\bar{\eta}]}{\delta\eta^{\alpha(2s+{{\cal
N}})}}=0~{}.$ (4.105)

Thus, self-duality of the action (4.101) is equivalent to the requirement that
$\mathfrak{S}^{(s)}_{\text{int}}[\eta,\bar{\eta}]$ is invariant under rigid
$\mathsf{U}(1)$ phase transformations

$\mathfrak{S}^{(s)}_{\text{int}}[{\rm e}^{{\rm i}\varphi}\eta,{\rm e}^{-{\rm
i}\varphi}\bar{\eta}]=\mathfrak{S}^{(s)}_{\text{int}}[\eta,\bar{\eta}]~{},\quad\varphi\in\mathbb{R}~{}.$
(4.106)

Having formulated the general theory for ${\cal N}$-extended duality invariant
models above, for the remainder of this subsection we employ this formalism to
derive some interesting supersymmetric duality-invariant higher-spin models.

#####  ${\cal N}=1$ duality-invariant models

Here we will construct some interesting duality-invariant models for ${\cal
N}=1$ superconformal gauge multiplets. To this end, we permit the action
(4.101) to depend on a dimension-2 conformal compensator $\Xi$111111A simple
realisation for the conformal compensator is $\Xi=S_{0}\bar{S}_{0}$, where
$S_{0}$ is the chiral compensator of old minimal supergravity, see section
3.5.2., allowing us to generate families of theories satisfying the self-
duality constraint (4.106). First, we consider the following interaction

$\displaystyle\mathfrak{S}^{(s)}_{\text{int}}[\eta,\bar{\eta};\Xi]$
$\displaystyle=$ $\displaystyle\frac{1}{2(2s+2)!}\int{\rm d}^{4}x{\rm
d}^{2}\theta{\rm
d}^{2}\bar{\theta}\,E\,\frac{(\eta^{2}\bar{\eta}^{2})^{s+1}}{\Xi^{3s+2}}\mathfrak{F}^{(s)}(v,\bar{v})~{},$
(4.107a) where $\displaystyle
v:=\frac{1}{8}\nabla^{2}\big{[}{\eta^{2}}{\Xi^{-2}}\big{]}~{}.$ (4.107b)

Applying (4.106), we find that our model is $\mathsf{U}(1)$ duality invariant
provided

$\displaystyle\mathfrak{F}^{(s)}(v,\bar{v})=\mathfrak{F}^{(s)}(v\bar{v})~{}.$
(4.108)

The present model is superconformal if the action is independent of $\Xi$.
This uniquely fixes the functional form of $\mathfrak{F}(v\bar{v})$ modulo a
single real parameter

$\mathfrak{F}^{(s)}_{\text{SC}}(v\bar{v})=\frac{\kappa}{(v\bar{v})^{\frac{1}{4}(3s+2)}}~{},\quad\kappa\in\mathbb{R}~{}.$
(4.109)

Employing (4.103), we arrive at

$\displaystyle\mathfrak{W}_{\alpha(2s+1)}=\eta_{\alpha(2s+1)}\bigg{\\{}1$
$\displaystyle+$
$\displaystyle\frac{(-1)^{s}\kappa}{8(2s+2)!}\bar{\nabla}^{2}\Big{[}(2s+2)\frac{\eta^{2s}\bar{\eta}^{2s+2}}{\Xi^{3s+2}}\mathfrak{F}^{(s)}_{\text{SC}}$
(4.110)
$\displaystyle\qquad+\frac{\bar{\eta}^{2s+2}}{4\Xi^{3s+2}}\nabla^{2}\Big{(}\eta^{2s+2}\partial_{v}\mathfrak{F}^{(s)}_{\text{SC}}\Big{)}\Big{]}\bigg{\\}}~{},$

which, along with its conjugate, allows us to integrate out the auxiliary
variables present in (4.107a). The final result, for $s=0$, is the model for
superconformal $\mathsf{U}(1)$ duality-invariant electrodynamics introduced in
[160, 159]

$\displaystyle{\cal S}^{(0)}[\mathfrak{W},\bar{\mathfrak{W}}]$
$\displaystyle=$ $\displaystyle\frac{1}{4}\cosh\gamma\int{\rm d}^{4}x{\rm
d}^{2}\theta\,{\cal E}\,\mathfrak{W}^{2}+{\rm c.c.}$ (4.111)
$\displaystyle+\frac{1}{4}\sinh\gamma\int{\rm d}^{4}x{\rm d}^{2}\theta{\rm
d}^{2}\bar{\theta}\,E\,\frac{\mathfrak{W}^{2}\,{\mathfrak{W}}^{2}}{\Xi^{2}\sqrt{u\bar{u}}}~{},$

where $\gamma$ is a real coupling constant. For $s>0$ the resulting model is

$\displaystyle{\cal S}^{(s>0)}[\mathfrak{W},\bar{\mathfrak{W}}]$
$\displaystyle=\frac{(-1)^{s}}{4}\int{\rm d}^{4}x{\rm
d}^{2}\theta\,\mathcal{E}\,\mathfrak{W}^{2}+\text{c.c.}$
$\displaystyle\qquad+\frac{\kappa}{2(2s+2)!}\int{\rm d}^{4}x{\rm
d}^{2}\theta{\rm
d}^{2}\bar{\theta}\,E\,\frac{(\mathfrak{W}^{2}\bar{\mathfrak{W}}^{2})^{s+1}}{\Xi^{3s+2}(u\bar{u})^{\frac{1}{4}(3s+2)}}~{}.$
(4.112)

In both cases we have made use of the shorthand

$\displaystyle
u=\frac{1}{8}\nabla^{2}\big{[}\mathfrak{W}^{2}\Xi^{-2}\big{]}~{}.$ (4.113)

We note that (4.3.2) is invariant under rescalings of the conformal
compensator

$\displaystyle\Xi\rightarrow{\rm e}^{2\sigma}\Xi~{},$ (4.114)

which implies that its dependence on $\Xi$ is superficial. Thus, (4.3.2) is
superconformal.

It is important to note that, at the component level, the purely bosonic
sector of the interaction (the $\kappa$-term) present in (4.3.2) identically
vanishes.121212The numerator in the $\kappa$-term in (4.3.2) contains a
product of $4(s+1)$ fermionic superfields. Thus, we do not have enough spinor
derivatives to convert all fermionic superfields into bosonic ones for $s>0$.
Thus, these actions describe different duality-invariant models than those
presented in the previous section. However, one may construct a supersymmetric
duality-invariant model that contains, for instance, the bosonic theory (4.86)
at the component level.

Consider the following supersymmetric duality-invariant model

$\displaystyle{\cal
S}^{(s)}[\mathfrak{W},\bar{\mathfrak{W}},\eta,\bar{\eta};\Xi]$
$\displaystyle=(-1)^{s}\int{\rm d}^{4}x{\rm
d}^{2}\theta\,\mathcal{E}\,\Big{\\{}\eta\mathfrak{W}-\frac{1}{2}\eta^{2}-\frac{1}{4}\mathfrak{W}^{2}\Big{\\}}+\text{c.c.}$
$\displaystyle\qquad+\frac{\beta}{8}\int{\rm d}^{4}x{\rm d}^{2}\theta{\rm
d}^{2}\bar{\theta}\,E\,\frac{\eta^{2}\bar{\eta}^{2}}{\Xi^{2}\sqrt{v\bar{v}}}~{},\qquad\beta\in\mathbb{R}~{}.$
(4.115)

It does not enjoy invariance under (4.114), hence it is not superconformal.
However, at the component level, this model contains certain conformal
duality-invariant actions. For this we restrict our attention to the bosonic
sector of (4.3.2). The bosonic field strengths contained in
$\mathfrak{W}_{\alpha(2s+1)}$ are:131313These field strengths can be related
to those of eq. (4.20) by a rescaling.

$\displaystyle\mathfrak{C}_{\alpha(2s+2)}$ $\displaystyle={\rm
i}\nabla_{\alpha}\mathfrak{W}_{\alpha(2s+1)}|_{\theta^{\alpha}=\bar{\theta}_{\dot{\alpha}}=0}~{},\qquad\mathfrak{C}_{\alpha(2s)}=-\frac{4(s+1)}{2s+1}\nabla^{\beta}\mathfrak{W}_{\beta\alpha(2s)}|_{\theta^{\alpha}=\bar{\theta}_{\dot{\alpha}}=0}~{}.$
(4.116a) Similarly, the bosonic component fields of $\eta_{\alpha(2s+1)}$ are:
$\displaystyle\rho_{\alpha(2s+2)}$ $\displaystyle={\rm
i}\nabla_{\alpha}\eta_{\alpha(2s+1)}|_{\theta^{\alpha}=\bar{\theta}_{\dot{\alpha}}=0}~{},\qquad\rho_{\alpha(2s)}=-\frac{4(s+1)}{2s+1}\nabla^{\beta}\eta_{\beta\alpha(2s)}|_{\theta^{\alpha}=\bar{\theta}_{\dot{\alpha}}=0}~{}.$
(4.116b)

Action (4.3.2) can be reduced to components by making use of the reduction
rule (2.89b). Since we are interested only in the bosonic sector, it suffices
to approximate

$\displaystyle
v|_{\theta^{\alpha}=\bar{\theta}_{\dot{\alpha}}=0}\approx-\frac{1}{\chi^{2}}\bigg{\\{}\frac{128(s+1)^{3}}{(2s+1)^{3}}\rho^{\alpha(2s+2)}\rho_{\alpha(2s+2)}+\frac{1}{2}\rho^{\alpha(2s)}\rho_{\alpha(2s)}\bigg{\\}}~{},$
(4.117)

where we have made the definition
$\chi:=\Xi|_{\theta^{\alpha}=\bar{\theta}_{\dot{\alpha}}=0}$. Further, it may
be checked that the interaction (the $\beta$-term) in (4.3.2) does not contain
contributions which are (i) linear in $\rho_{\alpha(2s)}$ and its conjugate;
and (ii) linear in $\rho_{\alpha(2s+2)}$ and its conjugate. Therefore, if we
switch off the spin-$s$ field,

$\displaystyle\mathfrak{C}_{\alpha(2s)}=0~{},$ (4.118a) then
$\displaystyle\rho_{\alpha(2s)}=0$ (4.118b)

is a solution of the corresponding equation of motion. Under these conditions,
the resulting bosonic action proves coincides with ${\cal
S}^{(s+1)}[\mathfrak{C},\bar{\mathfrak{C}},\rho,\bar{\rho}]$, which may be
obtained from (4.77) by performing the shift $s\rightarrow s+1$. We emphasise
that the compensator $\Xi$ does not contribute to this action, and therefore
it is locally conformal. Instead of considering the branch (4.118a), we may
switch off the spin-$(s+1)$ field, $\mathfrak{C}_{\alpha(2s+2)}=0$. Then it
follows that $\rho_{\alpha(2s+2)}=0$ is a solution of the corresponding
equation of motion, and the resulting bosonic action proves to coincide with
the conformal duality-invariant action (4.77).

To conclude this section, we derive a supersymmetric extension of the cubic
interaction present in (4.91). For this purpose, we will fix $s=1$ in (4.101).
The relevant interaction is constructed in terms of the following scalar
descendants of $\eta_{\alpha(3)}$:

$\displaystyle\Lambda=\frac{{\rm
i}}{2}\eta^{\alpha(2)\beta}\eta^{\alpha(2)}{}_{\beta}\nabla_{(\alpha_{1}}\eta_{\alpha_{2}\alpha_{3}\alpha_{4})}~{},\qquad
w=\nabla^{2}\big{(}\Lambda\Xi^{-3}\big{)}~{},$ (4.119)

and the conformal compensator $\Xi$. Using these, we construct the following
action

$\displaystyle{\cal
S}^{(1)}[\mathfrak{W},\bar{\mathfrak{W}},\eta,\bar{\eta};\Xi]$
$\displaystyle=-\int{\rm d}^{4}x{\rm
d}^{2}\theta\,\mathcal{E}\,\Big{\\{}\eta\mathfrak{W}-\frac{1}{2}\eta^{2}-\frac{1}{4}\mathfrak{W}^{2}\Big{\\}}+\text{c.c.}$
$\displaystyle\qquad+16\kappa\int{\rm d}^{4}x{\rm d}^{2}\theta{\rm
d}^{2}\bar{\theta}\,E\,\frac{\Lambda\bar{\Lambda}}{\Xi^{4}(w\bar{w})^{\frac{2}{3}}}~{}.$
(4.120)

We note that it is not superconformal as it lacks invariance under (4.114).
However, we will show that, at the component level, it contains the cubic
interaction present in (4.91). To demonstrate this, we restrict our attention
to the bosonic sector of (4.3.2), which allows us to approximate

$\displaystyle\quad
w|_{\theta^{\alpha}=\bar{\theta}_{\dot{\alpha}}=0}\approx-\frac{8192\sqrt{3}}{243\chi^{3}}\Big{(}\rho^{\alpha(2)}{}_{\beta(2)}\rho^{\beta(2)}{}_{\gamma(2)}\rho^{\gamma(2)}{}_{\alpha(2)}-\frac{9}{32}\rho^{\alpha(2)}\rho^{\alpha(2)}\rho_{\alpha(4)}\Big{)}~{}.$
(4.121)

Further, the interaction (the $\kappa$-term) present in (4.3.2) contains no
terms linear in either $\rho_{\alpha(4)}$ or $\rho_{\alpha(2)}$. Hence, if we
switch off the spin-$1$ field, $\mathfrak{C}_{\alpha(2)}=0$, then
$\rho_{\alpha(2)}=0$ solves the corresponding equations of motion. Under these
conditions, the resulting bosonic action takes the form

$\displaystyle{\cal S}^{(2)}[\mathfrak{C},\bar{\mathfrak{C}},\rho,\bar{\rho}]$
$\displaystyle=\int{\rm
d}^{4}x\,e\,\Big{\\{}2\rho\mathfrak{C}-\rho^{2}-\frac{1}{2}\mathfrak{C}^{2}\Big{\\}}+\text{c.c.}+\kappa\int{\rm
d}^{4}x\,e\,\big{(}\rho^{3}\bar{\rho}^{3}\big{)}^{\frac{1}{3}}~{}.$ (4.122)

In particular, we note that it is locally conformal as it is independent of
the compensator. Thus, (4.3.2) is a supersymmetric extension of the cubic
interaction of (4.91). In closing, we emphasise that if one instead attempts
to consider the branch $\mathfrak{C}_{\alpha(4)}=0$ and
$\mathfrak{C}_{\alpha(2)}\neq 0$, it is not legal to set $\rho_{\alpha(4)}=0$.
Such a choice would imply
$w|_{\theta^{\alpha}=\bar{\theta}_{\dot{\alpha}}=0}\approx 0$, making the
corresponding component action undefined.

