$(\tfrac{3}{2},\tfrac{1}{2},0)$ vertex in Section 6.3.

###  5.7 Component reduction of superfield equations of motions with source in
the spin 2 case

Here we consider the modification of equations of motion for the fields
$\Phi_{\rho\dot{\rho}}^{\alpha\dot{\alpha}}$,
$\mathcal{P}^{\alpha\dot{\alpha}}$ and
$\bar{\mathcal{P}}^{\alpha\dot{\alpha}}$ as compared to those found in the
spin $\mathbf{2}$ case without taking account of sources. We shall see that
the auxiliary $\mathcal{P}$-fields $\mathcal{P}^{\alpha\dot{\alpha}}$ and
$\bar{\mathcal{P}}^{\alpha\dot{\alpha}}$, unlike the free case, play a non-
trivial role in getting the correct modified equations of motion for the spin
2 fields 151515Note that in the first order in $\kappa$ the auxiliary fields
do not contribute to the component Lagrangian even in the interacting theory,
as will be explained in the next Section. The reason is that the auxiliary
fields $\mathcal{P}$ are vanishing on the free equations of motion. So in the
interacting theory their contribution starts with the order $\sim\kappa^{2}$..

The relevant parts of the spin 2 current superfields (in the bosonic sector)
read:

$J^{++}_{\alpha\dot{\alpha}}=\dots-\frac{i}{2}\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}\left(\partial_{\alpha\dot{\alpha}}\partial_{\rho\dot{\rho}}f^{i}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}f^{i}\partial_{\rho\dot{\rho}}\bar{f}_{i}-\partial_{\rho\dot{\rho}}f^{i}\partial_{\alpha\dot{\alpha}}\bar{f}_{i}+f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\rho\dot{\rho}}\bar{f}_{i}\right)+\dots\,,$
(5.112a)
$J^{+}_{\alpha}=\dots-\frac{i}{2}\bar{\theta}^{+\dot{\alpha}}\left(\partial_{\alpha\dot{\alpha}}f^{i}\bar{f}_{i}+f^{i}\partial_{\alpha\dot{\alpha}}\bar{f}_{i}\right)+\dots\,,$
(5.112b)
$J^{+}_{\dot{\alpha}}=\dots+\frac{i}{2}\theta^{+\alpha}\left(f^{i}\partial_{\alpha\dot{\alpha}}\bar{f}_{i}+\partial_{\alpha\dot{\alpha}}f^{i}\bar{f}_{i}\right)+\dots\,.$
(5.112c)

So in the presence of sources the free equations from Section 4.3.1 are
modified as:

$\begin{cases}8(\mathcal{P}_{\dot{\alpha}}^{\mu}-3\bar{\mathcal{P}}_{\dot{\alpha}}^{\mu})=\kappa_{2}\partial^{\mu}_{\dot{\alpha}}(f^{i}\bar{f}_{i})\\\
8(\bar{\mathcal{P}}^{\dot{\rho}}_{\alpha}-3\mathcal{P}^{\dot{\rho}}_{\alpha})=\kappa_{2}\partial_{\alpha}^{\dot{\rho}}(f^{i}\bar{f}_{i})\end{cases}\quad\Rightarrow\quad\mathcal{P}_{\dot{\alpha}}^{\mu}=\bar{\mathcal{P}}_{\dot{\alpha}}^{\mu}=-\frac{\kappa_{2}}{16}\partial_{\dot{\alpha}}^{\mu}\left(f^{i}\bar{f}_{i}\right).$
(5.113)

As a consequence, there appear important additional terms in the linearized
Einstein equations:

$\displaystyle\mathcal{G}_{\rho\dot{\rho}\alpha\dot{\alpha}}-\frac{1}{2}\partial_{\rho\dot{\alpha}}\mathcal{P}_{\alpha\dot{\rho}}-\frac{1}{2}\partial_{\alpha\dot{\rho}}\bar{\mathcal{P}}_{\rho\dot{\alpha}}-\frac{1}{2}\epsilon_{\dot{\alpha}\dot{\rho}}\partial^{\dot{\mu}}_{\rho}\mathcal{P}_{\alpha\dot{\mu}}-\frac{1}{2}\epsilon_{\alpha\rho}\partial^{\mu}_{\dot{\rho}}\bar{\mathcal{P}}_{\mu\dot{\alpha}}$
$\displaystyle=\,\frac{\kappa_{2}}{16}\left(\partial_{\alpha\dot{\alpha}}\partial_{\rho\dot{\rho}}f^{i}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}f^{i}\partial_{\rho\dot{\rho}}\bar{f}_{i}-\partial_{\rho\dot{\rho}}f^{i}\partial_{\alpha\dot{\alpha}}\bar{f}_{i}+f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\rho\dot{\rho}}\bar{f}_{i}\right).$
(5.114)

In this equation we made use of symmetry under the index permutation
$(\alpha\rho)(\dot{\alpha}\dot{\rho})\leftrightarrow(\rho\alpha)(\dot{\rho}\dot{\alpha})$.
Using eq. (5.113) and permuting the indices as
$(\alpha\rho)(\dot{\alpha}\dot{\rho})\leftrightarrow(\rho\alpha)(\dot{\rho}\dot{\alpha})$,
we obtain:

$\begin{split}\mathcal{G}_{\rho\dot{\rho}\alpha\dot{\alpha}}=&\,\frac{\kappa_{2}}{16}\left(\partial_{\alpha\dot{\alpha}}\partial_{\rho\dot{\rho}}f^{i}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}f^{i}\partial_{\rho\dot{\rho}}\bar{f}_{i}-\partial_{\rho\dot{\rho}}f^{i}\partial_{\alpha\dot{\alpha}}\bar{f}_{i}+f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\rho\dot{\rho}}\bar{f}_{i}\right)\\\
&-\frac{\kappa_{2}}{16}\left(\partial_{\alpha\dot{\rho}}\partial_{\rho\dot{\alpha}}-\epsilon_{\alpha\rho}\epsilon_{\dot{\alpha}\dot{\rho}}\Box\right)\left(f^{i}\bar{f}_{i}\right).\end{split}$
(5.115)

This is just the linearized Einstein equations with a source. Thus we have
shown that the auxiliary fields have a non-trivial impact on the eventual form
of the physical equations. In particular, the energy-momentum tensor that
stands in the right hand side of eq. (5.115) differs from the tensor (5.91)
entering $\mathcal{N}=2$ supercurrent (5.81). A similar situation takes place
for other higher spins, but we will not present here the corresponding
technical details.

Also, eq. (5.115) ensures the on-shell compatibility of the first equation in
the set (5.76) (at least, in its bosonic sector) with the source conservation
law $\partial_{\alpha\dot{\alpha}}J^{++\alpha\dot{\alpha}}=0$ fulfilled on the
free hypermultiplet equations of motion.

Relevant to the consideration in this Section is also the component reduction
of coupling in Section 6.3, where we analyze the effect of elimination of the
hypermultiplet auxiliary fields and discuss the structure of the on-shell
cubic vertex Lagrangian.

###  5.8 Spin 3

The spin 3 theory is the first non-trivial new example. Also the structure of
the superfield currents in the spin $\mathbf{3}$ case is of particular
interest, since it helps to clarify the role of the $\xi$-interaction.

Varying the cubic $(\mathbf{3},\mathbf{\tfrac{1}{2}},\mathbf{\tfrac{1}{2}})$
coupling with respect to the unconstrained analytical $\mathcal{N}=2$ spin
$\mathbf{3}$ potentials

$h^{++(\alpha\beta)(\dot{\alpha}\dot{\beta})}\,,\quad
h^{++5\alpha\dot{\alpha}}\,,\quad h^{++(\alpha\beta)\dot{\alpha}+}\,,\quad
h^{++(\dot{\alpha}\dot{\beta})\alpha+}$

yields the analytic current superfields:

$J^{++}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}=-\frac{1}{2}q^{+a}\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}Jq^{+}_{a}-\frac{1}{2}\xi\,\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}\left(q^{+a}Jq^{+}_{a}\right),$
(5.116a)
$J^{++}_{\alpha\dot{\alpha}}=-\frac{1}{2}q^{+a}\partial_{\alpha\dot{\alpha}}J\partial_{5}q^{+}_{a},$
(5.116b)
$J^{+}_{(\alpha\beta)\dot{\alpha}}=-\frac{1}{2}q^{+a}\partial_{(\alpha\dot{\alpha}}\partial^{-}_{\beta)}Jq^{+}_{a}-\frac{1}{2}\xi\partial_{(\alpha\dot{\alpha}}\partial^{-}_{\beta)}\left(q^{+a}Jq^{+}_{a}\right),$
(5.116c)
$J^{+}_{(\dot{\alpha}\dot{\beta})\alpha}=-\frac{1}{2}q^{+a}\partial_{\alpha(\dot{\alpha}}\partial^{-}_{\dot{\beta})}Jq^{+}_{a}-\frac{1}{2}\xi\partial_{\alpha(\dot{\alpha}}\partial^{-}_{\dot{\beta})}\left(q^{+a}Jq^{+}_{a}\right).$
(5.116d)

Like in the ${\bf s}=2$ case, these current superfields can be related to a
single “master” current superfield, this time the vector one,

$\mathcal{J}_{\alpha\dot{\alpha}}:=-\frac{1}{2}q^{+a}\mathcal{D}^{--}\partial_{\alpha\dot{\alpha}}Jq^{+}_{a}-\frac{1}{2}\xi\,\mathcal{D}^{--}\partial_{\alpha\dot{\alpha}}\left(q^{+a}Jq^{+}_{a}\right),$
(5.117)

obeying, on the hypermultiplet mass shell, the conservation law:

$\mathcal{D}^{++}\mathcal{J}_{\alpha\dot{\alpha}}=-\frac{1}{2}q^{+a}\partial_{\alpha\dot{\alpha}}Jq^{+}_{a}-\xi\,\partial_{\alpha\dot{\alpha}}\left(q^{+a}Jq^{+}_{a}\right).$
(5.118)

We find

$\mathcal{J}^{+}_{(\alpha\beta)\dot{\alpha}}=D^{+}_{(\alpha}\mathcal{J}_{\beta)\dot{\alpha}}\,,\qquad\mathcal{J}^{+}_{(\dot{\alpha}\dot{\beta})\alpha}=-\bar{D}^{+}_{(\dot{\alpha}}\mathcal{J}_{\dot{\beta})\alpha}\,,$
(5.119)
$J^{++}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}=-\frac{i}{2}D^{+}_{(\alpha}\bar{D}^{+}_{(\dot{\alpha}}\mathcal{J}_{\beta)\dot{\beta})}\,,\qquad
J_{\alpha\dot{\alpha}}^{++}=\frac{i}{4}(D^{+})^{2}\mathcal{J}_{\alpha\dot{\alpha}}=-\frac{i}{4}(\bar{D}^{+})^{2}\mathcal{J}_{\alpha\dot{\alpha}}\,,$
(5.120)

where

$\mathcal{J}^{+}_{(\dot{\alpha}\dot{\beta})\alpha}=J^{+}_{(\dot{\alpha}\dot{\beta})\alpha}+2i\theta^{-\mu}J^{++}_{(\alpha\mu)(\dot{\alpha}\dot{\beta})}+2i\bar{\theta}^{-}_{(\dot{\alpha}}J^{++}_{\alpha\dot{\beta})},$
(5.121a)
$\mathcal{J}^{+}_{(\alpha\beta)\dot{\alpha}}=J^{+}_{(\alpha\beta)\dot{\alpha}}-2i\bar{\theta}^{-\dot{\mu}}J^{++}_{(\alpha\beta)(\dot{\alpha}\dot{\mu})}+2i\theta^{-}_{(\alpha}J^{++}_{\beta)\dot{\alpha}}.$
(5.121b)

Since all the current superfields are obtained by action of analytic spinor
derivatives on ${\cal J}^{\alpha\dot{\alpha}}$, and the right hand side of the
harmonic constraint (5.118) is obviously analytic, these superfields
automatically satisfy the standard harmonic conservation laws,
$\mathcal{D}^{++}J^{++}_{\alpha\dot{\alpha}}=0\,,$ etc. 161616 This
construction can straightforwardly be generalized to an arbitrary ${\cal N}=2$
spin ${\bf s}$, with the appropriate non-analytic master supercurrent..

The current superfields (5.116) provide sources in the relevant superfield
equations of motion:

$\begin{cases}(D^{+})^{4}G^{--(\alpha\beta)(\dot{\alpha}\dot{\beta})}=-\frac{1}{2}\kappa_{3}J^{++(\alpha\beta)(\dot{\alpha}\dot{\beta})}\,,\\\
(D^{+})^{4}G^{--5\alpha\dot{\alpha}}=-\frac{1}{2}\kappa_{3}J^{++\alpha\dot{\alpha}}\,,\\\
(\bar{D}^{+})^{2}D^{+\beta}G^{--}_{(\beta\alpha)(\dot{\alpha}\dot{\beta})}-(D^{+})^{2}\bar{D}^{+}_{(\dot{\alpha}}G^{--5}_{\alpha\dot{\beta})}=-2i\kappa_{3}\mathcal{J}^{+}_{\alpha(\dot{\alpha}\dot{\beta})}\,,\\\
(D^{+})^{2}\bar{D}^{+\dot{\beta}}G^{--}_{(\alpha\beta)(\dot{\beta}\dot{\alpha})}+(\bar{D}^{+})^{2}D^{+}_{(\alpha}G^{--5}_{\beta)\dot{\alpha}}=-2i\kappa_{3}\mathcal{J}^{+}_{(\alpha\beta)\dot{\alpha}}\,.\end{cases}$
(5.122)

The component content of spin 3 current superfields can be analyzed in full
analogy with the spin 2 case.

1. The spin $\mathbf{3}$ current superfield has the form:

$\begin{split}J^{++}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}=&-\frac{1}{2}q^{+a}\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}Jq^{+}_{a}-\frac{1}{2}\xi\,\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}\left(q^{+a}Jq^{+}_{a}\right)\\\
=&-i\frac{1}{2}q^{+a}\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}\tau^{3}_{ab}q^{+b}-i\frac{1}{2}\xi\,\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}\left(\tau^{3}_{ab}q^{+a}q^{+b}\right).\end{split}$
(5.123)

To figure out its component structure, it is convenient to rewrite it in terms
of $q^{+}$ and $\tilde{q}^{+}$:

$J^{++}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}=i\frac{1}{2}\left(\tilde{q}^{+}\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}q^{+}+q^{+}\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}\tilde{q}^{+}\right)+i\xi\,\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}\left(\tilde{q}^{+}q^{+}\right).$
(5.124)

The divergence of the superfield (5.123) does not vanish,

$\partial^{\alpha\dot{\alpha}}J^{++}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\neq
0\,.$ (5.125)

So, in contrast to the spin $\mathbf{2}$ case, the spin $\mathbf{3}$ current
superfield reveals no any additional conservation condition beyond ${\cal
D}^{++}J^{++}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}=0\,$. We will be
interested in the following components in the $\theta$-expansion of the
supefield (5.123):

$J^{++}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}=\dots+\theta^{+\gamma}S^{i}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})\gamma}u^{+}_{i}+\bar{\theta}^{+\dot{\gamma}}S^{i}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})\dot{\gamma}}u^{+}_{i}+2i\theta^{+}\sigma^{n}\bar{\theta}^{+}j^{n}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}+\dots.$
(5.126)

We shall express them through the hypermultiplet fields from eq. (5.124)

  * •

The spinor component of (5.126) reads:

$\begin{split}S^{i}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})\rho}=&-\frac{i}{2}\left(\kappa_{\rho}\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}f^{i}+\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}\kappa_{\rho}f^{i}\right)\\\
&-\frac{i}{2}\left(\psi_{\rho}\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}\bar{f}^{i}+\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}\psi_{\rho}\bar{f}^{i}\right)\\\
&+i\xi\,\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}\left[\bar{f}^{i}\psi_{\rho}+f^{i}\kappa_{\rho}\right].\end{split}$
(5.127)

In contrast to the spin $\mathbf{2}$ case, this component does not satisfy any
conservation law. However, this does not mean that the spin $\tfrac{5}{2}$
fields interacts with a non-conserved current. The corresponding conserved
spin $\tfrac{5}{2}$ current consists of several constituents and will be
explicitly presented below (eq. (5.134)).

  * •

The conserved spin 3 current component of the superfield (5.123) reads:

$\begin{split}j^{n}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}=&-i\frac{1}{4}\Bigr{(}f^{i}\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}\partial^{n}\bar{f}_{i}-\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}\partial^{n}f^{i}\bar{f}_{i}\\\
&\qquad-\,\partial^{n}f^{i}\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}\bar{f}_{i}+\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}f^{i}\partial^{n}\bar{f}_{i}\Bigr{)}\\\
&-i\frac{\xi}{2}\,\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}\left(f^{i}\partial^{n}\bar{f}_{i}-\partial^{n}f^{i}\bar{f}_{i}\right).\end{split}$
(5.128)

We observe that the $\xi$-term is just the derivative of the ${\rm U}(1)$
current. The conservation law for the spin 3 current has the standard form:

$\partial_{n}j^{n}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}=0\,.$ (5.129)

The current (5.128) (with $\xi=0$) exactly coincides with the scalar field
current elaborated in Berends:1985xx .

2. Spin 2 current superfield

$J^{++}_{\alpha\dot{\alpha}}=-\frac{1}{2}q^{+a}\partial_{\alpha\dot{\alpha}}J\partial_{5}q^{+}_{a}=-m\frac{1}{2}q^{+a}\partial_{\alpha\dot{\alpha}}q^{+}_{a}$
(5.130)

coincides (up to the multiplier $m$) with the $\mathcal{N}=2$ free
hypermultiplet supercurrent (5.81) introduced in Section 5.6. The
corresponding spin 2 conserved current in the bosonic sector is:

$j_{\alpha\beta\dot{\alpha}\dot{\beta}}=\frac{m}{4}\Big{(}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}f^{i}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}f^{i}\partial_{\beta\dot{\beta}}\bar{f}_{i}-\partial_{\beta\dot{\beta}}f^{i}\partial_{\alpha\dot{\alpha}}\bar{f}_{i}+f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\bar{f}_{i}\Big{)}.$
(5.131)

3. The spin $\mathbf{3}$ fermionic current superfield (5.60) reads:

$\begin{split}J^{+}_{(\alpha\beta)\dot{\alpha}}=&-\frac{1}{2}q^{+a}\partial_{(\alpha\dot{\alpha}}\partial^{-}_{\beta)}Jq^{+}_{a}-\frac{1}{2}\xi\partial_{(\alpha\dot{\alpha}}\partial^{-}_{\beta)}\left(q^{+a}Jq^{+}_{a}\right)\\\
=&\;\frac{i}{2}\tilde{q}^{+}\partial_{(\alpha\dot{\alpha}}\partial^{-}_{\beta)}q^{+}+\frac{i}{2}q^{+}\partial_{(\alpha\dot{\alpha}}\partial^{-}_{\beta)}\tilde{q}^{+}-i\xi\partial_{(\alpha\dot{\alpha}}\partial^{-}_{\beta)}\left(\tilde{q}^{+}q^{+}\right).\end{split}$
(5.132)

As in the spin 2 case, it satisfies the modified conservation law (5.62):

$\mathcal{D}^{++}J^{+}_{(\alpha\beta)\dot{\alpha}}=2i\bar{\theta}^{+\dot{\rho}}J^{++}_{(\alpha\beta)(\dot{\alpha}\dot{\rho})}-2i\theta^{+}_{(\alpha}J^{++}_{\beta)\dot{\alpha}}\,.$
(5.133)

The corresponding conserved spin $\tfrac{5}{2}$ fermionic current, according
to the general discussion in Appendix C.1, is defined by the following
combination of various spinorial components:

$\mathfrak{j}^{i}_{\rho\dot{\rho}(\alpha\beta)\dot{\alpha}}:=j^{i}_{\rho\dot{\rho}(\alpha\beta)\dot{\alpha}}-S^{i}_{\rho\dot{\rho}(\alpha\beta)\dot{\alpha}}+m\epsilon_{\rho(\alpha}S^{i}_{\beta)\dot{\rho}\dot{\alpha}}\,.$
(5.134)

Here $j^{i}_{\rho\dot{\rho}(\alpha\beta)\dot{\alpha}}$ appears as a component
of $J^{+}_{(\alpha\beta)\dot{\alpha}}$,

$J^{+}_{(\alpha\beta)\dot{\alpha}}=\dots+2i\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}j^{i}_{\rho\dot{\rho}(\alpha\beta)\dot{\alpha}}+\dots\,.$
(5.135)

Using the on-shell hypermultiplet superfield (5.9) and the definition (5.132)
one can express it in terms of the free hypermultiplet fields:

$\begin{split}j^{i}_{\rho\dot{\rho}(\alpha\beta)\dot{\alpha}}=&\frac{i}{2}\left(\chi_{\rho}\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\rho}}f^{i}-\partial_{\rho\dot{\rho}}f^{i}\partial_{(\alpha\dot{\alpha}}\chi_{\beta)}\right)+\frac{i}{2}\left(\psi_{\rho}\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\rho}}\bar{f}^{i}-\partial_{\rho\dot{\rho}}\bar{f}^{i}\partial_{(\alpha\dot{\alpha}}\psi_{\beta)}\right)\\\
&+i\xi\partial_{(\alpha\dot{\alpha}}\left(\chi_{\beta)}\partial_{\rho\dot{\rho}}f^{i}-\chi_{\rho}\partial_{\beta)\dot{\rho}}f^{i}\right)+i\xi\partial_{(\alpha\dot{\alpha}}\left(\psi_{\beta)}\partial_{\rho\dot{\rho}}\bar{f}^{i}-\psi_{\rho}\partial_{\beta)\dot{\rho}}\bar{f}^{i}\right).\end{split}$
(5.136)

The term $S^{i}_{\rho\dot{\rho}(\alpha\beta)\dot{\alpha}}$ in (5.134) is the
fermionic component (5.127) of the spin 3 current superfield
$J^{++}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}$ (5.126). The last component
$S^{i}_{\beta\dot{\rho}\dot{\alpha}}$ in (5.134) is the $\mathcal{N}=2$
supersymmetry fermionic current defined in (5.87).

