$\begin{split}[\delta^{(s)}_{\lambda_{1}},\delta^{(s)}_{\lambda_{2}}]q^{+a}=&2i\left(\lambda_{1}^{\alpha(s-1)\dot{\alpha}(s-2)+}\lambda_{2}^{\beta(s-2)\dot{\beta}(s-1)-}-\lambda_{1}^{\alpha(s-1)\dot{\alpha}(s-2)-}\lambda_{2}^{\beta(s-2)\dot{\beta}(s-1)+}\right)\\\
&\times\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\partial^{s-2}_{\beta(s-2)\dot{\beta}(s-2)}\partial_{\alpha\dot{\beta}}q^{+a}+{\rm
c.c.}\end{split}$ (D.6)

Using the completeness relation

$u^{+}_{i}u^{-}_{j}-u^{-}_{i}u^{+}_{j}=\epsilon_{ij}\,,$ (D.7)

we obtain:

$[\delta^{(s)}_{\lambda_{1}},\delta^{(s)}_{\lambda_{2}}]q^{+a}=2i\lambda_{1}^{\alpha(s-1)\dot{\alpha}(s-2)j}\left(\lambda_{2}^{\beta(s-2)\dot{\beta}(s-1)}\right)_{j}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\partial^{s-2}_{\beta(s-2)\dot{\beta}(s-2)}\partial_{\alpha\dot{\beta}}q^{+a}+{\rm
c.c.}$ (D.8)

For $s=2$ case, we reproduce the standard result that commutator of two rigid
supersymmetries leads to translations. In fact, the transformations (D.1) are
closed only in the $s=2$ case, while for $s>2$ new transformations are
inevitably generated (as is expected for higher-spin algebras).

Let us adduce more details for the simplest non-trivial case of $s=3$:

$[\delta^{(3)}_{\lambda_{1}},\delta^{(3)}_{\lambda_{2}}]q^{+a}=2i\lambda_{1}^{(\alpha\beta)\dot{\alpha}j}\left(\lambda_{2}^{\gamma(\dot{\beta}\dot{\gamma})}\right)_{j}\partial_{\alpha\dot{\alpha}}\partial_{\gamma\dot{\gamma}}\partial_{\beta\dot{\beta}}q^{+a}+{\rm
c.c.}$ (D.9)

The transformation in the right hand side consists of the two irreducible
pieces:

1\. Totally symmetric one:

$[\delta^{(3)}_{\lambda_{1}},\delta^{(3)}_{\lambda_{2}}]_{sym}q^{+a}=\lambda^{(\alpha\beta\gamma)(\dot{\alpha}\dot{\beta}\dot{\gamma})}\partial_{\alpha\dot{\alpha}}\partial_{\gamma\dot{\gamma}}\partial_{\beta\dot{\beta}}q^{+a},$
(D.10)

where:

$\lambda^{(\alpha\beta\gamma)(\dot{\alpha}\dot{\beta}\dot{\gamma})}=2i\lambda_{1}^{(\alpha\beta)\dot{\alpha}j}\left(\lambda_{2}^{\gamma(\dot{\beta}\dot{\gamma})}\right)_{j}+{\rm
symm}(\alpha,\beta,\gamma)+{\rm c.c.}\,.$ (D.11)

2\. Antisymmetric in $[\alpha\gamma]$, $[\dot{\alpha}\dot{\gamma}]$:

$[\delta^{(3)}_{\lambda_{1}},\delta^{(3)}_{\lambda_{2}}]_{antisym}q^{+a}=\lambda^{\beta\dot{\beta}}\partial_{\beta\dot{\beta}}\Box
q^{+a},$ (D.12)

where:

$\lambda^{\beta\dot{\beta}}=-i\left(\lambda_{1}\right)^{(\alpha\beta)j}_{\dot{\gamma}}\left(\lambda_{2}\right)^{(\dot{\beta}\dot{\gamma})}_{\alpha
j}\,+{\rm c.c.}$ (D.13)

where we used the relation:

$\partial_{[\alpha\dot{\alpha}}\partial_{\gamma]\dot{\gamma}}=\frac{1}{2}\epsilon_{\alpha\gamma}\epsilon_{\dot{\alpha}\dot{\gamma}}\Box.$
(D.14)

Thus the commutator of two rigid spin $\tfrac{3}{2}$ supersymmetries generates
a rigid bosonic spin $s=4$ transformation associated with the parameter
$\lambda^{(\alpha\beta\gamma)(\dot{\alpha}\dot{\beta}\dot{\gamma})}$ and the
spin $s=2$-like transformation with a vector parameter and one $\Box$ operator
acting on $q^{+a}$. This clearly demonstrates that the algebra of
transformations (D.1) is not closed for $s\geq 3$ 242424It is a crucial
difference of the higher spin $3/2$ supersymmetry (and its higher-rank
cousins) from those closed $3/2$ superalgebras (and their higher-rank analogs)
which were considered, e.g., in 32SUSY ..

In the general case, we see that the commutator of two symmetry
transformations with spin $(s-\tfrac{3}{2})$ parameters results in a bosonic
symmetry transformation with a spin $2s-3$ parameter $(s\geq 2)$:

$\lambda^{(\alpha(s-1)\beta(s-2))(\dot{\alpha}(s-2)\dot{\beta}(s-1))}=2i\lambda_{1}^{(\alpha(s-1)(\dot{\alpha}(s-2)j}\left(\lambda_{2}^{\beta(s-2))\dot{\beta}(s-1))}\right)_{j}+{\rm
symm}(\alpha(s-1),\beta(s-2))+{\rm c.c.},$ (D.15)

plus some extra parts (for $(s\geq 3)$) which include powers of $\Box$ acting
on $q^{+a}$ and generalize those found in the $s=3$ example.

As a simple generalization, one can also calculate the bracket of two
different fermionic symmetries:

$\begin{split}[\delta^{(s_{1})}_{\lambda_{1}},\delta^{(s_{2})}_{\lambda_{2}}]q^{+a}=&2i\left(\lambda_{1}^{\alpha(s_{1}-1)\dot{\alpha}(s_{1}-2)+}\lambda_{2}^{\beta(s_{2}-2)\dot{\beta}(s_{2}-1)-}-\lambda_{1}^{\alpha(s_{1}-1)\dot{\alpha}(s_{1}-2)-}\lambda_{2}^{\beta(s_{2}-2)\dot{\beta}(s_{2}-1)+}\right)\\\
&\times\partial^{s_{1}-2}_{\alpha(s_{1}-2)\dot{\alpha}(s_{1}-2)}\partial^{s_{2}-2}_{\beta(s_{2}-2)\dot{\beta}(s_{2}-2)}\partial_{\alpha\dot{\beta}}\left(J\right)^{P(s_{1})+P(s_{2})}q^{+a}\\\
=&2i\,\lambda_{1}^{\alpha(s_{1}-1)\dot{\alpha}(s_{1}-2)j}\left(\lambda_{2}^{\beta(s_{2}-2)\dot{\beta}(s_{2}-1)}\right)_{j}\\\
&\times\partial^{s_{1}-2}_{\alpha(s_{1}-2)\dot{\alpha}(s_{1}-2)}\partial^{s_{2}-2}_{\beta(s_{2}-2)\dot{\beta}(s_{2}-2)}\partial_{\alpha\dot{\beta}}\left(J\right)^{P(s_{1}+s_{2})}q^{+a}.\end{split}$
(D.16)

It leads to the bosonic rigid spin $s=s_{1}+s_{2}-2$ symmetry transformation
with the spin $s=s_{1}+s_{2}-3$ constant parameter and some additional
symmetries involving powers of $\Box$ acting on $q^{+a}$. For example, in the
simplest nontrivial case ($s_{1}=2,\;s_{2}=3$) the bracket of rigid spin
$\tfrac{1}{2}$ ${\cal N}=2$ supersymmetry with a rigid spin $\tfrac{3}{2}$
supersymmetry in the right hand side yields, as a leading term, the bosonic
spin 3 symmetry transformation with the rank 2 symmetric traceless tensor
constant parameter.