#####  ${\cal N}=2$ duality-invariant models

Building on the ${\cal N}=1$ story above, here we derive some $\mathsf{U}(1)$
duality-invariant models in ${\cal N}=2$ superspace. As in the ${\cal N}=1$
case, we begin by allowing (4.101) to depend on a dimension-2 conformal
compensator $\Xi$.141414One may, for instance, choose $\Xi=W_{0}\bar{W}_{0}$,
where $W_{0}$ is the field strength of a compensating vector multiplet, see
section 3.5.3. This greatly simplifies the procedure of constructing possible
interactions satisfying condition (4.106), for instance

$\displaystyle\mathfrak{S}^{(s)}_{\text{int}}[{\eta},\bar{{\eta}};\Xi]=\int{\rm
d}^{4}x{\rm d}^{4}\theta{\rm
d}^{4}\bar{\theta}\,{E}\,{\mathfrak{F}}\Big{(}\frac{{\eta}^{2}\bar{{\eta}}^{2}}{\Xi^{2}}\Big{)}~{}.$
(4.123)

Here ${\mathfrak{F}}(x)$ is a real analytic function of a real variable.

It is not difficult to construct explicit examples of ${\cal N}=2$
superconformal and $\mathsf{U}(1)$ duality-invariant higher-spin theories, as
a generalisation of the ${\cal N}=2$ vector multiplet models presented in
[160]. We consider the self-interaction

$\displaystyle\mathfrak{S}^{(s)}_{\text{int}}[{\eta},\bar{{\eta}}]=c\int{\rm
d}^{4}x{\rm d}^{4}\theta{\rm
d}^{4}\bar{\theta}\,\ln{\eta}^{2}\ln\bar{{\eta}}^{2}~{},$ (4.124)

with $c$ a coupling constant. It is clearly invariant under rigid
$\mathsf{U}(1)$ phase transformations, eq. (4.106), and therefore it generates
a $\mathsf{U}(1)$ duality-invariant model. Additionally, (4.124) is ${\cal
N}=2$ superconformal. The above functional is well defined provided the
auxiliary variables ${\eta}_{\alpha(2s+2)}$ are chosen to belong to the open
domain ${\eta}^{2}\neq 0$.

###  4.4 (S)CHS fields interacting with matter

The theory of an infinite tower of interacting CHS fields is most easily
defined as an induced action, see [104, 105, 178]. For example, given the
gauge-invariant action ${\cal S}[\varphi,\bar{\varphi},h]$ for a massless
complex scalar $\varphi$ coupled to background CHS fields
$h_{\alpha(s){\dot{\alpha}}(s)}$, $s\geq 1$, an effective action may be
defined according to

$\displaystyle e^{{\rm
i}\Gamma_{\text{eff}}[h]}=\int\mathcal{D}\varphi\mathcal{D}\bar{\varphi}\,e^{{\rm
i}\mathcal{S}[\varphi,\bar{\varphi},h]}~{}.$ (4.125)

As $\mathcal{S}[\varphi,\bar{\varphi},h]$ is bilinear in $\varphi$, the latter
may be integrated out in (4.125), and the CHS action may be identified with
the logarithmically divergent part of $\Gamma_{\text{eff}}[h]$, which is local
and invariant under the gauge transformations of the higher-spin fields
$h_{\alpha(s){\dot{\alpha}}(s)}$. The ${\cal N}=1$ supersymmetric extension of
this story is described in [101, 106].

Hence, it is clear that the gauge-invariant action for some (super)conformal
matter multiplets coupled to background (S)CHS fields plays a central role in
this construction. Therefore, it is the objective of this section to derive
such gauge-invariant models where the (super)conformal symmetry is
manifest.151515In the non-supersymmetric case, the action for a complex scalar
coupled to background bosonic CHS fields was considered in [104, 178], though
conformal symmetry was neither discussed nor manifest in their construction.

####  4.4.1 Complex scalar field coupled to bosonic CHS fields

The results of “Closing the gauge algebra” and “Inclusion of auxiliary gauge
fields” below were obtained by Michael Ponds [7].

This subsection is devoted to the study of interactions between a massless
complex scalar $\varphi$ and a tower of background CHS fields
$h_{\alpha(s){\dot{\alpha}}(s)}$, $s\geq 1$. Further, by requiring off-shell
gauge-invariance of the corresponding action, we deduce the non-abelian
extension of the gauge transformations of $h_{\alpha(s){\dot{\alpha}}(s)}$.

##### Differential operators on the space of primary scalar fields

As a first step in building interactions involving $\varphi$, we analyse the
possible transformation rules consistent with its kinematic properties. To
begin, we propose the following linear transformation rule for $\varphi$

$\displaystyle\delta\varphi=-{\mathcal{U}}\varphi~{},\qquad{\mathcal{U}}=\sum_{s=0}^{\infty}{\mathcal{U}}^{(s)},$
(4.126a) where ${\mathcal{U}}^{(s)}$ is a differential operator of maximal
order $s$
$\displaystyle{\mathcal{U}}^{(s)}=\sum_{k=0}^{s}\zeta_{(s)}^{\alpha(k){\dot{\alpha}}(k)}(\nabla_{\alpha{\dot{\alpha}}})^{k}~{}.$
(4.126b)

At this stage, the coefficients $\zeta_{(s)}^{\alpha(k){\dot{\alpha}}(k)}$ are
complex. Operator ${\mathcal{U}}$ must preserve the off-shell properties of
$\varphi$ (3.38), and hence so too must each individual ${\mathcal{U}}^{(s)}$.
As we now show, this implies that the coefficients of each operator (4.126b)
may be expressed in terms of a single parameter, which may ultimately be
identified with $\ell_{\alpha(s){\dot{\alpha}}(s)}$, the gauge parameter
associated with $h_{\alpha(s+1){\dot{\alpha}}(s+1)}$. Keeping in mind the
analysis of section 3.4.1, it is clear that ${\mathcal{U}}^{(s)}$ takes the
form

$\displaystyle{\mathcal{U}}^{(s)}$ $\displaystyle={\rm
i}^{s}\sum_{k=0}^{s}\binom{s}{k}^{2}\binom{2s+2}{s-k}^{-1}(\nabla_{{\beta{\dot{\beta}}}})^{s-k}\zeta^{\alpha(k)\beta(s-k){\dot{\alpha}}(k){\dot{\beta}}(s-k)}(\nabla_{\alpha{\dot{\alpha}}})^{k}~{},$
(4.127)

where we have identified
$\zeta_{(s)}^{\alpha(s){\dot{\alpha}}(s)}\equiv\zeta^{\alpha(s){\dot{\alpha}}(s)}$,
which is an unconstrained complex parameter with the conformal properties:

$\displaystyle\mathbb{D}\zeta^{\alpha(s){\dot{\alpha}}(s)}=-s\zeta^{\alpha(s){\dot{\alpha}}(s)}~{},\qquad
K^{{\beta{\dot{\beta}}}}\zeta^{\alpha(s){\dot{\alpha}}(s)}=0~{}.$ (4.128)

The operator (4.127) is the unique one (modulo normalisation) of the
functional form (4.126b) preserving all off-shell properties of $\varphi$.
However, it is not the most general operator with this property, which would
include terms involving powers of the conformal d’Alembertian in the ansatz
(4.126b). It will be shown below we will see that it is possible to achieve
gauge invariance to order $\mathcal{O}(h)$ without such terms. However, they
turn out to be necessary to close the gauge algebra, hence we will revisit
them shortly.

##### Noether procedure to order $\mathcal{O}(h)$

From $\varphi$ and its conjugate $\bar{\varphi}$ one may construct the rank
$s\geq 0$ descendants

$\displaystyle j_{\alpha(s){\dot{\alpha}}(s)}={\rm
i}^{s}\sum_{k=0}^{s}(-1)^{k}\binom{s}{k}^{2}(\nabla_{\alpha{\dot{\alpha}}})^{k}\varphi(\nabla_{\alpha{\dot{\alpha}}})^{s-k}\bar{\varphi}~{}$
(4.129)

It may be shown that $j_{\alpha(s){\dot{\alpha}}(s)}$ possesses the following
crucial features:

  1. (i)

Reality

$\displaystyle
j_{\alpha(s){\dot{\alpha}}(s)}=\bar{j}_{\alpha(s){\dot{\alpha}}(s)}~{};$
(4.130a)

  2. (ii)

Conformal covariance on arbitrary gravitational backgrounds

$\displaystyle
K^{\beta{\dot{\beta}}}j_{\alpha(s){\dot{\alpha}}(s)}=0~{},\qquad\mathbb{D}j_{\alpha(s){\dot{\alpha}}(s)}=(s+2)j_{\alpha(s){\dot{\alpha}}(s)}~{};$
(4.130b)

  3. (iii)

Transverse on-shell when restricted to conformally flat backgrounds (for
$s>0$)

$\displaystyle
C_{abcd}=0\qquad\implies\qquad\nabla^{\beta{\dot{\beta}}}j_{\beta\alpha(s-1){\dot{\beta}}{\dot{\alpha}}(s-1)}\approx
0~{},$ (4.130c)

where we have made use of the symbol $\approx$ to denote equality on-shell.

The conformal currents (4.129) in Minkowski space were introduced for the
first time by Craigie, Dobrev and Todorov [179], although the method to derive
these currents had been described earlier by Makeenko [180].

The currents (4.129) with $s=1$ and $s=2$ are actually conserved on-shell in
arbitrary gravitational backgrounds. However, it was shown in [181] (building
on the analysis in [182]) that the Weyl tensor $C_{abcd}$ is the obstruction
for this property to hold in the case $s=3$. This is also true for spin $s>3$,
hence for the remainder of this section we restrict ourselves to conformally
flat backgrounds.

The conformal currents $j_{\alpha(s){\dot{\alpha}}(s)}$ naturally couple to
the integer spin-$s$ conformal gauge fields $h_{\alpha(s){\dot{\alpha}}(s)}$
via the cubic Noether coupling

$\displaystyle\mathcal{S}_{\rm{N.C.}}[\varphi,\bar{\varphi},h]$
$\displaystyle=\sum_{s=0}^{\infty}c_{s}\,\mathcal{S}^{(s)}_{\rm{N.C.}}[\varphi,\bar{\varphi},h_{(s)}]~{},$
(4.131a)
$\displaystyle\mathcal{S}^{(s)}_{\rm{N.C.}}[\varphi,\bar{\varphi},h_{(s)}]$
$\displaystyle=\int\text{d}^{4}x\,e\,h_{\alpha(s){\dot{\alpha}}(s)}j^{\alpha(s){\dot{\alpha}}(s)}~{},$
(4.131b)

for undetermined coefficients $c_{s}$. On account of (4.130c), the action
(4.131) is invariant under the gauge transformations

$\displaystyle\delta_{\ell}h_{\alpha(s){\dot{\alpha}}(s)}=\nabla_{\alpha{\dot{\alpha}}}\ell_{\alpha(s-1){\dot{\alpha}}(s-1)}~{},\qquad
s\geq 1~{},$ (4.132)

provided $\varphi$ is on-shell,
$\delta_{\ell}\mathcal{S}_{\rm{N.C.}}[\varphi,\bar{\varphi},h]\approx 0$.
Further, owing to the conformal properties (4.19), the cubic action (see
(3.37) for the kinetic action for $\varphi$)

$\displaystyle\mathcal{S}_{\text{Cubic}}[\varphi,\bar{\varphi},h]=\mathcal{S}[\varphi,\bar{\varphi}]+\mathcal{S}_{\rm{N.C.}}[\varphi,\bar{\varphi},h]$
(4.133)

is clearly conformal.161616It follows from (4.130b) that (4.133) is actually
conformal on arbitrary backgrounds.