The total conserved fermionic current (5.134) (for simplicity, we present it
for the $m=0$ case) is then given by the expression:

$\begin{split}\mathfrak{j}^{i}_{\rho\dot{\rho}(\alpha\beta)\dot{\alpha}}=&\;\;\;\;\frac{i}{2}\left(\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\rho}}\chi_{\rho}f^{i}+2\chi_{\rho}\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\rho}}f^{i}-\partial_{(\alpha\dot{\alpha}}\chi_{\beta)}\partial_{\rho\dot{\rho}}f^{i}\right)\\\
&+\frac{i}{2}\left(\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\rho}}\psi_{\rho}\bar{f}^{i}+2\psi_{\rho}\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\rho}}\bar{f}^{i}-\partial_{(\alpha\dot{\alpha}}\psi_{\beta)}\partial_{\rho\dot{\rho}}\bar{f}^{i}\right)\\\
&+i\xi\partial_{(\alpha\dot{\alpha}}\left(\chi_{\beta)}\partial_{\rho\dot{\rho}}f^{i}-\chi_{\rho}\partial_{\beta)\dot{\rho}}f^{i}\right)+i\xi\partial_{(\alpha\dot{\alpha}}\left(\psi_{\beta)}\partial_{\rho\dot{\rho}}\bar{f}^{i}-\psi_{\rho}\partial_{\beta)\dot{\rho}}\bar{f}^{i}\right)\\\
&-i\xi\,\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\rho}}\left(\chi_{\rho}f^{i}+\psi_{\rho}\bar{f}^{i}\right).\end{split}$
(5.137)

This current differs from the Noether spin $\tfrac{5}{2}$ current (D.19).
However, two conserved currents of the same spin can differ only by trivially
conserved terms. Let us explicitly give how these currents are related to each
other.

The $\xi$-terms in the third and fourth lines of (5.137), modulo free massless
equations of motions,

$\partial_{\rho\dot{\rho}}\chi_{\beta}-\partial_{\beta\dot{\rho}}\chi_{\rho}=0,\qquad\partial_{\rho\dot{\rho}}\psi_{\beta}-\partial_{\beta\dot{\rho}}\psi_{\rho}=0,$
(5.138)

can be reduced to the form:

$i\xi\partial_{(\alpha\dot{\alpha}}\partial_{\rho\dot{\rho}}\left(\chi_{\beta)}f^{i}+\psi_{\beta)}\bar{f}^{i}\right)-2i\xi\,\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\rho}}\left(\chi_{\rho}f^{i}+\psi_{\rho}\bar{f}^{i}\right).$
(5.139)

It is straightforward to see that this $\xi$ contribution to
$\mathfrak{j}$-current is trivially conserved. Moreover, it is a total
derivative and so do not contribute to the corresponding conserved charge.

Using the free equations (5.138) one can further reduce the current
$\mathfrak{j}$ to the form:

$\begin{split}\mathfrak{j}^{i}_{\rho\dot{\rho}(\alpha\beta)\dot{\alpha}}=&-2j^{i}_{\rho\dot{\rho}(\alpha\beta)\dot{\alpha}}\\\
&+\frac{i}{2}\Big{[}2\partial_{(\beta\dot{\rho}}\left(\chi_{\rho}\partial_{\alpha)\dot{\alpha}}f^{i}+\psi_{\rho}\partial_{\alpha)\dot{\alpha}}\bar{f}^{i}\right)-\partial_{\rho\dot{\rho}}\left(\chi_{(\beta}\partial_{\alpha)\dot{\alpha}}f^{i}+\psi_{(\beta}\partial_{\alpha)\dot{\alpha}}\bar{f}^{i}\right)\Big{]}\\\
&+\frac{i}{2}\partial_{\alpha\dot{\alpha}}\left(\chi_{\beta}\partial_{\rho\dot{\rho}}f^{i}-\partial_{\rho\dot{\rho}}\chi_{\beta}f^{i}\right)+\frac{i}{2}\partial_{\alpha\dot{\alpha}}\left(\psi_{\beta}\partial_{\rho\dot{\rho}}\bar{f}^{i}-\partial_{\rho\dot{\rho}}\psi_{\beta}\bar{f}^{i}\right).\end{split}$
(5.140)

The second line in this expression is trivially conserved. The third line are
derivatives of the independently conserved $\psi$ and $\chi$ parts of the
$\mathcal{N}=2$ supersymmetry fermionic current (5.87). Since these terms are
total derivatives, they do not give contributions to the conserved charges.

Thus we see that, up to numerical factors, the trivially conserved current and
derivatives of the supersymmetric current, the current
$\mathfrak{j}^{i}_{\rho\dot{\rho}(\alpha\beta)\dot{\alpha}}$ coincides with
the canonical Noether spin $\tfrac{5}{2}$ current.

The conserved spin $\tfrac{5}{2}$ current of hypermultiplet can also be
represented in the form:

$\mathcal{S}^{i}_{\rho\dot{\rho}(\alpha\beta)\dot{\alpha}}:=i\chi_{\rho}\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\rho}}f^{i}+i\psi_{\rho}\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\rho}}\bar{f}^{i},\qquad\partial^{\rho\dot{\rho}}\mathcal{S}^{i}_{\rho\dot{\rho}(\alpha\beta)\dot{\alpha}}=0.$
(5.141)

The various spin $\tfrac{5}{2}$ currents discussed in this section are related
by the equation:

$\mathcal{S}^{i}_{\rho\dot{\rho}(\alpha\beta)\dot{\alpha}}=\mathfrak{j}^{i}_{\rho\dot{\rho}(\alpha\beta)\dot{\alpha}}+j^{i}_{\rho\dot{\rho}(\alpha\beta)\dot{\alpha}}$
(5.142)

Thus, just as in the case of the conserved $\tfrac{3}{2}$ currents of rigid
$\mathcal{N}=2$ supersymmetry, the various spin $\tfrac{5}{2}$ conserved
currents coincide up to numerical factors, trivially conserved terms and
derivatives of the lower spin currents. In Section 6.4 we shall see that the
current $\mathcal{S}^{i}_{\rho\dot{\rho}(\alpha\beta)\dot{\alpha}}$ reappears
in the $(\tfrac{5}{2},\tfrac{1}{2},0)$ cubic vertices.

###  5.9 General spin $\mathbf{s}$

For an arbitrary spin, the calculations can be performed quite analogously to
the previous Subsections. As an example, we quote here how the conserved
bosonic spin s currents look for an arbitrary higher spin.

In the case of even spins the current superfield (5.46) has the form:

$J^{++}_{\alpha(s-1)\dot{\alpha}(s-1)}=-\frac{1}{2}q^{+a}\partial^{s-1}_{\alpha(s-1)\dot{\alpha}(s-1)}q^{+}_{a}\,.$
(5.143)

The corresponding conserved spin s current in the bosonic sector is expressed
as:

$\begin{split}j^{n}_{\alpha(s-1)\dot{\alpha}(s-1)}=&-\frac{1}{4}\partial^{n}\bar{f}^{i}\partial^{s-1}_{\alpha(s-1)\dot{\alpha}(s-1)}f_{i}-\frac{1}{4}\partial^{s-1}_{\alpha(s-1)\dot{\alpha}(s-1)}\bar{f}^{i}\partial^{n}f_{i}\\\
&+\frac{1}{4}\bar{f}^{i}\partial^{s-1}_{\alpha(s-1)\dot{\alpha}(s-1)}\partial^{n}f_{i}+\frac{1}{4}\partial^{s-1}_{\alpha(s-1)\partial^{n}\dot{\alpha}(s-1)}\bar{f}^{i}f_{i}\,.\end{split}$
(5.144)

In the odd spin case we have also a contribution from the $\xi$-term:

$J^{++}_{\alpha(s-1)\dot{\alpha}(s-1)}=-\frac{1}{2}q^{+a}\partial^{s-1}_{\alpha(s-1)\dot{\alpha}(s-1)}Jq^{+}_{a}-\frac{1}{2}\xi\,\partial^{s-1}_{\alpha(s-1)\dot{\alpha}(s-1)}\left(q^{+a}Jq^{+}_{a}\right).$
(5.145)

In this case the conserved current in the bosonic sector reads:

$\begin{split}j^{n}_{\alpha(s-1)\dot{\alpha}(s-1)}=&-i\frac{1}{4}\Bigr{(}f^{i}\partial^{s-1}_{\alpha(s-1)\dot{\alpha}(s-1)}\partial^{n}\bar{f}_{i}-\partial^{s-1}_{\alpha(s-1)\dot{\alpha}(s-1)}\partial^{n}f^{i}\bar{f}_{i}\\\
&\qquad\qquad-\partial^{n}f^{i}\partial^{s-1}_{\alpha(s-1)\dot{\alpha}(s-1)}\bar{f}_{i}+\partial^{s-1}_{\alpha(s-1)\dot{\alpha}(s-1)}f^{i}\partial^{n}\bar{f}_{i}\Bigr{)}\\\
&-i\frac{\xi}{2}\,\partial^{s-1}_{\alpha(s-1)\dot{\alpha}(s-1)}\left(f^{i}\partial^{n}\bar{f}_{i}-\partial^{n}f^{i}\bar{f}_{i}\right).\end{split}$
(5.146)

It is easy to check that the currents (5.144) and (5.146) are conserved on the
free hypermultiplet equations of motion (5.12):

$\partial_{n}j^{n}_{\alpha(s-1)\dot{\alpha}(s-1)}=0\,.$ (5.147)

##  6 Hypermultiplet equations of motion and the component reduction of cubic
couplings

In this Section, we shall study the component structure of $\mathcal{N}=2$
supersymmetric cubic interactions of hypermultiplet with higher spins. One of
our goals will be elucidation of the role of the $\xi$-interaction that
appears for odd spins $\mathbf{s}=\mathbf{3},\mathbf{5},\mathbf{7},\dots$. We
shall consider the simplest examples of the spin $\mathbf{1}$, $\mathbf{2}$
and $\mathbf{3}$ multiplets.

###  6.1 Hypermultiplet equations of motion

The presence of the interaction of a hypermultiplet with fields of higher
spins modifies the equations of motion for the hypermultiplet. This leads to a
change in the on-shell component expansion of the hypermultiplet and, as a
result, affects the component structure of the on-shell cubic vertex. Varying
the action (5.31) yields:

$\begin{split}\delta_{q}S^{(s)}_{gauge}=&-\frac{1}{2}\int
d\zeta^{(-4)}\;\Big{[}\delta
q^{+a}\left(\mathcal{D}^{++}+\kappa_{s}\hat{\mathcal{H}}^{++}_{(s)}(J)^{P(s)}+\kappa_{s}\xi\Gamma^{++}_{(s)}(J)^{P(s)}\right)q^{+}_{a}\\\
&+q^{+a}\left(\mathcal{D}^{++}+\kappa_{s}\hat{\mathcal{H}}^{++}_{(s)}(J)^{P(s)}+\kappa_{s}\xi\Gamma^{++}_{(s)}(J)^{P(s)}\right)\delta
q^{+}_{a}\Big{]}.\end{split}$ (6.1)

After integration by parts, the variation becomes:

$\begin{split}\delta_{q}S^{(s)}_{gauge}=&-\frac{1}{2}\int
d\zeta^{(-4)}\;\delta
q^{+a}\Bigr{[}2\mathcal{D}^{++}+\kappa_{s}\hat{\mathcal{H}}^{++}_{(s)}(J)^{P(s)}+2P(s)\kappa_{s}\xi\Gamma^{++}_{(s)}(J)^{P(s)}\\\
&+(-1)^{s}\kappa_{s}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\big{(}\hat{\mathcal{H}}^{++\alpha(s-2)\dot{\alpha}(s-2)}+\Gamma^{++\alpha(s-2)\dot{\alpha}(s-2)}\big{)}(J)^{P(s)}\Bigr{]}q^{+}_{a}\,.\end{split}$
(6.2)

Here, all derivatives act freely to the right. So the hypermultiplet equation
of motion on the higher spin gauge background reads:

$\begin{split}&\Bigr{[}2\mathcal{D}^{++}+\kappa_{s}\hat{\mathcal{H}}^{++}_{(s)}(J)^{P(s)}+2P(s)\kappa_{s}\xi\Gamma^{++}_{(s)}(J)^{P(s)}\\\
&+(-1)^{s+P(s)}\kappa_{s}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\big{(}\hat{\mathcal{H}}^{++\alpha(s-2)\dot{\alpha}(s-2)}+\Gamma^{++\alpha(s-2)\dot{\alpha}(s-2)}\big{)}(J)^{P(s)}\Bigr{]}q^{+}_{a}=0\,.\end{split}$
(6.3)

As a characteristic example, consider the case of spin $\mathbf{2}$, for which
the equation (6.3) is greatly simplified:

$\Bigr{(}\mathcal{D}^{++}+\kappa_{2}\hat{\mathcal{H}}^{++}_{(s=2)}+\frac{\kappa_{2}}{2}\Gamma^{++}_{(s=2)}\Bigr{)}q^{+}_{a}=0\,.$
(6.4)

It is seen that the new terms appearing in this equation include a coupling
constant. Thus, in the presence of coupling with $\mathcal{N}=2$ gauge
superfields, the on-shell component content of the hypermultiplet differs from
that in the non-interacting case (5.9) by $\kappa_{2}$ corrections. This will
play an important role in performing the component reduction of the cubic
vertex.

In the spin $\mathbf{3}$ case the superfield equation is more complicated :

$\begin{split}\Bigr{[}\mathcal{D}^{++}+\kappa_{3}\hat{\mathcal{H}}^{++}_{(s=3)}J+&\frac{\kappa_{3}}{2}\left(\partial_{\alpha\dot{\alpha}}\hat{\mathcal{H}}^{++\alpha\dot{\alpha}}\right)J\\\
&+\kappa_{3}\left(\xi+\frac{1}{2}\right)\Gamma^{++}_{(s=3)}J+\frac{\kappa_{3}}{2}\Gamma^{++\alpha\dot{\alpha}}\partial_{\alpha\dot{\alpha}}J\Bigr{]}\;q^{+}_{a}=0\,.\end{split}$
(6.5)

Now we shall proceed to a detailed discussion of these two simplest cases,
with an emphasis on the role of the auxiliary fields. As a preliminary step,
we once again first consider the simplest example of the spin ${\bf 1}$ ${\cal
N}=2$ multiplet.

###  6.2 Spin 1 coupling

The equation of motion for the hypermultiplet in the spin $\mathbf{1}$
background has the form:

$\left(\mathcal{D}^{++}+i\kappa_{1}h^{++}\right)q^{+}=0\,.$ (6.6)

To analyze the component structure of the cubic vertex, one needs to know how
the non-dynamical part of the constraint (6.6) modifies the free on-shell
expression (5.9) for the hypermultiplet superfield due to the presence of
interaction with the spin $\mathbf{1}$ multiplet. Before proceeding to
calculations, it is useful to note that only those auxiliary fields which are
non-zero in the free $q^{+}$ theory can contribute to a cubic vertex. In
particular, this explains why the auxiliary fields of the gauge spin
$\mathbf{1}$ multiplet do not contribute.

In the gauge sector, the spin $\mathbf{1}$ analytic potential has the form:

$h^{++}_{A}=-2i\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}A_{\rho\dot{\rho}}.$
(6.7)

Eliminating the auxiliary fields on such a background by the non-dynamical
part of (6.6) (terms up to $\sim(\theta^{+})^{2}$) leads to the following
modified on-shell hypermultiplet superfield:

$q^{+}_{(s=1)}=f^{i}u^{+}_{i}+\theta^{+\alpha}\psi_{\alpha}+\bar{\theta}^{+}_{\dot{\alpha}}\bar{\kappa}^{+\dot{\alpha}}+(\theta^{\hat{+}})^{2}mf^{i}u^{-}_{i}+2i\theta^{+}\sigma^{m}\bar{\theta}^{+}\nabla^{(s=1)}_{m}f^{i}u^{-}_{i}\,,$
(6.8)

where

$\nabla^{(s=1)}_{m}f^{i}:=\left(\partial_{m}+i\kappa_{1}A_{m}\right)f^{i}.$
(6.9)

Thus we observe the appearance of $\kappa_{1}$-correction to the free on-shell
hypermultiplet (5.9).

It will be useful to divide the on-shell hypermultiplet superfield as:

$q^{+}_{(s=1)}=q^{+}_{(0)}+\kappa_{1}q^{+}_{(\kappa_{1})},$ (6.10)

where the zeroth-order term $q_{(0)}$ is still given by the expression (5.9).

In this notations, the action to the leading order in $\kappa_{1}$ reads:

$\begin{split}S_{gauge}^{(s=1)}=&-\int
d\zeta^{(-4)}\;\tilde{q}^{+}\left(\mathcal{D}^{++}+i\kappa_{1}h^{++}\right)q^{+}\\\
=&-\int d\zeta^{(-4)}\;\tilde{q}_{(0)}^{+}\mathcal{D}^{++}q^{+}_{(0)}\\\
&-\kappa_{1}\int
d\zeta^{(-4)}\;\left(\tilde{q}_{(0)}^{+}\mathcal{D}^{++}q_{(\kappa_{1})}^{+}+\tilde{q}_{(\kappa_{1})}^{+}\mathcal{D}^{++}q_{(0)}^{+}\right)\\\
&-i\kappa_{1}\int
d\zeta^{(-4)}\;h^{++}\tilde{q}_{(0)}^{+}q^{+}_{(0)}.\end{split}$ (6.11)

Using the on-shell hypermultiplet expression (6.8) one can easily calculate
the bosonic contribution from the first two lines:

$\begin{split}&-\int
d\zeta^{(-4)}\;\tilde{q}^{+}_{(s=1)}\mathcal{D}^{++}q^{+}_{(s=1)}\\\
&\Rightarrow\int
d^{4}x\;\left(-\eta^{nm}\nabla^{(s=1)}_{n}f^{i}\nabla^{(s=1)}_{m}\bar{f}_{i}+\partial^{m}f^{i}\nabla^{(s=1)}_{m}\bar{f}_{i}+\nabla^{(s=1)}_{m}f^{i}\partial^{m}\bar{f}_{i}-m^{2}f^{i}\bar{f}_{i}\right)\\\
&=\int
d^{4}x\;\left(\partial^{m}f^{i}\partial_{m}\bar{f}_{i}-m^{2}f^{i}\bar{f}_{i}+\kappa_{1}^{2}A^{m}A_{m}f^{i}\bar{f}_{i}\right)\\\
&=\int
d^{4}x\;\left(\partial^{m}f^{i}\partial_{m}\bar{f}_{i}-m^{2}f^{i}\bar{f}_{i}\right)+\mathcal{O}(\kappa_{1}^{2}).\end{split}$
(6.12)

Thus these terms do not contribute to the cubic component interaction (but
contribute to quartic $(1,1,0,0)$ vertex!). So we reduce the cubic gauge
coupling in (6.11) to the form:

$S_{int}^{(s=1)}=-i\kappa_{1}\int
d\zeta^{(-4)}\;h^{++}\tilde{q}_{(0)}^{+}q^{+}_{(0)}=\kappa_{1}\int
d\zeta^{(-4)}h^{++}J^{++}\,.$ (6.13)

Here we used the expression (5.19) for the spin $\mathbf{1}$ current
superfield. We need to leave in it only the bosonic term relevant to our
study:

$J^{++}=-i\tilde{q}_{(0)}^{+}q^{+}_{(0)}=...+\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}\left(\partial_{\rho\dot{\rho}}f^{i}\bar{f}_{i}-f^{i}\partial_{\rho\dot{\rho}}\bar{f}_{i}\right).$
(6.14)

As a net result, the gauge field coupling in the first order in $\kappa_{1}$
is found to be:

$S_{int}^{(s=1)}=-\frac{i\kappa_{1}}{2}\int
d^{4}x\;A^{\rho\dot{\rho}}\left(\partial_{\rho\dot{\rho}}f^{i}\bar{f}_{i}-f^{i}\partial_{\rho\dot{\rho}}\bar{f}_{i}\right).$
(6.15)

This is the standard minimal Noether interaction of the spin 1 field with the
${\rm U}(1)$ current171717As distinct from the cases of ${\bf s}\geq 3$, in
the spin ${\bf 1}$ and ${\bf 2}$ cases one can restore the gauge field -
hypermultiplet couplings to all orders in $\kappa_{(s)}$ since the relevant
theories are well known in full generality: these are ${\cal N}=2$
supersymmetric Maxwell theory and ${\cal N}=2$ Einstein supergravity. For
instance, the evident completion of the minimal bosonic coupling (6.15) reads
$S^{(s=1)}_{int}{}^{\prime}=S^{(s=1)}_{int}+\kappa_{1}^{2}\int
d^{4}x\,(A^{m}A_{m})(f^{i}\bar{f}_{i})\,.$ However, keeping in mind the
construction of the minimal vertices for the non-trivial ${\bf s}\geq 3$ cases
as our main goal here, we limit our attention altogether only to such minimal
vertices..

###  6.3 Spin 2 coupling

Like in the previous example, we consider the equations of motion (6.4) for a
hypermultiplet interacting with the spin $\mathbf{2}$ gauge superfield. Once
again, the interaction modifies the component expansion of the hypermultiplet
as compared to the free case (5.9).

We will consider the cubic interaction to the first order in the coupling
constant $\kappa_{2}$:

$q^{+}_{(s=2)}=q^{+}_{(0)}+\kappa_{2}q^{+}_{(\kappa_{2})}+\mathcal{O}(\kappa_{2}^{2})\,,$
(6.16)

where $q^{+}_{(0)}$ was determined in (5.9) and $q^{+}_{(\kappa_{2})}$ is a
correction induced by the interaction.

The component cubic coupling then comes from the $\kappa_{2}$-terms of the
hypermultiplet action:

$\begin{split}S^{(s=2)}_{gauge}=&-\frac{1}{2}\int
d\zeta^{(-4)}\;q^{+a}\left(\mathcal{D}^{++}+\kappa_{2}\hat{\mathcal{H}}^{++}_{(s=2)}\right)q^{+}_{a}\\\
=&-\frac{1}{2}\int
d\zeta^{(-4)}\;q^{+a}_{(0)}\mathcal{D}^{++}q^{+}_{(0)a}-\kappa_{2}\int
d\zeta^{(-4)}\;q^{+a}_{(\kappa_{2})}\mathcal{D}^{++}q^{+}_{(0)a}\\\
&-\frac{\kappa_{2}}{2}\int
d\zeta^{(-4)}\;q^{+a}_{(0)}\hat{\mathcal{H}}^{++}_{(s=2)}q^{+}_{(0)a}+\mathcal{O}(\kappa^{2}_{2})\,,\end{split}$
(6.17)

where we sill retained the free action for $q^{+a}_{(0)}$.