Here we considered the global symmetries of a free hypermultiplet. If we
switch to the corresponding gauge transformations, then a similar reasoning
leads to the conclusion that in the right hand sides of the commutation
relations, gauge transformations of an increasing higher spins will appear.
This implies that in the hypothetical complete consistent interacting theory
of ${\cal N}=2$ higher spins, the entire infinite towers of all higher spins
should simultaneously participate, in agreement with a general belief.

###  D.2 Fermionic symmetry with spin $\tfrac{3}{2}$ parameter

It is instructive to see how the above transformations look at the component
level. Let us consider the $s=3$ rigid transformations. It is easy to verify
that the bosonic symmetry is the well-known spin $3$ symmetry of the free
bosonic action Berends:1985xx . We shall pay our attention to the less trivial
fermionic rigid symmetries:

$\begin{split}\delta^{(3)}_{rig}q^{+a}=&\quad
2i\left(\lambda^{(\alpha\beta)(\dot{\alpha}-}\bar{\theta}^{+\dot{\beta})}+\theta^{+(\alpha}\lambda^{\beta)(\dot{\alpha}\dot{\beta})-}\right)\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}i(\tau_{3})^{a}_{\;b}q^{+b}\\\
&-2i\left(\lambda^{(\alpha\beta)\dot{\alpha}-}\theta^{+}_{\beta}+\lambda^{\alpha(\dot{\alpha}\dot{\beta})-}\bar{\theta}^{+}_{\dot{\beta}}\right)\partial_{\alpha\dot{\alpha}}\partial_{5}i(\tau_{3})^{a}_{\;b}q^{+b}\\\
&+\left(\lambda^{(\alpha\beta)\dot{\alpha}+}\partial^{-}_{\beta}+\lambda^{(\dot{\alpha}\dot{\beta})\alpha+}\partial^{-}_{\dot{\alpha}}\right)\partial_{\alpha\dot{\beta}}i(\tau_{3})^{a}_{\;b}q^{+b}\,.\end{split}$
(D.17)

They are characterized by the constant spin $\tfrac{3}{2}$ Grassmann
parameters $\lambda^{(\alpha\beta)\dot{\alpha}i}$ and
$\bar{\lambda}^{(\dot{\alpha}\dot{\beta})\alpha i}$. These transformations are
realized on the on-shell hypermultiplet fields as:

$\begin{cases}\delta
f^{i}=i\lambda^{(\alpha\beta)\dot{\alpha}i}\partial_{\alpha\dot{\alpha}}\chi_{\beta}+i\bar{\lambda}^{(\dot{\alpha}\dot{\beta})\alpha
i}\partial_{\alpha\dot{\alpha}}\bar{\psi}_{\dot{\beta}}\,,\\\
\delta\bar{f}_{i}=-i\lambda_{i}^{(\alpha\beta)\dot{\alpha}}\partial_{\alpha\dot{\alpha}}\psi_{\beta}-i\bar{\lambda}_{i}^{(\dot{\alpha}\dot{\beta})\alpha}\partial_{\alpha\dot{\alpha}}\bar{\chi}_{\dot{\beta}}\,,\\\
\delta\psi_{\alpha}=-2\bar{\lambda}^{\beta(\dot{\alpha}\dot{\beta})i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\bar{f}_{i}-2im\lambda_{\alpha}^{\beta\dot{\beta}i}\partial_{\beta\dot{\beta}}\bar{f}_{i}\,,\\\
\delta\bar{\psi}_{\dot{\alpha}}=2\lambda^{(\alpha\beta)\dot{\beta}i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}f_{i}-2im\bar{\lambda}_{\dot{\alpha}}^{\beta\dot{\beta}i}\partial_{\beta\dot{\beta}}f_{i}\,,\\\
\delta\chi_{\alpha}=-2\bar{\lambda}^{\beta(\dot{\alpha}\dot{\beta})i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}f_{i}-2im\lambda_{\alpha}^{\beta\dot{\beta}i}\partial_{\beta\dot{\beta}}f_{i}\,,\\\
\delta\bar{\chi}_{\dot{\alpha}}=2\lambda^{(\alpha\beta)\dot{\beta}i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\bar{f}_{i}-2im\bar{\lambda}_{\dot{\alpha}}^{\dot{\beta}\beta
i}\partial_{\beta\dot{\beta}}\bar{f}_{i}\,.\end{cases}$ (D.18)

The resulting transformations look similar to $\mathcal{N}=2$ supersymmetry
transformations (5.11) of the free hypermultiplet, but involve more
derivatives. One can easily check that on-shell hypermultiplet action (5.13)
is invariant under these transformations (up to total derivatives). So we
observe that among symmetries of free hypermultiplet (D.1) there is a kind of
$\mathcal{N}=2$ “higher spin supersymmetry” transformations. The corresponding
conserved Noether current is given by the expression

$\begin{split}J^{\alpha\dot{\alpha}}_{(\mu\nu)\dot{\mu}i}=&\frac{i}{2}\left[\partial_{(\mu\dot{\mu}}\chi_{\nu)}\partial^{\alpha\dot{\alpha}}\bar{f}_{i}-\partial^{\alpha\dot{\alpha}}\partial_{(\mu\dot{\mu}}\chi_{\nu)}\bar{f}_{i}\right]+\frac{i}{2}\left[\partial_{\mu\dot{\mu}}\psi_{\beta}\partial^{\alpha\dot{\alpha}}f_{i}-\partial^{\alpha\dot{\alpha}}\partial_{\mu\dot{\mu}}\psi_{\beta}f_{i}\right]\\\
&-\delta_{(\mu}^{\alpha}\Big{[}\partial_{\nu)\dot{\mu}}\bar{f}_{i}\left(i\partial^{\dot{\alpha}\rho}\chi_{\rho}+m\bar{\psi}^{\dot{\alpha}}\right)+\partial_{\nu)\dot{\mu}}f_{i}\left(i\partial^{\dot{\alpha}\rho}\phi_{\rho}+m\bar{\chi}^{\dot{\alpha}}\right)\Big{]}\\\
&+\delta^{\alpha}_{(\nu}\delta^{\dot{\alpha}}_{\dot{\mu}}\Bigr{[}if_{i}\left(\Box\psi_{\mu}+m^{2}\psi_{\mu}\right)+i\bar{f}_{i}\left(\Box\chi_{\mu}+m^{2}\chi_{\mu}\right)\Bigr{]},\end{split}$
(D.19)
$\partial_{\alpha\dot{\alpha}}J^{\alpha\dot{\alpha}}_{(\mu\nu)\dot{\mu}i}=0\,.$
(D.20)

Note that the second and third lines of (D.19) are vanishing on the free
hypermultiplet equations of motion (5.12).