In order to elevate (4.132) to an off-shell symmetry, it is necessary to endow
$\varphi$ with its own transformation rule. For this we use the rule (4.126)
with ${\cal U}^{(s)}$ given by (4.127) and impose the reality condition
$\bar{\zeta}^{\alpha(s){\dot{\alpha}}(s)}=-\zeta^{\alpha(s){\dot{\alpha}}(s)}$.
Then, the properties (4.128) allow us to make the field identification
$\zeta^{\alpha(s){\dot{\alpha}}(s)}\equiv{\rm
i}\ell^{\alpha(s){\dot{\alpha}}(s)}$, where
$\ell_{\alpha(s){\dot{\alpha}}(s)}$ is the gauge parameter associated with
$h_{\alpha(s+1){\dot{\alpha}}(s+1)}$. Now we may fix the coefficients $c_{s}$
in (4.131) by requiring the cubic action (4.133) to be invariant up to terms
linear in gauge fields,

$\displaystyle
c_{s}=\frac{(-1)^{s}}{2}\binom{2s}{s}^{-1}\quad\implies\quad\delta_{\ell}\mathcal{S}_{\text{Cubic}}[\varphi,\bar{\varphi},h]=\mathcal{O}\big{(}h\big{)}~{}.$
(4.134)

It should be emphasised that this statement holds off-shell.

##### Closing the gauge algebra

Fundamental to our analysis above was the family of linear operators (4.126),
which, together with (4.132), constitute gauge transformations for the action
(4.131). It may be verified, however, that they do not form a closed algebra
(modulo trivial symmetries171717Given a functional $\mathcal{S}[\Upsilon^{I}]$
of some fields $\Upsilon^{I}$, a transformation $\delta\Upsilon^{I}$ is said
to be a trivial symmetry if it may be expressed in the form
$\delta\Upsilon^{I}=A^{IJ}\frac{\delta\mathcal{S}}{\delta\Upsilon^{J}}$ (here
we are using DeWitt’s condensed notation). This is possible if and only if
$\delta\mathcal{S}=0$ and $\delta\Upsilon^{I}\approx 0$ (see e.g. [183]). Here
$A^{IJ}$ is some differential operator satisfying $A^{IJ}=-A^{JI}$. ), as
their commutator involves terms proportional to $\square$. Hence, the latter
cannot be expressed in the form (4.126). Consequently, the cubic coupling of
$\varphi$ to the tower of CHS fields $h_{\alpha(s){\dot{\alpha}}(s)}$ is
already inconsistent at order $\mathcal{O}\big{(}h\big{)}$.

Therefore, it is necessary to consider more general matter transformations
than (4.126) to ensure that we obtain a closed algebra. Such transformations
are non-trivially constrained by the requirement that they must be consistent
with the conformal properties of $\varphi$. Naively, one may consider the
ansatz

$\displaystyle\delta\varphi=-\sum_{s=0}^{\infty}\sum_{l=0}^{\lfloor
s/2\rfloor}\sum_{k=0}^{s-2l}\zeta_{(s,l)}^{\alpha(k){\dot{\alpha}}(k)}(\nabla_{\alpha{\dot{\alpha}}})^{k}\square^{l}\varphi~{},$
(4.135)

and require that it preserves the kinematic properties of $\varphi$. Such an
approach proves to be computationally difficult, hence we will not pursue it
further. Instead, we will make use of a nowhere-vanishing real scalar field
$\eta$ with the conformal properties

$\displaystyle\mathbb{D}\eta=-2\eta~{},\qquad
K^{\alpha{\dot{\alpha}}}\eta=0~{}.$ (4.136)

It is clear to see that the operator $\hat{\square}:=\eta\square$ takes a
primary scalar of unit dimension to a primary scalar of unit dimension.

Employing this observation, we postulate the following matter transformations
for $\varphi$

$\displaystyle\delta\varphi=-{\mathcal{V}}\varphi=\sum_{s=0}^{\infty}{\mathcal{V}}^{(s)}\varphi~{},\qquad{\mathcal{V}}^{(s)}=\sum_{l=0}^{\lfloor
s/2\rfloor}{\mathcal{U}}^{(s-2l)}_{[s,l]}\hat{\square}^{l}~{}.$ (4.137a) where
the operator ${\cal U}_{[s,l]}^{(s-2l)}$ takes the same functional form as
(4.127) $\displaystyle{\mathcal{U}}^{(s-2l)}_{[s,l]}={\rm
i}^{s-2l}\sum_{k=0}^{s-2l}\binom{s-2l}{k}^{2}\binom{2s-4l+2}{s-2l-k}^{-1}(\nabla_{{\alpha{\dot{\alpha}}}})^{s-2l-k}\zeta_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}(\nabla_{\alpha{\dot{\alpha}}})^{k}~{}.$
(4.137b)

Requiring the transformation (4.137) to preserve the conformal properties of
$\varphi$, we find that the highest rank parameter of (4.137b) obeys

$\displaystyle
K^{\beta{\dot{\beta}}}\zeta_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}=0~{},\qquad\mathbb{D}\zeta_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}=-(s-2l)\zeta_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}~{}.$
(4.138)

In particular, the gauge parameters of (4.127) correspond to the $l=0$ case,
that is,
$\zeta_{[s,0]}^{\alpha(s){\dot{\alpha}}(s)}=\zeta^{\alpha(s){\dot{\alpha}}(s)}$.
In closing, we emphasise that the matter transformations (4.137) do indeed
form a closed algebra.

##### Inclusion of auxiliary gauge fields

In the previous subsection we extended the gauge group such that the matter
transformations (4.137) form a closed algebra. They were shown to be
parametrised by the primary fields
$\zeta_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}$, where
$\zeta_{[s,0]}^{\alpha(s){\dot{\alpha}}(s)}=\zeta^{\alpha(s){\dot{\alpha}}(s)}$.
In order to gauge this symmetry, we introduce new gauge fields181818For $l>0$,
they correspond to the traceful component of the CHS fields of [104, 178].
$h_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}$ associated with these
parameters. Their conformal properties are as follows:

$\displaystyle
K^{\beta{\dot{\beta}}}h_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}=0~{},\qquad\mathbb{D}h_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}=(2-s+2l)h_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}~{}.$
(4.139)

It should be emphasised that the original CHS gauge fields correspond to those
with $l=0$, $h_{[s,0]}^{\alpha(s){\dot{\alpha}}(s)}\equiv
h^{\alpha(s){\dot{\alpha}}(s)}$.

Omitting the techincal details, it may be shown that the cubic couplings
between $\varphi$ and $h_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}$ are
described by the action:

$\displaystyle\mathcal{S}_{\text{Cubic}}[\varphi,\bar{\varphi},h]$
$\displaystyle=\int\text{d}^{4}x\,e\,\Big{\\{}\bar{\varphi}\square\varphi+\sum_{s=0}^{\infty}\sum_{l=0}^{\lfloor
s/2\rfloor}h_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}j_{\alpha(s-2l){\dot{\alpha}}(s-2l)}^{[s,l]}\Big{\\}}~{},$
(4.140a) $\displaystyle j_{\alpha(s-2l){\dot{\alpha}}(s-2l)}^{[s,l]}$
$\displaystyle=\frac{{\rm
i}^{s}}{2}\sum_{k=0}^{s-2l}(-1)^{k}\binom{s-2l}{k}^{2}\Big{(}(\nabla_{\alpha{\dot{\alpha}}})^{k}\hat{\square}^{l}\varphi(\nabla_{\alpha{\dot{\alpha}}})^{s-2l-k}\bar{\varphi}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+(\nabla_{\alpha{\dot{\alpha}}})^{k}\varphi(\nabla_{\alpha{\dot{\alpha}}})^{s-2l-k}\hat{\square}^{l}\bar{\varphi}\Big{)}~{},$
(4.140b)

By construction $j_{\alpha(s-2l){\dot{\alpha}}(s-2l)}^{[s,l]}$ is a real
primary field of dimension $(2+s-2l)$. Additionally, for $l=0$ it coincides
with (4.129). At the same time, it is easily seen that for $l>0$ the current
vanishes on-shell.

We now spell out the gauge transformations of the fields
$h_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}$ which, together with the matter
transformations (4.137), ensure off-shell gauge invariance of (4.140). First,
the CHS fields transform as follows:191919The additional factor of $c_{s}$ in
(4.141b) occurs as we have absorbed this factor in the definition of
$h_{\alpha(s){\dot{\alpha}}(s)}$, c.f. (4.134).

$\displaystyle\delta h_{\alpha(s){\dot{\alpha}}(s)}$
$\displaystyle=\nabla_{\alpha{\dot{\alpha}}}\ell_{\alpha(s-1){\dot{\alpha}}(s-1)}+\mathcal{O}(h)~{},$
(4.141a) $\displaystyle\ell_{\alpha(s-1){\dot{\alpha}}(s-1)}$
$\displaystyle\equiv c_{s}{\rm
i}\big{(}\zeta_{\alpha(s-1){\dot{\alpha}}(s-1)}-\bar{\zeta}_{\alpha(s-1){\dot{\alpha}}(s-1)}\big{)}~{},$
(4.141b)

In particular, we note that (4.141a) contains a non-abelian component (whose
explicit form we do not give here). On the other hand, the gauge fields
$h_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}$ with $l>0$ begin their
transformations algebraically:

$\displaystyle\delta
h_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}\propto\eta^{-1}\big{(}\zeta_{[s-2,l-1]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}+\bar{\zeta}_{[s-2,l-1]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}\big{)}+\cdots~{},\qquad
1\leq l\leq\lfloor s/2\rfloor~{}.$ (4.142)

Hence, the real component of each
$\zeta_{[s-2,l-1]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}$ may be utilised to fix
the gauge

$\displaystyle h_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}=0~{},\qquad s\geq
0~{}\text{ and }~{}1\leq l\leq\lfloor s/2\rfloor~{}.$ (4.143)

Consequently, the only remaining gauge parameters are the imaginary parts of
$\zeta^{\alpha(s-1){\dot{\alpha}}(s-1)}$.202020For $l>0$, the imaginary
components of $\zeta_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}$ do not appear
in (4.141). Thus, for the variations of both the matter and gauge fields to
preserve this gauge (4.143), the former should be supplemented with a
compensating $h^{\alpha(s){\dot{\alpha}}(s)}$ and
$\ell^{\alpha(s-1){\dot{\alpha}}(s-1)}$ dependent transformation. However, due
to its complexity, we omit this analysis.

##### Rigid symmetries of free Klein-Gordon action

As is well known, the reducibility parameters of
$h_{\alpha(s){\dot{\alpha}}(s)}$ define rigid symmetries of the matter action
${\cal S}[\varphi,\bar{\varphi}]$, see e.g. [95]. We recall that the former
are those gauge parameters $\ell^{(0)}_{\alpha(s-1){\dot{\alpha}}(s-1)}$
leaving $h_{\alpha(s){\dot{\alpha}}(s)}$ unchanged

$\displaystyle\delta_{\ell^{(0)}}h_{\alpha(s){\dot{\alpha}}(s)}=\nabla_{\alpha{\dot{\alpha}}}\ell^{(0)}_{\alpha(s-1){\dot{\alpha}}(s-1)}=0~{}.$
(4.144)

In particular, this implies that $\ell^{(0)}_{\alpha(s-1){\dot{\alpha}}(s-1)}$
constitutes a conformal Killing tensor field (3.27). It trivially follows from
eq. (4.165) that

$\displaystyle\delta_{\ell^{(0)}}\mathcal{S}[\varphi,\bar{\varphi}]=-\delta_{\ell^{(0)}}\mathcal{S}_{\text{N.C.}}[\varphi,\bar{\varphi},h]\big{|}_{h=0}=0$
(4.145)

Hence, the matter action ${\cal S}[\varphi,\bar{\varphi}]$ is invariant under
transformations (4.127) upon requiring that
$\zeta_{\alpha(s-1){\dot{\alpha}}(s-1)}={\rm
i}\ell_{\alpha(s-1){\dot{\alpha}}(s-1)}^{(0)}$. We note that such matter
transformations may be obtained from the higher symmetries of the conformal
d’Alembertian, eq. (3.46), by fixing
$\xi_{\alpha(s-1){\dot{\alpha}}(s-1)}={\rm
i}^{s+1}\ell^{(0)}_{\alpha(s-1){\dot{\alpha}}(s-1)}$.

####  4.4.2 Chiral multiplet coupled to superspin-($s+\frac{1}{2}$) SCHS
fields

The results of this subsection, excluding part of the calculation to obtain
(4.162b) and those of “Rigid symmetries of the free massless Wess-Zumino
action” were obtained by Michael Ponds.

As an extension of the non-supersymmetric analysis above, this subsection is
devoted to the construction of a superspace action describing the interactions
between a massless chiral multiplet $\Phi$ and the superspin-($s+\frac{1}{2}$)
SCHS prepotentials $H_{\alpha(s){\dot{\alpha}}(s)}$ in conformally flat
superspace backgrounds.

##### Differential operators on the space of primary chiral superfields

As in the non-supersymmetric case, we begin our analysis by constructing the
permissible higher-derivative transformation laws for the chiral scalar $\Phi$
consistent with its kinematics, specifically its chirality and superconformal
properties. We propose the transformation

$\displaystyle\delta\Phi=-{\mathcal{U}}\Phi~{},\qquad{\mathcal{U}}=\sum_{s=0}^{\infty}{\mathcal{U}}^{(s)},$
(4.146a) where ${\mathcal{U}}^{(s)}$ is a linear differential operator of
order $s$
$\displaystyle{\mathcal{U}}^{(s)}=\sum_{k=0}^{s}\Omega_{(s)}^{\alpha(k){\dot{\alpha}}(k)}(\nabla_{\alpha{\dot{\alpha}}})^{k}+\sum_{k=1}^{s}\Omega_{(s)}^{\alpha(k){\dot{\alpha}}(k-1)}(\nabla_{\alpha{\dot{\alpha}}})^{k-1}\nabla_{\alpha}~{}.$
(4.146b)

It is clear that each ${\cal U}^{(s)}$ must preserve the kinematic properties
of $\Phi$. These conditions uniquely determine all coefficients of (4.146b)
(up to overall normalisation) in terms of a single superfield, which will be
shown to be proportional to the gauge parameter of
$H_{\alpha(s){\dot{\alpha}}(s)}$.