Below we shall show that in the sectors of spins 2 and 1, after eliminating
the auxiliary fields, the cubic vertex in the first order in $\kappa_{2}$ is
obtained as:

$\begin{split}S^{(s=2)}_{int}=&-\frac{\kappa_{2}}{2}\int
d\zeta^{(-4)}\;q^{+a}_{(0)}\hat{\mathcal{H}}^{++}_{(s=2)}q^{+}_{(0)a}\\\
=&-\frac{\kappa_{2}}{2}\int
d\zeta^{(-4)}\;\left[\tilde{q}_{(0)}^{+}\hat{\mathcal{H}}^{++}_{(s=2)}q_{(0)}^{+}-q_{(0)}^{+}\hat{\mathcal{H}}^{++}_{(s=2)}\tilde{q}_{(0)}^{+}\right],\end{split}$
(6.18)

Thus we have written the minimal cubic interaction in terms of the on-shell
free hypermultiplet $q^{+a}_{(0)}$. We now turn to considering the component
structure of the bosonic sector of the spin $\mathbf{2}$ theory, also in the
first order in $\kappa_{2}$. For this we should make use of the representation
(5.9) and (5.10) for $q^{+a}_{(0)}$.

Spin 2 sector

The spin 2 sector of the gauge potentials is spanned by the following terms in
the relevant $\theta$-expansions:

$h^{++\alpha\dot{\alpha}}_{(\Phi,\mathcal{P})}=-2i\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}\Phi_{\rho\dot{\rho}}^{\alpha\dot{\alpha}}\,,$
(6.19a)
$h^{++\alpha+}_{(\Phi,\mathcal{P})}=-2i(\theta^{+})^{2}\bar{\theta}^{+\dot{\beta}}B_{\dot{\beta}}^{\alpha}-2i(\theta^{+})^{2}\bar{\theta}^{+\dot{\beta}}\mathcal{P}_{\dot{\beta}}^{\alpha}\,,$
(6.19b)
$h^{++\dot{\alpha}+}_{(\Phi,\mathcal{P})}=-2i(\bar{\theta}^{+})^{2}\theta^{+\beta}B_{\beta}^{\dot{\alpha}}-2i(\bar{\theta}^{+})^{2}\theta^{+\beta}\bar{\mathcal{P}}_{\beta}^{\dot{\alpha}}\,.$
(6.19c)

After elimination of the auxiliary fields by the non-dynamical part of the
constraint (6.4) one obtains the following expression for the on-shell
hypermultiplet on the spin 2 background (6.19):

$q^{+}_{(s=2)}=f^{i}u^{+}_{i}+\theta^{+\alpha}\psi_{\alpha}+\bar{\theta}^{+}_{\dot{\alpha}}\bar{\kappa}^{+\dot{\alpha}}+(\theta^{\hat{+}})^{2}mf^{i}u^{-}_{i}+2i\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}\nabla^{(s=2)}_{\rho\dot{\rho}}f^{i}u^{-}_{i},$
(6.20)

where we have introduced the derivative covariant under the linearized gauge
spin 2 transformations:

$\nabla^{(s=2)}_{\rho\dot{\rho}}f^{i}:=\left(\partial_{\rho\dot{\rho}}+\kappa_{2}\Phi^{\alpha\dot{\alpha}}_{\rho\dot{\rho}}\partial_{\alpha\dot{\alpha}}+\frac{\kappa_{2}}{2}\left[\partial_{\alpha\dot{\alpha}}\Phi_{\rho\dot{\rho}}^{\alpha\dot{\alpha}}+4B_{\rho\dot{\rho}}+2\mathcal{P}_{\rho\dot{\rho}}+2\bar{\mathcal{P}}_{\rho\dot{\rho}}\right]\right)f^{i}.$
(6.21)

One can unfold the second line of (6.17), in full analogy with the spin
$\mathbf{1}$ case:

$\begin{split}&-\int
d\zeta^{(-4)}\;\tilde{q}^{+}_{(s=2)}\mathcal{D}^{++}q^{+}_{(s=2)}\\\ &=\int
d^{4}x\;\left(-\eta^{nm}\nabla^{(s=2)}_{n}f^{i}\nabla^{(s=2)}_{m}\bar{f}_{i}+\partial^{m}f^{i}\nabla^{(s=2)}_{m}\bar{f}_{i}+\nabla^{(s=2)}_{m}f^{i}\partial^{m}\bar{f}_{i}-m^{2}f^{i}\bar{f}_{i}\
\right)\\\ &=\int
d^{4}x\;\left(\partial^{m}f^{i}\partial_{m}\bar{f}_{i}-m^{2}f^{i}\bar{f}_{i}\right)+\mathcal{O}(\kappa_{2}^{2}).\end{split}$
(6.22)

So this line does not contribute to the cubic coupling.

Substituting the expressions (6.19) into (6.18) and using the $\theta$
-expansions (5.9) and (5.10), we obtain in the bosonic sector:

$\begin{split}S^{(s=2)}_{int}=&-\frac{\kappa_{2}}{4}\int
d^{4}x\;\Phi^{\alpha\beta\dot{\alpha}\dot{\beta}}\left(\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}f^{i}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}f^{i}\partial_{\beta\dot{\beta}}\bar{f}_{i}-\partial_{\beta\dot{\beta}}f^{i}\partial_{\alpha\dot{\alpha}}\bar{f}_{i}+f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\bar{f}_{i}\right)\\\
&\;+\frac{\kappa_{2}}{2}\int
d^{4}x\left(2B^{\alpha\dot{\alpha}}+\mathcal{P}^{\alpha\dot{\alpha}}+\bar{\mathcal{P}}^{\alpha\dot{\alpha}}\right)\partial_{\alpha\dot{\alpha}}\left(f^{i}\bar{f}_{i}\right).\end{split}$
(6.23)

The second line contains an interaction with the linearized scalar curvature
due to the relation
$\partial_{\alpha\dot{\alpha}}B^{\alpha\dot{\alpha}}=-2\mathcal{R}$.

After using the expression (A.7) for $B^{\alpha\dot{\alpha}}$ and eliminating
the auxiliary fields $\mathcal{P}^{\alpha\dot{\alpha}}$ and
$\bar{\mathcal{P}}^{\alpha\dot{\alpha}}$ by eqs. (A.19), the full spin 2
action involving the cubic coupling to scalars takes the form:

$\begin{split}S^{(s=2)}_{{}_{(\Phi)}}+S_{int}^{(s=2)}=&\int
d^{4}x\;2\Phi^{\alpha\beta\dot{\alpha}\dot{\beta}}\mathcal{G}_{\alpha\beta\dot{\alpha}\dot{\beta}}\\\
&-\frac{\kappa_{2}}{4}\int
d^{4}x\;\Phi^{\alpha\beta\dot{\alpha}\dot{\beta}}\Big{[}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}f^{i}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}f^{i}\partial_{\beta\dot{\beta}}\bar{f}_{i}-\partial_{\beta\dot{\beta}}f^{i}\partial_{\alpha\dot{\alpha}}\bar{f}_{i}+f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\bar{f}_{i}\\\
&+\left(\partial_{\alpha\dot{\beta}}\partial_{\beta\dot{\alpha}}-\epsilon_{\alpha\beta}\epsilon_{\dot{\alpha}\dot{\beta}}\Box\right)\left(f^{i}\bar{f}_{i}\right)\Big{]}+\mathcal{O}(\kappa_{2}^{2})\,.\end{split}$
(6.24)

The equation of motion for $\Phi^{\alpha\beta\dot{\alpha}\dot{\beta}}$ yields:

$\begin{split}\mathcal{G}_{\alpha\beta\dot{\alpha}\dot{\beta}}=&\frac{\kappa_{2}}{16}\Big{[}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}f^{i}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}f^{i}\partial_{\beta\dot{\beta}}\bar{f}_{i}-\partial_{\beta\dot{\beta}}f^{i}\partial_{\alpha\dot{\alpha}}\bar{f}_{i}+f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\bar{f}_{i}\\\
&-\left(\partial_{\alpha\dot{\beta}}\partial_{\beta\dot{\alpha}}-\epsilon_{\alpha\beta}\epsilon_{\dot{\alpha}\dot{\beta}}\Box\right)\left(f^{i}\bar{f}_{i}\right)\Big{]},\end{split}$
(6.25)

which coincides with (5.115). The interaction of spin 2 with a scalar field
can now be read off from eq. (6.24) and cast in the generic Noether form
(5.68):

$S^{(s=2)}_{int}=\kappa_{2}\int
d^{4}x\,\Phi^{\alpha\dot{\alpha}\beta\dot{\beta}}Y_{\alpha\dot{\alpha}\beta\dot{\beta}}\,.$
(6.26)

The corresponding $Y$ current is:

$\begin{split}Y_{\alpha\dot{\alpha}\beta\dot{\beta}}=-\frac{1}{4}\Big{[}&\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}f^{i}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}f^{i}\partial_{\beta\dot{\beta}}\bar{f}_{i}-\partial_{\beta\dot{\beta}}f^{i}\partial_{\alpha\dot{\alpha}}\bar{f}_{i}+f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\bar{f}_{i}\\\
&-\left(\partial_{\alpha\dot{\beta}}\partial_{\beta\dot{\alpha}}-\epsilon_{\alpha\beta}\epsilon_{\dot{\alpha}\dot{\beta}}\Box\right)\left(f^{i}\bar{f}_{i}\right)\Big{]}.\end{split}$
(6.27)

Up to the last term (automatically satisfying the conservation law), the
$Y$-current obtained coincides with the energy-momentum tensor (5.91).

In Appendix E.1, using the proper field redefinitions, we prove that the
coupling constructed is equivalent to the minimal linearized gravity coupling
of scalar fields. It is also possible to remove the last line in (6.24) (and
the last line in the corresponding $Y$ current (6.27)) through a redefinition
of the spin 2 field, as described in Appendix E.2. After such a redefinition
we reproduce the standard linearized spin 2 coupling to a complex scalar, as
it is given, e.g., in Bekaert:2009ud .

Spin 1 sector

After elimination of the auxiliary fields, we are left with the following spin
1 sector:

$h^{++5}_{(C)}=-2i\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}C_{\rho\dot{\rho}}\,,$
(6.28)
$h^{++\alpha+}_{(C)}=i(\bar{\theta}^{+})^{2}\theta^{+}_{\nu}\partial^{(\alpha}_{\dot{\rho}}C^{\dot{\rho}\nu)}\,,$
(6.29)
$h^{++\dot{\alpha}+}_{(C)}=-i(\theta^{+})^{2}\bar{\theta}^{+}_{\dot{\nu}}\partial^{(\dot{\alpha}}_{\rho}C^{\rho\dot{\nu})}\,.$
(6.30)

In this spin 1 sector of $\mathcal{N}=2$ spin $\mathbf{2}$ theory, the
interaction term (6.4) deforms the free on-shell hypermultiplet (5.9) as:

$q^{+}_{(s=2)}=f^{i}u^{+}_{i}+\theta^{+\alpha}\psi_{\alpha}+\bar{\theta}^{+}_{\dot{\alpha}}\bar{\kappa}^{+\dot{\alpha}}+(\theta^{\hat{+}})^{2}mf^{i}u^{-}_{i}+2i\theta^{+}\sigma^{n}\bar{\theta}^{+}\nabla^{(s=2)}_{n}f^{i}u^{-}_{i}.$
(6.31)

Here we have defined the spin 1 covariant derivative in the spin $\mathbf{2}$
${\cal N}=2$ theory as:

$\nabla^{(s=2)}_{n}f^{i}:=\left(\partial_{n}+im\kappa_{2}C_{n}\right)f^{i}.$
(6.32)

Using the same argument as in (6.12), we conclude that term

$-\kappa_{2}\int
d\zeta^{(-4)}\;q^{+a}_{(\kappa_{2})}\mathcal{D}^{++}q^{+}_{(0)a}$

does not contribute to $(1,0,0)$ vertex. There is only one contribution to
cubical $(1,0,0)$ vertex and it is given by (6.18).

The actual contribution to the interaction in the first order in $\kappa_{2}$
comes only from the prepotential $h^{++5}$,
$\hat{\mathcal{H}}^{++}_{(s=2)}\rightarrow h^{++5}\partial_{5}$:

$\begin{split}S^{(s=2)}_{int}=&-\frac{\kappa_{2}}{2}\int
d\zeta^{(-4)}\;\left[\tilde{q}_{(0)}^{+}\hat{\mathcal{H}}^{++}_{(s=2)}q_{(0)}^{+}-q_{(0)}^{+}\hat{\mathcal{H}}^{++}_{(s=2)}\tilde{q}_{(0)}^{+}\right]\\\
\Rightarrow&-\frac{im\kappa_{2}}{2}\int
d^{4}x\;C^{\rho\dot{\rho}}\left(f^{i}\partial_{\rho\dot{\rho}}\bar{f}_{i}-\partial_{\rho\dot{\rho}}f^{i}\bar{f}_{i}\right).\end{split}$
(6.33)

So, the resulting spin 1 coupling is proportional to the mass $m$ of the
hypermultiplet and has the standard Noether form (5.68):

$S^{(s=2)}_{int}=\kappa_{2}\int d^{4}x\,C^{\rho\dot{\rho}}Y_{\rho\dot{\rho}},$
(6.34)

where :

$Y_{\rho\dot{\rho}}=-\frac{im}{2}\left(f^{i}\partial_{\rho\dot{\rho}}\bar{f}_{i}-\partial_{\rho\dot{\rho}}f^{i}\bar{f}_{i}\right).$
(6.35)

This current coincides with the appropriate conserved bosonic component of the
current superfield (5.101).

Spin $3/2$ sector

A similar analysis can also be carried out for the spinor sector. The
corresponding cubic interactions also has the Noether form (5.68):

$S^{(s=2)}_{int}=\kappa_{2}\int
d^{4}x\,\left(\psi^{\alpha\beta\dot{\alpha}i}Y_{\alpha\beta\dot{\alpha}i}+\bar{\psi}^{\alpha\dot{\alpha}\dot{\beta}i}\bar{Y}_{\alpha\dot{\alpha}\dot{\beta}i}\right).$
(6.36)

The corresponding spin $3/2$ currents coincides with the conserved fermionic
current of $\mathcal{N}=2$ supersymmetry (5.110), up to terms which are
trivially conserved without making use of the equations of motion:

$\begin{split}Y^{i}_{\rho\dot{\rho}\alpha}=-\frac{1}{4}\mathcal{S}^{i}_{\rho\dot{\rho}\alpha}=-\frac{1}{4}\Big{[}\psi_{\rho}\partial_{\alpha\dot{\rho}}\bar{f}^{i}-\chi_{\rho}\partial_{\alpha\dot{\rho}}f^{i}-im\epsilon_{\rho\alpha}\left(\bar{\chi}_{\dot{\rho}}\bar{f}^{i}-\bar{\psi}_{\dot{\rho}}f^{i}\right)\Big{]}.\end{split}$
(6.37)

The overall coefficient is uniquely fixed by inspecting the interaction of the
symmetric part $\psi^{(\alpha\rho)\dot{\rho}i}$ of the spin $\tfrac{3}{2}$
gauge field with the hypermultiplet fields. The contribution to this part of
vertex comes from a single term in the WZ gauge of the analytic potential
$h^{++\alpha\dot{\alpha}}$.

The terms by which the various conserved spin $\tfrac{3}{2}$ currents defined
in Section 5.6 differ from each other yield fake cubic
$(\frac{3}{2},\frac{1}{2},0)$ interactions. Such terms can be eliminated by
means of some field redefinition, according to the general prescription of
Appendix E. We do not provide the relevant details here.

###  6.4 Spin 3 coupling

Like in the spin $\mathbf{2}$ case, in order to obtain the cubic coupling to
the leading order in $\kappa_{3}$, one needs to partly solve the equation of
motion for the hypermultiplet in the spin $\mathbf{3}$ background:

$\begin{split}\Big{[}\mathcal{D}^{++}+\kappa_{3}\hat{\mathcal{H}}^{++}_{(s=3)}J&+\frac{\kappa_{3}}{2}\left(\partial_{\alpha\dot{\alpha}}\hat{\mathcal{H}}^{++\alpha\dot{\alpha}}\right)J\\\
&+\kappa_{3}\left(\xi+\frac{1}{2}\right)\Gamma^{++}_{(s=3)}J+\frac{\kappa_{3}}{2}\Gamma^{++\alpha\dot{\alpha}}\partial_{\alpha\dot{\alpha}}J\Big{]}\,q^{+}_{a}=0\,.\end{split}$
(6.38)

We represent the solution as a perturbative series in $\kappa_{3}$:

$q^{+}=q^{+}_{(0)}+\kappa_{3}q^{+}_{(\kappa_{3})}+\mathcal{O}(\kappa_{3}^{2})\,,$
(6.39)

where $q^{+}_{0}$ is described by the free hypermultiplet action and is given
in (5.9). The action, up to the first order in $\kappa_{3}$, is written as:

$\begin{split}S^{(s=3)}_{gauge}=&-\frac{1}{2}\int
d\zeta^{(-4)}\;q^{+a}\left(\mathcal{D}^{++}+\kappa_{3}\hat{\mathcal{H}}^{++}_{(s=3)}J+\kappa_{3}\xi\Gamma^{++}_{(s=3)}J\right)q^{+}_{a}\\\
=&-\frac{1}{2}\int d\zeta^{(-4)}\;q^{+a}_{(0)}\mathcal{D}^{++}q^{+}_{(0)a}\\\
&-\kappa_{3}\int
d\zeta^{(-4)}\;q^{+a}_{(\kappa_{3})}\mathcal{D}^{++}q^{+}_{(0)a}\\\
&-\frac{\kappa_{3}}{2}\int
d\zeta^{(-4)}\;q^{+a}_{(0)}\left(\hat{\mathcal{H}}^{++}_{(s=3)}J+\xi\Gamma^{++}_{(s=3)}J\right)q^{+}_{(0)a}+\mathcal{O}(\kappa_{3}^{2})\,.\end{split}$
(6.40)

Below we will show that after elimination of the auxiliary fields the third
line does not contribute to the cubic vertex, and the latter takes the form:

$\begin{split}S^{(s=3)}_{int}=&-\frac{\kappa_{3}}{2}\int
d\zeta^{(-4)}\;q^{+a}_{(0)}\left(\hat{\mathcal{H}}^{++}_{(s=3)}J+\xi\Gamma^{++}_{(s=3)}J\right)q^{+}_{(0)a}\\\
=&\;\frac{i\kappa_{3}}{2}\int
d\zeta^{(-4)}\;\left[q_{(0)}^{+}\hat{\mathcal{H}}^{++}_{(s=3)}\tilde{q}_{(0)}^{+}+\tilde{q}_{(0)}^{+}\hat{\mathcal{H}}^{++}_{(s=3)}q_{(0)}^{+}\right]\\\
&+\xi\cdot i\kappa_{3}\int
d\zeta^{(-4)}\;\Gamma^{++}_{(s=3)}\;\tilde{q}^{+}_{(0)}q^{+}_{(0)}\,.\end{split}$
(6.41)

Another convenient form of the cubic interaction related to (6.41) by a total
derivative is as follows :

$\begin{split}S^{(s=3)}_{int}=&-\frac{i\kappa_{3}}{2}\int
d\zeta^{(-4)}\;\left[\partial_{\alpha\dot{\alpha}}\tilde{q}_{(0)}^{+}\hat{\mathcal{H}}^{++\alpha\dot{\alpha}}_{(s=3)}q_{(0)}^{+}+\partial_{\alpha\dot{\alpha}}q_{(0)}^{+}\hat{\mathcal{H}}^{++\alpha\dot{\alpha}}_{(s=3)}\tilde{q}_{(0)}^{+}\right]\\\
&-\frac{i\kappa_{3}}{2}\int
d\zeta^{(-4)}\;\left[\partial_{\alpha\dot{\alpha}}\tilde{q}_{(0)}^{+}\Gamma^{++\alpha\dot{\alpha}}_{(s=3)}q_{(0)}^{+}+\partial_{\alpha\dot{\alpha}}q_{(0)}^{+}\Gamma^{++\alpha\dot{\alpha}}_{(s=3)}\tilde{q}_{(0)}^{+}\right]\\\
&+\xi\cdot i\kappa_{3}\int
d\zeta^{(-4)}\;\Gamma^{++}_{(s=3)}\;\tilde{q}^{+}_{(0)}q^{+}_{(0)}\,.\end{split}$
(6.42)

Using these expressions for the cubic vertex, it is easy to find the component
structure of the $({3},{0},{0})$ interaction.

Spin 3 sector

Since the spin ${\bf 3}$ coupling is an essentially new subject as compared to
the previous cases, we consider it with more intermediate details.