One can calculate Lie bracket of such transformations, for example:

$\begin{split}[\delta_{\lambda_{1}},\delta_{\bar{\lambda}_{2}}]f^{i}=&-2i\left(\lambda_{1}^{(\alpha\beta)\dot{\alpha}j}\bar{\lambda}_{2}^{(\dot{\rho}\dot{\sigma})\rho
i}-\lambda_{1}^{(\alpha\beta)\dot{\alpha}i}\bar{\lambda}_{2}^{(\dot{\rho}\dot{\sigma})\rho
j}\right)\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\sigma}}\partial_{\rho\dot{\rho}}f_{j}\\\
=&-2i\lambda_{1}^{(\alpha\beta)\dot{\alpha}j}\left(\bar{\lambda}_{2}^{(\dot{\rho}\dot{\sigma})\rho}\right)_{j}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\sigma}}\partial_{\rho\dot{\rho}}f^{i}.\end{split}$
(D.21)

So in the full agreement with the discussion of off-shell $s=3$ transformation
(D.9) in the previous Subsection D.1, we observe that the commutator of two
spin $\tfrac{3}{2}$ “supersymmetries” yields spin $s=4$ symmetry with a spin 3
parameter and an extra $\Box$ term.

###  D.3 Noether theorem

In this subsection we derive the current superfields corresponding to the
rigid symmetries (D.1) of the free hypermultiplet action.

We will use Noether theorem in the form which can be simply extended to the
superspace setting. If some continuous transformations with parameters $a^{A}$
($A$ is an index enumerating symmetries) form a rigid symmetry, then the
variation of the action with the coordinate-dependent parameters $a^{A}(x)$
has the universal form:

$\delta S=\int d^{4}x\;\partial^{\mu}a^{A}(x)J^{A}_{\mu}(x)\,.$ (D.22)

If $a^{A}(x)=const$ we obtain that $\delta S=0$ in agreement with the
assumption that the given transformation is a symmetry of the action. On the
equation of motion, any field variation is vanishing:

$\delta S=0\,,$ (D.23)

and we conclude, that on equations of motions, due to an arbitrariness of the
parameters $a^{A}(x)$, the current in (D.22) is subject to the conservation
law:

$\partial^{\mu}J^{A}_{\mu}(x)=0\,.$ (D.24)

Now we shall apply this general procedure to rigid symmetries of the free
hypermultiplet. Varying the action (5.1) by the transformations (D.1) with
arbitrary analytic parameters, which constitute a local version of the rigid
symmetry transformations from the previous subsection, we obtain:

$\begin{split}\delta S=&\frac{1}{2}\int
d\zeta^{(-4)}\;\Bigr{[}\lambda^{\alpha(s-2)\dot{\alpha}(s-2)M}\left(\partial_{M}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}(J)^{P(s)}q^{+a}\right)\mathcal{D}^{++}q^{+}_{a}\\\
&\qquad\qquad\qquad+q^{+a}\left(\mathcal{D}^{++}\lambda^{\alpha(s-2)\dot{\alpha}(s-2)M}\partial_{M}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}(J)^{P(s)}q^{+}_{a}\right)\Bigr{]}.\end{split}$
(D.25)

One can rewrite the second term using the relation:

$\begin{split}\Bigr{(}\mathcal{D}^{++}\lambda^{\alpha(s-2)\dot{\alpha}(s-2)M}\partial_{M}&\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}(J)^{P(s)}q^{+}_{a}\Bigr{)}\\\
=&\left(\mathcal{D}^{++}\lambda^{\alpha(s-2)\dot{\alpha}(s-2)M}\right)\partial_{M}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}(J)^{P(s)}q^{+}_{a}\\\
&+\lambda^{\alpha(s-2)\dot{\alpha}(s-2)M}\left[\mathcal{D}^{++},\partial_{M}\right]\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}(J)^{P(s)}q^{+}_{a}\\\
&+\lambda^{\alpha(s-2)\dot{\alpha}(s-2)M}\partial_{M}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}(J)^{P(s)}\mathcal{D}^{++}q^{+}_{a}\,.\end{split}$
(D.26)

To obtain the conserved currents it is convenient to make use of the free
equations of motion to eliminate terms involving $\mathcal{D}^{++}q^{+}_{a}$
which vanish on the hypermultiplet mass shell. It is also useful to separate,
in the variation (D.25), the contributions with different index structures:

$\delta S\Bigr{|}_{M=\alpha\dot{\alpha}}=\frac{1}{2}\int
d\zeta^{(-4)}\;\left(\mathcal{D}^{++}\lambda^{\alpha(s-1)\dot{\alpha}(s-1)}\right)q^{+a}\partial^{s-1}_{\alpha(s-1)\dot{\alpha}(s-1)}(J)^{P(s)}q^{+}_{a}\,,$
(D.27) $\delta S\Bigr{|}_{M=5}=\frac{1}{2}\int
d\zeta^{(-4)}\;\left(\mathcal{D}^{++}\lambda^{\alpha(s-2)\dot{\alpha}(s-2)}\right)q^{+a}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\partial_{5}(J)^{P(s)}q^{+}_{a}\,,$
(D.28) $\begin{split}\delta S\Bigr{|}_{M=\alpha}=\frac{1}{2}\int
d\zeta^{(-4)}\;&\left(\mathcal{D}^{++}\lambda^{\alpha(s-1)\dot{\alpha}(s-2)+}\right)q^{+a}\partial^{-}_{\alpha}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}(J)^{P(s)}q^{+}_{a}\\\
&+\lambda^{\alpha(s-1)\dot{\alpha}(s-2)}q^{+a}[\mathcal{D}^{++},\partial^{-}_{\alpha}]\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}(J)^{P(s)}q^{+}_{a}\,,\end{split}$
(D.29) $\begin{split}\delta S\Bigr{|}_{M=\dot{\alpha}}=\frac{1}{2}\int
d\zeta^{(-4)}\;&\left(\mathcal{D}^{++}\lambda^{\alpha(s-2)\dot{\alpha}(s-1)+}\right)q^{+a}\partial^{-}_{\dot{\alpha}}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}(J)^{P(s)}q^{+}_{a}\\\
&+\lambda^{\alpha(s-2)\dot{\alpha}(s-1)+}q^{+a}[\mathcal{D}^{++},\partial^{-}_{\dot{\alpha}}]\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}(J)^{P(s)}q^{+}_{a}\,.\end{split}$
(D.30)

One immediately observes that, if the parameters are rigid as in (D.3), i.e.,
satisfy the condition $[\mathcal{D}^{++},\lambda^{M}\partial_{M}]=0$, all
these variations vanish in full agreement with the general statement of
Noether theorem. In this way we derive the current superfield conservation
laws:

$\mathcal{D}^{++}\left(q^{+a}\partial^{s-1}_{\alpha(s-1)\dot{\alpha}(s-1)}(J)^{P(s)}q^{+}_{a}\right)=0\,,$
(D.31)
$\mathcal{D}^{++}\left(q^{+a}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\partial_{5}(J)^{P(s)}q^{+}_{a}\right)=0\,,$
(D.32)
$\mathcal{D}^{++}\left(q^{+a}\partial^{-}_{\alpha}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}(J)^{P(s)}q^{+}_{a}\right)=q^{+a}[\mathcal{D}^{++},\partial^{-}_{\alpha}]\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}(J)^{P(s)}q^{+}_{a}\,,$
(D.33)
$\mathcal{D}^{++}\left(q^{+a}\partial^{-}_{\dot{\alpha}}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}(J)^{P(s)}q^{+}_{a}\right)=q^{+a}[\mathcal{D}^{++},\partial^{-}_{\dot{\alpha}}]\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}(J)^{P(s)}q^{+}_{a}\,.$
(D.34)

The first two equations (D.31) and (D.32) directly give the laws of
conservation of current superfields (5.46) and (5.49).

Taking into account the equations

$[\mathcal{D}^{++},\partial^{-}_{\alpha}]=-\partial^{+}_{\alpha}+2i\bar{\theta}^{+\dot{\alpha}}\partial_{\alpha\dot{\alpha}}+2i\theta^{+}_{\alpha}\partial_{5},$
(D.35a)
$[\mathcal{D}^{++},\partial^{-}_{\dot{\alpha}}]=-\partial^{+}_{\dot{\alpha}}-2i\theta^{+\alpha}\partial_{\alpha\dot{\alpha}}+2i\bar{\theta}^{+}_{\dot{\alpha}}\partial_{5},$
(D.35b)

we can immediately identify eqs. (D.33) and (D.34) with the modified
conservation laws (5.55) and (5.62).

So we conclude that the above equations fully reproduce the current superfield
conservation conditions from Section 5.4 (we restricted our consideration to
the $\xi=0$ case, because $\xi\neq 0$ gives a trivial contribution to the
current superfield) and give the explicit form of the current superfields.
Thus we have established the direct connection between the rigid symmetries
(D.1) and the conserved current superfields. This Noether theorem can be
considered as a generalization of Noether theorem in ordinary superspace
Magro:2001aj  to the HSS case.

In full agreement with the results of Appendix C.1, the superfield conditions
derived above lead to the conservation of the bosonic spin $s$ current, the
doublet of fermionic spin $s-\tfrac{1}{2}$ currents and the bosonic spin $s-1$
current. These currents correspond to the global spin $\mathbf{s}$ symmetry
(D.1) of the free hypermultiplet.

In conclusion of this Section, note that the conservation law (5.82)

$\partial_{\alpha\dot{\alpha}}J^{++\alpha\dot{\alpha}}=0$ (D.36)

does not directly follow from Noether theorem for rigid symmetries with the
structure (D.1). Presumably, it is related to some other symmetry of the free
hypermultiplet. It would be interesting to explicitly identify this extra
bosonic symmetry.

##  Appendix E Field redefinitions

In this Appendix we describe various field redefinitions capable to simplify
the structure of the linearized cubic bosonic vertices in the spin 2 and spin
3 cases.

Before proceeding to concrete examples, it is important to note a simple
observation that if a cubic vertex is equal to zero on free equations of
motion, then such a vertex can be removed by appropriate redefinition of
fields. This observation is based on the property that, when redefining
fields:

$\phi\to\phi+F(\phi)\,,$ (E.1)

the action is changing as

$S[\phi]\to S[\phi]+\int d^{4}x\frac{\delta S[\phi]}{\delta\phi}F(\phi).$
(E.2)

The variation of the action on the equations of motion is zero, which leads to
the vanishing of the vertex on the equations of motion. Such interactions are
called "fake interactions", see e.g. Ponomarev:2022vjb .

###  E.1 Minimal gravity coupling of scalar fields and scalar field
redefinitions

For comparing the results of Section 6.3 with those known in literature, it is
more convenient to pass from the spinor notation for spin 2 fields to the more
accustomed vector notation:

$\Phi^{\alpha\dot{\alpha}\beta\dot{\beta}}:=\sigma^{\alpha\dot{\alpha}}_{n}\sigma^{\beta\dot{\beta}}_{m}h^{(nm)}\,,$
(E.3)
$\mathcal{R}=\frac{1}{2}R=\frac{1}{2}\left(\partial^{n}\partial^{m}-\eta^{nm}\Box\right)h_{mn}\,.$
(E.4)

With this notation, the scalar part of on-shell hypermultiplet action (5.13)
and the (2,0,0) coupling derived in (6.24) looks as:

$\begin{split}S_{free}+S^{(s=2)}_{int}=&\int
d^{4}x\,\left(\partial_{n}f^{i}\partial^{n}\bar{f}_{i}-m^{2}f^{i}\bar{f}_{i}\right)\\\
&-\kappa_{2}\int
d^{4}x\;h^{(mn)}\left(\partial_{m}\partial_{n}f^{i}\bar{f}_{i}-2\partial_{m}f^{i}\partial_{n}\bar{f}_{i}+f^{i}\partial_{m}\partial_{n}\bar{f}_{i}\right)\\\
&+\kappa_{2}\int
d^{4}x\;R\left(f^{i}\bar{f}_{i}\right)+\mathcal{O}(\kappa_{2}^{2})\,.\end{split}$
(E.5)

It will be convenient to integrate by parts in the second line to bring the
interacting action in the form:

$S^{(s=2)}_{int}=\kappa_{2}\int
d^{4}x\;\left(4h^{(mn)}\partial_{m}f^{i}\partial_{n}\bar{f}_{i}-\Box
hf^{i}\bar{f}_{i}\right)+\mathcal{O}(\kappa_{2}^{2}).$ (E.6)

On the other hand, the standard (non-linear) minimal $(2,0,0)$ interaction has
a different form:

$S^{(s=2)}_{min}=\int
d^{4}x\sqrt{-g}\;\left(g^{mn}\partial_{m}f^{i}\partial_{n}\bar{f}_{i}-m^{2}f^{i}\bar{f}_{i}\right).$
(E.7)

To compare it with our result, we need to consider the linearized cubic
vertices that follow from this Lagrangian. To do this, it is necessary to
establish a relationship between the metric $g_{mn}$ and the fields (E.3) used
for spin 2. According to Galperin:1987ek ; Galperin:1987em  the full analytic
$\mathcal{N}=2$ supergravity potential $H^{++m}$ in Wess-Zumino gauge has the
form:

$H^{++m}=-2i\theta^{+}\sigma^{a}\bar{\theta}^{+}e_{a}^{m}+\dots,$ (E.8)

where $e_{a}^{m}$ is the “inverse vierbein”, i.e.:

$g_{mn}=\eta_{ab}e^{a}_{m}e^{b}_{n}.$ (E.9)

Using the relation

$H^{++m}=-2i\theta^{+}\sigma^{m}\bar{\theta}^{+}+\kappa_{2}h^{++\alpha\dot{\alpha}}\sigma^{m}_{\alpha\dot{\alpha}},$
(E.10)

the spin $\mathbf{2}$ Wess-Zumino gauge (3.13) and the relation (E.3), we
obtain:

$e_{a}^{m}=\delta_{a}^{m}+2\kappa_{2}h_{a}^{m}.$ (E.11)

So we have:

$g_{mn}=\eta_{mn}-4\kappa_{2}h_{mn}+\mathcal{O}(h^{2})\,,\qquad\sqrt{-g}=1-2\kappa_{2}h+\mathcal{O}(\kappa_{2}^{2})\,,$
(E.12)

and obtain that the action (E.7), to the linear order in the spin 2 field
$h^{mn}$, reads:

$\begin{split}S^{(s=2)}_{int-min}=&\int
d^{4}x\;\left(\eta^{mn}\partial_{m}f^{i}\partial_{n}\bar{f}_{i}-m^{2}f^{i}\bar{f}_{i}\right)\\\
&+4\kappa_{2}\int
d^{4}x\;h^{mn}\partial_{m}f^{i}\partial_{n}\bar{f}_{i}-2\kappa_{2}\int
d^{4}x\;h\left(\eta^{mn}\partial_{m}f^{i}\partial_{n}\bar{f}_{i}-m^{2}f^{i}\bar{f}_{i}\right)+\mathcal{O}(\kappa_{2}^{2})\,.\end{split}$
(E.13)

Now we redefine the scalar field as

$f^{i}=\left(-g\right)^{-\frac{1}{4}}\mathfrak{f}^{i}=\mathfrak{f}^{i}+\kappa_{2}h\mathfrak{f}^{i}+\mathcal{O}(\kappa_{2}^{2})\,.$
(E.14)

Then the free kinetic term of $f^{i}$ is transformed to

$\begin{split}S_{free}=&\int
d^{4}x\;\left(\eta^{mn}\partial_{m}f^{i}\partial_{n}\bar{f}_{i}-m^{2}f^{i}\bar{f}_{i}\right)\\\
=&\int
d^{4}x\;\left(\eta^{mn}\partial_{m}\mathfrak{f}^{i}\partial_{n}\bar{\mathfrak{f}}_{i}-m^{2}\mathfrak{f}^{i}\bar{\mathfrak{f}}_{i}\right)\\\
&+2\kappa_{2}\int
d^{4}x\;h\left(\eta^{mn}\partial_{m}\mathfrak{f}^{i}\partial_{n}\bar{\mathfrak{f}}_{i}-m^{2}\mathfrak{f}^{i}\bar{\mathfrak{f}}_{i}\right)-\kappa_{2}\int
d^{4}x\;\Box
h\left(\mathfrak{f}^{i}\bar{\mathfrak{f}}_{i}\right)+\mathcal{O}(\kappa_{2}^{2})\,,\end{split}$
(E.15)

which modifies the linearized cubic interaction. The action (E.13) takes the
form:

$\begin{split}S^{(s=2)}_{int-min}=&\int
d^{4}x\;\left(\eta^{mn}\partial_{m}\mathfrak{f}^{i}\partial_{n}\bar{\mathfrak{f}}_{i}-m^{2}\mathfrak{f}^{i}\bar{\mathfrak{f}}_{i}\right)\\\
&+4\kappa_{2}\int
d^{4}x\;h^{mn}\partial_{m}\mathfrak{f}^{i}\partial_{n}\bar{\mathfrak{f}}_{i}-\kappa_{2}\int
d^{4}x\;\Box
h\left(\mathfrak{f}^{i}\bar{\mathfrak{f}}_{i}\right)+\mathcal{O}(\kappa_{2}^{2})\,.\end{split}$
(E.16)

Thus the interaction of spin 2 with scalars (E.6) found in Section 6.3,
actually coincides with the minimal (E.7) up to the field redefinitions
(E.14).

###  E.2 Redefinition of the spin 2 gauge field and $(2,0,0)$ coupling

In this Subsection we study how a redefinition of the spin 2 gauge field can
affect the structure of the (2, 0, 0) vertex.

Recall the Pauli-Fierz action (A.20) in the spinor notations:

$\begin{split}S_{PF}(\Phi^{(\alpha\beta)(\dot{\alpha}\dot{\beta})},\Upsilon)&=-\int
d^{4}x\;\Big{[}\Phi^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\Box\Phi_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}-\Phi^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\partial_{\alpha\dot{\alpha}}\partial^{\rho\dot{\rho}}\Phi_{(\rho\beta)(\dot{\rho}\dot{\beta})}\\\
&\;\;\;\;\;\;\;\;\;\;\;+2\,\Phi\partial^{\alpha\dot{\alpha}}\partial^{\beta\dot{\beta}}\Phi_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}-6\Phi\Box\Phi\Big{]}.\end{split}$
(E.17)

Redefining the field $\Phi$,

$\Phi\,\Rightarrow\,\Phi+af^{i}\bar{f}_{i}\,,$ (E.18)

leads to:

$\begin{split}S_{PF}(\Phi^{(\alpha\beta)(\dot{\alpha}\dot{\beta})},\Phi)\,\Rightarrow\,&S_{PF}(\Phi^{(\alpha\beta)(\dot{\alpha}\dot{\beta})},\Phi)\\\
&-a\int
d^{4}x\;\left[2\Phi^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\left(f^{i}\bar{f}_{i}\right)-12\Phi\Box\left(f^{i}\bar{f}_{i}\right)\right]+\mathcal{O}(a^{2}).\end{split}$
(E.19)

After integration by parts one can rewrite the second line in terms of scalar
curvature $\mathcal{R}$ introduced in (A.11):

$S_{PF}(\Phi^{(\alpha\beta)(\dot{\alpha}\dot{\beta})},\Phi)\;\Rightarrow\;S_{PF}(\Phi^{(\alpha\beta)(\dot{\alpha}\dot{\beta})},\Phi)-16a\int
d^{4}x\;\mathcal{R}\left(f^{i}\bar{f}_{i}\right)+\mathcal{O}(a^{2})\,.$ (E.20)

Thus we have shown that at the linearized level, the $(2,0,0)$ vertex of form
$\mathcal{R}\bar{f}f$ can be eliminated by the redefinition (E.18) of the spin
2 field trace. This property was used, e.g., in the discussion in the end of
Section 6.4.

###  E.3 Redefinition of scalar fields and $(3,0,0)$ coupling

1. In a direct analogy with (E.14) it is possible to redefine the complex scalar fields as:

$f^{i}=\mathfrak{f}^{i}-ib\left(\partial_{\alpha\dot{\alpha}}\Phi^{\alpha\dot{\alpha}}\right)\mathfrak{f}^{i}\,,\qquad\bar{f}_{i}=\bar{\mathfrak{f}}_{i}+ib\left(\partial_{\alpha\dot{\alpha}}\Phi^{\alpha\dot{\alpha}}\right)\bar{\mathfrak{f}}_{i}\,.$
(E.21)

Here $b$ is some real parameter and the field $\Phi^{\alpha\dot{\alpha}}$ was
defined in (A.29). Under this redefinition the free scalar Lagrangian changes
as:

$S_{free}(f^{i})=S_{free}(\mathfrak{f}^{i})-\frac{ib}{2}\Phi^{\alpha\dot{\alpha}}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\left(\mathfrak{f}^{i}\partial^{\beta\dot{\beta}}\bar{\mathfrak{f}}_{i}-\partial^{\beta\dot{\beta}}\mathfrak{f}^{i}\bar{\mathfrak{f}}_{i}\right)+\mathcal{O}(b^{2})\,.$
(E.22)

So, making the redefinition (E.21), one can eliminate the cubic vertex of the
form appearing, e.g., in (6.53).