We begin by imposing the chirality condition
$\bar{\nabla}_{\dot{\alpha}}\mathcal{U}^{(s)}\Phi=0$, which yields the
constraints:

$\displaystyle\bar{\nabla}^{\dot{\alpha}}\Omega^{\alpha(k){\dot{\alpha}}(k)}_{(s)}$
$\displaystyle=0~{},\qquad$ $\displaystyle 0\leq k\leq s~{},$ (4.147a)
$\displaystyle\Omega^{\alpha(k){\dot{\alpha}}(k-1)}_{(s)}$
$\displaystyle=\frac{{\rm
i}k}{2(k+1)}\bar{\nabla}_{\dot{\beta}}\Omega^{\alpha(k){\dot{\alpha}}(k-1){\dot{\beta}}}_{(s)}~{},\qquad$
$\displaystyle~{}1\leq k\leq s~{}.$ (4.147b)

In particular, (4.147a) implies that
$\Omega_{(s)}^{\alpha(k){\dot{\alpha}}(k)}$ is longitudinal linear for $s>0$
and chiral for $s=0$. The former condition may be used to solve for
$\Omega_{(s)}^{\alpha(s){\dot{\alpha}}(s)}$ in terms an unconstrained
parameter $\omega$ by making the definition

$\displaystyle s>0:$
$\displaystyle\qquad\Omega_{(s)}^{\alpha(s){\dot{\alpha}}(s)}={\rm
i}^{s}\bar{\nabla}^{{\dot{\alpha}}}\omega^{\alpha(s){\dot{\alpha}}(s-1)}~{},$
(4.148a) $\displaystyle s=0:$
$\displaystyle\qquad\Omega_{(0)}=\bar{\nabla}^{2}\omega~{}.$ (4.148b)

Requiring the transformed field to be primary,
$K^{B}{\mathcal{U}}^{(s)}\Phi=0$, we find that this parameter is a primary
superfield

$\displaystyle s>0:$ $\displaystyle\qquad
K^{B}\omega^{\alpha(s){\dot{\alpha}}(s-1)}=0~{},\qquad\mathbb{D}\omega^{\alpha(s){\dot{\alpha}}(s-1)}=-\big{(}s+1/2\big{)}\omega^{\alpha(s){\dot{\alpha}}(s-1)}~{},$
(4.149a) $\displaystyle s=0:$ $\displaystyle\qquad
K^{B}\omega=0~{},\qquad\mathbb{D}\omega=-1~{}.$ (4.149b)

and that the remaining parameters are uniquely determined as its descendants.
Specifically, it may be shown that (4.146b) takes the form

$\displaystyle s>0:$ $\displaystyle\qquad\mathcal{U}^{(s)}=\sum_{k=1}^{s}{\rm
i}^{s}\binom{s}{k}\binom{s-1}{k-1}\binom{2s+1}{s-k}^{-1}\Big{\\{}$
$\displaystyle\qquad~{}(\nabla_{\beta{\dot{\beta}}})^{s-k}\bar{\nabla}^{{\dot{\alpha}}}\omega^{\alpha(k)\beta(s-k){\dot{\beta}}(s-k){\dot{\alpha}}(k-1)}(\nabla_{\alpha{\dot{\alpha}}})^{k}$
$\displaystyle\qquad+\frac{{\rm
i}}{4}\bar{\nabla}^{2}(\nabla_{\beta{\dot{\beta}}})^{s-k}\omega^{\alpha(k)\beta(s-k){\dot{\beta}}(s-k){\dot{\alpha}}(k-1)}(\nabla_{\alpha{\dot{\alpha}}})^{k-1}\nabla_{\alpha}$
$\displaystyle\qquad-\frac{{\rm
i}}{4}\frac{k}{s+k+1}\bar{\nabla}^{2}\nabla_{\beta}(\nabla_{\beta{\dot{\beta}}})^{s-k}\omega^{\alpha(k-1)\beta(s-k+1){\dot{\beta}}(s-k){\dot{\alpha}}(k-1)}(\nabla_{\alpha{\dot{\alpha}}})^{k-1}\Big{\\}}~{},$
(4.150a) $\displaystyle s=0:$
$\displaystyle\qquad\mathcal{U}^{(0)}=\bar{\nabla}^{2}\omega~{}.$ (4.150b)

It should be noted that did not include contributions to $\delta\Phi$ which
vanish on-shell. As will be shown below, it is possible for the action for
$\Phi$ coupled to the infinite tower of SCHS fields
$H_{\alpha(s){\dot{\alpha}}(s)}$, $s\geq 0$, to be gauge-invariant off-shell
at order $\mathcal{O}(H)$ in their absence. However, such on-shell vanishing
terms prove to be necessary for the algebra of matter transformations to
close, hence they will be revisited below.

##### Noether procedure to order $\mathcal{O}(H)$

From $\Phi$ and $\bar{\Phi}$ we construct the rank-$s\geq 0$ descendants

$\displaystyle J_{\alpha(s){\dot{\alpha}}(s)}$ $\displaystyle={\rm
i}^{s}\sum_{k=0}^{s}(-1)^{k}\binom{s}{k}^{2}\Big{(}(\nabla_{\alpha{\dot{\alpha}}})^{k}\Phi(\nabla_{\alpha{\dot{\alpha}}})^{s-k}\bar{\Phi}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-\frac{\text{i}}{2}\frac{s-k}{k+1}(\nabla_{\alpha{\dot{\alpha}}})^{k}\nabla_{\alpha}\Phi(\nabla_{\alpha{\dot{\alpha}}})^{s-k-1}\bar{\nabla}_{{\dot{\alpha}}}\bar{\Phi}\Big{)}~{},$
(4.151)

It may be shown that $J_{\alpha(s){\dot{\alpha}}(s)}$ possesses the following
crucial features:

  1. (i)

Reality

$\displaystyle
J_{\alpha(s){\dot{\alpha}}(s)}=\bar{J}_{\alpha(s){\dot{\alpha}}(s)}~{};$
(4.152a)

  2. (ii)

Superconformal covariance in an arbitrary supergravity background

$\displaystyle
K^{B}J_{\alpha(s){\dot{\alpha}}(s)}=0~{},\qquad\mathbb{D}J_{\alpha(s){\dot{\alpha}}(s)}=(s+2)J_{\alpha(s){\dot{\alpha}}(s)}~{};$
(4.152b)

  3. (iii)

When restricted to a conformally flat background, $W_{\alpha\beta\gamma}=0$,
$J_{\alpha(s){\dot{\alpha}}(s)}$ constitutes a conserved supercurrent if
$\Phi$ is on-shell212121The current multiplets (4.129) with $s=0$ and $s=1$
are actually conserved on-shell on arbitrary backgrounds. However, the super-
Weyl tensor $W_{\alpha\beta\gamma}$ proves to be the obstruction for this
property to hold for case $s\geq 2$.

$\displaystyle s\geq 1:$
$\displaystyle\qquad\nabla_{\beta}J^{\alpha(s-1)\beta{\dot{\alpha}}(s)}=0~{},\quad\bar{\nabla}_{\dot{\beta}}J^{\alpha(s){\dot{\alpha}}(s-1){\dot{\beta}}}=0~{},$
(4.152c) $\displaystyle s=0:$
$\displaystyle\qquad\nabla^{2}J=0~{},\quad\bar{\nabla}^{2}J=0~{}.$ (4.152d)

For $s=0$ and $s=1$, (4.151) coincides with the flavour current multiplet
[150] and the Ferrara-Zumino supercurrent [184], respectively. In the higher-
spin case, $s\geq 2$, the corresponding supercurrents were introduced in [101]
in Minkowksi superspace, and shortly afterwards they were extended to AdS
superspace [151].

The above supercurrents naturally couple to the half-integer
superspin-$(s+\frac{1}{2})$ conformal prepotentials
$H_{\alpha(s){\dot{\alpha}}(s)}$ via the cubic Noether coupling

$\displaystyle\mathcal{S}_{\text{N.C.}}[\Phi,\bar{\Phi},H]$
$\displaystyle=\sum_{s=0}^{\infty}c_{s}\mathcal{S}^{(s)}_{\text{N.C.}}[\Phi,\bar{\Phi},H]~{},$
(4.153a) $\displaystyle\mathcal{S}^{(s)}_{\text{N.C.}}[\Phi,\bar{\Phi},H]$
$\displaystyle=\int\text{d}^{4|4}z\,E\,H^{\alpha(s){\dot{\alpha}}(s)}J_{\alpha(s){\dot{\alpha}}(s)}~{},$
(4.153b)

for undetermined coefficients $c_{s}\in\mathbb{R}$. It is clear to see that
(4.153) is invariant under the gauge transformations

$\displaystyle s\geq 1:$
$\displaystyle\qquad\qquad\qquad\delta_{\zeta}H_{\alpha(s){\dot{\alpha}}(s)}=\bar{\nabla}_{{\dot{\alpha}}}\zeta_{\alpha(s){\dot{\alpha}}(s-1)}-\nabla_{\alpha}\bar{\zeta}_{\alpha(s-1){\dot{\alpha}}(s)}~{},\qquad\qquad$
(4.154a) $\displaystyle s=0:$
$\displaystyle\qquad\qquad\qquad\qquad~{}~{}\delta_{\zeta}H=\bar{\nabla}^{2}\zeta+\nabla^{2}\bar{\zeta}~{},\qquad\qquad$
(4.154b)

provided $\Phi$ is on-shell,
$\delta_{\zeta}\mathcal{S}_{\text{N.C.}}[\Phi,\bar{\Phi},H]\approx 0$.
Further, the cubic action

$\displaystyle\mathcal{S}_{\text{Cubic}}[\Phi,\bar{\Phi},H]=\mathcal{S}[\Phi,\bar{\Phi}]+\mathcal{S}_{\text{N.C.}}[\Phi,\bar{\Phi},H]$
(4.155)

is manifestly superconformal.222222Action (4.155) is actually invariant under
the superconformal group on an arbitrary backgrounds.

In order to elevate (4.154) to an off-shell symmetry, it is necessary to endow
$\Phi$ with its own gauge transformation rule. For this we employ the
operators (4.150) derived above with the identifications
$\zeta^{\alpha(s){\dot{\alpha}}(s-1)}\equiv\omega^{\alpha(s){\dot{\alpha}}(s-1)}$
and $\zeta\equiv\omega$. Now, requiring (4.153) to be invariant to order
$\mathcal{O}(H)$, we determine the coefficients $c_{s}$

$\displaystyle
c_{s}=(-1)^{s}\binom{2s+1}{s}^{-1}~{}\quad\implies\quad\delta_{\zeta}\mathcal{S}_{\text{Cubic}}[\Phi,\bar{\Phi},H]=\mathcal{O}\big{(}H\big{)}~{}.$
(4.156)

Thus, we have shown a correspondence between the matter
transformations232323As will be shown at the end of this subsection, the
former yield rigid symmetries of the free massless Wess-Zumino model. (4.150)
and the conformal supercurrents (4.151). This superspace Noether procedure may
be understood as a generalisation of the one described in [185] to the
superconformal higher-spin case.

It should be noted that, in the rigid supersymmetric case, cubic vertices
(4.153) were introduced in [101]. However, they took $\Phi$ to be on-shell,
hence the coefficients (4.156) were not fixed.

##### Consistency to all orders

As was the case in the non-supersymmetric model, the gauge transformations
(4.146) with $\mathcal{U}^{(s)}$ given by (4.150) do not form a closed
algebra. This is because we have restricted our attention to transformation
operators which, when acting on $\Phi$, do not vanish on the free equations of
motion. This will be done by utilising a superconformal extension of the
operator $\hat{\square}$ introduced in the previous subsection.

To this end, we introduce a nowhere-vanishing chiral scalar $\Upsilon$

$\displaystyle\bar{\nabla}_{{\dot{\alpha}}}\Upsilon=0~{},$ (4.157a) with the
superconformal properties $\displaystyle\mathbb{D}\Upsilon=-\Upsilon~{},\qquad
K^{B}\Upsilon=0\qquad\implies\qquad\mathbb{Y}\Upsilon=\frac{2}{3}\Upsilon~{}.$
(4.157b)

From $\Upsilon$ we construct the following two operators

$\displaystyle\Delta:=\Upsilon\bar{\nabla}^{2}~{},\qquad\bar{\Delta}:=\bar{\Upsilon}\nabla^{2}~{}.$
(4.158)

It then follows that their product,
$\mbox{\boldmath$\Delta$}:=\Delta\bar{\Delta}$, preserves all kinematic
properties of $\Phi$:

$\displaystyle\mathbb{D}\mbox{\boldmath$\Delta$}\Phi=\mbox{\boldmath$\Delta$}\Phi~{},\qquad$
$\displaystyle\mathbb{Y}\mbox{\boldmath$\Delta$}\Phi=-\frac{2}{3}\mbox{\boldmath$\Delta$}\Phi~{},\qquad
K^{B}\mbox{\boldmath$\Delta$}\Phi=0~{},\qquad\bar{\nabla}_{{\dot{\alpha}}}\mbox{\boldmath$\Delta$}\Phi=0~{}.$
(4.159)

Thus, $\Delta$ is a superconformal extension of $\hat{\square}$.