The spin 3 sector of the gauge $\mathcal{N}=2$ spin $\mathbf{3}$ potentials is
encompassed by the following contributions:

$h^{++(\alpha\beta)(\dot{\alpha}\dot{\beta})}=-2i\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}\Phi^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}_{\rho\dot{\rho}}\,,$
(6.43a)
$h^{++(\alpha\beta)\dot{\alpha}+}=-2i(\theta^{+})^{2}\bar{\theta}^{+\dot{\beta}}B^{(\alpha\beta)\dot{\alpha}}_{\dot{\beta}}\,,$
(6.43b)
$h^{++(\dot{\alpha}\dot{\beta})\alpha+}=-2i(\bar{\theta}^{+})^{2}\theta^{+\beta}B_{\beta}^{(\dot{\alpha}\dot{\beta})\alpha}\,.$
(6.43c)

Here we used the notations from Appendix A.2:

$\Phi_{(\alpha\beta)\gamma(\dot{\alpha}\dot{\beta})\dot{\gamma}}=\Phi_{(\alpha\beta\gamma)(\dot{\alpha}\dot{\beta}\dot{\gamma})}+\Phi_{(\alpha(\dot{\alpha}}\epsilon_{\beta)\gamma}\epsilon_{\dot{\beta})\dot{\gamma}}\,,$
(6.44) $\displaystyle
B_{(\alpha\beta)\dot{\alpha}\dot{\beta}}=-\frac{1}{2}\Big{[}\partial^{\gamma\dot{\gamma}}\Phi_{(\alpha\beta\gamma)(\dot{\alpha}\dot{\beta}\dot{\gamma})}-\partial_{(\alpha(\dot{\alpha}}\Phi_{\beta)\dot{\beta})}-\partial_{(\alpha\dot{\alpha}}\Phi_{\beta)\dot{\beta}}\Big{]},$
(6.45) $\displaystyle
B_{\alpha\beta(\dot{\alpha}\dot{\beta})}=-\frac{1}{2}\Big{[}\partial^{\gamma\dot{\gamma}}\Phi_{(\alpha\beta\gamma)(\dot{\alpha}\dot{\beta}\dot{\gamma})}-\partial_{(\alpha(\dot{\alpha}}\Phi_{\beta)\dot{\beta})}-\partial_{\alpha(\dot{\alpha}}\Phi_{\beta\dot{\beta})}\Big{]}.$
(6.46)

Using these fields and hypermultiplet equation in the spin 3 background (6.5),
we obtain for the on-shell hypermultiplet:

$q^{+}_{(s=3)}=f^{i}u^{+}_{i}+\theta^{+\alpha}\psi_{\alpha}+\bar{\theta}^{+}_{\dot{\alpha}}\bar{\kappa}^{+\dot{\alpha}}+(\theta^{\hat{+}})^{2}mf^{i}u^{-}_{i}+2i\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}\nabla^{(s=3)}_{\rho\dot{\rho}}f^{i}u^{-}_{i}.$
(6.47)

Here we introduced the spin 3 gauge-covariant derivative:

$\nabla^{(s=3)}_{\rho\dot{\rho}}f^{i}=\Big{\\{}\partial_{\rho\dot{\rho}}-i\kappa_{3}\Phi_{\rho\dot{\rho}}^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}-i\frac{\kappa_{3}}{2}\left(\partial_{\alpha\dot{\alpha}}\Phi_{\rho\dot{\rho}}^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\right)\partial_{\beta\dot{\beta}}\\\
-i\frac{\kappa_{3}}{2}(1+2\xi)\left[\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\Phi^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}_{\rho\dot{\rho}}+2\partial^{\alpha\dot{\alpha}}B_{(\alpha\rho)\dot{\rho}\dot{\alpha}}+2\partial^{\alpha\dot{\alpha}}B_{\rho\alpha(\dot{\alpha}\dot{\rho})}\right]\\\
-i\frac{\kappa_{3}}{2}\left[\partial_{\beta\dot{\beta}}\Phi^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}_{\rho\dot{\rho}}+2B^{(\alpha\dot{\alpha}}_{\rho)\dot{\rho}}+2B^{\alpha(\dot{\alpha}}_{\rho\dot{\rho})}\right]\partial_{\alpha\dot{\alpha}}\Big{\\}}f^{i}.$
(6.48)

Using the expression (6.47), one can easily unfold the second and third lines
of (6.40) in full analogy with the spin $\mathbf{1}$ and spin $\mathbf{2}$
cases:

$\begin{split}&-\int
d\zeta^{(-4)}\;\tilde{q}^{+}_{(s=3)}\mathcal{D}^{++}q^{+}_{(s=3)}\\\ &=\int
d^{4}x\;\left(-\eta^{nm}\nabla^{(s=3)}_{n}f^{i}\nabla^{(s=3}_{m}\bar{f}_{i}+\partial^{m}f^{i}\nabla^{(s=3)}_{m}\bar{f}_{i}+\nabla^{(s=3)}_{m}f^{i}\partial^{m}\bar{f}_{i}-m^{2}f^{i}\bar{f}_{i}\right)\\\
&=\int
d^{4}x\;\left(\partial^{m}f^{i}\partial_{m}\bar{f}_{i}-m^{2}f^{i}\bar{f}_{i}\right)+\mathcal{O}(\kappa_{3}^{2}).\end{split}$
(6.49)

So these terms of (6.40) doe not contribute to the first-order cubic gauge
$(3,0,0)$ coupling.

Substituting the expressions (6.43) into the action (6.41), we obtain the
component bosonic core of the $(3,0,0)$ cubic coupling as:

$\begin{split}S^{(s=3)}_{int}=&-\frac{i\kappa_{3}}{4}\;\int
d^{4}x\;\Phi^{\rho\dot{\rho}(\alpha\beta)(\dot{\alpha}\dot{\beta})}\Big{(}f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\partial_{\rho\dot{\rho}}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\partial_{\rho\dot{\rho}}f^{i}\bar{f}_{i}\\\
&\qquad\qquad\;\;\;\;-\partial_{\rho\dot{\rho}}f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\bar{f}_{i}+\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}f^{i}\partial_{\rho\dot{\rho}}\bar{f}_{i}\Big{)}\\\
&+\frac{i\kappa_{3}}{2}\int
d^{4}x\;\Big{(}B^{(\alpha\beta)\dot{\alpha}\dot{\beta}}+B^{(\dot{\alpha}\dot{\beta})\alpha\beta}\Big{)}\Big{(}f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}f^{i}\bar{f}_{i}\Big{)}\\\
&-\frac{i\xi\kappa_{3}}{2}\;\int
d^{4}x\;\partial_{\alpha\dot{\alpha}}\Big{[}\partial_{\beta\dot{\beta}}\Phi^{\rho\dot{\rho}(\alpha\beta)(\dot{\alpha}\dot{\beta})}+2\Big{(}B^{(\alpha\beta)\dot{\alpha}\dot{\beta}}+B^{(\dot{\alpha}\dot{\beta})\alpha\beta}\Big{)}\Big{]}\Big{(}f^{i}\partial_{\rho\dot{\rho}}\bar{f}_{i}-\partial_{\rho\dot{\rho}}f^{i}\bar{f}_{i}\Big{)}.\end{split}$
(6.50)

We now turn to the analysis of various terms in this expression. Some of such
terms can be removed by a field redefinition, essentially simplifying the
resulting vertex.

1. One of the basic relations we use is the following:

$\displaystyle\partial^{\alpha\dot{\alpha}}B_{(\alpha\beta)\dot{\alpha}\dot{\beta}}=\partial^{\alpha\dot{\alpha}}B_{\alpha\beta(\dot{\alpha}\dot{\beta})}=-\frac{1}{2}\mathcal{R}_{\beta\dot{\beta}}$
$\displaystyle=\,-\frac{1}{2}\Big{(}\partial^{\alpha\dot{\alpha}}\partial^{\gamma\dot{\gamma}}\Phi_{(\alpha\beta\gamma)(\dot{\alpha}\dot{\beta}\dot{\gamma})}-\frac{1}{4}\partial_{\beta\dot{\beta}}\partial^{\alpha\dot{\alpha}}\Phi_{\alpha\dot{\alpha}}-\frac{5}{2}\Box\Phi_{\beta\dot{\beta}}\Big{)}.$
(6.51)

This is the linearized tensor (spin 3 “scalar curvature”) defined in (A.39).
So the last term in (6.50) is reduced to

$i\xi\kappa_{3}\int
d^{4}x\;\mathcal{R}^{\beta\dot{\beta}}\Big{(}f^{i}\partial_{\beta\dot{\beta}}\bar{f}_{i}-\partial_{\beta\dot{\beta}}f^{i}\bar{f}_{i}\Big{)}.$
(6.52)

As explained in Appendix E.4, it can be eliminated by a field redefinition
181818One can also note that on the free equation of motion for the spin 3
gauge field one can eliminate this term thanks to the relation (4.68). .

2. The second line in (6.50) can also be partially rewritten in the more convenient form, integrating by parts and once again using (6.4):

$\begin{split}&\int
d^{4}x\;B^{(\dot{\alpha}\dot{\beta})(\alpha\beta)}\Big{(}f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}f^{i}\bar{f}_{i}\Big{)}\\\
&\qquad=\frac{1}{2}\;\int
d^{4}x\;\left(\mathcal{R}^{\alpha\dot{\alpha}}-\frac{1}{4}\partial^{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\Phi^{\beta\dot{\beta}}+\frac{1}{2}\Box\Phi^{\alpha\dot{\alpha}}\right)\Big{(}f^{i}\partial_{\alpha\dot{\alpha}}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}f^{i}\bar{f}_{i}\Big{)}.\end{split}$
(6.53)

This term can be eliminated through a field redefinition described in
Appendices E.3 and E.4.

3. Now it becomes possible to clarify the role of the $\xi$-interaction. The remaining $\xi$-term with $\Phi$ in the last line of (6.50) can be reduced to the form:

$\begin{split}&\int
d^{4}x\;\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\Phi^{\rho\dot{\rho}(\alpha\beta)(\dot{\alpha}\dot{\beta})}\Big{(}f^{i}\partial_{\rho\dot{\rho}}\bar{f}_{i}-\partial_{\rho\dot{\rho}}f^{i}\bar{f}_{i}\Big{)}\\\
&=\int
d^{4}x\;\Big{[}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\Phi^{(\alpha\beta\rho)(\dot{\alpha}\dot{\beta}\dot{\rho})}+\frac{1}{2}\partial^{\rho\dot{\rho}}\partial_{\alpha\dot{\alpha}}\Phi^{\alpha\dot{\alpha}}+\frac{1}{2}\partial^{\rho\dot{\alpha}}\partial^{\alpha\dot{\rho}}\Phi_{\alpha\dot{\alpha}}\Big{]}\Big{(}f^{i}\partial_{\rho\dot{\rho}}\bar{f}_{i}-\partial_{\rho\dot{\rho}}f^{i}\bar{f}_{i}\Big{)}.\end{split}$
(6.54)

The first and second terms in this expression can be eliminated in the way
described in Appendices E.3 and E.4. Let us consider the third term:

$\begin{split}&\int
d^{4}x\;\Phi_{\alpha\dot{\alpha}}\,\partial^{\rho\dot{\alpha}}\partial^{\alpha\dot{\rho}}\,\Big{(}f^{i}\partial_{\rho\dot{\rho}}\bar{f}_{i}-\partial_{\rho\dot{\rho}}f^{i}\bar{f}_{i}\Big{)}\\\
&=\int
d^{4}x\;\Phi^{\alpha\dot{\alpha}}\Box\Big{(}f^{i}\partial_{\alpha\dot{\alpha}}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}f^{i}\bar{f}_{i}\Big{)}+2\int
d^{4}x\;\partial_{\alpha\dot{\alpha}}\Phi^{\alpha\dot{\alpha}}\Big{(}\Box
f^{i}\bar{f}_{i}-f^{i}\Box\bar{f}_{i}\Big{)}.\end{split}$ (6.55)

The first piece here can be eliminated by the scalar field redefinition
described in Appendix E.3. On shell, using the equation $(\Box+m^{2})f^{i}=0$,
one can also eliminate the second piece.

As the result of our analysis, we conclude that the $\xi$-term can be
completely removed on shell through various field redefinitions. Finally, the
$(3,0,0)$ coupling is reduced to the form:

$\begin{split}S^{(s=3)}_{int}=&-\frac{i\kappa_{3}}{4}\;\int
d^{4}x\;\Phi^{(\alpha\beta)\rho(\dot{\alpha}\dot{\beta})\dot{\rho}}\Bigr{(}f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\partial_{\rho\dot{\rho}}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\partial_{\rho\dot{\rho}}f^{i}\bar{f}_{i}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\quad\;\;\;\;-\partial_{\rho\dot{\rho}}f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\bar{f}_{i}+\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}f^{i}\partial_{\rho\dot{\rho}}\bar{f}_{i}\Bigr{)}.\end{split}$
(6.56)

It has the standard Noether form (5.68):

$S^{(s=3)}_{int}=\kappa_{3}\int
d^{4}x\,\Phi^{(\alpha\beta)\rho(\dot{\alpha}\dot{\beta})\dot{\rho}}Y_{(\alpha\beta)\rho(\dot{\alpha}\dot{\beta})\dot{\rho}},$
(6.57)

where the corresponding conserved current (in massless case) is:

$Y_{(\alpha\beta)\rho(\dot{\alpha}\dot{\beta})\dot{\rho}}=-\frac{i}{4}\Bigr{(}f^{i}\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}\partial_{\rho\dot{\rho}}\bar{f}_{i}-\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}\partial_{\rho\dot{\rho}}f^{i}\bar{f}_{i}-\partial_{\rho\dot{\rho}}f^{i}\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}\bar{f}_{i}+\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\beta}}f^{i}\partial_{\rho\dot{\rho}}\bar{f}_{i}\Bigr{)}.$
(6.58)

This current coincides with the conserved component (5.128) of the spin
$\mathbf{3}$ current superfield (5.123) for the $\xi=0$ case.

It is useful to further transform the action (6.56) with the help of the
relation (valid up to total derivatives):

$\displaystyle\int
d^{4}x\;\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\Phi^{\rho\dot{\rho}(\alpha\beta)(\dot{\alpha}\dot{\beta})}\Big{(}f^{i}\partial_{\rho\dot{\rho}}\bar{f}_{i}-\partial_{\rho\dot{\rho}}f^{i}\bar{f}_{i}\Big{)}$
$\displaystyle\Rightarrow\,\int
d^{4}x\;\Phi^{\rho\dot{\rho}(\alpha\beta)(\dot{\alpha}\dot{\beta})}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\Big{(}f^{i}\partial_{\rho\dot{\rho}}\bar{f}_{i}-\partial_{\rho\dot{\rho}}f^{i}\bar{f}_{i}\Big{)}$
$\displaystyle=\,\int
d^{4}x\;\Phi^{\rho\dot{\rho}(\alpha\beta)(\dot{\alpha}\dot{\beta})}\Bigr{(}f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\partial_{\rho\dot{\rho}}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\partial_{\rho\dot{\rho}}f^{i}\bar{f}_{i}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-\partial_{\rho\dot{\rho}}f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\bar{f}_{i}+\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}f^{i}\partial_{\rho\dot{\rho}}\bar{f}_{i}\Bigr{)}$
$\displaystyle\qquad+2\int
d^{4}x\;\Phi^{\rho\dot{\rho}(\alpha\beta)(\dot{\alpha}\dot{\beta})}\Big{(}\partial_{\alpha\dot{\alpha}}f^{i}\partial_{\beta\dot{\beta}}\partial_{\rho\dot{\rho}}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}\partial_{\rho\dot{\rho}}f^{i}\partial_{\beta\dot{\beta}}\bar{f}_{i}\Big{)}.$
(6.59)

On the one hand, this term as a whole can be omitted by properly redefining
the fields and using the equations of motion. On the other hand, using this
relation, one can cast the cubic $(3,0,0)$ vertex in the form:

$\begin{split}S^{(s=3)}_{int}=&\;\frac{i\kappa_{3}}{2}\;\int
d^{4}x\;\Phi^{(\alpha\beta)\rho(\dot{\alpha}\dot{\beta})\dot{\rho}}\left(\partial_{\alpha\dot{\alpha}}f^{i}\partial_{\beta\dot{\beta}}\partial_{\rho\dot{\rho}}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}\partial_{\rho\dot{\rho}}f^{i}\partial_{\beta\dot{\beta}}\bar{f}_{i}\right)\\\
=&\;\frac{i\kappa_{3}}{2}\;\int
d^{4}x\;\Biggr{[}\Phi^{(\alpha\beta\rho)(\dot{\alpha}\dot{\beta}\dot{\rho})}\left(\partial_{\alpha\dot{\alpha}}f^{i}\partial_{\beta\dot{\beta}}\partial_{\rho\dot{\rho}}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}\partial_{\rho\dot{\rho}}f^{i}\partial_{\beta\dot{\beta}}\bar{f}_{i}\right)\\\
&\qquad\qquad\qquad\qquad+\Phi^{\alpha\dot{\alpha}}\left(\partial_{\alpha\dot{\alpha}}f^{i}\Box\bar{f}_{i}-\Box
f^{i}\partial_{\alpha\dot{\alpha}}\bar{f}_{i}\right)\\\
&\qquad\qquad\qquad\qquad+\frac{1}{4}\Phi^{\alpha\dot{\alpha}}\left(\partial^{\beta\dot{\beta}}f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}f^{i}\partial^{\beta\dot{\beta}}\bar{f}_{i}\right)\Biggr{]}.\end{split}$
(6.60)

This vertex exactly coincides with the (3,0,0) vertex given in Zinoviev:2010cr
.

Spin 2 sector

The spin 2 sector is spanned by the following pieces in the analytic
potentials:

$h^{++5\alpha\dot{\alpha}}=-2i\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}C_{\rho\dot{\rho}}^{\alpha\dot{\alpha}}\,,$
(6.61a)
$h^{++\dot{\alpha}(\alpha\beta)+}=i(\bar{\theta}^{+})^{2}\theta^{+}_{\rho}H^{\dot{\alpha}\rho(\alpha\beta)}\,,$
(6.61b)
$h^{++\alpha(\dot{\alpha}\dot{\beta})}=i(\theta^{+})^{2}\bar{\theta}^{+}_{\dot{\rho}}\bar{H}^{\alpha\dot{\rho}(\dot{\alpha}\dot{\beta})}\,.$
(6.61c)

Here we used the notations (A.47), (A.48) and (A.49).

The relevant contribution to the on-shell hypermultiplet is:

$\begin{split}q^{+}_{(s=3)}=&f^{i}u^{+}_{i}+\theta^{+\alpha}\psi_{\alpha}+\bar{\theta}^{+}_{\dot{\alpha}}\bar{\kappa}^{+\dot{\alpha}}+(\theta^{+})^{2}\left[mf^{i}u^{-}_{i}+\kappa_{3}M^{-}\right]\\\
&-(\bar{\theta}^{+})^{2}\left[mf^{i}u^{-}_{i}+\kappa_{3}N^{-}\right]+2i\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}\nabla^{(s=3)}_{\rho\dot{\rho}}f^{i}u^{-}_{i}.\end{split}$
(6.62)

Here we introduced the spin 2 covariant derivative in the spin $\mathbf{3}$
${\cal N}=2$ theory:

$\nabla^{(s=3)}_{\rho\dot{\rho}}f^{i}=\Bigr{(}\partial_{\rho\dot{\rho}}+m\kappa_{3}C_{\rho\dot{\rho}}^{\alpha\dot{\alpha}}\partial_{\alpha\dot{\alpha}}F^{+}+\frac{1}{2}m\kappa_{3}(\partial_{\alpha\dot{\alpha}}C_{\rho\dot{\rho}}^{\alpha\dot{\alpha}})\Bigr{)}f^{i}\,,$
(6.63)

as well as, defined:

$N^{-}=-M^{-}=-\xi\mathcal{R}f^{i}u^{-}_{i}+\frac{3}{2}\partial_{\alpha\dot{\alpha}}(H^{\alpha\dot{\alpha}}f^{i}u^{-}_{i}).$
(6.64)

Here we used the notation $\mathcal{R}$ (A.11), with replacing $\Phi\to C$.

Now it is direct to show that these terms do not contribute to the cubic
$(2,0,0)$ vertex:

$-\int d\zeta^{(-4)}\,\tilde{q}^{+}_{(s=3)}\mathcal{D}^{++}q^{+}_{(s=3)}=\int
d^{4}x\;\left(\partial^{m}f^{i}\partial_{m}\bar{f}_{i}-m^{2}f^{i}\bar{f}_{i}\right)+\mathcal{O}(\kappa_{3}^{2}).$
(6.65)

Thus the minimal coupling $(2,0,0)$ in the spin 2 sector (up to a total
derivative and discarding fermionic terms) is generated as:

$\begin{split}S_{int-min}^{(s=3)}=&\frac{i\kappa_{3}}{2}\int
d\zeta^{(-4)}\;\left(q_{(0)}^{+}\hat{\mathcal{H}}^{++}_{(s=3)}\tilde{q}_{(0)}^{+}+\tilde{q}_{(0)}^{+}\hat{\mathcal{H}}^{++}_{(s=3)}q_{(0)}^{+}\right)\\\
=&-\frac{\kappa_{3}m}{4}\int
d^{4}x\;C^{\alpha\beta\dot{\alpha}\dot{\beta}}\left(\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}f^{i}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}f^{i}\partial_{\beta\dot{\beta}}\bar{f}_{i}-\partial_{\beta\dot{\beta}}f^{i}\partial_{\alpha\dot{\alpha}}\bar{f}_{i}+f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\bar{f}_{i}\right)\\\
&+\frac{\kappa_{3}m}{4}\int
d^{4}x\;\underbrace{\left(\partial_{\alpha\dot{\alpha}}\bar{H}^{\alpha(\dot{\alpha}\dot{\beta})}_{\dot{\beta}}-\partial_{\alpha\dot{\alpha}}H^{\dot{\alpha}(\alpha\beta)}_{\beta}\right)}_{8\mathcal{R}}f^{i}\bar{f}_{i}\,.\end{split}$
(6.66)

The superfield form of $\Gamma^{++}_{(s=3)}$ on the background (6.61b) and
(6.61c) is reduced to

$\begin{split}\Gamma^{++}_{(s=3)}=&\partial_{\alpha\dot{\alpha}}\left(-\partial^{-}_{\beta}h^{++(\alpha\beta)\dot{\alpha}}-\partial^{-}_{\dot{\beta}}h^{++(\dot{\alpha}\dot{\beta})\alpha}\right)\\\
=&i(\bar{\theta}^{+})^{2}\partial_{\alpha\dot{\alpha}}H^{\dot{\alpha}(\alpha\beta)}_{\beta}+i(\theta^{+})^{2}\partial_{\alpha\dot{\alpha}}\bar{H}^{\alpha(\dot{\alpha}\dot{\beta})}_{\dot{\beta}}\\\
=&-4i(\bar{\theta}^{+})^{2}\mathcal{R}+4i(\theta^{+})^{2}\mathcal{R}=4i(\theta^{\hat{+}})^{2}\mathcal{R}\,.\end{split}$
(6.67)

From this expression one can easily deduce the contribution of $\xi$-term to
the spin 2 sector:

$\begin{split}S_{int-\xi}^{(s=3)}\sim\int
d^{4}xd^{4}\theta^{+}\;\Gamma^{++}_{s=3}\;\tilde{q}^{+}_{(0)}q^{+}_{(0)}&=4i\int
d^{4}xd^{4}\theta^{+}\;(\theta^{\hat{+}})^{2}\mathcal{R}\cdot
m\,(\theta^{\hat{+}})^{2}f^{i}\bar{f}_{i}\\\ &=-8i\int
d^{4}x\;m\,\mathcal{R}\,f^{i}\bar{f}_{i}\,.\end{split}$ (6.68)

Putting together all the above expressions, we can write the total $(2,0,0)$
vertex in the spin 2 sector of $\mathcal{N}=2$ spin $\mathbf{3}$ theory:

$\begin{split}S_{int}^{(s=3)}=&-\frac{\kappa_{3}m}{4}\int
d^{4}x\;C^{\alpha\beta\dot{\alpha}\dot{\beta}}\left(\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}f^{i}\bar{f}_{i}-2\partial_{\alpha\dot{\alpha}}f^{i}\partial_{\beta\dot{\beta}}\bar{f}_{i}+f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\bar{f}_{i}\right)\\\
&+2\kappa_{3}m\int
d^{4}x\;\mathcal{R}f^{i}\bar{f}_{i}-8\kappa_{3}m\cdot\xi\int
d^{4}x\;\mathcal{R}f^{i}\bar{f}_{i}\,.\end{split}$ (6.69)

Comparing it with the results of Appendix E.1, we conclude that (6.69)
contains a non-minimal interaction of the scalar field with the spin 2 field
(2nd line) and this deviation from the minimal interaction includes, in
particular, the term proportional to the parameter $\xi$. However, using the
results of Appendix E.2, one can remove on shell both non-minimal terms in
(6.69) by a proper field redefinition.