2. There is one more admissible scalar fields redefinition:

$f^{i}=\mathfrak{f}^{i}-ic\,\Phi^{\alpha\dot{\alpha}}\partial_{\alpha\dot{\alpha}}\mathfrak{f}^{i}\,,\qquad\bar{f}_{i}=\bar{\mathfrak{f}}_{i}+ic\,\Phi^{\alpha\dot{\alpha}}\partial_{\alpha\dot{\alpha}}\bar{\mathfrak{f}}_{i}\,.$
(E.23)

This redefinition leads to:

$\begin{split}S_{free}(f^{i})=&\;S_{free}(\mathfrak{f}^{i})-\frac{ic}{2}\Phi^{\beta\dot{\beta}}\left(\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\mathfrak{f}^{i}\partial^{\alpha\dot{\alpha}}\bar{\mathfrak{f}}_{i}-\partial_{\alpha\dot{\alpha}}\mathfrak{f}^{i}\partial^{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\bar{\mathfrak{f}}_{i}\right)\\\
&+ic\;m^{2}\Phi^{\beta\dot{\beta}}\left(\partial_{\beta\dot{\beta}}\mathfrak{f}^{i}\bar{\mathfrak{f}}_{i}-\mathfrak{f}^{i}\partial_{\beta\dot{\beta}}\bar{\mathfrak{f}}_{i}\right)+\mathcal{O}(c^{2}).\end{split}$
(E.24)

Further manipulations lead to the expression:

$\begin{split}S_{free}(f^{i})=&\;S_{free}(\mathfrak{f}^{i})-\frac{ic}{2}\int
d^{4}x\,\Phi^{\beta\dot{\beta}}\Box\left(\partial_{\beta\dot{\beta}}\mathfrak{f}^{i}\bar{\mathfrak{f}}_{i}-\mathfrak{f}^{i}\partial_{\beta\dot{\beta}}\bar{\mathfrak{f}}_{i}\right)\\\
&+\frac{ic}{2}\;\int
d^{4}x\,\Phi^{\beta\dot{\beta}}\Bigr{\\{}\partial_{\beta\dot{\beta}}(\Box+m^{2})\mathfrak{f}^{i}\bar{\mathfrak{f}}_{i}+\partial_{\beta\dot{\beta}}\mathfrak{f}^{i}(\Box+m^{2})\bar{\mathfrak{f}}_{i}\\\
&\qquad\qquad\qquad\qquad-(\Box+m^{2})\mathfrak{f}^{i}\partial_{\beta\dot{\beta}}\bar{\mathfrak{f}}_{i}-\mathfrak{f}^{i}\partial_{\beta\dot{\beta}}(\Box+m^{2})\bar{\mathfrak{f}}_{i}\Bigr{\\}}+\mathcal{O}(c^{2}).\end{split}$
(E.25)

So on the free equations of motions for $\mathfrak{f}^{i}$ one can eliminate
the cubic $(3,0,0)$ coupling of the form:

$\int
d^{4}x\,\Phi^{\beta\dot{\beta}}\,\Box\,\left(\partial_{\beta\dot{\beta}}\mathfrak{f}^{i}\bar{\mathfrak{f}}_{i}-\mathfrak{f}^{i}\partial_{\beta\dot{\beta}}\bar{\mathfrak{f}}_{i}\right).$
(E.26)

###  E.4 Redefinition of spin 3 gauge field and $(3,0,0)$ coupling

The spin 3 action (A.42) has the form:

$\displaystyle
S^{(s=3)}\Big{(}\Phi^{(\alpha_{1}\alpha_{2}\alpha_{3})(\dot{\alpha}_{1}\dot{\alpha}_{2}\dot{\alpha}_{3})},\Phi^{\alpha\dot{\alpha}}\Big{)}=\int
d^{4}x\,\Big{[}\Phi^{(\alpha_{1}\alpha_{2}\alpha_{3})(\dot{\alpha}_{1}\dot{\alpha}_{2}\dot{\alpha}_{3})}\Box\Phi_{(\alpha_{1}\alpha_{2}\alpha_{3})(\dot{\alpha}_{1}\dot{\alpha}_{2}\dot{\alpha}_{3})}$
$\displaystyle-\,\frac{3}{2}\Phi^{(\alpha_{1}\alpha_{2}\alpha_{3})(\dot{\alpha}_{1}\dot{\alpha}_{2}\dot{\alpha}_{3})}\partial_{\alpha_{1}\dot{\alpha}_{1}}\partial^{\rho\dot{\rho}}\Phi_{(\rho\alpha_{2}\alpha_{3})(\dot{\rho}\dot{\alpha}_{2}\dot{\alpha}_{3})}+3\Phi^{(\alpha_{1}\alpha_{2}\alpha_{3})(\dot{\alpha}_{1}\dot{\alpha}_{2}\dot{\alpha}_{3})}\partial_{\alpha_{1}\dot{\alpha}_{1}}\partial_{\alpha_{2}\dot{\alpha}_{2}}\Phi_{\alpha_{3}\dot{\alpha}_{3}}$
$\displaystyle\,-\frac{15}{4}\Phi^{\alpha\dot{\alpha}}\Box\Phi_{\alpha\dot{\alpha}}+\frac{3}{8}\partial_{\alpha_{1}\dot{\alpha}_{1}}\Phi^{\alpha_{1}\dot{\alpha}_{1}}\partial_{\alpha_{2}\dot{\alpha}_{2}}\Phi^{\alpha_{2}\dot{\alpha}_{2}}\Big{]}.$
(E.27)

After the field redefinition,

$\Phi^{\alpha\dot{\alpha}}\Rightarrow\Phi^{\alpha\dot{\alpha}}+iu\left(f^{i}\partial^{\alpha\dot{\alpha}}\bar{f}_{i}-\partial^{\alpha\dot{\alpha}}f^{i}\bar{f}_{i}\right),$
(E.28)

the action is redefined as:

$\begin{split}S^{(s=3)}\Big{(}&\Phi^{(\alpha_{1}\alpha_{2}\alpha_{3})(\dot{\alpha}_{1}\dot{\alpha}_{2}\dot{\alpha}_{3})},\Phi^{\alpha\dot{\alpha}}\Big{)}\Rightarrow\,S^{(s=3)}\Big{(}\Phi^{(\alpha_{1}\alpha_{2}\alpha_{3})(\dot{\alpha}_{1}\dot{\alpha}_{2}\dot{\alpha}_{3})},\Phi^{\alpha\dot{\alpha}}\Big{)}\\\
&+iu\int
d^{4}x\;\Bigr{[}3\Phi^{(\alpha_{1}\alpha_{2}\alpha_{3})(\dot{\alpha}_{1}\dot{\alpha}_{2}\dot{\alpha}_{3})}\partial_{\alpha_{1}\dot{\alpha}_{1}}\partial_{\alpha_{2}\dot{\alpha}_{2}}\left(f^{i}\partial_{\alpha_{3}\dot{\alpha}_{3}}\bar{f}_{i}-\partial_{\alpha_{3}\dot{\alpha}_{3}}f^{i}\bar{f}_{i}\right)\\\
&\qquad\qquad\qquad-\frac{15}{2}\Phi^{\alpha\dot{\alpha}}\Box\left(f^{i}\partial^{\alpha\dot{\alpha}}\bar{f}_{i}-\partial^{\alpha\dot{\alpha}}f^{i}\bar{f}_{i}\right)\\\
&\qquad\qquad\qquad-\frac{3}{4}\Phi^{\alpha_{1}\dot{\alpha}_{1}}\partial_{\alpha_{1}\dot{\alpha}_{1}}\partial_{\alpha_{2}\dot{\alpha}_{2}}\left(f^{i}\partial^{\alpha_{2}\dot{\alpha}_{2}}\bar{f}_{i}-\partial^{\alpha_{2}\dot{\alpha}_{2}}f^{i}\bar{f}_{i}\right)\Bigr{]}+\mathcal{O}(u^{2})\,.\end{split}$
(E.29)

Using the definition of the linearized spin 3 curvature (A.39) and integrating
by parts, one can rewrite the transformation (E.29) as

$\begin{split}S^{(s=3)}\Big{(}&\Phi^{(\alpha_{1}\alpha_{2}\alpha_{3})(\dot{\alpha}_{1}\dot{\alpha}_{2}\dot{\alpha}_{3})},\Phi^{\alpha\dot{\alpha}}\Big{)}\;\Rightarrow\;S^{(s=3)}\Big{(}\Phi^{(\alpha_{1}\alpha_{2}\alpha_{3})(\dot{\alpha}_{1}\dot{\alpha}_{2}\dot{\alpha}_{3})},\Phi^{\alpha\dot{\alpha}}\Big{)}\\\
&+iu\int
d^{4}x\;\Bigr{[}3\mathcal{R}^{\alpha\dot{\alpha}}\left(f^{i}\partial_{\alpha\dot{\alpha}}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}f^{i}\bar{f}_{i}\right)\Bigr{]}+\mathcal{O}(u^{2})\,.\end{split}$
(E.30)

So, the field redefinition (E.28) ensures eliminating the $(3,0,0)$ cubic
couplings of the form:

$i\int
d^{4}x\;\mathcal{R}^{\alpha\dot{\alpha}}\left(f^{i}\partial_{\alpha\dot{\alpha}}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}f^{i}\bar{f}_{i}\right).$
(E.31)

##  Appendix F Gauge-fixing

The aim of this Section is to rewrite the superfield equations of motions for
gauge superfields in terms of unconstrained analytic potentials $h^{++\dots}$
subject to the proper manifestly supersymmetric gauge conditions. Equations
(4.43) are written in terms of the superfields $G^{--\dots}$ defined through
the zero curvature equations (3.4), (3.5) and related to $h^{++\dots}$ by eqs.
(3.2) and (3.3). It would be very convenient to write the equations directly
through the analytic gauge potentials.

###  F.1 Spin $\mathbf{1}$ theory

$\mathcal{N}=2$ Maxwell multiplet is described by the analytic superfield
$h^{++}(\zeta)$ with the gauge transformation:

$\delta h^{++}=\mathcal{D}^{++}\lambda\,.$ (F.1)

and the equation of motion

$(D^{+})^{4}h^{--}=0\,,$ (F.2)

where $h^{--}$ is a solution of the zero-curvature equation,

$\mathcal{D}^{++}h^{--}=\mathcal{D}^{--}h^{++}\,.$ (F.3)

Let us consider the superfield transformation law of $\mathcal{D}^{++}h^{++}$:

$\delta\left(\mathcal{D}^{++}h^{++}\right)=\mathcal{D}^{++}\mathcal{D}^{++}\lambda\,.$
(F.4)

This law prompts that, in addition to the Wess-Zumino gauge (4.4), one can
gauge away one more component from $\mathcal{D}^{++}h^{++}$. Indeed, keeping
in mind that

$\left(\mathcal{D}^{++}h^{++}\right)_{gauge}=-2(\theta^{+})^{2}(\bar{\theta}^{+})^{2}\partial_{m}A^{m}\,,$
(F.5)

and making use of the residual gauge freedom,

$\lambda(\zeta)=a(x)\,,$ (F.6)

we can choose Lorentz gauge,

$\delta\left(\partial_{m}A^{m}\right)=\Box
a\qquad\Rightarrow\qquad\partial_{m}A^{m}=0\,,$ (F.7)

that further restricts the residual gauge freedom to

$\lambda(\zeta)=a(x)\,,\qquad\Box a(x)=0\,.$ (F.8)

Gauge-fixing $\Big{(}\mathcal{D}^{++}h^{++}\Big{)}_{gauge}=0$ is just the
$\mathcal{N}=2$ superfield generalization of Lorentz gauge in Maxwell theory.

It is possible from the very beginning to fix the gauge in the manifestly
supersymmetric way as

$\begin{split}\mathcal{D}^{++}h^{++}=0\;\;\;\;\;&\Rightarrow\;\;\;\;\;h^{++}=h^{(ij)}(Z)u^{+}_{i}u^{+}_{j}\;\;\;\;\;\\\
&\Rightarrow\;\;\;\;\;h^{--}=h^{(ij)}(Z)u^{-}_{i}u^{-}_{j}=\frac{1}{2}\mathcal{D}^{--}\mathcal{D}^{--}h^{++}\,.\end{split}$
(F.9)

Here we used the cental basis in HSS and, at the last step, rewrote the result
of solving the harmonic zero-curvature condition in the covariant basis-
independent form.

It is straightforward to check that, taking into account the analyticity of
$h^{++}$, we can transform the equation of motion (F.2) to the form

$(D^{+})^{4}h^{--}=\frac{1}{2}(D^{+})^{4}(D^{--})^{2}h^{++}\sim\Box h^{++}\,.$
(F.10)

So we conclude that $\mathcal{N}=2$ superspin 1 can be described by a pair of
equations:

$\begin{cases}\Box h^{++}=0\,,\\\ \mathcal{D}^{++}h^{++}=0\,.\end{cases}$
(F.11)

The first equation is the massless Klein-Gordon equation, the second one is
the gauge-fixing condition.

###  F.2 Spin $\mathbf{2}$ theory

The spin $\mathbf{2}$ equations (4.46) are more complicated:

$\begin{cases}(D^{+})^{4}G^{--\alpha\dot{\alpha}}=0\,,\\\
(D^{+})^{4}G^{--5}=0\,,\\\
(\bar{D}^{+})^{2}D^{+\alpha}G^{--}_{\alpha\dot{\alpha}}-(D^{+})^{2}\bar{D}^{+}_{\dot{\alpha}}G^{--5}=0\,,\\\
(D^{+})^{2}\bar{D}^{+\dot{\alpha}}G^{--}_{\alpha\dot{\alpha}}+(\bar{D}^{+})^{2}D^{+}_{\alpha}G^{--5}=0\,.\end{cases}$
(F.12)

It is convenient to choose the covariant gauge in full analogy to the spin
$\mathbf{1}$ case:

$\mathcal{D}^{++}G^{++\alpha\dot{\alpha}}=0\,,$ (F.13a)
$\mathcal{D}^{++}G^{++5}=0\,.$ (F.13b)

Note, that this gauge-fixing condition explicitly involves the spinor
coordinates $\theta^{+\hat{\alpha}}$. In terms of the analytic potentials
these conditions amount to the set:

$\mathcal{D}^{++}h^{++\alpha+}=0\,,\qquad\mathcal{D}^{++}h^{++\dot{\alpha}+}=0\,,$
(F.14a)
$\mathcal{D}^{++}h^{++\alpha\dot{\alpha}}+2ih^{++\alpha+}\bar{\theta}^{+\dot{\alpha}}-2ih^{++\dot{\alpha}+}\theta^{+\alpha}=0\,,$
(F.14b)
$\mathcal{D}^{++}h^{++5}-2ih^{++\alpha+}\theta^{+}_{\alpha}-2ih^{++\dot{\alpha}+}\bar{\theta}^{+}_{\dot{\alpha}}=0\,.$
(F.14c)

This gauge is just the superfield form of de-Donder gauge:

$\partial_{m}\bar{h}^{mn}=0\,,\qquad\text{where}\qquad\bar{h}^{mn}:=h^{mn}-\frac{1}{2}\eta^{mn}h\,.$
(F.15)

In this gauge, one can resolve the zero-curvature conditions in analogy with
(F.9):

$G^{--\alpha\dot{\alpha}}=\frac{1}{2}(\mathcal{D}^{--})^{2}G^{++\alpha\dot{\alpha}}\,,$
(F.16) $G^{--5}=\frac{1}{2}(\mathcal{D}^{--})^{2}G^{++5}\,.$ (F.17)

Using these solutions, one arrives at the equations of motions written in
terms of the analytic potentials only:

$\begin{cases}\Box
h^{++\alpha\dot{\alpha}}-4\partial^{\mu(\dot{\alpha}}\partial^{-}_{\mu}h^{++\alpha\dot{)}+}+4\partial^{(\alpha\dot{\mu}}\bar{\partial}^{-}_{\dot{\mu}}h^{++\dot{\alpha})+}=0,\\\
\Box h^{++\alpha+}=0,\\\ \Box h^{++\dot{\alpha}+}=0\,,\\\ \Box
h^{++5}+2\partial^{\mu\dot{\mu}}\partial^{-}_{\dot{\mu}}h^{+++}_{\mu}-2\partial^{\mu\dot{\mu}}\partial^{-}_{\mu}h^{+++}_{\dot{\mu}}=0\,.\end{cases}$
(F.18)

We see that, in addition to the $\Box$ terms, the first and fourth equations
contain additional terms linear in the derivatives. The first equation, in
particular, yields:

$\Box\bar{h}^{mn}=0\,.$ (F.19)

Thus the equations of motion for the spin ${\bf 2}$ case amount to the
Lorentz-type covariant gauge conditions (F.14a) - (F.14c) and eqs. (F.18), all
written in terms of analytic potentials.

###  F.3 General spin $\mathbf{s}$ theory

One can easily repeat the same steps as in spin $\mathbf{2}$ case for the
generic spin ${\bf s}$ superfield equation (4.43):

$(D^{+})^{4}G^{--\alpha(s-1)\dot{\alpha}(s-1)}=0\,.$ (F.20)

It is possible to show that this equation is equivalent to the system of
equations:

$\begin{cases}\Box
G^{++\alpha(s-1)\dot{\alpha}(s-1)}+i\partial^{\mu\dot{\mu}}\left(D^{-}_{\mu}\bar{D}^{+}_{\dot{\mu}}-\bar{D}^{-}_{\dot{\mu}}D^{+}_{\mu}\right)G^{++\alpha(s-1)\dot{\alpha}(s-1)}=0\,,\\\
D^{++}G^{++\alpha(s-1)\dot{\alpha}(s-1)}=0\,.\end{cases}$ (F.21)

Using the relation (3.2), this system can be represented as a system of
equations for the analytic potentials.

The gauge conditions in terms of the analytic potentials read:

$\begin{cases}D^{++}h^{++\alpha(s-1)\dot{\alpha}(s-1)}+2i\big{[}h^{++\alpha(s-1)(\dot{\alpha}(s-2)+}\bar{\theta}^{+\dot{\alpha}_{s-1})}+h^{++\dot{\alpha}(s-1)(\alpha(s-2)+}\,\theta^{+\alpha_{s-1})}\big{]}=0\,,\\\
D^{++}h^{++\alpha(s-1)\dot{\alpha}(s-2)+}=0\,,\\\
D^{++}h^{++\dot{\alpha}(s-1)\alpha(s-2)+}=0\,.\end{cases}$ (F.22)

The superfield equations of motion are written as

$\begin{cases}\Box
h^{++\alpha(s-1)\dot{\alpha}(s-1)}-4\partial^{\mu(\dot{\alpha}(s-1)}\partial^{-}_{\mu}h^{++\alpha(s-1)\dot{\alpha}(s-2))+}+4\partial^{(\alpha(s-1)\dot{\mu}}\bar{\partial}^{-}_{\dot{\mu}}h^{++\dot{\alpha}(s-1)\alpha(s-2))+}=0\,,\\\
\Box h^{++\alpha(s-1)\dot{\alpha}(s-2)+}=0\,,\\\ \Box
h^{++\dot{\alpha}(s-1)\alpha(s-2)+}=0\,.\end{cases}$ (F.23)

The equation

$(D^{+})^{4}G^{--5\alpha(s-2)\dot{\alpha}(s-2)}=0$ (F.24)

can also be restructured in the same way and it amounts to the gauge condition

$\mathcal{D}^{++}h^{++5\alpha(s-2)\dot{\alpha}(s-2)}-2ih^{++(\alpha(s-2)\rho)\dot{\alpha}(s-2)}\theta^{+}_{\rho}-2ih^{++\alpha(s-2)(\dot{\alpha}(s-2)\dot{\rho})}\bar{\theta}^{+}_{\dot{\rho}}=0$
(F.25)

and the following equation for the analytic potentials:

$\Box
h^{++5\alpha(s-2)\dot{\alpha}(s-2)}+2\partial^{\mu\dot{\mu}}\partial^{-}_{\dot{\mu}}h^{++\alpha(s-2)\dot{\alpha}(s-2)+}_{\mu}-2\partial^{\mu\dot{\mu}}\partial^{-}_{\mu}h^{++\alpha(s-2)\dot{\alpha}(s-2)+}_{\dot{\mu}}=0\,.$
(F.26)

The results presented in this Appendix should be considered as first steps in
the study of representations of $\mathcal{N}=2$ supersymmetry algebra on
harmonic superfields from the group-theory point of view. More complete
analysis of the group-theoretical origin of these equations will be a subject
of separate publication.

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