It follows that we may generate higher-order primary operators by combining
powers of $\Delta$ with (4.150). In particular, the most general
transformation rule for $\Phi$ preserving all its kinematic properties is

$\displaystyle\delta\Phi=-{\mathcal{V}}\Phi~{},\qquad{\mathcal{V}}=\sum_{s=0}^{\infty}{\mathcal{V}}^{(s)}\Phi~{}.$
(4.160a) where we have introduced the order-$s$ operator
$\displaystyle{\mathcal{V}}^{(s)}=\sum_{l=0}^{\lfloor
s/2\rfloor}{\mathcal{U}}^{(s-2l)}_{[s,l]}\mbox{\boldmath$\Delta$}^{l}$
(4.160b) and the notation $\displaystyle{\mathcal{U}}^{(s-2l)}_{[s,l]}$
$\displaystyle=\sum_{k=1}^{s-2l}{\rm
i}^{s}\binom{s}{k}\binom{s-1}{k-1}\binom{2s+1}{s-k}^{-1}\Big{\\{}\bar{\nabla}^{{\dot{\alpha}}}\nabla^{s-2l-k}_{\beta{\dot{\beta}}}\omega_{[s,l]}^{\alpha(k)\beta(s-2l-k){\dot{\beta}}(s-2l-k){\dot{\alpha}}(k-1)}\nabla^{k}_{\alpha{\dot{\alpha}}}$
$\displaystyle\phantom{=}+\frac{{\rm
i}}{4}\bar{\nabla}^{2}\nabla^{s-2l-k}_{\beta{\dot{\beta}}}\omega_{[s,l]}^{\alpha(k)\beta(s-2l-k){\dot{\beta}}(s-2l-k){\dot{\alpha}}(k-1)}\nabla^{k-1}_{\alpha{\dot{\alpha}}}\nabla_{\alpha}$
$\displaystyle\phantom{=}-\frac{{\rm
i}}{4}\frac{k}{s-2l+k+1}\bar{\nabla}^{2}\nabla_{\beta}\nabla^{s-2l-k}_{\beta{\dot{\beta}}}\omega_{[s,l]}^{\alpha(k-1)\beta(s-2l-k+1){\dot{\beta}}(s-2l-k){\dot{\alpha}}(k-1)}\nabla^{k-1}_{\alpha{\dot{\alpha}}}\Big{\\}}~{}.$
(4.160c)

The gauge parameters introduced above are primary with weights and charges
given by

$\displaystyle\mathbb{D}\omega_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l-1)}$
$\displaystyle=-\big{(}s-2l+1/2\big{)}\omega_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l-1)}~{},$
(4.161a)
$\displaystyle\mathbb{Y}\omega_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l-1)}$
$\displaystyle=\omega_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l-1)}~{}.$
(4.161b)

It is clear that the algebra of gauge transformations (4.160) now closes.

Omitting the technical details, it may be shown that the cubic couplings
between $\Phi$ and $H_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}$ are
described by the action

$\displaystyle\mathcal{S}_{\text{Cubic}}[\Phi,\bar{\Phi},H]$
$\displaystyle=\int\text{d}^{4|4}z\,E\,\Big{\\{}\bar{\Phi}\Phi+\sum_{s=0}^{\infty}\sum_{l=0}^{\lfloor
s/2\rfloor}H_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}J_{\alpha(s-2l){\dot{\alpha}}(s-2l)}^{[s,l]}\Big{\\}}~{},$
(4.162a) $\displaystyle J^{[s,l]}_{\alpha(s-2l){\dot{\alpha}}(s-2l)}$
$\displaystyle=\frac{{\rm
i}^{s}}{2}\sum_{k=0}^{s-2l}(-1)^{k}\binom{s-2l}{k}^{2}\Big{\\{}\nabla_{\alpha{\dot{\alpha}}}^{k}\Phi\nabla_{\alpha{\dot{\alpha}}}^{s-k-2l}\mbox{\boldmath$\Delta$}^{l}\bar{\Phi}+\nabla_{\alpha{\dot{\alpha}}}^{k}\mbox{\boldmath$\Delta$}^{l}\Phi\nabla_{\alpha{\dot{\alpha}}}^{s-k-2l}\bar{\Phi}$
$\displaystyle-\frac{\text{i}}{2}$
$\displaystyle\frac{s-2l-k}{k+1}\Big{(}\nabla_{\alpha{\dot{\alpha}}}^{k}\nabla_{\alpha}\Phi\nabla_{\alpha{\dot{\alpha}}}^{s-2l-k-1}\bar{\nabla}_{{\dot{\alpha}}}\mbox{\boldmath$\Delta$}^{l}\bar{\Phi}+\nabla_{\alpha{\dot{\alpha}}}^{k}\nabla_{\alpha}\mbox{\boldmath$\Delta$}^{l}\Phi\nabla_{\alpha{\dot{\alpha}}}^{s-2l-k-1}\bar{\nabla}_{{\dot{\alpha}}}\bar{\Phi}\Big{)}\Big{\\}}~{}.$
(4.162b)

For $l=0$, the supercurrents $J_{\alpha(s-2l){\dot{\alpha}}(s-2l)}^{[s,l]}$
coincide with (4.151), while for $l>0$ they vanish on-shell. The gauge
transformations of SCHS prepotentials $H^{\alpha(s){\dot{\alpha}}(s)}\equiv
H^{\alpha(s){\dot{\alpha}}(s)}_{[s,0]}$ may be shown to take the form

$\displaystyle\delta H_{\alpha(s){\dot{\alpha}}(s)}$
$\displaystyle=\bar{\nabla}_{{\dot{\alpha}}}\zeta_{\alpha(s){\dot{\alpha}}(s-1)}-\nabla_{\alpha}\bar{\zeta}_{\alpha(s-1){\dot{\alpha}}(s)}+\mathcal{O}(H)~{},$
(4.163a) $\displaystyle\zeta_{\alpha(s){\dot{\alpha}}(s-1)}$
$\displaystyle\equiv c_{s}\omega^{[s,0]}_{\alpha(s){\dot{\alpha}}(s-1)}~{},$
(4.163b)

where $c_{s}$ are defined in (4.156). In particular, it should be noted that
(4.163a) contains a non-abelian component (whose explicit form we do not
determine here). We expect that the ‘auxiliary’ superfields
$H_{[s,l]}^{\alpha(s-2l){\dot{\alpha}}(s-2l)}$ with $l>0$ begin their gauge
transformations algebraically, allowing them to be gauged away (such a fixing
will lead to new non-trivial contributions to the gauge transformation law of
$H_{\alpha(s){\dot{\alpha}}(s)}$).

##### Rigid symmetries of the free massless Wess-Zumino model

It remains to provide some comments on the rigid symmetries of the free matter
action ${\cal S}[\Phi,\bar{\Phi}]$. From (4.156) one may deduce that
$\delta_{\zeta}\mathcal{S}[\Phi,\bar{\Phi}]=-\delta_{\zeta}\mathcal{S}_{\text{N.C.}}[\Phi,\bar{\Phi},H]\big{|}_{H=0}$,
from which it follows that

$\displaystyle\delta_{\zeta}H_{\alpha(s){\dot{\alpha}}(s)}=0\qquad\implies\qquad\delta_{\zeta}\mathcal{S}[\Phi,\bar{\Phi}]=0~{}.$
(4.164)

Hence, the reducibility parameters of $H_{\alpha(s){\dot{\alpha}}(s)}$ define
rigid symmetries of $\mathcal{S}[\Phi,\bar{\Phi}]$. The former are those gauge
parameters $\zeta^{(0)}_{\alpha(s){\dot{\alpha}}(s-1)}$ satisfying the
constraint

$\displaystyle\delta_{\zeta^{(0)}}H_{\alpha(s){\dot{\alpha}}(s)}=\bar{\nabla}_{\dot{\alpha}}\zeta^{(0)}_{\alpha(s){\dot{\alpha}}(s-1)}-{\nabla}_{\alpha}\bar{\zeta}^{(0)}_{\alpha(s-1){\dot{\alpha}}(s)}=0~{}.$
(4.165)

The above constraint implies that $\chi_{\alpha(s){\dot{\alpha}}(s)}={\rm
i}\bar{\nabla}_{{\dot{\alpha}}}\zeta^{(0)}_{\alpha(s){\dot{\alpha}}(s-1)}=\bar{\chi}_{\alpha(s){\dot{\alpha}}(s)}$
constitutes a real conformal Killing tensor superfield of the background
(3.27)

$\displaystyle\bar{\nabla}_{{\dot{\alpha}}}\chi_{\alpha(s){\dot{\alpha}}(s)}=0\qquad\Longleftrightarrow\qquad\nabla_{\alpha}\chi_{\alpha(s){\dot{\alpha}}(s)}=0~{}.$
(4.166)

It should be noted that, when (4.166) is imposed, the operators ${\cal
U}^{(s)}$, which define the gauge transformations for the matter multiplet,
may be written solely in terms of $\chi_{\alpha(s){\dot{\alpha}}(s)}$

$\displaystyle{\cal U}^{(s)}$ $\displaystyle=$ $\displaystyle-{\rm
i}^{s}\sum_{k=0}^{n}{n\choose k}^{2}{2n+1\choose
n-k}^{-1}\Big{\\{}(\nabla_{\beta{\dot{\beta}}})^{n-k}-\frac{{\rm
i}(n-k)}{2(n+1)}(\nabla_{\beta{\dot{\beta}}})^{n-k-1}\nabla_{\beta}\bar{\nabla}_{\dot{\beta}}\Big{\\}}\chi^{\alpha(k)\beta(n-k){\dot{\alpha}}(k){\dot{\beta}}(n-k)}$
(4.167) $\displaystyle\times(\nabla_{\alpha{\dot{\alpha}}})^{k}-{\rm
i}^{s+1}\sum_{k=1}^{n}\frac{n}{2(n+1)}{n\choose k}{n-1\choose
k-1}{{2n+1\choose
n-k}}^{-1}(\nabla_{\beta{\dot{\beta}}})^{n-k}\bar{\nabla}_{\dot{\beta}}\chi^{\alpha(k)\beta(n-k){\dot{\alpha}}(k-1){\dot{\beta}}(n-k+1)}$
$\displaystyle\times(\nabla_{\alpha{\dot{\alpha}}})^{k-1}\nabla_{\alpha}~{},$

We emphasise that the transformations they induce constitute rigid symmetries
of the free action, and consequently of its equation of motion. Further, we
note that such transformations may be obtained from the higher symmetries of
the massless Wess-Zumino operator, eq. (3.59), by fixing
$\xi_{\alpha(s){\dot{\alpha}}(s)}={\rm
i}^{s}\chi_{\alpha(s){\dot{\alpha}}(s)}$.

###  4.5 Summary of results

This chapter was devoted to the study of models for (super)conformal higher-
spin gauge multiplets in curved backgrounds. Their associated gauge
(super)fields may be readily deduced by employing the method of supercurrent
multiplets. To this end, in section 4.1 we classified the conformal higher-
spin (super)current multiplets. Following this, in section 4.2, by virtue of
the Noether coupling (4.17) we obtained the (S)CHS gauge multiplets and their
corresponding linearised gauge transformations. Utilising this result, we
obtained the (super)conformal and gauge-invariant free action for each gauge
(super)field in conformally flat backgrounds. Additionally, in appendix 4.A we
performed such an analysis for the ${\cal N}=2$ superconformal gravitino
multiplet and identified the gauge prepotential describing its dynamics in
addition to its associated gauge-invariant action. These results greatly
extend the existing literature242424For the appropriate references, we refer
the reader to the main body of this chapter. of superconformal higher-spin
gauge multiplets in the ${\cal N}\geq 2$ case.

Building on the above result, we then proceeded to study $\mathsf{U}(1)$
duality-invariant models for the (S)CHS fields described above in section 4.3.
Specifically, we deduced the self-duality equation for real gauge
(super)fields.252525In the non-supersymmetric case the extension of this self-
duality equation for general CHS fields was obtained in appendix 4.A. It was
also shown that such models are self-dual under Legendre transformations.
Next, we generalised the Ivanov-Zupnik auxiliary variable formulation [169,
170, 171] and its ${\cal N}=1$ supersymmetric extension [176, 177] to the case
of higher-spins with ${\cal N}$-extended superconformal symmetry. This
formulation allowed us to derive several interesting self-dual models,
including (${\cal N}=1$ supersymmetric) higher-spin generalisations of ModMax
electrodynamics [157, 159, 160].

To conclude this chapter, we constructed manifestly (super)conformal gauge-
invariant models for conformal matter (super)fields interacting with an
infinite tower of background (S)CHS gauge multiplets in section 4.4.
Specifically, we coupled a massless complex scalar to the spin-$s$ gauge
fields $h_{\alpha(s){\dot{\alpha}}(s)}$, $s\geq 1$, and also considered
interactions between a massless chiral multiplet and the
superspin-$(s+\frac{1}{2})$ gauge multiplets $H_{\alpha(s){\dot{\alpha}}(s)}$,
$s\geq 0$. To ensure off-shell gauge-invariance of the corresponding actions,
it was necessary for each matter multiplet to transform under gauge
transformations of the (S)CHS fields. Further, upon considering the class of
gauge transformations preserving the latter, we obtained the rigid symmetries
for the free matter theory. The latter are shown to be intimately connected to
the higher symmetries described in section 3.4.

## Appendix

##  Appendix 4.A The ${\cal N}=2$ superconformal vector and gravitino
multiplets

In sections 4.1, we derived the ${\cal N}$-extended (S)CHS gauge (super)fields
by first identifying all possible conformal (super)current multiplets and then
utilising an appropriate Noether coupling to deduce the gauge transformations
of the former. Since we restricted our analysis in section 4.1 to
(super)currents inert under $\mathsf{SU}({\cal N})$ rotations, we have ignored
the study of several important gauge multiplets. This appendix serves to
bridge this gap, in the ${\cal N}=2$ case.

####  4.A.1 The ${\cal N}=2$ vector multiplet

In rigid supersymmetry, the off-shell ${\cal N}=2$ vector multiplet was
formulated by Grimm, Sohnius and Wess [186]. The multiplet of currents
associated with its prepotential is the so-called linear multiplet
$J^{ij}=J^{ji}$ [187, 188]. It is a primary, real iso-triplet of dimension $2$
subject to the conservation equations:

$\displaystyle\nabla_{\alpha}^{(i}J^{jk)}=0~{},\qquad\bar{\nabla}_{\dot{\alpha}}^{(i}J^{jk)}=0~{}.$
(4.168)

Hence, by making use of the Noether coupling

$\displaystyle\mathcal{S}_{\rm N.C.}=\int\text{d}^{4|8}z\,E\,H_{ij}J^{ij}~{},$
(4.169)

we see that the gauge superfield $H_{ij}$ is real, primary and of dimension
$-2$

$\displaystyle K^{C}H_{ij}=0~{},\qquad\mathbb{D}H_{ij}=-2H_{ij}~{},$ (4.170)

and is defined only up to gauge transformations of the form

$\displaystyle\delta_{\Lambda}H_{ij}$ $\displaystyle=\nabla^{\alpha
k}\Lambda_{\alpha}{}_{kij}+\text{c.c.}~{},\qquad\Lambda_{\alpha}{}_{kij}=\Lambda_{\alpha}{}_{(kij)}~{},$
(4.171)

where $\Lambda_{\alpha kij}$ is arbitrary modulo the algebraic condition
given. The superfield $H_{ij}$ is a curved superspace extension of
Mezincescu’s prepotential [189].

From $H_{ij}$, we construct the field strength

$\displaystyle\mathfrak{W}(H)=\frac{1}{48}\bar{\nabla}^{4}\nabla^{ij}H_{ij}~{},$
(4.172)

which is a primary chiral superfield of unit dimension

$\displaystyle
K^{A}\mathfrak{W}(H)=0~{},\quad\bar{\nabla}^{\dot{\alpha}}_{i}\mathfrak{W}(H)=0~{},\quad\mathbb{D}\mathfrak{W}(H)=\mathfrak{W}(H)~{},$
(4.173)

and is gauge-invariant (4.171) on arbitrary curved backgrounds;
$\delta_{\Lambda}\mathfrak{W}(H)=0$. Further, owing to the Bianchi identity

$\displaystyle\nabla^{ij}\mathfrak{W}(H)$ $\displaystyle=$
$\displaystyle\bar{\nabla}^{ij}\bar{\mathfrak{W}}(H)~{},$ (4.174a) the
following is a total derivative $\displaystyle{\rm i}\int{\rm d}^{4}x{\rm
d}^{2{{\cal N}}}\theta\,{\cal E}\,\mathfrak{W}(H)^{2}+\text{c.c.}=0~{}.$
(4.174b)

Hence, the free action for $H_{ij}$ is simply

$\displaystyle{\cal S}_{\text{VM}}[H]=\int{\rm d}^{4}x{\rm d}^{2{{\cal
N}}}\theta\,{\cal E}\,\mathfrak{W}(H)^{2}~{},$ (4.175)

where it should be emphasised that (4.175) is gauge-invariant and
superconformal on arbitrary curved backgrounds.

####  4.A.2 The ${\cal N}=2$ superconformal gravitino multiplet

The supercurrent multiplet which couples to the ${\cal N}=2$ superconformal
gravitino multiplet takes the form of a primary, complex isospinor ${J}^{i}$.
It is subject to the conservation equations:

$\displaystyle{\nabla}_{\alpha}^{(i}{J}^{j)}=0~{},\qquad\bar{\nabla}^{(ij}{J}^{k)}=0~{},$
(4.176)

and characterised by the superconformal properties

$\displaystyle
K^{M}J^{i}=0~{},\qquad\mathbb{D}J^{i}=\mathbb{Y}J^{i}=2J^{i}~{}.$ (4.177)

The present multiplet naturally arises via an ${\cal N}=3\rightarrow 2$
superspace reduction of the ${\cal N}=3$ conformal supercurrent multiplet
[99].

We now utilise a Noether coupling to determine the gauge prepotential
describing the superconformal gravitino multiplet, which we denote $H_{i}$.
Specifically, the latter may be determined by requiring that

$\displaystyle\mathcal{S}_{\text{N.C.}}=\int{\rm
d}^{4|8}z\,E\,H_{i}\,{J}^{i}+\text{c.c.}~{},$ (4.178)

is locally superconformal and gauge-invariant. It is clear to see that $H_{i}$
is defined modulo the gauge freedom

$\displaystyle\delta H_{i}=\nabla^{\alpha j}\zeta_{\alpha
ij}+\bar{\nabla}^{jk}\omega_{ijk}~{},$ (4.179)

and its superconformal transformation law is characterised by the properties:

$\displaystyle
K^{B}H_{i}=0~{},\qquad\mathbb{D}H_{i}=\mathbb{Y}H_{i}=-2H_{i}~{}.$ (4.180)

It should be emphasised that the superconformal properties (4.180) imply that
the gauge transformations (4.179) are superconformal.

Having determined the new gauge prepotential $H_{i}$ above, it remains to
determine its kinetic action. For simplicity, we perform this analysis in
conformally flat superspace backgrounds. In particular, one may show that the
following chiral descendants

$\displaystyle\hat{\mathfrak{W}}_{\alpha}(H)=\frac{1}{48}\bar{\nabla}^{4}\nabla^{ij}\nabla_{\alpha
i}H_{j}~{},\qquad\check{\mathfrak{W}}_{\alpha}(H)=\frac{1}{48}\bar{\nabla}^{4}\nabla_{\alpha}^{i}\bar{H}_{i}~{},$
(4.181)

are gauge-invariant in conformally flat superspaces and possess the
superconformal properties:

$\displaystyle K^{M}\hat{\mathfrak{W}}_{\alpha}(H)$
$\displaystyle=0~{},\qquad\mathbb{D}\hat{\mathfrak{W}}_{\alpha}(H)=\frac{3}{2}\hat{\mathfrak{W}}_{\alpha}(H)~{},$
(4.182a) $\displaystyle K^{M}\check{\mathfrak{W}}_{\alpha}(H)$
$\displaystyle=0~{},\qquad\mathbb{D}\check{\mathfrak{W}}_{\alpha}(H)=\frac{1}{2}\check{\mathfrak{W}}_{\alpha}(H)~{},$
(4.182b)

which are valid in generic backgrounds. It then follows that

$\displaystyle\mathcal{S}_{\text{GM}}[H,\bar{H}]=\int{\rm d}^{4}x{\rm
d}^{4}\theta\,{\cal
E}\,\hat{\mathfrak{W}}^{\alpha}(H)\check{\mathfrak{W}}_{\alpha}(H)+\text{c.c.}~{},$
(4.183)

is the locally superconformal and gauge-invariant action for the ${\cal N}=2$
superconformal gravitino multiplet in conformally flat backgrounds. The
overall coefficient in (4.183) has been chosen due to the identity

$\displaystyle{\rm i}\int{\rm d}^{4}x{\rm d}^{4}\theta\,{\cal
E}\,\hat{\mathfrak{W}}^{\alpha}(H)\check{\mathfrak{W}}_{\alpha}(H)+\text{c.c.}=0~{},$
(4.184)

which holds up to a total derivative on conformally flat backgrounds.

##  Appendix 4.B $\mathsf{U}(1)$ duality rotations for complex CHS fields

In section 4.3.1 we restricted our attention to the analysis of duality-
invariant models described by real gauge fields. However, our construction
readily extends itself to the complex case. In particular, one may consider a
complex CHS gauge field $h_{\alpha(m){\dot{\alpha}}(n)}$, $m\neq n$ and
$m,n\geq 1$. The action functional describing its dynamics is required to
depend only on the field strengths (4.20), that is ${\cal
S}^{(m,n)}[\hat{\mathfrak{C}},\check{\mathfrak{C}},\bar{\hat{\mathfrak{C}}},\bar{\check{\mathfrak{C}}}]$,
where we have made the definitions. It is important to note that the field
strengths (4.20) also obey the Bianchi identity

$(\nabla^{\beta}{}_{\dot{\alpha}})^{m}\hat{\mathfrak{C}}_{\alpha(n)\beta(m)}=(\nabla_{\alpha}{}^{{\dot{\beta}}})^{n}\bar{\check{\mathfrak{C}}}_{{\dot{\alpha}}(m){\dot{\beta}}(n)}~{}.$
(4.185)

Now, considering ${\cal
S}^{(m,n)}[\hat{\mathfrak{C}},\check{\mathfrak{C}},\bar{\hat{\mathfrak{C}}},\bar{\check{\mathfrak{C}}}]$
as a functional of the unconstrained fields
$\hat{\mathfrak{C}}_{\alpha(m+n)}$, $\check{\mathfrak{C}}_{\alpha(m+n)}$ and
their conjugates, we may introduce the primary fields

${\rm i}^{m+n+1}\hat{\mathfrak{M}}_{\alpha(m+n)}:=\frac{\delta{\cal
S}^{(m,n)}[\hat{\mathfrak{C}},\check{\mathfrak{C}},\bar{\hat{\mathfrak{C}}},\bar{\check{\mathfrak{C}}}]}{\delta\check{\mathfrak{C}}^{\alpha(m+n)}}~{},\quad{\rm
i}^{m+n+1}\check{\mathfrak{M}}_{\alpha(m+n)}:=\frac{\delta{\cal
S}^{(m,n)}[\hat{\mathfrak{C}},\check{\mathfrak{C}}~{},\bar{\hat{\mathfrak{C}}},\bar{\check{\mathfrak{C}}}]}{\delta\hat{\mathfrak{C}}^{\alpha(m+n)}}~{},$
(4.186)

where we have made the definition

$\displaystyle\delta{\cal
S}^{(m,n)}[\hat{\mathfrak{C}},\check{\mathfrak{C}},\bar{\hat{\mathfrak{C}}},\bar{\check{\mathfrak{C}}}]$
$\displaystyle=\int{\rm
d}^{4}x\,e\,\Big{\\{}\delta\hat{\mathfrak{C}}^{\alpha(m+n)}\frac{\delta{\cal
S}^{(m,n)}[\hat{\mathfrak{C}},\check{\mathfrak{C}},\bar{\hat{\mathfrak{C}}},\bar{\check{\mathfrak{C}}}]}{\delta\hat{\mathfrak{C}}^{\alpha(m+n)}}$
$\displaystyle\qquad\qquad\qquad\qquad+\delta\check{\mathfrak{C}}^{\alpha(m+n)}\frac{\delta{\cal
S}^{(m,n)}[\hat{\mathfrak{C}},\check{\mathfrak{C}},\bar{\hat{\mathfrak{C}}},\bar{\check{\mathfrak{C}}}]}{\delta\check{\mathfrak{C}}^{\alpha(m+n)}}\Big{\\}}+\text{c.c.}$
(4.187)

The conformal properties of the dual fields (4.186) are:

$\displaystyle K^{\beta{\dot{\beta}}}\hat{\mathfrak{M}}_{\alpha(m+n)}$
$\displaystyle=$ $\displaystyle
0~{},\qquad\mathbb{D}\hat{\mathfrak{M}}_{\alpha(m+n)}=\Big{(}2+\frac{1}{2}(n-m)\Big{)}\hat{\mathfrak{M}}_{\alpha(m+n)}~{},$
(4.188a) $\displaystyle
K^{{\beta{\dot{\beta}}}}\check{{\mathfrak{M}}}_{\alpha(m+n)}$ $\displaystyle=$
$\displaystyle
0~{},\qquad\mathbb{D}\check{{\mathfrak{M}}}_{\alpha(m+n)}=\Big{(}2+\frac{1}{2}(m-n)\Big{)}\check{{\mathfrak{M}}}_{\alpha(m+n)}~{}.$
(4.188b)

Varying ${\cal
S}^{(m,n)}[\hat{\mathfrak{C}},\check{\mathfrak{C}},\bar{\hat{\mathfrak{C}}},\bar{\check{\mathfrak{C}}}]$
with respect to $h_{\alpha(m){\dot{\alpha}}(n)}$ yields the equation of motion

$\displaystyle(\nabla^{\beta}{}_{\dot{\alpha}})^{m}\hat{\mathfrak{M}}_{\alpha(n)\beta(m)}=(\nabla_{\alpha}{}^{{\dot{\beta}}})^{n}\bar{\check{\mathfrak{M}}}_{{\dot{\alpha}}(m){\dot{\beta}}(n)}~{}.$
(4.189)

It is clear from the discussion above that the system of equations (4.185) and
(4.189) is invariant under the $\mathsf{U}(1)$ duality rotations

$\displaystyle\delta_{\lambda}\hat{\mathfrak{C}}_{\alpha(m+n)}=\lambda\hat{\mathfrak{M}}_{\alpha(m+n)}~{},\quad\delta_{\lambda}\check{\mathfrak{C}}_{\alpha(m+n)}=\lambda\check{\mathfrak{M}}_{\alpha(m+n)}~{},$
(4.190a)
$\displaystyle\delta_{\lambda}\hat{\mathfrak{M}}_{\alpha(m+n)}=-\lambda\hat{\mathfrak{C}}_{\alpha(m+n)}~{},\quad\delta_{\lambda}\check{\mathfrak{M}}_{\alpha(m+n)}=-\lambda\check{\mathfrak{C}}_{\alpha(m+n)}~{}.$
(4.190b)

One may then perform similar analyses to those undertaken in section 4.3.1 and
construct $\mathsf{U}(1)$ duality-invariant nonlinear models for such
fields.262626We mention in passing that such a construction can also be
uplifted to the case of a SCHS theory described by a complex prepotential.
They satisfy the self-duality equation

$\displaystyle{\rm i}^{m+n+1}\int{\rm
d}^{4}x\,e\,\Big{\\{}\hat{\mathfrak{C}}^{\alpha(m+n)}\check{\mathfrak{C}}_{\alpha(m+n)}+\hat{\mathfrak{M}}^{\alpha(m+n)}\check{\mathfrak{M}}_{\alpha(m+n)}\Big{\\}}+\text{c.c.}=0~{},$
(4.191)

which must hold for unconstrained fields $\hat{\mathfrak{C}}_{\alpha(m+n)}$
and $\check{\mathfrak{C}}_{\alpha(m+n)}$. The simplest solution of this
equation is the free CHS action (4.24).