Thus we conclude that in the cubic bosonic vertices which come from the
superfield cubic vertex involving the spin ${\bf 3}$ ${\cal N}=2$ gauge
superfield, the dependence on $\xi$ can be fully eliminated on shell.

Finally, the spin 2 coupling to scalars can be rewritten in the Noether form
(5.68):

$\begin{split}S_{int}^{(s=3)}=&-\frac{\kappa_{3}m}{4}\int
d^{4}x\;C^{\alpha\beta\dot{\alpha}\dot{\beta}}\left(\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}f^{i}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}f^{i}\partial_{\beta\dot{\beta}}\bar{f}_{i}-\partial_{\beta\dot{\beta}}f^{i}\partial_{\alpha\dot{\alpha}}\bar{f}_{i}+f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\bar{f}_{i}\right),\end{split}$
(6.70)

with

$Y_{\alpha\beta\dot{\alpha}\dot{\beta}}=-\frac{m}{4}\left(\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}f^{i}\bar{f}_{i}-\partial_{\alpha\dot{\alpha}}f^{i}\partial_{\beta\dot{\beta}}\bar{f}_{i}-\partial_{\beta\dot{\beta}}f^{i}\partial_{\alpha\dot{\alpha}}\bar{f}_{i}+f^{i}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\bar{f}_{i}\right).$
(6.71)

This current coincides with the conserved current (5.131) (modulo the mass
coefficient) entering the current superfield (5.130).

Spin $5/2$ sector

The analysis of the spinor sector can be performed in exactly the same way and
here we do not give the relevant details. The interaction has also the generic
Noether form (5.68):

$S_{int}^{(s=3)}=\kappa_{3}\int
d^{4}x\,\left(\psi^{(\alpha\beta)(\dot{\alpha}\dot{\beta})\gamma
i}Y_{(\alpha\beta)(\dot{\alpha}\dot{\beta})\gamma
i}+\bar{\psi}^{(\alpha\beta)(\dot{\alpha}\dot{\beta})\dot{\gamma}i}\bar{Y}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})\dot{\gamma}i}\right),$
(6.72)

where the corresponding fermionic spin $\tfrac{5}{2}$ $Y$-current (in massless
case), up to numerical factor, coincides with the conserved spin
$\tfrac{5}{2}$ $\mathcal{S}$-current, defined in (5.141):

$\begin{split}Y^{i}_{(\alpha\beta)(\dot{\alpha}\dot{\rho})\rho}=-\frac{1}{4}\mathcal{S}^{i}_{(\alpha\beta)(\dot{\alpha}\dot{\rho})\rho}=-\frac{i}{4}\chi_{\rho}\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\rho}}\bar{f}^{i}-\frac{i}{4}\psi_{\rho}\partial_{(\alpha\dot{\alpha}}\partial_{\beta)\dot{\rho}}f^{i}.\end{split}$
(6.73)

The overall coefficient is fixed by inspecting the contribution of symmetric
part of the spin $\tfrac{5}{2}$ gauge field
$\psi^{(\alpha\beta\gamma)(\dot{\alpha}\dot{\beta})i}$. It solely comes from
the analytic potential $h^{++(\alpha\beta)(\dot{\alpha}\dot{\beta})}$.

Other spin $\tfrac{5}{2}$ currents defined in Section 5.8 differ by terms
which are conserved automatically, without use of the equations of motion, and
by terms which are derivatives of the conserved spin $\tfrac{3}{2}$ currents.
This type of terms results in fake cubic $(\tfrac{5}{2},\tfrac{1}{2},0)$
vertices which can be eliminated by tricks much similar to those employed in
Appendix E for bosonic vertices. We do not give here any details.

##  7 Summary and outlook

Here we briefly summarize the basic results and discuss some further problems.

  1. 1.

In Section 4, we have derived superfield equations of motion for an arbitrary
$\mathcal{N}=2$ spin $\mathbf{s}$. They have the universal form:

$\begin{cases}(D^{+})^{2}\bar{D}^{+\dot{\beta}}G^{--}_{\alpha(s-1)(\dot{\beta}\dot{\alpha}(s-2))}+(\bar{D}^{+})^{2}D^{+}_{(\alpha}G^{--}_{\alpha(s-2))\dot{\alpha}(s-2)}=0\,,\\\
(\bar{D}^{+})^{2}D^{+\beta}G^{--}_{(\beta\alpha(s-2))\dot{\alpha}(s-1)}-(D^{+})^{2}\bar{D}^{+}_{(\dot{\alpha}}G^{--}_{\alpha(s-2)\dot{\alpha}(s-2))}=0\,.\end{cases}$
(7.1)

We have derived these equations by varying with respect either to the analytic
potentials $h^{++\dots}(\zeta)$ or to unconstrained Mezincescu-type
prepotentials $\Psi^{-\dots}(\zeta,\theta^{-})$. These prepotentials are
defined as191919Here we assume the complete symmetry between dotted and
undotted spinor indices.

$\begin{split}\hat{\mathcal{H}}^{++}_{(s)}=&\;h^{++\alpha(s-2)\dot{\alpha}(s-2)M}\partial_{M}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\\\
:=&\;(D^{+})^{4}\left(\Psi^{-\alpha(s-2)\dot{\alpha}(s-2)\hat{\mu}}D^{-}_{\hat{\mu}}\right)\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\end{split}$
(7.2)

and provide alternative representations for $\mathcal{N}=2$ integer spin
$\mathbf{s}$ supermultiplets. We have also tracked in detail how these
superfield equations yield the ordinary equations of motion in the bosonic
sector.

  2. 2.

In the presence of cubic couplings (5.31) of hypermultiplet to $\mathcal{N}=2$
spin $\mathbf{s}$ supermultiplets the equations (7.1) acquire the superfield
sources constructed as bi-linear combinations of the hypermultiplet
superfield:

$\begin{cases}(D^{+})^{2}\bar{D}^{+\dot{\beta}}G^{--}_{\alpha(s-1)(\dot{\beta}\dot{\alpha}(s-2))}+(\bar{D}^{+})^{2}D^{+}_{(\alpha}G^{--}_{\alpha(s-2))\dot{\alpha}(s-2)}=i\kappa_{s}(-1)^{s}\mathcal{J}^{+}_{\dot{\alpha}(s-2)\alpha(s-1)}\,,\\\
(\bar{D}^{+})^{2}D^{+\beta}G^{--}_{(\beta\alpha(s-2))\dot{\alpha}(s-1)}-(D^{+})^{2}\bar{D}^{+}_{(\dot{\alpha}}G^{--}_{\alpha(s-2)\dot{\alpha}(s-2))}=i\kappa_{s}(-1)^{s}\mathcal{J}^{+}_{\alpha(s-2)\dot{\alpha}(s-1)}\,.\end{cases}$
(7.3)

The conserved current superfieds $\mathcal{J}^{+}_{\dots}$ are defined as

$\mathcal{J}^{+}_{\alpha(s-2)\dot{\alpha}(s-2)\hat{\mu}}=-\frac{1}{2}q^{+a}D^{-}_{\hat{\mu}}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}(J)^{P(s)}q^{+}_{a}-\frac{1}{2}\xi
D^{-}_{\hat{\mu}}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\left[q^{+a}(J)^{P(s)}q^{+}_{a}\right].$
(7.4)

We have analyzed the bosonic component structure of these currents and found
the explicit expressions for the components in terms of the on-shell
hypermultiplet fields.

  3. 3.

In Section 6 we have derived the equations of motion for the hypermultiplet
coupled to $\mathcal{N}=2$ spin $\mathbf{s}$ gauge supermultiplet. We analyzed
the component contents of the spin $\mathbf{2}$ and spin $\mathbf{3}$ models
in the bosonic sectors and found the full agreement with the standard results
for the corresponding cubic vertices. In particular, we demonstrated that on
shell and after proper field redefinitions, one can eliminate the
$\xi$-interactions which are present in the cubic vertices involving odd spin
${\cal N}=2$ gauge superfields. It is still an open question whether it is
possible to eliminate such $\xi$ interaction at the full off-shell superfield
level. We have explicitly presented, in the leading order in the coupling
constants, the on-shell component form of the superfield cubic vertices
involving the spin ${\bf 2}$ and ${\bf 3}$ gauge supermultiplets.

Some further possible extensions of the results obtained can be summarized as
follows:

  * •

One of the urgent problems is a generalization to the superconformal case. It
is likely that in the superconformal case the relevant structures cannot
significantly differ from the non-conformal case, as implied by
$\mathcal{N}=2$ conformal supergravity in the HSS approach. Indeed,
$\mathcal{N}=2$ Weyl multiplet Galperin:1987ek  is described in a way very
similar to the formulation of $\mathcal{N}=2$ Einstein supergravity
Galperin:1987em ; one just needs to further covariantize the analyticity-
preserving harmonic derivative by an extra vielbein associated with local
${\rm SU}(2)_{aut}$ symmetry. The concrete generalization of such a picture to
higher spins will of course require some efforts. Once this is done, a
generalization of $\mathcal{N}=2$ higher spin theory to superconformally flat
backgrounds, including the most interesting (A)dSd option, will hopefully be
performed rather straightforwardly.

  * •

The task of generalizing the results obtained to other non-trivial
supergravity backgrounds is indeed of significant interest. In particular,
such theories in (A)dSd could display a number of features characteristic of
the fully consistent higher spin theories. How these structures are changed
and constrained in the presence of $\mathcal{N}=2$ supersymmetry is an
important open question.

It is worth noting that such theories have already been discussed in ref.
Kuzenko:2021pqm  (see also Buchbinder:2018nkp  and a recent paper KPR ),
without use of the harmonic superspace approach. These theories seemingly
correspond to the choice of special gauge for the analytic potentials. We hope
to address these questions elsewhere 202020Let us point out once more that in
all these papers the current superfields were constructed within the on-shell
description of the hypermultiplet in terms of the properly constrained
conventional ${\cal N}=2$ (or ${\cal N}=1$) superfields. The basic distinction
of our approach is that we use the off-shell description of both higher spin
gauge ${\cal N}=2$ multiplets and the matter hypermultiplets in terms of
unconstrained harmonic analytic superfields..

  * •

Quartic interactions (see, e.g. Karapetyan:2021wdc  and the references
therein) also constitute a very interesting area for application of the
strategy adopted here and in our previous work Buchbinder:2022kzl . This
circle of problems would involve, in particular, finding out nonlinear
corrections to the linearized higher spin $\mathcal{N}=2$ superfield actions.
Note that the question of the structure of quartic higher spin interactions
may be closely related to the generic problem of locality in higher spin
theory (see a recent discussion of this issue in Vasiliev , Didenko1 ).

###### Acknowledgements.

We thank X. Bekaert, V.A. Krykhtin, T.V. Snegirev, M. Tsulaia and Y.M.
Zinoviev for useful discussions and S.M. Kuzenko for valuable comments. Work
of I.B. was supported in part by the Ministry of Education of the Russian
Federation, project QZOY-2023-0003. Work of N.Z. was partially supported by
the grant 22-1-1-42-2 from the Foundation for the Advancement of Theoretical
Physics and Mathematics “BASIS”.

##  Appendix A Component reduction of bosonic gauge sector

In this Appendix we present details of the component structure of the theories
of ${\cal N}=2$ spins $\mathbf{2}$ and $\mathbf{3}$. We will use the results
of our previous work Buchbinder:2021ite .

###  A.1 ${\cal N}=2$ spin $\mathbf{2}$ theory: bosonic sector

We start with collecting all the necessary details related to the bosonic
sector of the spin $\mathbf{2}$ ${\cal N}=2$ theory. It consists of the spin 2
and spin 1 sectors. Particular attention is paid to the role of auxiliary
fields in deriving the correct component equations from the superfield
equations.

Spin 2 sector

The bosonic spin 2 sector involves three relevant fields: the gauge spin 2
field $\Phi^{\alpha\dot{\alpha}}_{\beta\dot{\beta}}$ and auxiliary fields
$P_{\alpha\dot{\alpha}}$, $\bar{P}_{\alpha\dot{\alpha}}$. In the spinor
notation, the irreducible parts of the spin 2 field are defined by the
decomposition:

$\Phi_{\alpha\beta\dot{\alpha}\dot{\beta}}=\Phi_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}+\epsilon_{\alpha\beta}\epsilon_{\dot{\alpha}\dot{\beta}}\Phi\,.$
(A.1)

The spin 2 gauge transformations with parameters $a_{\alpha\dot{\alpha}}$ have
the form:

$\delta\Phi_{\alpha\beta\dot{\alpha}\dot{\beta}}=\frac{1}{2}\left(\partial_{\alpha\dot{\alpha}}a_{\beta\dot{\beta}}+\partial_{\beta\dot{\beta}}a_{\alpha\dot{\alpha}}\right)\,\qquad\Leftrightarrow\qquad\begin{cases}\delta\Phi_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}=\partial_{(\alpha(\dot{\alpha}}a_{\beta)\dot{\beta})}\,,\\\
\delta\Phi=\frac{1}{4}\partial_{\alpha\dot{\alpha}}a^{\alpha\dot{\alpha}}\,.\end{cases}$
(A.2)

These transformations define the residual bosonic symmetries of the Wess-
Zumino gauge (3.13) and fully agree with the spin 2 gauge transformations in
Fronsdal theory.

The $\Phi,P$ sector of the superfield $G^{++\alpha\dot{\alpha}}$ is as
follows:

$G^{++\alpha\dot{\alpha}}_{(\Phi,P)}=-2i\theta^{+\beta}\bar{\theta}^{+\dot{\beta}}\Phi_{\beta\dot{\beta}}^{\alpha\dot{\alpha}}+4(\theta^{+})^{2}\bar{\theta}^{+\dot{\beta}}\bar{\theta}^{-\dot{\alpha}}P_{\dot{\beta}}^{\alpha}-4(\bar{\theta}^{+})^{2}\theta^{+\beta}\theta^{-\alpha}\bar{P}_{\beta}^{\dot{\alpha}}\,.$
(A.3)

The corresponding solution of the zero-curvature equation (3.4) is given by:

$\begin{split}G^{--\alpha\dot{\alpha}}_{(\Phi,P)}=&-2i\theta^{-\beta}\bar{\theta}^{-\dot{\beta}}\Phi_{\beta\dot{\beta}}^{\alpha\dot{\alpha}}\\\
&+2(\theta^{-})^{2}\bar{\theta}^{-(\dot{\rho}}\bar{\theta}^{+\dot{\beta})}\partial_{\dot{\rho}}^{\beta}\Phi_{\beta\dot{\beta}}^{\alpha\dot{\alpha}}-2(\bar{\theta}^{-})^{2}\theta^{-(\rho}\bar{\theta}^{+\beta)}\partial_{\rho}^{\dot{\beta}}\Phi_{\beta\dot{\beta}}^{\alpha\dot{\alpha}}\\\
&+2i(\theta^{-})^{2}(\bar{\theta}^{-})^{2}\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}\left(\Box\Phi^{\alpha\dot{\alpha}}_{\rho\dot{\rho}}-\frac{1}{2}\partial_{\rho\dot{\rho}}\partial^{\beta\dot{\beta}}\Phi_{\beta\dot{\beta}}^{\alpha\dot{\alpha}}\right)\\\
&+4(\theta^{-})^{2}\bar{\theta}^{+(\dot{\alpha}}\bar{\theta}^{-\dot{\beta})}P_{\dot{\beta}}^{\alpha}-2(\theta^{-})^{2}(\bar{\theta}^{+}\bar{\theta}^{-})\bar{P}^{\alpha\dot{\alpha}}\\\
&-4(\bar{\theta}^{-})^{2}\theta^{+(\alpha}\theta^{-\beta)}\bar{P}_{\beta}^{\dot{\alpha}}-2(\bar{\theta}^{-})^{2}(\theta^{+}\theta^{-})P^{\alpha\dot{\alpha}}\\\
&+4i(\theta^{-})^{2}(\bar{\theta}^{-})^{2}\theta^{+\rho}\bar{\theta}^{+(\dot{\alpha}}\partial^{\dot{\rho})}_{\rho}P^{\alpha}_{\dot{\rho}}-4i(\theta^{-})^{2}(\bar{\theta}^{-})^{2}\bar{\theta}^{+\dot{\rho}}\theta^{+(\alpha}\partial^{\rho)}_{\dot{\rho}}\bar{P}^{\dot{\alpha}}_{\rho}\,.\end{split}$
(A.4)

Passing to the auxiliary fields
$\mathcal{P}_{\mu\dot{\mu}},\bar{\mathcal{P}}_{\mu\dot{\mu}}$ which are
invariant under the gauge transformation (A.2) is accomplished by the
following shifts Buchbinder:2021ite

$P_{\mu\dot{\mu}}=B_{\mu\dot{\mu}}+\mathcal{P}_{\mu\dot{\mu}}\,,$ (A.5)
$\bar{P}_{\mu\dot{\mu}}=B_{\mu\dot{\mu}}+\bar{\mathcal{P}}_{\mu\dot{\mu}}\,,$
(A.6)

where

$B_{\mu\dot{\mu}}=\frac{1}{4}\left(3\partial_{\mu\dot{\mu}}\Phi-\partial^{\rho\dot{\rho}}\Phi_{(\mu\rho)(\dot{\mu}\dot{\rho})}\right)=\frac{1}{4}\left(\partial_{\mu\dot{\mu}}\Phi-\partial^{\rho\dot{\rho}}\Phi_{\mu\rho\dot{\mu}\dot{\rho}}\right)$
(A.7)

is the spin 2-dependent part. Using these expressions, one can express the
spin 2 part of the coefficient of the monomial
$2i(\theta^{-})^{2}(\bar{\theta}^{-})^{2}$ in (A.4) through the linearized
Einstein tensor $\mathcal{G}_{\rho\dot{\rho}}^{\alpha\dot{\alpha}}$:

$\displaystyle\left(\Box\Phi^{\alpha\dot{\alpha}}_{\rho\dot{\rho}}-\frac{1}{2}\partial_{\rho\dot{\rho}}\partial^{\beta\dot{\beta}}\Phi_{\beta\dot{\beta}}^{\alpha\dot{\alpha}}\right)+\partial^{\dot{\alpha}}_{\rho}B_{\dot{\rho}}^{\alpha}+\partial^{\alpha}_{\dot{\rho}}B_{\rho}^{\dot{\alpha}}+\delta^{\dot{\alpha}}_{\dot{\rho}}\partial^{\dot{\mu}}_{\rho}B_{\dot{\mu}}^{\alpha}+\delta^{\alpha}_{\rho}\partial^{\mu}_{\dot{\rho}}B^{\dot{\alpha}}_{\mu}=-2\mathcal{G}_{\rho\dot{\rho}}^{\alpha\dot{\alpha}}\,.$
(A.8)

Here

$\mathcal{G}_{\alpha\beta\dot{\alpha}\dot{\beta}}=\mathcal{R}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}-\varepsilon_{\alpha\beta}\varepsilon_{\dot{\alpha}\dot{\beta}}\mathcal{R}\,,$
(A.9)

and

$\mathcal{R}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}=\frac{1}{2}\partial_{(\alpha(\dot{\alpha}}\partial^{\sigma\dot{\sigma}}\Phi_{\beta)\sigma\dot{\beta})\dot{\sigma}}-\frac{1}{2}\Box\Phi_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}-\frac{1}{2}\partial_{(\alpha(\dot{\alpha}}\partial_{\beta)\dot{\beta})}\Phi\,,$
(A.10)
$\mathcal{R}=\frac{1}{8}\partial^{\alpha\dot{\alpha}}\partial^{\beta\dot{\beta}}\Phi_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}-\frac{3}{4}\Box\Phi\,.$
(A.11)

In the first term of the r.h.s. of eq. (A.10), to simplify the notation, we
omitted symmetrization of indices in
$\Phi_{(\beta\sigma)(\dot{\beta}\dot{\sigma})}$. All these objects are
linearized tensors which are invariant with respect to the gauge
transformations (A.2). It is useful to note that the linearized Einstein
tensor (A.9) by construction satisfies the transversality condition:

$\partial^{\alpha\dot{\alpha}}\mathcal{G}_{\alpha\beta\dot{\alpha}\dot{\beta}}=0\,.$
(A.12)

With the above relations at hand, the complete expression for the highest
component of $G^{--\alpha\dot{\alpha}}_{(\Phi,P)}$ in (A.4) reads:

$G^{--\alpha\dot{\alpha}}_{(\Phi,P)}=\dots+4i(\theta^{-})^{2}(\bar{\theta}^{-})^{2}\left[-\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}\mathcal{G}^{\alpha\dot{\alpha}}_{\rho\dot{\rho}}+\theta^{+\rho}\bar{\theta}^{+(\dot{\alpha}}\partial^{\dot{\rho})}_{\rho}\mathcal{P}^{\alpha}_{\dot{\rho}}-\bar{\theta}^{+\dot{\rho}}\theta^{+(\alpha}\partial^{\rho)}_{\dot{\rho}}\bar{\mathcal{P}}^{\dot{\alpha}}_{\rho}\right].$
(A.13)

We used this result in the analysis of the superfield equations of motion in
Sections 4.3.1 and 5.7. It is worth recalling that this leading component is a
linearized gauge invariant.

One more object in the spin 2 sector is the superfield $G^{++5}$:

$G_{(\Phi,P)}^{++5}=-4(\theta^{+})^{2}\bar{\theta}^{+\dot{\rho}}\theta^{-}_{\mu}P^{\mu}_{\dot{\rho}}-4(\bar{\theta}^{+})^{2}\theta^{+\mu}\bar{\theta}^{-}_{\dot{\rho}}\bar{P}_{\mu}^{\dot{\rho}}\,.$
(A.14)

The corresponding solution of the zero-curvature equation (3.5) reads:

$\begin{split}G_{(\Phi,P)}^{--5}&=4(\theta^{-})^{2}\bar{\theta}^{-\dot{\rho}}\theta^{+}_{\mu}P^{\mu}_{\dot{\rho}}+2i(\theta^{+})^{2}(\theta^{-})^{2}(\bar{\theta}^{-})^{2}\partial_{\rho\dot{\rho}}P^{\rho\dot{\rho}}\\\
&\,+4(\bar{\theta}^{-})^{2}\theta^{-\mu}\bar{\theta}^{+}_{\dot{\rho}}\bar{P}_{\mu}^{\dot{\dot{\rho}}}-2i(\bar{\theta}^{+})^{2}(\theta^{-})^{2}(\bar{\theta}^{-})^{2}\partial_{\rho\dot{\rho}}\bar{P}^{\rho\dot{\rho}}\,.\end{split}$
(A.15)

Using (A.3), (A.14) and the zero-curvature solutions (A.4), (A.15)), we deduce
from the general superfield action (3.1) the following spin $\mathbf{2}$
component action:

$\begin{split}S^{(s=2)}_{(\Phi,P)}=-\int
d^{4}x\;\Bigr{[}&\Phi^{\alpha\beta\dot{\alpha}\dot{\beta}}\left(\Box\Phi_{\alpha\beta\dot{\alpha}\dot{\beta}}-\frac{1}{2}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\Phi\right)\\\
&+8(P^{\alpha\dot{\alpha}}+\bar{P}^{\alpha\dot{\alpha}})B_{\alpha\dot{\alpha}}-6P^{\alpha\dot{\alpha}}P_{\alpha\dot{\alpha}}-6\bar{P}^{\alpha\dot{\alpha}}\bar{P}_{\alpha\dot{\alpha}}+4P^{\alpha\dot{\alpha}}\bar{P}_{\alpha\dot{\alpha}}\Bigr{]}.\end{split}$
(A.16)

Now, singling out the gravity-dependent parts from $P$-fields as in (A.5) and
(A.6), one can expose the pure spin $2$ part of this Lagrangian. In
particular, using the expression (A.7) one finds

$8\int d^{4}x\;B^{\alpha\dot{\alpha}}B_{\alpha\dot{\alpha}}=16\int
d^{4}x\;\Phi\mathcal{R}+\int
d^{4}x\;\Phi^{\alpha\beta\dot{\alpha}\dot{\beta}}\left(\frac{3}{2}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\Phi-\frac{1}{2}\partial_{\alpha\dot{\alpha}}\partial^{\rho\dot{\rho}}\Phi_{(\rho\beta)(\dot{\rho}\dot{\beta})}\right).$
(A.17)

The final form of the action is as follows:

$\begin{split}S^{(s=2)}_{(\Phi,P)}=\int
d^{4}x\;\Bigr{[}&2\Phi^{\alpha\beta\dot{\alpha}\dot{\beta}}\mathcal{G}_{\alpha\beta\dot{\alpha}\dot{\beta}}+6\mathcal{P}^{\alpha\dot{\alpha}}\mathcal{P}_{\alpha\dot{\alpha}}+6\bar{\mathcal{P}}^{\alpha\dot{\alpha}}\bar{\mathcal{P}}_{\alpha\dot{\alpha}}-4\mathcal{P}^{\alpha\dot{\alpha}}\bar{\mathcal{P}}_{\alpha\dot{\alpha}}\Bigr{]}.\end{split}$
(A.18)

Here, the first term is just the linearized action of the spin 2 field (the
Pauli-Fierz action). In the free spin $\mathbf{2}$ theory, the algebraic
equations of motion for the auxiliary $\mathcal{P}$-fields are trivial:

$\mathcal{P}_{\alpha\dot{\alpha}}=\bar{\mathcal{P}}_{\alpha\dot{\alpha}}=0\,,$
(A.19)

and after elimination of these fields we are left with the linearized spin 2
theory:

$\begin{split}S^{(s=2)}_{(\Phi)}&=2\int
d^{4}x\;\Phi^{\alpha\beta\dot{\alpha}\dot{\beta}}\mathcal{G}_{\alpha\beta\dot{\alpha}\dot{\beta}}\\\
&=-\int
d^{4}x\;\Big{[}\Phi^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\Box\Phi_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}-\Phi^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\partial_{\alpha\dot{\alpha}}\partial^{\rho\dot{\rho}}\Phi_{(\rho\beta)(\dot{\rho}\dot{\beta})}\\\
&\;\;\;\;\;\;\;\;\;\;\;\qquad\quad+2\,\Phi\partial^{\alpha\dot{\alpha}}\partial^{\beta\dot{\beta}}\Phi_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}-6\Phi\Box\Phi\Big{]}.\end{split}$
(A.20)

In the theory including an interaction with matter the auxiliary fields will
no longer vanish and play an important role in forming the final matter-
modified gauge field action. We discuss this in detail below.

Spin 1 sector

The bosonic spin 1 sector of ${\cal N}=2$ spin $\mathbf{2}$ theory involves
three relevant fields: the gauge field $C_{\alpha\dot{\alpha}}$ and the
auxiliary fields $T^{(\alpha\beta)}$, $T^{(\dot{\alpha}\dot{\beta})}$. The
gauge transformations of $C_{\alpha\dot{\alpha}}$ have the standard form:

$\delta C_{\alpha\dot{\alpha}}=\partial_{\alpha\dot{\alpha}}b\,.$ (A.21)

The fields $C$ and $T$ make the following contributions to the superfields
$G^{++\alpha\dot{\alpha}}$ and $G^{++5}$:

$G^{++\alpha\dot{\alpha}}_{(C,T)}=2i(\bar{\theta}^{+})^{2}\theta^{+}_{\nu}\bar{\theta}^{-\dot{\alpha}}T^{(\alpha\nu)}-2i(\theta^{+})^{2}\bar{\theta}^{+}_{\dot{\nu}}\theta^{-\alpha}\bar{T}^{(\dot{\alpha}\dot{\nu})}\,,$
(A.22)
$G^{++5}_{(C,T)}=-2i\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}C_{\rho\dot{\rho}}-2i(\bar{\theta}^{+})^{2}\theta^{+}_{\nu}\theta^{-}_{\alpha}T^{(\alpha\nu)}-2i(\theta^{+})^{2}\bar{\theta}^{+}_{\dot{\nu}}\bar{\theta}^{-}_{\dot{\alpha}}\bar{T}^{(\dot{\alpha}\dot{\nu})}\,.$
(A.23)

The corresponding solutions of the zero-curvature equations (3.4) and (3.5)
read:

$\begin{split}G^{--\alpha\dot{\alpha}}_{(C,T)}=&-2i(\bar{\theta}^{-})^{2}\theta^{-}_{\nu}\bar{\theta}^{+\dot{\alpha}}T^{(\alpha\nu)}+2i(\theta^{-})^{2}\bar{\theta}^{-}_{\dot{\nu}}\theta^{+\alpha}\bar{T}^{(\dot{\alpha}\dot{\nu})}\\\
&+(\bar{\theta}^{+})^{2}(\bar{\theta}^{-})^{2}(\theta^{-})^{2}\partial^{\dot{\alpha}}_{\nu}T^{(\alpha\nu)}-(\theta^{+})^{2}(\theta^{-})^{2}(\bar{\theta}^{-})^{2}\partial^{\alpha}_{\dot{\nu}}\bar{T}^{(\dot{\alpha}\dot{\nu})}\,,\end{split}$
(A.24)

$\begin{split}G^{--5}_{(C,T)}=&-2i\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}C_{\rho\dot{\rho}}\\\
&+2(\theta^{-})^{2}\bar{\theta}^{-(\dot{\rho}}\bar{\theta}^{+\dot{\beta})}\partial_{\dot{\rho}}^{\beta}C_{\beta\dot{\beta}}-2(\bar{\theta}^{-})^{2}\theta^{-(\rho}\theta^{+\beta)}\partial_{\rho}^{\dot{\beta}}C_{\beta\dot{\beta}}\\\
&+2i(\theta^{-})^{2}(\bar{\theta}^{-})^{2}\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}\left[\Box
C_{\rho\dot{\rho}}-\frac{1}{2}\partial_{\rho\dot{\rho}}\partial^{\beta\dot{\beta}}C_{\beta\dot{\beta}}\right]\\\
&-2i(\bar{\theta}^{-})^{2}\theta^{+}_{\nu}\theta^{-}_{\alpha}T^{(\alpha\nu)}-2i(\theta^{-})^{2}\bar{\theta}^{+}_{\dot{\nu}}\bar{\theta}^{-}_{\dot{\alpha}}\bar{T}^{(\dot{\alpha}\dot{\nu})}\\\
&+2(\theta^{-})^{2}(\bar{\theta}^{-})^{2}\bar{\theta}^{+\dot{\rho}}\theta^{+}_{\mu}\partial_{\rho\dot{\rho}}T^{(\mu\rho)}+2(\theta^{-})^{2}(\bar{\theta}^{-})^{2}\theta^{+\rho}\bar{\theta}^{+}_{\dot{\mu}}\partial_{\rho\dot{\rho}}\bar{T}^{(\dot{\mu}\dot{\rho})}\,.\end{split}$
(A.25)

The spin 1 sector of the total action is expressed as:

$\begin{split}S^{(s=2)}_{(C,T)}=-\int
d^{4}x\;\Bigr{[}&C^{\rho\dot{\rho}}\left(\Box
C_{\rho\dot{\rho}}-\frac{1}{2}\partial_{\rho\dot{\rho}}\partial^{\beta\dot{\beta}}C_{\beta\dot{\beta}}\right)\\\
&+2iC^{\dot{\rho}}_{\mu}\partial_{\rho\dot{\rho}}T^{(\mu\rho)}-2iC^{\rho}_{\dot{\mu}}\partial_{\rho\dot{\rho}}\bar{T}^{(\dot{\mu}\dot{\rho})}-T^{(\mu\nu)}T_{(\mu\nu)}-\bar{T}^{(\dot{\mu}\dot{\nu})}\bar{T}_{(\dot{\mu}\dot{\nu})}\Bigr{]}.\end{split}$
(A.26)

The equations of motion for the auxiliary fields yield:

$T_{(\mu\nu)}=-i\partial_{(\mu\dot{\rho}}C^{\dot{\rho}}_{\nu)}\,,\qquad\qquad\bar{T}_{(\dot{\mu}\dot{\nu})}=i\partial_{(\dot{\mu}\rho}C^{\rho}_{\dot{\nu})}\,.$
(A.27)

After elimination of the auxiliary fields we reproduce the standard Maxwell
action in the spinor formalism:

$\begin{split}S^{(s=2)}_{(C)}&=\int d^{4}x\;C^{\rho\dot{\rho}}\left(\Box
C_{\rho\dot{\rho}}-\frac{1}{2}\partial_{\rho\dot{\rho}}\partial^{\beta\dot{\beta}}C_{\beta\dot{\beta}}\right)\\\
&=-\frac{1}{2}\int
d^{4}x\;\partial^{\beta\dot{\beta}}C^{\rho\dot{\rho}}\left(\partial_{\beta\dot{\beta}}C_{\rho\dot{\rho}}-\partial_{\rho\dot{\rho}}C_{\beta\dot{\beta}}\right).\end{split}$
(A.28)

We see that the role of the tensorial auxiliary fields
$T_{(\mu\nu)},\bar{T}_{(\dot{\mu}\dot{\nu})}$ in the present case is just to
ensure the correct sign of the final spin ${\bf 1}$ gauge action.

###  A.2 ${\cal N}=2$ spin $\mathbf{3}$ theory: bosonic sector

Here we perform the corresponding bosonic component reduction of the spin
$\mathbf{3}$ ${\cal N}=2$ theory Buchbinder:2021ite . As distinct from the
spin $\mathbf{2}$ case, here we do not discuss the auxiliary fields
$\mathcal{P}^{(\alpha\beta)\dot{\alpha}\dot{\beta}}$,
$\bar{\mathcal{P}}^{(\dot{\alpha}\dot{\beta})\alpha\beta}$ since they give a
trivial contribution in the free spin $\mathbf{3}$ theory. Like in the spin
$\mathbf{2}$ case, they still play an essential role in the interacting
theory, when passing to the superfield equations with sources. The bosonic
gauge field sector of the ${\cal N}=2$ spin $\mathbf{3}$ theory involves the
spin 3 and spin 2 fields.

Spin 3 sector

The gauge fields of the spin 3 sector are collected in:

$\Phi_{(\alpha\beta)\gamma(\dot{\alpha}\dot{\beta})\dot{\gamma}}=\Phi_{(\alpha\beta\gamma)(\dot{\alpha}\dot{\beta}\dot{\gamma})}+\Phi_{(\alpha(\dot{\alpha}}\epsilon_{\beta)\gamma}\epsilon_{\dot{\beta})\dot{\gamma}}\,.$
(A.29)

They are subject to gauge transformations with the gauge parameters
$a_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}$:

$\delta\Phi_{(\alpha\beta\gamma)(\dot{\alpha}\dot{\beta}\dot{\gamma})}=\partial_{(\alpha(\dot{\alpha}}a_{\beta\gamma)\dot{\beta}\dot{\gamma})}\,,\qquad\qquad\delta\Phi_{\alpha\dot{\alpha}}=\frac{4}{9}\partial^{\beta\dot{\beta}}a_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\,.$
(A.30)

For the component contents of relevant $G^{++\dots}$-superfields we have:

$\begin{split}G_{(\Phi,B)}^{++(\alpha\beta)(\dot{\alpha}\dot{\beta})}=&-2i\theta^{+\gamma}\bar{\theta}^{+\dot{\gamma}}\Phi_{\gamma\dot{\gamma}}^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\\\
&+4(\theta^{+})^{2}\bar{\theta}^{+\dot{\gamma}}\bar{\theta}^{-(\dot{\alpha}}B^{\dot{\beta})(\alpha\beta)}_{\dot{\gamma}}-\,4(\bar{\theta}^{+})^{2}\theta^{+\gamma}\theta^{-(\alpha}\bar{B}^{\beta)(\dot{\alpha}\dot{\beta})}_{\gamma}\,,\end{split}$
(A.31)
$G_{(\Phi,B)}^{++5\alpha\dot{\alpha}}=-2(\theta^{+})^{2}\bar{\theta}^{+\dot{\rho}}\theta^{-}_{\mu}B^{(\mu\alpha)\dot{\alpha}}_{\dot{\rho}}-2(\bar{\theta}^{+})^{2}\theta^{+\beta}\bar{\theta}^{-}_{\dot{\rho}}\bar{B}^{\alpha(\dot{\rho}\dot{\alpha})}_{\beta}\,.$
(A.32)

Here we employed the notations:

$B_{(\alpha\beta)\dot{\alpha}\dot{\beta}}=-\frac{1}{2}\left[\partial^{\gamma\dot{\gamma}}\Phi_{(\alpha\beta\gamma)(\dot{\alpha}\dot{\beta}\dot{\gamma})}-\partial_{(\alpha(\dot{\alpha}}\Phi_{\beta)\dot{\beta})}-\partial_{(\alpha\dot{\alpha}}\Phi_{\beta)\dot{\beta}}\right],$
(A.33)
$B_{\alpha\beta(\dot{\alpha}\dot{\beta})}=-\frac{1}{2}\left[\partial^{\gamma\dot{\gamma}}\Phi_{(\alpha\beta\gamma)(\dot{\alpha}\dot{\beta}\dot{\gamma})}-\partial_{(\alpha(\dot{\alpha}}\Phi_{\beta)\dot{\beta})}-\partial_{\alpha(\dot{\alpha}}\Phi_{\beta\dot{\beta})}\right].$
(A.34)

Note useful identities for the derivatives of (A.33) and (A.34):

$\partial_{\rho\dot{\rho}}B^{(\rho\alpha)\dot{\rho}\dot{\alpha}}=-\frac{1}{2}\mathcal{R}^{\alpha\dot{\alpha}}\,,\qquad\partial_{\rho\dot{\rho}}\bar{B}^{\rho\alpha(\dot{\rho\dot{\alpha}})}=-\frac{1}{2}\mathcal{R}^{\alpha\dot{\alpha}}\,.$
(A.35)

The corresponding solutions of the zero-curvature equations (3.4) and (3.5)
read:

$\begin{split}G_{(\Phi,B)}^{--(\alpha\beta)(\dot{\alpha}\dot{\beta})}=&-2i\theta^{-\beta}\bar{\theta}^{-\dot{\beta}}\Phi_{\beta\dot{\beta}}^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\\\
&+2(\theta^{-})^{2}\bar{\theta}^{-(\dot{\rho}}\bar{\theta}^{+\dot{\beta})}\partial^{\beta}_{\dot{\rho}}\Phi_{\beta\dot{\beta}}^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}-2(\bar{\theta}^{-})^{2}\theta^{-(\rho}\theta^{+\beta)}\partial_{\rho}^{\dot{\beta}}\Phi_{\beta\dot{\beta}}^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\\\
&+\,4(\theta^{-})^{2}\bar{\theta}^{+(\dot{\alpha}}\bar{\theta}^{-\dot{\rho}}B_{\dot{\rho}}^{\dot{\beta})(\alpha\beta)}-4(\bar{\theta}^{-})^{2}\theta^{+\rho}\theta^{-(\alpha}\bar{B}^{\beta)(\dot{\alpha}\dot{\beta})}_{\rho}\\\
&-3i(\theta^{-})^{2}(\bar{\theta}^{-})^{2}\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}\mathcal{G}_{\rho\dot{\rho}}^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\,,\end{split}$
(A.36)
$\begin{split}G_{(\Phi,B)}^{--5\alpha\dot{\alpha}}=&2(\theta^{-})^{2}\bar{\theta}^{-\dot{\rho}}\theta^{+}_{\mu}B^{(\mu\alpha)\dot{\alpha}}_{\dot{\rho}}+i(\theta^{+})^{2}(\bar{\theta}^{-})^{2}(\theta^{-})^{2}\partial_{\rho\dot{\rho}}B^{(\rho\alpha)\dot{\rho}\dot{\alpha}}\\\
&+2\,(\bar{\theta}^{-})^{2}\theta^{-\beta}\bar{\theta}^{+}_{\dot{\rho}}\bar{B}_{\beta}^{\alpha(\dot{\alpha}\dot{\rho})}-i(\bar{\theta}^{+})^{2}(\bar{\theta}^{-})^{2}(\theta^{-})^{2}\partial_{\rho\dot{\rho}}\bar{B}^{\rho\alpha(\dot{\rho}\dot{\alpha})}\,.\end{split}$
(A.37)

In eq. (A.36) the following expression for the linearized Fronsdal tensor was
used:

$\displaystyle\mathcal{G}_{(\alpha_{1}\alpha_{2})\alpha_{3}(\dot{\alpha}_{1}\dot{\alpha}_{2})\dot{\alpha}_{3}}=\mathcal{R}_{(\alpha_{1}\alpha_{2}\alpha_{3})(\dot{\alpha}_{1}\dot{\alpha}_{2}\dot{\alpha}_{3})}-\frac{4}{9}\mathcal{R}_{(\alpha_{1}(\dot{\alpha}_{1}}\epsilon_{\dot{\alpha}_{2})\dot{\beta}}\epsilon_{\alpha_{2})\beta}\,,$
(A.38)

where the linearized spin 3 curvatures are defined as:

$\mathcal{R}_{\alpha_{1}\dot{\alpha}_{1}}=\partial^{\alpha_{2}\dot{\alpha}_{2}}\partial^{\alpha_{3}\dot{\alpha}_{3}}\Phi_{(\alpha_{1}\alpha_{2}\alpha_{3})(\dot{\alpha}_{1}\dot{\alpha}_{2}\dot{\alpha}_{3})}-\frac{1}{4}\partial_{\alpha_{1}\dot{\alpha}_{1}}\partial^{\alpha_{2}\dot{\alpha}_{2}}\Phi_{\alpha_{2}\dot{\alpha}_{2}}-\frac{5}{2}\Box\Phi_{\alpha_{1}\dot{\alpha}_{1}}\,,$
(A.39)
$\displaystyle\mathcal{R}_{(\alpha_{1}\alpha_{2}\alpha_{3})(\dot{\alpha}_{1}\dot{\alpha}_{2}\dot{\alpha}_{3})}$
$\displaystyle=$
$\displaystyle\partial_{(\alpha_{1}(\dot{\alpha}_{1}}\partial^{\rho\dot{\rho}}\Phi_{\alpha_{2}\alpha_{3})\rho\dot{\alpha}_{2}\dot{\alpha}_{3})\dot{\rho}}-\frac{2}{3}\Box\Phi_{(\alpha_{1}\alpha_{2}\alpha_{3})(\dot{\alpha}_{1}\dot{\alpha}_{2}\dot{\alpha}_{3})}$
(A.40)
$\displaystyle-\,\partial_{(\alpha_{1}(\dot{\alpha}_{1}}\partial_{\alpha_{2}\dot{\alpha}_{2}}\Phi_{\alpha_{3})\dot{\alpha}_{3})}\,.$

In the first term in eq. (A.40) we omitted the symmetrization of the indices
in the involved field. The curvatures $\mathcal{R}_{\alpha\dot{\alpha}}$ and
$\mathcal{R}_{(\alpha_{1}\alpha_{2}\alpha_{3})(\dot{\alpha}_{1}\dot{\alpha}_{2}\dot{\alpha}_{3})}$
are invariant under the spin 3 gauge transformations (A.30). The linearized
Fronsdal tensor satisfies the spin-3 transversality condition:

$\partial^{\alpha_{3}\dot{\alpha}_{3}}\mathcal{G}_{(\alpha_{1}\alpha_{2})\alpha_{3}(\dot{\alpha}_{1}\dot{\alpha}_{2})\dot{\alpha}_{3}}=0\,.$
(A.41)

The resulting action for the spin 3 gauge field is written as:

$\begin{split}&S^{(s=3)}_{(\Phi)}=-\frac{3}{2}\int
d^{4}x\;\Phi^{(\alpha_{1}\alpha_{2})\alpha_{3}(\dot{\alpha}_{1}\dot{\alpha}_{2})\dot{\alpha}_{3}}\mathcal{G}_{(\alpha_{1}\alpha_{2})\alpha_{3}(\dot{\alpha}_{1}\dot{\alpha}_{2})\dot{\alpha}_{3}}\\\
&\;\;\;\;\;\;\;\;\;\;=\int
d^{4}x\;\Big{[}\Phi^{(\alpha_{1}\alpha_{2}\alpha_{3})(\dot{\alpha}_{1}\dot{\alpha}_{2}\dot{\alpha}_{3})}\Box\Phi_{(\alpha_{1}\alpha_{2}\alpha_{3})(\dot{\alpha}_{1}\dot{\alpha}_{2}\dot{\alpha}_{3})}\\\
&\;\;\;\;\;\;\;\;\;\qquad\qquad\qquad-\,\frac{3}{2}\Phi^{(\alpha_{1}\alpha_{2}\alpha_{3})(\dot{\alpha}_{1}\dot{\alpha}_{2}\dot{\alpha}_{3})}\partial_{\alpha_{1}\dot{\alpha}_{1}}\partial^{\rho\dot{\rho}}\Phi_{(\rho\alpha_{2}\alpha_{3})(\dot{\rho}\dot{\alpha}_{2}\dot{\alpha}_{3})}\\\
&\;\;\;\;\;\;\;\;\;\qquad\qquad\qquad+\,3\Phi^{(\alpha_{1}\alpha_{2}\alpha_{3})(\dot{\alpha}_{1}\dot{\alpha}_{2}\dot{\alpha}_{3})}\partial_{\alpha_{1}\dot{\alpha}_{1}}\partial_{\alpha_{2}\dot{\alpha}_{2}}\Phi_{\alpha_{3}\dot{\alpha}_{3}}-\frac{15}{4}\Phi^{\alpha\dot{\alpha}}\Box\Phi_{\alpha\dot{\alpha}}\\\
&\;\;\;\;\;\;\;\;\;\qquad\qquad\qquad+\,\frac{3}{8}\partial_{\alpha_{1}\dot{\alpha}_{1}}\Phi^{\alpha_{1}\dot{\alpha}_{1}}\partial_{\alpha_{2}\dot{\alpha}_{2}}\Phi^{\alpha_{2}\dot{\alpha}_{2}}\Big{]}.\end{split}$
(A.42)

This is the spin 3 Fronsdal action in spinor notation. It is invariant under
the gauge transformations (A.30).