##  Chapter 5 Conformal supergeometries in two dimensions

In the preceeding chapters, a central role was played by the framework of
conformal superspace. In particular, it proved to be a natural formalism to
study (conformal) isometries of a given background, higher derivative
symmetries of matter multiplets and various models for (super)conformal
higher-spin fields. It should be emphasised that, while our attention was
restricted to the four-dimensional case, conformal superspace formulations
have previously been constructed for $3\leq d\leq 6$ spacetime dimensions [70,
71, 72, 73, 74] and have lead to a plethora of applications, see chapter 1 for
more details. More recently, the $d=2$ case was constructed in [10] and is the
subject of this chapter.

This chapter is organised as follows. Section 5.1 is devoted to a derivation
of the infinite-dimensional superconformal algebra of Minkowski superspace
$\mathbb{M}^{(2|p,q)}$ by employing its conformal Killing supervector fields.
We then describe the additional constraints on the conformal Killing
supervector fields which single out the finite-dimensional superconformal
algebra $\mathfrak{osp}(p|2;\mathbb{R})\oplus\mathfrak{osp}(q|2;\mathbb{R})$.
In section 5.2 we review the formulation of conformal gravity in two
dimensions as the gauge theory of the conformal group
$\mathsf{SL}(2,{\mathbb{R}})\times\mathsf{SL}(2,{\mathbb{R}})$. Building on
the construction of conformal gravity, in section 5.3 we formulate conformal
$(p,q)$ supergravity as a gauge theory of the superconformal group
$\mathsf{OSp}_{0}(p|2;{\mathbb{R}})\times\mathsf{OSp}_{0}(q|2;{\mathbb{R}})$
with a flat connection. The procedure of ‘degauging’ from this superconformal
formulation to the $\mathsf{SO}(p)\times\mathsf{SO}(q)$ superspaces is
described in section 5.4. Section 5.5 is mostly devoted to generalisations and
applications of the results derived in section 5.4. The main body of this
chapter is accompanied by two technical appendices. In appendix 5.A we
construct $\mathcal{N}=(1,0)$ conformal superspace with non-vanishing
curvature. Appendix 5.B is devoted to the supertwistor realisation of
compactified Minkowski superspace $\overline{\mathbb{M}}^{(2|2n,q)}$ as a
homogeneous space of the superconformal group
$\mathsf{SU}(1,1|n)\times{\mathsf{OSp}}_{0}(q|2;{\mathbb{R}})$.

Throughout this chapter we will make use of two types of notation, $(p,q)$ and
${\cal N}=(p,q)$, to denote superspaces with $p$ left and $q$ right real
spinor coordinates. Additionally, the special case $p=q$ is also referred to
as ${\cal N}=p$.

###  5.1 Rigid superconformal transformations of $\mathbb{M}^{(2|p,q)}$

This section is devoted the description of the conformal Killing supervector
fields of $\mathbb{M}^{(2|p,q)}$, the $(p,q)$ Minkowski superspace in two
dimensions [190]. Our presentation is similar to that of section 2.1, though
there are some features specific to the two-dimensional case, which will be
emphasised.

####  5.1.1 The conformal Killing supervector fields of $\mathbb{M}^{(2|p,q)}$

Minkowski superspace $\mathbb{M}^{(2|p,q)}$ is parametrised by the real
coordinates $z^{A}=(x^{a},\theta^{+{\overline{I}}},\theta^{-\underline{I}})$,
where $x^{a}=(x^{++},x^{--})=\frac{1}{\sqrt{2}}(x^{0}+x^{1},x^{0}-x^{1})$,
${\overline{I}}=\overline{1},\dots,\overline{p}$ and
$\underline{I}={\underline{1}},\dots,\underline{q}$. Its covariant derivatives
$D_{A}=(\partial_{a},D_{+}^{{\overline{I}}},D_{-}^{\underline{I}})$ take the
form

$\displaystyle\partial_{a}:=\frac{\partial}{\partial
x^{a}}=(\partial_{++},\partial_{--})~{},\quad
D_{+}^{{\overline{I}}}:=\frac{\partial}{\partial\theta^{+{\overline{I}}}}+{\rm
i}\theta^{+{\overline{I}}}\partial_{++}~{},\quad
D_{-}^{\underline{I}}:=\frac{\partial}{\partial\theta^{-\underline{I}}}+{\rm
i}\theta^{-\underline{I}}\partial_{--}~{},$ (5.1)

and satisfy the algebra:

$\displaystyle\\{D_{+}^{{\overline{I}}},D_{+}^{\overline{J}}\\}=2{\rm
i}\delta^{{\overline{I}}\overline{J}}\partial_{++}~{},\qquad\\{D_{-}^{\underline{I}},D_{-}^{\underline{J}}\\}=2{\rm
i}\delta^{\underline{I}\underline{J}}\partial_{--}~{}.$ (5.2)

We emphasise that for $p=0$ the left spinor covariant derivative
$D_{+}^{{\overline{I}}}$ does not appear, similarly $D_{-}^{\underline{I}}$ is
not present for $q=0$.

The conformal Killing supervector fields of $\mathbb{M}^{(2|p,q)}$,

$\xi=\xi^{a}\partial_{a}+\xi^{+{\overline{I}}}D_{+}^{{\overline{I}}}+\xi^{-\underline{I}}D_{-}^{\underline{I}}=\xi^{++}\partial_{++}+\xi^{--}\partial_{--}+\xi^{+{\overline{I}}}D_{+}^{{\overline{I}}}+\xi^{-\underline{I}}D_{-}^{\underline{I}}=\bar{\xi}~{},$
(5.3)

may be defined to satisfy the constraints

$[\xi,D_{+}^{{\overline{I}}}]=-(D_{+}^{{\overline{I}}}\xi^{+\overline{J}})D_{+}^{\overline{J}}~{},\qquad[\xi,D_{-}^{\underline{I}}]=-(D_{-}^{\underline{I}}\xi^{-\underline{J}})D_{-}^{\underline{J}}~{}.$
(5.4)

We note that for vanishing $p$ ($q$), the spinor $\xi^{+{\overline{I}}}$
($\xi^{-{\underline{I}}}$) must be turned off. From $\eqref{5.4}$ we obtain
the fundamental equations

$\displaystyle D_{+}^{{\overline{I}}}\xi^{--}$ $\displaystyle=$ $\displaystyle
0\quad\implies\quad\partial_{++}\xi^{--}=0~{},$ (5.5a) $\displaystyle
D_{-}^{\underline{I}}\xi^{++}$ $\displaystyle=$ $\displaystyle
0\quad\implies\quad\partial_{--}\xi^{++}=0~{},$ (5.5b)

and expressions for the spinor parameters

$\displaystyle\xi^{+{\overline{I}}}=-\frac{{\rm
i}}{2}D_{+}^{{\overline{I}}}\xi^{++}~{},\qquad\xi^{-\underline{I}}=-\frac{{\rm
i}}{2}D_{-}^{\underline{I}}\xi^{--}~{}.$ (5.6)

Hence, we see that every conformal Killing supervector field (5.3) is
completely determined by its vector parameter $\xi^{a}$. Additionally,
equations (5.5) tell us that

$\displaystyle\xi^{++}=\xi^{++}(x^{++},\theta^{+})\equiv\xi^{++}(\zeta_{L})~{},\qquad\xi^{--}=\xi^{--}(x^{--},\theta^{-})\equiv\xi^{--}(\zeta_{R})~{}.$
(5.7)

Here $\xi^{++}(\zeta_{L})$ and $\xi^{--}(\zeta_{R})$ are arbitrary functions
of $\zeta_{L}$ and $\zeta_{R}$, respectively.

Taking (5.5) into account, the equations (5.4) can be rewritten in the form

$\displaystyle[\xi,D_{+}^{{\overline{I}}}]$
$\displaystyle=-\frac{1}{2}(\sigma[\xi]+K[\xi])D_{+}^{{\overline{I}}}-\rho[\xi]^{{\overline{I}}\overline{J}}D_{+}^{\overline{J}}~{},$
(5.8a) $\displaystyle[\xi,D_{-}^{\underline{I}}]$
$\displaystyle=-\frac{1}{2}(\sigma[\xi]-K[\xi])D_{-}^{\underline{I}}-\rho[\xi]^{\underline{I}\underline{J}}D_{-}^{\underline{J}}~{},$
(5.8b)

where we have defined the following parameters:

$\displaystyle\sigma[\xi]$
$\displaystyle:=\frac{1}{2}\big{(}\partial_{++}\xi^{++}+\partial_{--}\xi^{--}\big{)}~{},$
(5.9a) $\displaystyle K[\xi]$
$\displaystyle:=\frac{1}{2}\big{(}\partial_{++}\xi^{++}-\partial_{--}\xi^{--}\big{)}~{},$
(5.9b) $\displaystyle\rho[\xi]^{{\overline{I}}\overline{J}}$
$\displaystyle:=-\frac{{\rm
i}}{4}\big{[}D_{+}^{{\overline{I}}},D_{+}^{\overline{J}}\big{]}\xi^{++}~{},$
(5.9c) $\displaystyle\rho[\xi]^{\underline{I}\underline{J}}$
$\displaystyle:=-\frac{{\rm
i}}{4}\big{[}D_{-}^{\underline{I}},D_{-}^{\underline{J}}\big{]}\xi^{--}~{}.$
(5.9d)

Their $z$-independent components generate scale, Lorentz, $\mathfrak{so}(p)$
and $\mathfrak{so}(q)$ transformations, respectively. We point out that the
$\mathfrak{so}(p)$ parameter $\rho[\xi]^{{\overline{I}}\overline{J}}$ is a
function of the left variables $\zeta_{L}=(x^{++},\theta^{+})$, while the
$\mathfrak{so}(q)$ parameter $\rho[\xi]^{\underline{I}\underline{J}}$ is a
function of the right variables $\zeta_{R}=(x^{--},\theta^{-})$. Further, the
former (latter) identically vanishes when $p<2$ ($q<2$).

Given two conformal Killing supervectors $\xi_{1}$ and $\xi_{2}$, their
commutator is another conformal Killing supervector $\xi_{3}$

$\displaystyle[\xi_{1},\xi_{2}]=\xi_{3}^{A}D_{A}=\xi_{3}~{},$ (5.10a) with the
definitions $\displaystyle\xi_{3}^{++}$
$\displaystyle=\xi_{1}^{++}\partial_{++}\xi_{2}^{++}-\xi_{2}^{++}\partial_{++}\xi_{1}^{++}+2{\rm
i}\xi_{1}^{+{\overline{I}}}\xi_{2}^{+{\overline{I}}}~{}\implies~{}D_{-}^{{\underline{I}}}\xi_{3}^{++}=0~{},$
(5.10b) $\displaystyle\xi_{3}^{--}$
$\displaystyle=\xi_{1}^{--}\partial_{--}\xi_{2}^{--}-\xi_{2}^{--}\partial_{--}\xi_{1}^{--}+2{\rm
i}\xi_{1}^{-{\underline{I}}}\xi_{2}^{-{\underline{I}}}~{}\implies~{}D_{+}^{{\overline{I}}}\xi_{3}^{--}=0~{},$
(5.10c)

and the spinor parameters are determined by eq. (5.6). Equations (5.5) imply
that the algebra of conformal Killing supervector fields of
$\mathbb{M}^{(2|p,q)}$ is infinite dimensional. It may be referred to as a
$(p,q)$ super Virasoro algebra. For $p=q$, such superalgebras were studied in
[191, 192, 193].

The superconformal transformation law of a primary tensor superfield $U$ (with
suppressed Lorentz, $\mathsf{SO}(p)$ and $\mathsf{SO}(q)$ indices) is

$\displaystyle\delta_{\xi}U$ $\displaystyle=$
$\displaystyle\Big{\\{}\xi+\lambda_{U}K[\xi]+\Delta_{U}\sigma[\xi]+\frac{1}{2}\rho[\xi]^{{\overline{I}}\overline{J}}\mathfrak{L}^{{\overline{I}}{\overline{J}}}+\frac{1}{2}\rho[\xi]^{\underline{I}\underline{J}}\mathfrak{R}^{{\underline{I}}{\underline{J}}}\Big{\\}}U~{},$
(5.11)

where $\mathfrak{L}^{{\overline{I}}{\overline{J}}}$ and
$\mathfrak{R}^{{\underline{I}}{\underline{J}}}$ are the generators of the
groups $\mathsf{SO}(p)$ and $\mathsf{SO}(q)$, respectively. The parameters
$\lambda_{U}$ and $\Delta_{U}$ are called the Lorentz weight and the dimension
(or Weyl weight) of $U$, respectively. These weights are related if $U$
depends only on $\zeta_{L}$ or $\zeta_{R}$,

$\displaystyle U$ $\displaystyle=$ $\displaystyle
U(\zeta_{L})\quad\implies\quad\lambda_{U}=\Delta_{U}~{},\qquad\mathfrak{R}^{{\underline{I}}{\underline{J}}}U=0~{},$
(5.12a) $\displaystyle U$ $\displaystyle=$ $\displaystyle
U(\zeta_{R})\quad\implies\quad\lambda_{U}=-\Delta_{U}~{},\qquad\mathfrak{L}^{{\overline{I}}{\overline{J}}}U=0~{}.$
(5.12b)

As pointed out above, the algebra of conformal Killing supervector fields of
$\mathbb{M}^{(2|p,q)}$ is infinite dimensional. It contains a finite
dimensional subalgebra which is singled out by the constraints

$\displaystyle p=0:$
$\displaystyle\quad\partial_{++}\partial_{++}\partial_{++}\xi^{++}=0~{},$
(5.13a) $\displaystyle p=1:$
$\displaystyle\quad\partial_{++}\partial_{++}D_{+}\xi^{++}=0~{},$ (5.13b)
$\displaystyle p=2:$
$\displaystyle\quad\partial_{++}D_{+}^{[{\overline{I}}}D_{+}^{{\overline{J}}]}\xi^{++}=0~{},$
(5.13c) $\displaystyle p>2:$ $\displaystyle\quad
D_{+}^{[{\overline{I}}}D_{+}^{{\overline{J}}}D_{+}^{{\overline{K}}]}\xi^{++}=0~{},$
(5.13d)

and their counterparts in the right sector. Physically, these conditions mean
that $\xi$ generates those infinitesimal superconformal transformations that
belong to the superconformal algebra
$\mathfrak{osp}(p|2;\mathbb{R})\oplus\mathfrak{osp}(q|2;\mathbb{R})$. This is
the Lie algebra of the superconformal group
${\mathsf{OSp}}_{0}(p|2;{\mathbb{R}})\times{\mathsf{OSp}}_{0}(q|2;{\mathbb{R}})$,
which acts on the compactified Minkowski superspace (5.152). It is instructive
to check that (5.10) preserves the conditions (5.13). We will assume (5.13) in
what follows.

Employing (5.5), it is possible to show that the parameters (5.9a) and (5.9b)
satisfy the constraints

$\displaystyle D_{+}^{{\overline{I}}}\sigma[\xi]$
$\displaystyle=D_{+}^{{\overline{I}}}K[\xi]\quad\implies\quad\partial_{++}\sigma[\xi]=\partial_{++}K[\xi]~{},$
(5.14a) $\displaystyle D_{-}^{\underline{I}}\sigma[\xi]$
$\displaystyle=-D_{-}^{\underline{I}}K[\xi]\quad\implies\quad\partial_{--}\sigma[\xi]=-\partial_{--}K[\xi]~{}.$
(5.14b) Next, by using (5.5) in conjunction with (5.13), one obtains the
following constraints on the $R$-symmetry parameters: $\displaystyle
D_{+}^{{\overline{I}}}\rho[\xi]^{\overline{J}{\overline{K}}}$
$\displaystyle=2\delta^{{\overline{I}}[\overline{J}}D_{+}^{{\overline{K}}]}\sigma[\xi]\quad\implies\quad\partial_{++}\rho^{{\overline{I}}{\overline{J}}}[\xi]=0~{},$
(5.14c) $\displaystyle
D_{-}^{\underline{I}}\rho[\xi]^{\overline{J}{\overline{K}}}$
$\displaystyle=0\quad\implies\quad\partial_{--}\rho^{{\overline{I}}{\overline{J}}}[\xi]=0~{},$
(5.14d) $\displaystyle
D_{+}^{{\overline{I}}}\rho[\xi]^{{\underline{J}}{\underline{K}}}$
$\displaystyle=0\quad\implies\quad\partial_{++}\rho^{{\underline{I}}{\underline{J}}}[\xi]=0~{},$
(5.14e) $\displaystyle
D_{-}^{{\underline{I}}}\rho[\xi]^{{\underline{J}}{\underline{K}}}$
$\displaystyle=2\delta^{{\underline{I}}[{\underline{J}}}D_{-}^{{\underline{K}}]}\sigma[\xi]\quad\implies\quad\partial_{--}\rho^{{\underline{I}}{\underline{J}}}[\xi]=0~{},$
(5.14f) and on the scaling parameter: $\displaystyle
D_{+}^{{\overline{I}}}D_{+}^{{\overline{J}}}\sigma[\xi]$
$\displaystyle=\frac{{\rm
i}}{p}\delta^{{\overline{I}}{\overline{J}}}\partial_{++}\sigma[\xi]\quad\implies\quad\partial_{++}D_{+}^{\overline{I}}\sigma[\xi]=0~{},$
(5.14g) $\displaystyle
D_{-}^{{\underline{I}}}D_{-}^{{\underline{J}}}\sigma[\xi]$
$\displaystyle=\frac{{\rm
i}}{q}\delta^{{\underline{I}}{\underline{J}}}\partial_{--}\sigma[\xi]\quad\implies\quad\partial_{--}D_{-}^{\underline{I}}\sigma[\xi]=0~{}.$
(5.14h)

The above results mean that we may parametrise the conformal Killing
supervector fields obeying (5.13) as

$\xi\equiv\xi(\lambda(P)^{a},\lambda(Q)^{+{\overline{I}}},\lambda(Q)^{-{\underline{I}}},\lambda(M),\lambda(\mathbb{D}),\lambda(\mathfrak{L})^{{\overline{I}}{\overline{J}}},\lambda(\mathfrak{R})^{{\underline{I}}{\underline{J}}},\lambda(K)_{a},\lambda(S)_{+}^{{\overline{I}}},\lambda(S)_{-}^{{\underline{I}}})~{},$
(5.15)

where we have defined the parameters

$\displaystyle\lambda(P)^{a}$
$\displaystyle:=\xi^{a}|_{z=0}~{},\qquad\lambda(Q)^{+{\overline{I}}}:=\xi^{+{\overline{I}}}|_{z=0}~{},\qquad\lambda(Q)^{-{\underline{I}}}:=\xi^{-{\underline{I}}}|_{z=0}~{},$
(5.16a)
$\displaystyle\qquad\quad~{}\,\lambda(M):=K[\xi]|_{z=0}~{},\qquad\quad\,\lambda({\mathbb{D}}):=\sigma[\xi]|_{z=0}~{},$
(5.16b)
$\displaystyle\qquad\quad\lambda(\mathfrak{L})^{{\overline{I}}{\overline{J}}}:=\rho[\xi]^{{\overline{I}}{\overline{J}}}|_{z=0}~{},\;\,\,\quad\lambda(\mathfrak{R})^{{\underline{I}}{\underline{J}}}:=\rho[\xi]^{{\underline{I}}{\underline{J}}}|_{z=0}$
(5.16c) $\displaystyle\lambda(K)_{a}$
$\displaystyle:=\frac{1}{2}\partial_{a}\sigma[\xi]|_{z=0}~{},\quad\lambda(S)_{+}^{{\overline{I}}}:=\frac{1}{2}D_{+}^{\overline{I}}\sigma[\xi]|_{z=0}~{},\quad\lambda(S)_{-}^{{\underline{I}}}:=\frac{1}{2}D_{-}^{\underline{I}}\sigma[\xi]|_{z=0}~{}.$
(5.16d)

In particular, $\xi$ may be represented as

$\xi=\lambda(X)^{\tilde{A}}X_{\tilde{A}}~{},$ (5.17)

where we have introduced a condensed notation for the superconformal
parameters

$\displaystyle\lambda(X)^{\tilde{A}}$
$\displaystyle=(\lambda(P)^{A},\lambda(M),\lambda(\mathbb{D}),\lambda(\mathfrak{L})^{{\overline{I}}{\overline{J}}},\lambda(\mathfrak{R})^{{\underline{I}}{\underline{J}}},\lambda(K)_{A})~{},$
(5.18a) $\displaystyle\lambda(P)^{A}$
$\displaystyle=(\lambda(P)^{a},\lambda(Q)^{+{\overline{I}}},\lambda(Q)^{-{\underline{I}}}),\qquad\lambda(K)_{A}=(\lambda(K)_{a},\lambda(S)_{+}^{{\overline{I}}},\lambda(S)_{-}^{{\underline{I}}})~{},$
(5.18b)

and for generators of the superconformal algebra

$\displaystyle X_{\tilde{A}}$
$\displaystyle=(P_{A},M,\mathbb{D},\mathfrak{L}^{{\overline{I}}{\overline{J}}},\mathfrak{R}^{{\underline{I}}{\underline{J}}},K^{A})~{},$
(5.19a) $\displaystyle P_{A}$
$\displaystyle=(P_{a},Q^{{\overline{I}}}_{+},Q^{{\underline{I}}}_{-})~{},\qquad
K^{A}=(K^{a},S^{+{\overline{I}}},S^{-{\underline{I}}})~{}.$ (5.19b)

Making use of the above results allows us to derive the graded commutation
relations for the superconformal algebra. This may be achieved by utilising
the relation

$\displaystyle[\xi_{1},\xi_{2}]=-\lambda(X)_{2}^{\tilde{B}}\lambda(X)_{1}^{\tilde{A}}\big{[}X_{\tilde{A}},X_{\tilde{B}}\big{\\}}~{}.$
(5.20)

The commutation relations for the conformal algebra are as follows:111These
relations can be compared with (2.17).

$\displaystyle[M,P_{\pm\pm}]$ $\displaystyle=\pm
P_{\pm\pm}~{},\qquad~{}\,[\mathbb{D},P_{\pm\pm}]=P_{\pm\pm}~{},$ (5.21a)
$\displaystyle[M,K^{\pm\pm}]$ $\displaystyle=\mp
K^{\pm\pm}~{},\qquad[\mathbb{D},K^{\pm\pm}]=-K^{\pm\pm}~{},$ (5.21b)
$\displaystyle\;[K^{\pm\pm},P_{\pm\pm}]=2(\mathbb{D}\pm M)~{}.$ (5.21c)

The $R$-symmetry generators $\mathfrak{L}^{{\overline{I}}{\overline{J}}}$ and
$\mathfrak{R}^{{\underline{I}}{\underline{J}}}$ commute with all the
generators of the conformal group. Amongst themselves, they obey the algebra

$\displaystyle[\mathfrak{L}^{{\overline{I}}{\overline{J}}},\mathfrak{L}^{{\overline{K}}{\overline{L}}}]$
$\displaystyle=2\delta^{{\overline{K}}[{\overline{I}}}\mathfrak{L}^{{\overline{J}}]{\overline{L}}}-2\delta^{{\overline{L}}[{\overline{I}}}\mathfrak{L}^{{\overline{J}}]{\overline{K}}}~{},$
(5.22a)
$\displaystyle[\mathfrak{R}^{{\underline{I}}{\underline{J}}},\mathfrak{R}^{{\underline{K}}{\underline{L}}}]$
$\displaystyle=2\delta^{{\underline{K}}[{\underline{I}}}\mathfrak{R}^{{\underline{J}}]{\underline{L}}}-2\delta^{{\underline{L}}[{\underline{I}}}\mathfrak{R}^{{\underline{J}}]{\underline{K}}}~{}.$
(5.22b)

The superconformal algebra is then obtained by extending the translation
generator $P_{a}$ to $P_{A}$ and the special conformal generator $K^{a}$ to
$K^{A}$. The commutation relations involving the $Q$-supersymmetry generators
with the bosonic ones are:

$\displaystyle\big{[}M,Q_{+}^{\overline{I}}\big{]}$
$\displaystyle=\frac{1}{2}Q_{+}^{\overline{I}}~{},\qquad\qquad\quad\big{[}M,Q_{-}^{\underline{I}}\big{]}=-\frac{1}{2}Q_{-}^{\underline{I}}~{},$
(5.23a) $\displaystyle\big{[}\mathbb{D},Q_{+}^{\overline{I}}\big{]}$
$\displaystyle=\frac{1}{2}Q_{+}^{\overline{I}}~{},\qquad\qquad\quad\,\,\big{[}\mathbb{D},Q^{\underline{I}}_{-}\big{]}=\frac{1}{2}Q^{\underline{I}}_{-}~{},$
(5.23b)
$\displaystyle\big{[}\mathfrak{L}^{{\overline{I}}{\overline{J}}},Q_{+}^{\overline{K}}\big{]}$
$\displaystyle=2\delta^{{\overline{K}}[{\overline{I}}}Q_{+}^{{\overline{J}}]}~{},\qquad\;\;\big{[}\mathfrak{R}^{{\underline{I}}{\underline{J}}},Q_{-}^{\underline{K}}\big{]}=2\delta^{{\underline{K}}[{\underline{I}}}Q_{-}^{{\underline{J}}]}~{},$
(5.23c) $\displaystyle\big{[}K^{++},Q_{+}^{\overline{I}}\big{]}$
$\displaystyle={\rm
i}S^{+{\overline{I}}}~{},\qquad\quad\quad\,\big{[}K^{--},Q_{-}^{\underline{I}}\big{]}={\rm
i}S^{-{\underline{I}}}~{}.$ (5.23d)

The commutation relations involving the $S$-supersymmetry generators with the
bosonic operators are:

$\displaystyle\big{[}M,S^{+{\overline{I}}}\big{]}$