Spin 2 sector

The spin 2 sector of the spin $\mathbf{3}$ ${\cal N}=2$ multiplet involves the
field $C^{\alpha\dot{\alpha}}_{\beta\dot{\beta}}$

$C_{\alpha\beta\dot{\alpha}\dot{\beta}}=C_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}+\epsilon_{\alpha\beta}\epsilon_{\dot{\alpha}\dot{\beta}}C\,,$
(A.43)

with the gauge freedom:

$\delta
C_{\alpha\beta\dot{\alpha}\dot{\beta}}=\frac{1}{2}\left(\partial_{\alpha\dot{\alpha}}b_{\beta\dot{\beta}}+\partial_{\beta\dot{\beta}}b_{\alpha\dot{\alpha}}\right).$
(A.44)

Here $b_{\alpha\dot{\alpha}}$ is the spin 2 gauge parameter.

The relevant pieces of $G^{++\dots}$ superfields are expressed as:

$G_{(C)}^{++(\alpha\beta)(\dot{\alpha}\dot{\beta})}=-2(\bar{\theta}^{+})^{2}\theta^{+}_{\rho}\bar{\theta}^{-(\dot{\alpha}}H^{\dot{\beta})\rho(\alpha\beta)}-2(\theta^{+})^{2}\bar{\theta}^{+}_{\dot{\rho}}\theta^{-(\alpha}\bar{H}^{\beta)\dot{\rho}(\dot{\alpha}\dot{\beta})}\,,$
(A.45)
$G^{++5\alpha\dot{\alpha}}_{(C)}=-2i\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}C_{\rho\dot{\rho}}^{\alpha\dot{\alpha}}-2(\theta^{+})^{2}\bar{\theta}^{+}_{\dot{\rho}}\bar{\theta}^{-}_{\dot{\mu}}\bar{H}^{\alpha\dot{\rho}(\dot{\mu}\dot{\alpha})}+2(\bar{\theta}^{+})^{2}\theta^{+}_{\rho}\theta^{-}_{\mu}H^{\dot{\alpha}\rho(\alpha\mu)}\,.$
(A.46)

Here the following notations were used:

$\bar{H}^{\alpha\dot{\rho}(\dot{\alpha}\dot{\mu})}:=\bar{H}^{\alpha(\dot{\rho}\dot{\alpha}\dot{\mu})}+\epsilon^{\dot{\rho}(\dot{\alpha}}\bar{H}^{\alpha\dot{\mu})}\,,\;\;\;\;\;\;\;\;\;H^{\dot{\alpha}\rho(\mu\alpha)}:=H^{\dot{\alpha}(\rho\mu\alpha)}+\epsilon^{\rho(\alpha}H^{\dot{\alpha}\mu)}\,,$
(A.47)
$\bar{H}^{\alpha(\dot{\rho}\dot{\alpha}\dot{\mu})}=\partial_{\beta}^{(\dot{\rho}}C^{\dot{\alpha}\dot{\mu})(\alpha\beta)}\,,\;\;\;\;\;\;\;\;H^{\dot{\alpha}(\rho\mu\alpha)}=\partial^{(\alpha}_{\dot{\beta}}C^{\mu\rho)(\dot{\alpha}\dot{\beta})}\,,$
(A.48)
$H^{\mu\dot{\mu}}=\partial^{\mu\dot{\mu}}C-\frac{1}{3}\partial_{\rho\dot{\rho}}C^{(\mu\rho)(\dot{\mu}\dot{\rho})}=\bar{H}^{\mu\dot{\mu}}\,.$
(A.49)

These objects satisfy the relations:

$\partial_{\rho}^{(\dot{\alpha}}H^{\dot{\beta})\rho(\alpha\beta)}=-\frac{2}{3}\mathcal{R}^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\,,\qquad\partial^{(\alpha}_{\dot{\rho}}\bar{H}^{\beta)\dot{\rho}(\dot{\alpha}\dot{\beta})}=-\frac{2}{3}\mathcal{R}^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\,.$
(A.50)

Here we used $\mathcal{R}^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}$ defined in
(A.10), with the replacement $\Phi\to C$ .

The negatively charged partners of (A.45) and (A.46) can be obtained as
solutions of the zero-curvature equations (3.4) and (3.5):

$\begin{split}G_{(C)}^{--(\alpha\beta)(\dot{\alpha}\dot{\beta})}=&\;2(\bar{\theta}^{-})^{2}\theta^{-}_{\rho}\bar{\theta}^{+(\dot{\alpha}}H^{\dot{\beta})\rho(\alpha\beta)}+2(\theta^{-})^{2}\bar{\theta}^{-}_{\dot{\rho}}\theta^{+(\alpha}\bar{H}^{\beta)\dot{\rho}(\dot{\alpha}\dot{\beta})}\\\
&+i(\bar{\theta}^{+})^{2}(\theta^{-})^{2}(\bar{\theta}^{-})^{2}\partial_{\rho}^{(\dot{\alpha}}H^{\dot{\beta})\rho(\alpha\beta)}+i(\theta^{+})^{2}(\theta^{-})^{2}(\bar{\theta}^{-})^{2}\partial^{(\alpha}_{\dot{\rho}}H^{\beta)\dot{\rho}(\dot{\alpha}\dot{\beta})}\,,\end{split}$
(A.51)
$G_{(C)}^{--5\alpha\dot{\alpha}}=-2i\theta^{-\rho}\bar{\theta}^{-\dot{\rho}}C_{\rho\dot{\rho}}^{\alpha\dot{\alpha}}+2(\theta^{-})^{2}\bar{\theta}^{-(\dot{\rho}}\bar{\theta}^{+\dot{\beta})}\partial^{\beta}_{\dot{\rho}}C_{\beta\dot{\beta}}^{\alpha\dot{\alpha}}-2(\bar{\theta}^{-})^{2}\theta^{-(\rho}\theta^{+\beta)}\partial_{\rho}^{\dot{\beta}}C_{\beta\dot{\beta}}^{\alpha\dot{\alpha}}\\\
+2(\theta^{-})^{2}\bar{\theta}^{+}_{\dot{\rho}}\bar{\theta}^{-}_{\dot{\mu}}\bar{H}^{\alpha\dot{\rho}(\dot{\mu}\dot{\alpha})}+2(\bar{\theta}^{-})^{2}\theta^{+}_{\rho}\theta^{-}_{\mu}H^{\dot{\alpha}\rho(\mu\alpha)}+4i(\theta^{-})^{2}(\bar{\theta}^{-})^{2}\theta^{+}_{\rho}\bar{\theta}^{+}_{\dot{\rho}}\mathcal{G}^{\alpha\rho\dot{\alpha}\dot{\rho}}\,.$
(A.52)

Here, $\mathcal{G}^{\alpha\rho\dot{\alpha}\dot{\rho}}$ is the linearized form
of Einstein tensor introduced in (A.9).

Substituting all these expressions in the superfield action (3.1) for
$\mathbf{s}=3$, we obtain the spin 2 action of the $\mathcal{N}=2$ spin
$\mathbf{3}$ theory

$S^{(s=3)}_{(C)}=4\int
d^{4}x\;C^{\alpha\beta\dot{\alpha}\dot{\beta}}\mathcal{G}_{\alpha\beta\dot{\alpha}\dot{\beta}}=-2\int
d^{4}x\;\Big{[}C^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\Box
C_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\\\
-C^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\partial_{\alpha\dot{\alpha}}\partial^{\rho\dot{\rho}}C_{(\rho\beta)(\dot{\rho}\dot{\beta})}+2\,C\partial^{\alpha\dot{\alpha}}\partial^{\beta\dot{\beta}}C_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}-6C\Box
C\Big{]}.$ (A.53)

It is precisely the linearized Pauli-Fierz action for spin 2.

It is worth to note that in this case the auxiliary fields play no any role in
forming the correct final action, in contradistinction to the spin 1 sector of
the spin $\mathbf{2}$ ${\cal N}=2$ supermultiplet (recall eq. (A.27)).

##  Appendix B Unconstrained $\mathcal{N}=2$ higher spin prepotentials and an
alternative derivation of superfield equations

In this Appendix we introduce the Mezincescu-type $\mathcal{N}=2$ higher spin
prepotentials and describe an alternative derivation of the superfield
equations of motion.

###  B.1 Arbitrary $\mathcal{N}=2$ higher spin: unconstrained prepotential

In order to define prepotentials for $\mathcal{N}=2$ higher spins, it is
necessary to compactly collect all the analytic potentials
$h^{++\dots}(\zeta)$ into a single object. Beforehand, we need to introduce
some appropriate formalism.

In the previous paper Buchbinder:2022kzl  we introduced analytic differential
operators, constructed in terms of the analytic potentials
$h^{++\dots}(\zeta)$:

$\hat{\mathcal{H}}^{++\alpha(s-2)\dot{\alpha}(s-2)}:=h^{++\alpha(s-2)\dot{\alpha}(s-2)M}\partial_{M}\,,$
(B.1)
$[D^{+}_{\hat{\mu}},\hat{\mathcal{H}}^{++\alpha(s-2)\dot{\alpha}(s-2)}]=0\,.$
(B.2)

One can also construct higher order analytic differential operators

$\hat{\mathcal{H}}^{++}_{(s)}:=\hat{\mathcal{H}}^{++\alpha(s-2)\dot{\alpha}(s-2)}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\,,$
(B.3)

which are invariant under $\mathcal{N}=2$ supersymmetry and encode gauge
transformations for an arbitrary integer spin $\bf{s}$ $\mathcal{N}=2$
supermultiplet:

$\delta_{\lambda}\mathcal{H}^{++}_{(s)}=[\mathcal{D}^{++},\hat{\Lambda}_{(s)}]\,.$
(B.4)

Here we used the notations

$\hat{\Lambda}_{(s)}:=\hat{\Lambda}^{\alpha(s-2)\dot{\alpha}(s-2)}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\,,$
(B.5)
$\hat{\Lambda}^{\alpha(s-2)\dot{\alpha}(s-2)}:=\lambda^{\alpha(s-2)\dot{\alpha}(s-2)M}\partial_{M}\,.$
(B.6)

Note that the presence of the derivative $\partial_{5}$ in (B.1) plays an
important role: it ensures the invariance of $\hat{\mathcal{H}}^{++}_{(s)}$
with respect to $\mathcal{N}=2$ supersymmetry transformations. The operators
defined above are crucial for building interactions of higher spins with the
hypermultiplet, as discussed in Section 5.3. Recall that the spinor indices
hidden in the multi-index $M$ in both operators (B.1) and (B.6) are assumed to
be fully symmetrized with other indices of the same type.

The analyticity constraints (B.2) for the supermultiplet of spin ${\bf s}$ can
be solved in terms of unconstrained fermionic prepotentials
$\Psi^{-\alpha(s-2)\dot{\alpha}(s-2)\hat{\mu}}(Z)$ as:

$\begin{split}\hat{\mathcal{H}}^{++}_{(s)}&=\hat{\mathcal{H}}^{++\alpha(s-2)\dot{\alpha}(s-2)}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\\\
&=(D^{+})^{4}\left(\Psi^{-\alpha(s-2)\dot{\alpha}(s-2)\hat{\mu}}D^{-}_{\hat{\mu}}\right)\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\,,\end{split}$
(B.7)

where we have assumed that $\Psi^{-\alpha(s-2)\dot{\alpha}(s-2)\hat{\mu}}$ are
symmetric with respect to all indices of the same type as in (B.1) and (B.6).
Here we have also used the covariant spinor derivatives $D^{-}_{\hat{\mu}}$
defined in (2.13).

The relation (B.7) is a higher spin generalization of the Mezincescu
prepotential relation (4.15) of ${\cal N}=2$ Maxwell theory. For the spin
$\mathbf{2}$ case (the linearized $\mathcal{N}=2$ Einstein supergravity) it
was generalized in Gates:1981qq  and further used in ref. Zupnik:1998td .

Since the operator $\hat{\mathcal{H}}^{++}_{(s)}$ is invariant with respect to
supersymmetry transformations (3.11), the prepotentials introduced above are
$\mathcal{N}=2$ scalars by construction:

$\delta_{\epsilon}\Psi^{-\alpha(s-2)\dot{\alpha}(s-2)\hat{\mu}}=0\,.$ (B.8)

Using (B.7), one can deduce the explicit expressions for the potentials
$h^{++\dots}(\zeta)$ through the unconstrained prepotentials
$\Psi^{-\alpha(s-2)\dot{\alpha}(s-2)\hat{\mu}}(Z)$:

$\begin{cases}h^{++(\alpha(s-2)\beta)(\dot{\alpha}(s-2)\dot{\beta})}=2i\Bigr{[}(D^{+})^{2}\bar{D}^{+(\dot{\beta}}\Psi^{-(\alpha(s-2)\beta)\dot{\alpha}(s-2))}+(\bar{D}^{+})^{2}D^{+(\beta}\bar{\Psi}^{-\alpha(s-2))(\dot{\alpha}(s-2)\dot{\beta})}\Bigr{]}_{\theta^{-}=0}\,,\\\
h^{++\alpha(s-1)\dot{\alpha}(s-2)+}=-(D^{+})^{4}\Psi^{-\alpha(s-1)\dot{\alpha}(s-2)}\,,\\\
h^{++\alpha(s-2)\dot{\alpha}(s-1)+}=-(D^{+})^{4}\bar{\Psi}^{-\alpha(s-2)\dot{\alpha}(s-1)}\,,\\\
h^{++5\alpha(s-2)\dot{\alpha}(s-2)}=2i\Bigr{[}(\bar{D}^{+})^{2}D^{+}_{\beta}\Psi^{-(\beta\alpha(s-2))\dot{\alpha}(s-2)}-(D^{+})^{2}\bar{D}^{+}_{\dot{\beta}}\bar{\Psi}^{-\alpha(s-2)(\dot{\beta}\dot{\alpha}(s-2))}\Bigr{]}_{\theta^{-}=0}.\\\
\end{cases}$ (B.9)

The notation “$\theta^{-}=0$” means the restriction to the
$\theta^{-}$-independent parts.

The prepotentials give an alternative superfield representation for the higher
spin degrees of freedom, but at cost of losing analyticity and enlarging the
gauge group. We will not present these gauge transformations here, though they
can easily be written.

One can express $G^{++}$-superfields through $\Psi^{-\dots}$. In order to
accomplish this, it is useful and instructive to rewrite the differential
operator (5.33) in the basis of the $\mathcal{N}=2$ covariant derivatives,
$D_{M}=(\partial_{\alpha\dot{\alpha}},D^{-}_{\hat{\alpha}},\partial_{5})$:

$\hat{\mathcal{H}}^{++\alpha(s-2)\dot{\alpha}(s-2)}=G^{++\alpha(s-2)\dot{\alpha}(s-2)M}D_{M}\,.$
(B.10)

Derivatives $D_{M}$ commute with $\mathcal{N}=2$ supersymmetry, so the
coefficients $G^{++\dots}$ are invariant under supersymmetry by construction,
though they are not analytic. Their explicit expressions through the
$h^{++\dots}$-potentials is given by eqs. (3.2) and (3.3).

From eqs. (B.7) and (B.10) one can directly express $G^{++\dots}$ through
prepotentials:

$G^{++(\alpha(s-2)\beta)(\dot{\alpha}(s-2)\dot{\beta})}=4i(D^{+})^{2}\bar{D}^{+(\dot{\beta}}\Psi^{-(\alpha(s-2)\beta)\dot{\alpha}(s-2))}+4i(\bar{D}^{+})^{2}D^{+(\beta}\bar{\Psi}^{-\alpha(s-2))(\dot{\alpha}(s-1)\dot{\beta})}\,,$
(B.11)
$G^{++5\alpha(s-2)\dot{\alpha}(s-2)}=4i(\bar{D}^{+})^{2}D^{+}_{\beta}\Psi^{-(\beta\alpha(s-2))\dot{\alpha}(s-2)}-4i(D^{+})^{2}\bar{D}^{+}_{\dot{\beta}}\bar{\Psi}^{-\alpha(s-2)(\dot{\beta}\dot{\alpha}(s-2))}\,.$
(B.12)

Using these formulas, one can vary $G$-superfields with respect to the
prepotentials $\Psi^{-}$. Varying the action (4.26) in this way yields:

$\begin{split}\delta_{\Psi}S_{(s)}=&8i(-1)^{s+1}\int d^{4}xd^{8}\theta
du\;\Bigr{\\{}(D^{+})^{2}\bar{D}^{+(\dot{\beta}}\delta\Psi^{-\alpha(s-1)\dot{\alpha}(s-2))}G^{--}_{\alpha(s-1)(\dot{\beta}\dot{\alpha}(s-2))}\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+(\bar{D}^{+})^{2}D^{+}_{\beta}\delta\Psi^{-(\beta\alpha(s-2))\dot{\alpha}(s-2)}G^{--5}_{\alpha(s-2)\dot{\alpha}(s-2)}\Bigr{\\}},\end{split}$
(B.13) $\begin{split}\delta_{\bar{\Psi}}S_{(s)}=&8i(-1)^{s+1}\int
d^{4}xd^{8}\theta
du\;\Bigr{\\{}(\bar{D}^{+})^{2}D^{+(\beta}\delta\bar{\Psi}^{-\alpha(s-2))\dot{\alpha}(s-1)}G^{--}_{(\beta\alpha(s-2))\dot{\alpha}(s-1)}\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad-(D^{+})^{2}\bar{D}^{+}_{\dot{\beta}}\delta\bar{\Psi}^{-\alpha(s-2)(\dot{\beta}\dot{\alpha}(s-2))}G^{--5}_{\alpha(s-2)\dot{\alpha}(s-2)}\Bigr{\\}}.\end{split}$
(B.14)

Thus the variations of the actions were reduced to the variations of
prepotentials which are unconstrained harmonic superfields. After integration
by parts, we deduce the equations of motion:

$\begin{cases}(D^{+})^{2}\bar{D}^{+\dot{\beta}}G^{--}_{(\beta\alpha(s-2))(\dot{\beta}\dot{\alpha}(s-2))}+(\bar{D}^{+})^{2}D^{+}_{(\beta}G^{--5}_{\alpha(s-2))\dot{\alpha}(s-2)}=0\,,\\\
(\bar{D}^{+})^{2}D^{+\beta}G^{--}_{(\beta\alpha(s-2))(\dot{\beta}\dot{\alpha}(s-2))}-(D^{+})^{2}\bar{D}^{+}_{(\dot{\beta}}G^{--5}_{\alpha(s-2)\dot{\alpha}(s-2))}=0\,.\end{cases}$
(B.15)

These equations coincide with eqs. (4.41) and (4.42) obtained earlier by
varying with respect to the analytic potentials $h^{++\dots}$.

###  B.2 Current superfields for $\Psi^{-\dots}$ prepotentials

Varying the cubic coupling (5.31) with respect to the Mezincescu-type
$\Psi^{-\dots}$ prepotential yields:

$\begin{split}\delta_{\Psi}S^{(s)}_{gauge}=&-\frac{\kappa_{s}}{2}\int
d\zeta^{(-4)}\;\Biggr{\\{}q^{+a}(D^{+})^{4}\left(\delta\Psi^{-\alpha(s-2)\dot{\alpha}(s-2)\hat{\mu}}D^{-}_{\hat{\mu}}\right)\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}(J)^{P(s)}q^{+}_{a}\\\
&\qquad\qquad+\xi(D^{+})^{4}\left(\delta\Psi^{-\alpha(s-2)\dot{\alpha}(s-2)\hat{\mu}}D^{-}_{\hat{\mu}}\right)\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\left[q^{+a}(J)^{P(s)}q^{+}_{a}\right]\Biggr{\\}}\\\
=&-\frac{\kappa_{s}}{2}\int
d\zeta^{(-4)}\,(D^{+})^{4}\;\Biggr{\\{}q^{+a}\left(\delta\Psi^{-\alpha(s-2)\dot{\alpha}(s-2)\hat{\mu}}D^{-}_{\hat{\mu}}\right)\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}(J)^{P(s)}q^{+}_{a}\\\
&\qquad\qquad\qquad+\xi\left(\delta\Psi^{-\alpha(s-2)\dot{\alpha}(s-2)\hat{\mu}}D^{-}_{\hat{\mu}}\right)\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\left[q^{+a}(J)^{P(s)}q^{+}_{a}\right]\Biggr{\\}}.\end{split}$
(B.16)

Here we used the representation (B.7). In the last equation one can restore
the full harmonic superspace integration measure.

So we find that the superfield currents conjugated to the prepotentials
$\Psi^{-\dots}$ are given by the expression:

$\begin{split}\mathcal{J}^{+}_{\alpha(s-2)\dot{\alpha}(s-2)\hat{\mu}}:=&\;-\frac{1}{\kappa_{s}}\frac{\delta
S^{(s)}_{gauge}}{\delta\Psi^{-\alpha(s-2)\dot{\alpha}(s-2)\hat{\mu}}}\\\
=&\;\frac{1}{2}q^{+a}D^{-}_{\hat{\mu}}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}(J)^{P(s)}q^{+}_{a}+\frac{1}{2}\xi
D^{-}_{\hat{\mu}}\left[q^{+a}(J)^{P(s)}q^{+}_{a}\right]\,.\end{split}$ (B.17)

These currents coincide with the supersymmetry-invariant spinor current
superfields (5.57) and (5.63) found earlier. The conservation law has the
simple form:

$\mathcal{D}^{++}\mathcal{J}^{+}_{\alpha(s-2)\dot{\alpha}(s-2)\hat{\mu}}=0\,.$
(B.18)

It is a direct consequence of the analyticity of the hypermultiplet and its
equation of motion.

It is also important to note that the supercurrents constructed in this way
are manifestly invariant under ${\cal N}=2$ supersymmetry:

$\delta_{\epsilon}\mathcal{J}^{+}_{\alpha(s-2)\dot{\alpha}(s-2)\hat{\mu}}=0\,.$
(B.19)

In all these relations it is assumed that the index $\hat{\mu}(\mu,\dot{\mu})$
is fully symmetrized with all other spinorial indices of the same nature.

##  Appendix C Current superfields

Composite operators in a supersymmetric theory are members of some
supermultiplets. This means the existence of a superfield that contains
composite operators as its components related to each other by supersymmetry
transformations. We will be interested in superfields containing conserved
currents, since they appear as sources in the equations of motion for fields
of higher spins. Moreover, conserved currents furnish an important class of
local operators and can be used to study the dynamics of supersymmetric field
theories, supersymmetry breaking and coupling to gauge theories (including
higher spin theories) Dumitrescu:2011zz . In this Appendix we will discuss
general properties of supercurrents and present analytic harmonic superfields
which capture $\mathcal{N}=2$ conserved current supermultiplet212121Among the
first papers where supercurrents were defined and discussed we mention, e.g.,
works Ferrara:1974pz ; Ogievetsky:1976qc ..

There are two essentially different types of the current supermultiplets:

  * •

Symmetry current supermultiplet (for example, multiplets involving currents of
internal symmetry);

  * •

Supercurrent (it contains a fermionic current generating rigid supersymmetry
together with the stress tensor associated with space-time translations).

The structure of $\mathcal{N}=1$ current superfields is well known, see, e.g.,
Dumitrescu:2011zz ; Bertolini . The construction of higher spin conserved
$\mathcal{N}=1$ current superfields was presented in refs. Howe:1981qj ; HK1 ;
Buchbinder:2018wzq ; Buchbinder:2018nkp ; Buchbinder:2018gle ; Gates:2019cnl ;
Hutomo:2020wca . We shall describe in Section C.1 $\mathcal{N}=2$ harmonic
superfields that accommodate various current supermultiplets. In particular,
we shall present harmonic superfields containing higher spin currents.

The structure of $\mathcal{N}=1$ supercurrents is also well known, see for
example BK ; Kuzenko:2010am ; Bertolini . In $\mathcal{N}=2$ theories the
structure of supercurrents was elaborated in Butter:2010sc . However, to the
best of our knowledge, the supercurrent structure in harmonic superspace was
not systematically discussed so far.

###  C.1 $\mathcal{N}=2$ current superfields

1. We consider the real (with respect to the tilde conjugation, $\widetilde{J^{++A}}=J^{++A}$) analytic superfield $J^{++A}$, where $A$ is some multi-index of the form $A=\alpha(n)\dot{\alpha}(n)$. We impose the “harmonic conservation law” on the superfield $J^{++A}$ 222222This conservation law formally coincides with the harmonic constraint $\mathcal{D}^{++}L^{++}=0$ defining $\mathcal{N}=2$ tensor multiplet $L^{++}(\zeta)$ off shell. For details see Section 6.2 of 18 . :

$\mathcal{D}^{++}J^{++A}=\partial^{++}J^{++A}-2i\theta^{+\alpha}\bar{\theta}^{+\dot{\alpha}}\partial_{\alpha\dot{\alpha}}J^{++A}=0\,.$
(C.1)

Here we assumed that the supercurrent does not depend on $x^{5}$.

The general solution of the constraint $\mathcal{D}^{++}J^{++A}=0$ is given
by:

$\begin{split}J^{++A}(x,\theta^{+},u)=&j^{(ij)A}(x)u^{+}_{i}u^{+}_{j}+\theta^{+\alpha}j^{iA}_{\alpha}(x)u^{+}_{i}+\bar{\theta}^{+\dot{\alpha}}\bar{j}^{iA}_{\dot{\alpha}}(x)u^{+}_{i}\\\
&+(\theta^{+})^{2}j^{A}(x)+(\bar{\theta}^{+})^{2}\bar{j}^{A}(x)\\\
&+2i\theta^{+}\sigma^{n}\bar{\theta}^{+}\left[j_{n}^{A}(x)+\partial_{n}j^{(ij)A}(x)u^{+}_{i}u^{-}_{j}\right]\\\
&-i(\bar{\theta}^{+})^{2}\theta^{+\alpha}\partial^{\dot{\alpha}}_{\alpha}\bar{j}^{iA}_{\dot{\alpha}}u^{-}_{i}-i(\theta^{+})^{2}\bar{\theta}^{+\dot{\alpha}}\partial_{\dot{\alpha}}^{\alpha}j^{iA}_{\alpha}u^{-}_{i}\\\
&-(\theta^{+})^{2}(\bar{\theta}^{+})^{2}\Box
j^{(ij)A}u^{-}_{i}u^{-}_{j}\,.\end{split}$ (C.2)

As independent components, it involves the real field $j^{(ij)A}(x)$, the
complex field $j^{iA}_{\alpha}(x)$ and its complex conjugate
$\bar{j}^{iA}_{\dot{\alpha}}(x)$, the complex field $j^{A}(x)$ and the real
field $j_{n}^{A}(x)$. Equation (C.1) implies the current conservation
condition for the latter:

$\partial^{n}j_{n}^{A}=0\,.$ (C.3)

This allows us to identify the harmonic superfield $J^{++A}$ with the
supermultiplet containing the conserved current and the equation
$\mathcal{D}^{++}J^{++A}=0$ with the relevant superfield conservation
condition. All bosonic higher spin current superfields considered in Section 5
have such a structure.

2. Fermionic higher spin current superfields are also analytic, but, in contrast to bosonic higher spin current superfields, they satisfy different conditions (5.55):

$\begin{cases}\mathcal{D}^{++}J^{+}_{\alpha(s-2)\dot{\alpha}(s-1)}=-2i\theta^{+\rho}J^{++}_{(\alpha(s-2)\rho)\dot{\alpha}(s-1)}-2i\bar{\theta}^{+}_{(\dot{\alpha}}J^{++}_{\alpha(s-2)\dot{\alpha}(s-2))}\,,\\\
\mathcal{D}^{++}J^{++}_{\alpha(s-1)\dot{\alpha}(s-1)}=0\,,\\\
\mathcal{D}^{++}J^{++}_{\alpha(s-2)\dot{\alpha}(s-2)}=0\,.\end{cases}$ (C.4)

Using the general solution (C.2) of the second and third equations in (C.4)
and substituting it in the first equation o, one can obtain, for the
coefficient in the $\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}$ piece of the
fermionic current superfield,

$J^{+}_{\alpha(s-2)\dot{\alpha}(s-1)}=\dots+2i\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}j^{i}_{\rho\dot{\rho}\alpha(s-2)\dot{\alpha}(s-1)}u^{-}_{i}+\dots\,,$
(C.5)

the following conservation condition:

$\partial^{\rho\dot{\rho}}\left(j^{i}_{\rho\dot{\rho}\alpha(s-2)(\dot{\alpha}(s-2)\dot{\sigma})}-j^{i}_{\dot{\rho}(\alpha(s-2)\rho)(\dot{\alpha}(s-2)\dot{\sigma})}+\epsilon_{(\dot{\sigma}\dot{\rho}}j^{i}_{\rho\alpha(s-2)\dot{\alpha}(s-2))}\right)=0\,.$
(C.6)

Here, the fields $j^{i}_{\dot{\rho}\alpha(s-1)\dot{\alpha}(s-1)}$ and
$j^{i}_{\rho\alpha(s-2)\dot{\alpha}(s-2)}$ are the appropriate fermionic
components of the bosonic current superfields
$J^{++}_{\alpha(s-1)\dot{\alpha}(s-1)}$ and
$J^{++}_{\alpha(s-2)\dot{\alpha}(s-2)}$, respectively.

So we conclude that the fermionic current, defined as

$\mathfrak{j}^{i}_{{\rho\dot{\rho}\alpha(s-2)(\dot{\alpha}(s-2)\dot{\sigma})}}:=j^{i}_{\rho\dot{\rho}\alpha(s-2)(\dot{\alpha}(s-2)\dot{\sigma})}-j^{i}_{\dot{\rho}(\alpha(s-2)\rho)(\dot{\alpha}(s-2)\dot{\sigma})}-\epsilon_{\dot{\rho}(\dot{\sigma}}j^{i}_{\rho\alpha(s-2)\dot{\alpha}(s-2))}$
(C.7)

is conserved as a consequence of the superfield equations (C.4).

3. An analogous consideration for the complex conjugate current superfield (5.62),

$\begin{cases}\mathcal{D}^{++}J^{+}_{\alpha(s-1)\dot{\alpha}(s-2)}=2i\bar{\theta}^{+\dot{\rho}}J^{++}_{\alpha(s-1)(\dot{\alpha}(s-2)\dot{\rho})}-2i\theta^{+}_{(\alpha}J^{++}_{\alpha(s-2))\dot{\alpha}(s-2)}\,,\\\
\mathcal{D}^{++}J^{++}_{\alpha(s-1)\dot{\alpha}(s-1)}=0\,,\\\
\mathcal{D}^{++}J^{++}_{\alpha(s-2)\dot{\alpha}(s-2)}=0\,,\end{cases}$ (C.8)

evidently implies that the corresponding fermionic component,

$J^{+}_{\alpha(s-1)\dot{\alpha}(s-2)}=\dots+2i\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}j^{i}_{\rho\dot{\rho}\alpha(s-1)\dot{\alpha}(s-2)}+\dots\,,$
(C.9)

obeys the conservation law:

$\partial^{\rho\dot{\rho}}\left(j^{i}_{\rho\dot{\rho}(\alpha(s-2)\sigma)\dot{\alpha}(s-2)}-j^{i}_{\rho(\alpha(s-2)\sigma)(\dot{\alpha}(s-2)\dot{\rho})}+\epsilon_{\rho(\sigma}j^{i}_{\dot{\rho}\alpha(s-2))\dot{\alpha}(s-2)}\right)=0\,.$
(C.10)

Here, the fields $j^{i}_{\rho\alpha(s-1)\dot{\alpha}(s-1)}$ and
$j^{i}_{\dot{\rho}\alpha(s-2)\dot{\alpha}(s-2)}$ are proper fermionic
components of the bosonic current superfields
$J^{++}_{\alpha(s-1)\dot{\alpha}(s-1)}$ and
$J^{++}_{\alpha(s-2)\dot{\alpha}(s-2)}$, respectively. Thus the relevant
conserved fermionic current is defined by:

$\mathfrak{j}^{i}_{\rho\dot{\rho}(\alpha(s-2)\sigma)\dot{\alpha}(s-2)}:=j^{i}_{\rho\dot{\rho}(\alpha(s-2)\sigma)\dot{\alpha}(s-2)}-j^{i}_{\rho(\alpha(s-2)\sigma)(\dot{\alpha}(s-2)\dot{\rho})}+\epsilon_{\rho(\sigma}j^{i}_{\dot{\rho}\alpha(s-2))\dot{\alpha}(s-2)}\,.$
(C.11)

In Sections 5.6 and 5.8 we presented the explicit expressions of these current
superfields in terms of the free hypermultiplet in the spin $\mathbf{2}$ (see
also Appendix C.2) and the spin $\mathbf{3}$ cases.

In Appendix D the conservation conditions (C.1), (C.4) and (C.8) will be
independently derived by making use of $\mathcal{N}=2$ superfield Noether
theorem for rigid symmetries of the free hypermultiplet.

###  C.2 $\mathcal{N}=2$ supercurrent

Let us now consider the special supercurrent $J^{++\alpha\dot{\alpha}}$ having
an additional conservation law:

$\begin{cases}\mathcal{D}^{++}J^{++\alpha\dot{\alpha}}=0\,,\\\
\partial_{\alpha\dot{\alpha}}J^{++\alpha\dot{\alpha}}=0\,.\end{cases}$ (C.12)

The second condition implies the conservation of all the component currents in
its relevant $\theta$-expansion (C.2):

$\begin{cases}\partial_{\alpha\dot{\alpha}}j^{(ij)\alpha\dot{\alpha}}=0\,,\\\
\partial_{\alpha\dot{\alpha}}j^{i\alpha\dot{\alpha}}_{\beta}=0\,,\\\
\partial_{\alpha\dot{\alpha}}j^{\alpha\dot{\alpha}}=0\,,\\\
\partial_{m}j_{n}^{m}=0\,.\end{cases}$ (C.13)

These currents can be identified with the currents inherent to any
$\mathcal{N}=2$ super Poincaré invariant theory: $j_{n}^{m}$ with the energy-
momentum tensor, $j^{\alpha\dot{\alpha}}$ with ${\rm U}(1)$ R-symmetry
current, $j^{i\alpha\dot{\alpha}}_{\beta}$ with the doublet of supersymmetry
fermionic currents and $j^{(ij)\alpha\dot{\alpha}}$ with the ${\rm SU}(2)$
R-symmetry current.

In what follows we will use the notation:

$\begin{split}J^{++}_{\alpha\dot{\alpha}}(x,\theta^{+},u)=&R^{(ij)}_{\alpha\dot{\alpha}}(x)u^{+}_{i}u^{+}_{j}+\theta^{+\beta}S^{i}_{\alpha\dot{\alpha}\beta}(x)u^{+}_{i}+\bar{\theta}^{+\dot{\beta}}\bar{S}^{i}_{\alpha\dot{\alpha}\dot{\beta}}(x)u^{+}_{i}\\\
&+(\theta^{+})^{2}R_{\alpha\dot{\alpha}}(x)+(\bar{\theta}^{+})^{2}\bar{R}_{\alpha\dot{\alpha}}(x)\\\
&+2i\theta^{+}\sigma^{n}\bar{\theta}^{+}\left[T_{n\,\alpha\dot{\alpha}}(x)+\partial_{n}R^{(ij)}_{\alpha\dot{\alpha}}(x)u^{+}_{i}u^{-}_{j}\right]\\\
&-i(\bar{\theta}^{+})^{2}\theta^{+\beta}\partial^{\dot{\beta}}_{\beta}\bar{S}^{i}_{\alpha\dot{\alpha}\dot{\beta}}u^{-}_{i}-i(\theta^{+})^{2}\bar{\theta}^{+\dot{\beta}}\partial_{\dot{\beta}}^{\beta}S^{i}_{\alpha\dot{\alpha}\beta}u^{-}_{i}\\\
&-(\theta^{+})^{2}(\bar{\theta}^{+})^{2}\Box
R^{(ij)}_{\alpha\dot{\alpha}}u^{-}_{i}u^{-}_{j}\,.\end{split}$ (C.14)

For example, the analytic current superfield (5.81) constructed out of the
hypermultiplet superfield $q^{+}$, has such a structure and so can be
identified with the $\mathcal{N}=2$ supercurrent superfield of hypermultiplet,
as was already mentioned in Section 5.6.

##  Appendix D $\mathcal{N}=2$ Noether theorem and rigid symmetries of
hypermultiplet

In this Appendix, we briefly review the issue of higher-derivative rigid
symmetries of the free hypermultiplet action defined in Buchbinder:2022kzl
and relate these symmetries to the conserved current superfields.

###  D.1 Rigid symmetries

The rigid symmetries of hypermultiplet considered in Buchbinder:2022kzl  are
realized by the following transformations232323Here $s\geq 2$. For $s=1$, the
symmetry transformation is:  $\delta_{rig}^{(s=1)}q^{+a}=-\lambda Jq^{+a}.$
Unlike these transformations, all higher spin ($s\geq 2$) transformations
contain derivatives.:

$\delta^{(s)}_{rig}q^{+a}=-\hat{\Lambda}_{rig}^{\alpha(s-2)\dot{\alpha}(s-2)}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}(J)^{P(s)}q^{+a}\,,$
(D.1)

where the differential operator
$\hat{\Lambda}_{rig}^{\alpha(s-2)\dot{\alpha}(s-2)}$ is defined as

$\hat{\Lambda}_{rig}^{\alpha(s-2)\dot{\alpha}(s-2)}:=\Lambda^{\alpha(s-2)\dot{\alpha}(s-2)M}\partial_{M},\qquad
M=(\alpha\dot{\alpha},\alpha,\dot{\alpha},5),$ (D.2)

with the parameters of the transformations as coefficients:

$\Lambda^{(\alpha(s-2)\alpha)(\dot{\alpha}(s-2)\dot{\alpha})}=\lambda^{\alpha(s-1)\dot{\alpha}(s-1)}-2i\lambda^{\alpha(s-1)(\dot{\alpha}(s-2)-}\bar{\theta}^{+\dot{\alpha})}-2i\theta^{+(\alpha}\lambda^{\alpha(s-2))\dot{\alpha}(s-1)-}\,,$
(D.3a)
$\Lambda^{\alpha(s-2)\dot{\alpha}(s-2)5}=\lambda^{\alpha(s-2)\dot{\alpha}(s-2)5}+2i\lambda^{(\alpha(s-2)\beta)\dot{\alpha}(s-2)-}\theta^{+}_{{\beta}}+2i\lambda^{\alpha(s-2)(\dot{\alpha}(s-2)\dot{\beta})-}\bar{\theta}^{+}_{{\dot{\beta}}}\,,$
(D.3b)
$\Lambda^{\alpha(s-1)\dot{\alpha}(s-2)+}=\lambda^{\alpha(s-1)\dot{\alpha}(s-2)+}\,,\;\;\;\;\;\lambda^{\alpha(s-1)\dot{\alpha}(s-2)\pm}=\lambda^{\alpha(s-1)\dot{\alpha}(s-2)i}u_{i}^{\pm}\,,$
(D.3c)
$\Lambda^{\alpha(s-2)\dot{\alpha}(s-1)+}=\lambda^{\alpha(s-2)\dot{\alpha}(s-1)+}\,,\;\;\;\;\;\lambda^{\alpha(s-2)\dot{\alpha}(s-1)\pm}=\lambda^{\alpha(s-2)\dot{\alpha}(s-1)i}u_{i}^{\pm}\,.$
(D.3d)

In (D.2) the same convention concerning the spinor indices inside the index
$M$ is assumed as in the differential operators (5.33). All the parameters
$\lambda^{\alpha(s-1)\dot{\alpha}(s-1)}\,,\lambda^{\alpha(s-2)\dot{\alpha}(s-2)5}\,,$
$\lambda^{\alpha(s-1)\dot{\alpha}(s-2)i},\,\lambda^{\alpha(s-2)\dot{\alpha}(s-1)i}$
in the operator $\hat{\Lambda}_{rig}^{\alpha(s-2)\dot{\alpha}(s-2)}$ are
coordinate-independent. The analytic-superspace localization of these
parameters produces the spin $s$, spin $s-1$ and spin $s-\tfrac{1}{2}$ gauge
transformations, correspondingly. The analytic operator (D.2) satisfies the
condition:

$[\mathcal{D}^{++},\hat{\Lambda}_{rig}^{\alpha(s-2)\dot{\alpha}(s-2)}]\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}=0\,.$
(D.4)

This property is crucial for the invariance of the free hypermultiplet action
(5.1) under (D.1).

Let us consider the algebra of these symmetry transformations. For simplicity,
we shall limit our study to the massless case (i.e., with
$\partial_{5}q^{+a}=0$). The commutation relation of two rigid symmetries D.1
has the following generic form:

$\begin{split}[\delta^{(s)}_{\lambda_{1}},\delta^{(s)}_{{\lambda}_{2}}]q^{+a}=&\Lambda_{1}^{\alpha(s-2)\dot{\alpha}(s-2)N}\partial_{N}\Lambda_{2}^{\beta(s-2)\dot{\beta}(s-2)M}\partial_{M}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\partial^{s-2}_{\beta(s-2)\dot{\beta}(s-2)}\\\
&-\Lambda_{2}^{\alpha(s-2)\dot{\alpha}(s-2)M}\partial_{M}\Lambda_{1}^{\beta(s-2)\dot{\beta}(s-2)N}\partial_{N}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\partial^{s-2}_{\beta(s-2)\dot{\beta}(s-2)}.\end{split}$
(D.5)

Only transformations with fermionic parameters can give a nontrivial
contribution to this commutator. Keeping this in mind, we stick to the rigid
spinor parameters $\lambda_{1}^{\alpha(s-1)\dot{\alpha}(s-2)i}$,
$\lambda_{2}^{\alpha(s-2)\dot{\alpha}(s-1)i}$. A simple computation yields: