smalltableaux [a,b]Nikita Zaigraev

#  $\mathcal{N}=2$ higher-spin theories and harmonic superspace

Ioseph Buchbinder  Evgeny Ivanov

###### Abstract

A brief review of the harmonic superspace approach to the construction of
$\mathcal{N}=2$ supersymmetric higher spin theories is given. We define off-
shell analytic harmonic gauge potentials of $\mathcal{N}=2$ supersymmetric
higher-spin multiplets and of $\mathcal{N}=2$ superconformal higher-spin
multiplets for an arbitrary integer highest spin. The component contents of
these $\mathcal{N}=2$ higher-spin supermultiplets are explicitly presented. We
also construct their cubic couplings to the hypermultiplet. This short report
summarizes the basic results of the series of published papers [1, 2, 3] as
well as announces those of the forthcoming article [4].

##  1 Introduction

Construction of consistent interacting higher-spin theories is a long-standing
problem. Various no-go theorems [5] indicate that setting up self-consistent
couplings of Fronsdal fields on a flat background is a non-trivial task which
requires a significant revision of the principles and foundations of
relativistic field theory. On the other hand, there are well-known nonlinear
Vasiliev equations on an (A)dS background [6] and the conformal theory of
interacting higher spins on a flat background [7]. Of great interest are
supersymmetric generalizations of higher spins. Combining supersymmetry with
gauge symmetry of higher spins provides significant limitations on the
structure of admissible interactions. As is well known, it is a superfield
language that essentially simplifies the construction of supersymmetric
theories.

In four dimensions, off-shell superfield approaches are known for theories
with $\mathcal{N}\leq 3$. Most appropriate for construction of unconstrained
off-shell formulations of $\mathcal{N}=2$ supersymmetric theories is
$\mathcal{N}=2$ harmonic superspace (HSS) [8]. In a series of articles [1, 2,
3, 4], we showed that the HSS approach is very efficient for describing
$\mathcal{N}=2$ massless Fronsdal fields [1], their cubic interactions with a
hypermultiplet [2, 3] and higher-spin $\mathcal{N}=2$ superconformal
multiplets [4]. The purpose of this short article is to summarize the results
obtained, so as to help the interested reader to get familiar with these new
applications of the HSS method. We also announce a number of results on
$\mathcal{N}=2$ superconformal higher spins to be soon published.

##  2 ABC of harmonic superspace

The conventional $\mathcal{N}=2,4D$ superspace is parametrized by the
coordinates $\left(x^{\alpha\dot{\alpha}},\theta^{\alpha
i},\bar{\theta}^{\dot{\alpha}i}\right)$, $i=1,2$. In $\mathcal{N}=2$ harmonic
superspace these coordinates are supplemented by new auxiliary harmonic
variables $u^{\pm}_{i}$, which parametrize an internal sphere $S^{2}$ and
satisfy the condition $u^{+i}u^{-}_{i}=1$. Because of the presence of the
$S^{2}$ harmonics, any harmonic superfield has an infinite number of component
fields. One can consider the analytic basis in HSS,
$\theta^{\pm\alpha,\dot{\alpha}}:=\theta^{\alpha,\dot{\alpha}i}u^{\pm}_{i},$
$x^{\alpha\dot{\alpha}}_{A}:=x^{\alpha\dot{\alpha}}-4i\theta^{\alpha(i}\bar{\theta}^{\dot{\alpha}j)}u^{+}_{i}u^{-}_{j}$.
$\mathcal{N}=2$ supersymmetry with parameters $\epsilon^{\alpha
i},\bar{\epsilon}^{\dot{\alpha}i}$ ($\epsilon^{\pm\alpha}=\epsilon^{\alpha
i}u^{\pm}_{i}$,
$\bar{\epsilon}^{\pm\dot{\alpha}}=\bar{\epsilon}^{\dot{\alpha}i}u^{\pm}_{i}$)
is realized on the HSS coordinates as:

$\delta_{\epsilon}x_{A}^{\alpha\dot{\alpha}}=-4i\left(\epsilon^{-\alpha}\bar{\theta}^{+\dot{\alpha}}+\theta^{+\alpha}\bar{\epsilon}^{-\dot{\alpha}}\right),\quad\delta_{\epsilon}\theta^{\pm\mu}=\epsilon^{\pm\mu},\quad\delta_{\epsilon}\bar{\theta}^{\pm\dot{\mu}}=\bar{\epsilon}^{\pm\dot{\mu}},\qquad\delta_{\epsilon}u^{\pm}_{i}=0.$
(1)

It immediately follows from the form of these transformations that there is an
invariant subspace with half of Grassmann variables:

$\zeta:=\left(x_{A}^{\alpha\dot{\alpha}},\theta^{+\alpha},\bar{\theta}^{+\dot{\alpha}},u^{\pm}_{i}\right).$
(2)

This is called analytic harmonic superspace, and in $\mathcal{N}=2$ theories
it plays a fundamental role similar to $\mathcal{N}=1$ chiral superspace in
$\mathcal{N}=1$ theories. All $\mathcal{N}=2$ multiplets admit a nice
description by analytic superfields. For description of the matter
$\mathcal{N}=2$ hypermultiplets with non-zero mass (see below) it is also
necesary to add an auxiliary coordinate $x^{5}$, such that:

$\delta_{\epsilon}x^{5}=2i\left(\epsilon^{-\alpha}\theta^{+}_{\alpha}-\bar{\epsilon}^{-}_{\dot{\alpha}}\bar{\theta}^{+\dot{\alpha}}\right).$
(3)

The analogue of complex conjugation in HSS is the tilde conjugation. On
complex functions, it acts like ordinary complex conjugation. The action of
tilde-conjugation on the harmonic superspace coordinates is given by:

$\widetilde{x^{\alpha\dot{\alpha}}_{A}}=x^{\alpha\dot{\alpha}}_{A},\quad\widetilde{\theta^{\pm}_{\alpha}}=\bar{\theta}^{\pm}_{\dot{\alpha}},\quad\widetilde{\bar{\theta}^{\pm}_{\dot{\alpha}}}=-\theta^{\pm}_{\alpha},\quad\widetilde{u^{\pm
i}}=-u_{i}^{\pm},\quad\widetilde{u^{\pm}_{i}}=u^{\pm
i},\quad\widetilde{x^{5}}=x^{5}.$ (4)

The important ingredients of the harmonic superspace formalism are the
covariant harmonic derivatives. They are defined as follows:

$\mathcal{D}^{++}:=\partial^{++}-4i\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}\partial_{\rho\dot{\rho}}+\theta^{+\hat{\rho}}\partial^{+}_{\hat{\rho}}+\left[(\theta^{+})^{2}-(\bar{\theta}^{+})^{2}\right]\partial_{5},$
(5a)
$\mathcal{D}^{--}:=\partial^{--}-4i\theta^{-\rho}\bar{\theta}^{-\dot{\rho}}\partial_{\rho\dot{\rho}}+\theta^{-\hat{\rho}}\partial^{-}_{\hat{\rho}}+\left[(\theta^{-})^{2}-(\bar{\theta}^{-})^{2}\right]\partial_{5},$
(5b)
$\mathcal{D}^{0}=\partial^{0}+\theta^{+\hat{\rho}}\partial^{-}_{\hat{\rho}}-\theta^{-\hat{\rho}}\partial^{+}_{\hat{\rho}}$
(5c)

and satisfy $SU(2)$ algebraic relations:

$[\mathcal{D}^{++},\mathcal{D}^{--}]=\mathcal{D}^{0},\qquad[\mathcal{D}^{0},\mathcal{D}^{\pm\pm}]=\pm
2\mathcal{D}^{\pm\pm}.$ (6)

Here we used the standard notation for the partial harmonic derivatives:

$\partial^{++}=u^{+i}\frac{\partial}{\partial
u^{-i}},\qquad\partial^{--}=u^{-i}\frac{\partial}{\partial
u^{+i}},\qquad\partial^{0}=u^{+i}\frac{\partial}{\partial
u^{+i}}-u^{-i}\frac{\partial}{\partial u^{-i}}.$ (7)

All known $\mathcal{N}=2$ theories have a superfield formulation in the
harmonic superspace [8]. Here we present two important examples: the ultimate
$\mathcal{N}=2$ multiplet of matter - a hypermultiplet and the simplest
Maxwell gauge multiplet.

Hypermultiplet is described by an unconstrained analytic superfield
$q^{+}(\zeta)$. It contains a doublet of complex scalars $f^{i}$ and a pair of
singlet spinors $\psi_{\alpha},\kappa_{\alpha}$, as well as an infinite set of
auxiliary fields which comes from the harmonic $S^{2}$ expansions:

$q^{+}(\zeta)=f^{i}u^{+}_{i}+\theta^{+\alpha}\psi_{\alpha}+\bar{\theta}^{+}_{\dot{\alpha}}\bar{\kappa}^{\dot{\alpha}}+\text{auxiliary
fields}.$ (8)

So the off-shell hypermultiplet carries
$\mathbf{\boldsymbol{\infty}_{B}+\boldsymbol{\infty}_{F}}$ degrees of freedom.
For massive hypermultiplet (with mass equal to central charge) one must
introduce dependence on $x^{5}$ as
$q^{+}(\zeta,x_{5})=e^{imx^{5}}q^{+}(\zeta)$. The free hypermultiplet action
has the form111Integration measure in the analytic superspace is defined as
$d\zeta^{(-4)}:=d^{4}xd^{2}\theta^{+}d^{2}\bar{\theta}^{+}du$. Harmonic
integration is defined as $\int du\,1=1$, otherwise $0$.:

$S_{free}=-\int
d\zeta^{(-4)}\tilde{q}^{+}\mathcal{D}^{++}q^{+}=-\frac{1}{2}\int
d\zeta^{(-4)}\,q^{+a}\mathcal{D}^{++}q^{+}_{a}.$ (9)

Here we introduced the notation $q^{+a}=(\tilde{q}^{+},q^{+})$,
$q^{+}_{a}=\epsilon_{ab}q^{+b}=(q^{+},-\tilde{q}^{+})$. In the second form we
have manifest Pauli-Gürsey $SU(2)_{PG}$ symmetry (see e.g. [8]). This form is
most convenient for construction of higher-spin cubic vertices. After
eliminating an infinite number of auxiliary fields, the action (9) is reduced
to the sum of free actions for a doublet of scalars and a pair of fermions, so
on shell there survives $4_{B}+4_{F}$ degrees of freedom. Note that both
massive and massless hypermultiplets are described uniformly.

$\mathcal{N}=2$ Maxwell supermultiplet is described by an unconstrained
analytic gauge potential $V^{++}(\zeta)$ with the gauge freedom
$\delta_{\lambda}V^{++}(\zeta)=\mathcal{D}^{++}\lambda(\zeta)$. Using this
freedom one can impose Wess-Zumino gauge:

$V^{++}_{WZ}=-4i\theta^{+}_{\beta}\bar{\theta}^{+}_{\dot{\beta}}A^{\beta\dot{\beta}}+i(\bar{\theta}^{+})^{2}\phi-i(\theta^{+})^{2}\bar{\phi}+(\bar{\theta}^{+})^{2}\theta^{+\beta}\psi_{\beta}^{i}u^{-}_{i}+(\theta^{+})^{2}\bar{\theta}^{+\dot{\beta}}\bar{\psi}_{\dot{\beta}}^{i}u^{-}_{i}+(\theta^{+})^{2}(\bar{\theta}^{+})^{2}D^{(ij)}u^{-}_{i}u^{-}_{j}.$
(10)

Here we are left with the gauge spin $1$ field $A_{\beta\dot{\beta}}$ with the
gauge freedom $\delta A_{\beta\dot{\beta}}\sim\partial_{\beta\dot{\beta}}a$,
the doublet of fermions $\psi^{i}_{\beta}$, a complex scalar $\phi$ and an
auxiliary triplet of real scalar fields $D^{(ij)}$. So off shell there are
$\mathbf{8_{B}+8_{F}}$ degrees of freedom. The free supersymmetric and gauge-
invariant action reads:

$S_{(s=1)}=\int d^{4}xd^{8}\theta du\,V^{++}V^{--},$ (11)

where $V^{--}$ is a solution of zero-curvature condition:

$\mathcal{D}^{++}V^{--}=\mathcal{D}^{--}V^{++}.$ (12)

In Wess-Zumino gauge, after elimination of auxiliary fields, the action is
reduced to the sum of free Maxwell action, free action for doublet of fermions
and Klein-Gordon action for complex scalar.

##  3 $\mathcal{N}=2$ higher-spin multiplets

$\mathcal{N}=2$ higher-spin supermultiplet with highest spin $s\geq 2$ (we
will denote such multiplet as spin $\mathbf{s}$ supermultiplet) is described
by the set of unconstrained analytic gauge potentials 222Spin $\mathbf{1}$
also fits in this construction: we just have to omit all the superfields where
the number of indices is formally negative.:

$h^{++\alpha(s-1)\dot{\alpha}(s-1)}(\zeta),\;\;\;h^{++\alpha(s-2)\dot{\alpha}(s-2)}(\zeta),\;\;\;h^{++\alpha(s-1)\dot{\alpha}(s-2)+}(\zeta),\;\;\;h^{++\dot{\alpha}(s-1)\alpha(s-2)+}(\zeta),$
(13)

where we used notations for symmetrized combinations of indices
$\alpha(s):=(\alpha_{1}\ldots\alpha_{s}),\dot{\alpha}(s):=(\dot{\alpha}_{1}\ldots\dot{\alpha}_{s})$.
These superfields are related by rigid $\mathcal{N}=2$ supersymmetry
transformations:

$\begin{split}&\delta_{\epsilon}h^{++\alpha(s-1)\dot{\alpha}(s-1)}=-4i\big{[}h^{++\alpha(s-1)(\dot{\alpha}(s-2)+}\bar{\epsilon}^{-\dot{\alpha}_{s-1})}-h^{++\dot{\alpha}(s-1)(\alpha(s-2)+}\,\epsilon^{-\alpha_{s-1})}\big{]}\,,\\\
&\delta_{\epsilon}h^{++\alpha(s-2)\dot{\alpha}(s-2)}=2i\big{[}h^{++(\alpha(s-2)\alpha_{s-1})\dot{\alpha}(s-2)+}\epsilon^{-}_{\alpha_{s-1}}+h^{++\alpha(s-2)(\dot{\alpha}(s-2)\dot{\alpha}_{s-1})+}\,\bar{\epsilon}^{-}_{\dot{\alpha}_{s-1}}\big{]},\\\
&\delta_{\epsilon}h^{++\alpha(s-1)\dot{\alpha}(s-2)+}=0,\qquad\qquad\delta_{\epsilon}h^{++\dot{\alpha}(s-1)\alpha(s-2)+}=0.\end{split}$
(14)

The gauge potentials are defined up to a gauge freedom:

$\begin{split}&\delta_{\lambda}h^{++\alpha(s-1)\dot{\alpha}(s-1)}=\mathcal{D}^{++}\lambda^{\alpha(s-1)\dot{\alpha}(s-1)}+4i\big{[}\lambda^{+\alpha(s-1)(\dot{\alpha}(s-2)}\bar{\theta}^{+\dot{\alpha}_{s-1})}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\,\theta^{+(\alpha_{s-1}}\bar{\lambda}^{+\alpha(s-2))\dot{\alpha}(s-1)}\big{]},\\\
&\delta_{\lambda}h^{++\alpha(s-2)\dot{\alpha}(s-2)}=\mathcal{D}^{++}\lambda^{\alpha(s-2)\dot{\alpha}(s-2)}-2i\,\big{[}\lambda^{+(\alpha(s-2)\alpha_{s-1})\dot{\alpha}(s-2)}\theta^{+}_{\alpha_{s-1}}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\,\bar{\lambda}^{+(\dot{\alpha}(s-2)\dot{\alpha}_{s-1})\alpha(s-2)}\bar{\theta}^{+}_{\dot{\alpha}_{s-1}}\big{]},\\\
&\delta_{\lambda}h^{++\alpha(s-1)\dot{\alpha}(s-2)+}=\mathcal{D}^{++}\lambda^{\alpha(s-1)\dot{\alpha}(s-2)+},\\\
&\delta_{\lambda}h^{++\dot{\alpha}(s-1)\alpha(s-2)+}=\mathcal{D}^{++}\bar{\lambda}^{\dot{\alpha}(s-1)\alpha(s-2)+}.\end{split}$
(15)

Using this freedom, one can impose the Wess-Zumino gauge:

$\begin{split}&h_{WZ}^{++\alpha(s-1)\dot{\alpha}(s-1)}=-4i\theta^{+}_{\beta}\bar{\theta}^{+}_{\dot{\beta}}\Phi^{(\beta\alpha(s-1))(\dot{\beta}\dot{\alpha}(s-1))}-4i\theta^{+(\alpha}\bar{\theta}^{+(\dot{\alpha}}\Phi^{\alpha(s-2))\dot{\alpha}(s-2))}\\\
&\qquad\qquad\qquad\qquad+(\bar{\theta}^{+})^{2}\theta^{+\beta}\psi_{\beta}^{\alpha(s-1)\dot{\alpha}(s-1)i}u^{-}_{i}+\,(\theta^{+})^{2}\bar{\theta}^{+\dot{\beta}}\bar{\psi}_{\dot{\beta}}^{\alpha(s-1)\dot{\alpha}(s-1)i}u_{i}^{-}\\\
&\qquad\qquad\qquad\qquad+(\theta^{+})^{2}(\bar{\theta}^{+})^{2}V^{\alpha(s-1)\dot{\alpha}(s-1)(ij)}u^{-}_{i}u^{-}_{j}\,,\\\
&h_{WZ}^{++\alpha(s-2)\dot{\alpha}(s-2)}=-4i\theta^{+}_{\beta}\bar{\theta}^{+}_{\dot{\beta}}C^{(\beta\alpha(s-2))(\dot{\beta}\dot{\alpha}(s-2))}-4i\theta^{+(\alpha}\bar{\theta}^{+(\dot{\alpha}}C^{\alpha(s-3))\dot{\alpha}(s-3))}\\\
&\qquad\qquad\qquad\qquad+(\bar{\theta}^{+})^{2}\theta^{+\beta}\rho_{\beta}^{\alpha(s-2)\dot{\alpha}(s-2)i}u^{-}_{i}+(\theta^{+})^{2}\bar{\theta}^{+\dot{\beta}}\bar{\rho}_{\dot{\beta}}^{\alpha(s-2)\dot{\alpha}(s-2)i}u_{i}^{-}\\\
&\qquad\qquad\qquad\qquad+(\theta^{+})^{2}(\bar{\theta}^{+})^{2}S^{\alpha(s-2)\dot{\alpha}(s-2)(ij)}u^{-}_{i}u^{-}_{j}\,,\\\
&h_{WZ}^{++\alpha(s-1)\dot{\alpha}(s-2)+}=(\theta^{+})^{2}\bar{\theta}^{+}_{\dot{\beta}}P^{\alpha(s-1)\dot{\alpha}(s-2)\dot{\beta}}+\left(\bar{\theta}^{+}\right)^{2}\theta^{+}_{\beta}T^{\dot{\alpha}(s-2)\alpha(s-1)\beta}\\\
&\qquad\qquad\qquad\qquad+(\theta^{+})^{2}(\bar{\theta}^{+})^{2}\chi^{\alpha(s-1)\dot{\alpha}(s-2)i}u^{-}_{i}\,,\\\
&h_{WZ}^{++\dot{\alpha}(s-1)\alpha(s-2)+}=\widetilde{\left(h_{WZ}^{++\alpha(s-1)\dot{\alpha}(s-2)+}\right)}\,.\end{split}$
(16)

These remaining fields form the off-shell $\mathcal{N}=2$ supersymmetric spin
$\mathbf{s}$ multiplet. The residual gauge freedom, which preserves the WZ
gauge form (16), implies the appropriate gauge freedom on the component
fields. As a result, we obtain a set of physical massless spin fields 333For a
review of description of higher-spin fields in the spinor notation see, e.g.,
[9]. $(s,s-1/2,s-1/2,s-1)$ and a set of auxiliary fields444Note that some
auxiliary fields must be properly redefined in terms of physical ones for
ensuring them to be gauge group scalars.:

$\bullet$ Fields
$(\Phi^{\alpha(s)\dot{\alpha}(s)},\Phi^{\alpha(s-2)\dot{\alpha}(s-2)})$ and
$(C^{\alpha(s-1)\dot{\alpha}(s-1)},C^{\alpha(s-3)\dot{\alpha}(s-3)})$
correspond to the massless Fronsdal spin $s$ and $s-1$ fields:

$\begin{split}&\delta\Phi^{\alpha(s)\dot{\alpha}(s)}\sim\partial^{(\alpha(\dot{\alpha}}a^{\alpha(s-1))\dot{\alpha}(s-1))},\qquad\delta\Phi^{\alpha(s-2)\dot{\alpha}(s-2)}\sim\partial_{\beta\dot{\beta}}a^{(\beta\alpha(s-2))(\dot{\beta}\dot{\alpha}(s-2))};\\\
&\delta
C^{\alpha(s-1)\dot{\alpha}(s-1)}\sim\partial^{(\alpha(\dot{\alpha}}b^{\alpha(s-2))\dot{\alpha}(s-2))},\qquad\delta
C^{\alpha(s-3)\dot{\alpha}(s-3)}\sim\partial_{\beta\dot{\beta}}a^{(\beta\alpha(s-3))(\dot{\beta}\dot{\alpha}(s-3))}.\end{split}$
(17)

$\bullet$ Fields
$V^{\alpha(s-1)\dot{\alpha}(s-1)(ij)},S^{\alpha(s-2)\dot{\alpha}(s-2)(ij)}$
are real bosonic auxiliary fields,
$P^{\alpha(s-1)\dot{\alpha}(s-2)\dot{\mu}},$
$T^{\alpha(s-1)\nu\dot{\alpha}(s-2)}$ are complex bosonic auxiliary fields.

$\bullet$ Fields
$\left(\psi^{\alpha(s-1)\dot{\alpha}(s-1)}_{\beta},\bar{\rho}_{\dot{\beta}}^{\alpha(s-2)(\dot{\alpha}(s-3)\dot{\beta})}\right)$
possess gauge freedom characteristic of the doublet of massless spin
$s-\frac{1}{2}$ Fang-Fronsdal fields:

$\delta\psi_{\beta}^{\alpha(s-1)\dot{\alpha}(s-1)}\sim\partial_{\beta}^{(\dot{\alpha}}\xi^{\alpha(s-1)\dot{\alpha}(s-1))},\qquad\delta\bar{\rho}^{\alpha(s-2)(\dot{\alpha}(s-3)\dot{\beta})}_{\dot{\beta}}\sim\partial_{\beta\dot{\beta}}\xi^{(\alpha(s-2)\beta)(\dot{\alpha}(s-3)\dot{\beta})}.$
(18)

$\bullet$ Fields
$\rho^{\alpha(s-1)\dot{\alpha}(s-2)i},\chi^{\alpha(s-1)\dot{\alpha}(s-2)i}$
are auxiliary fermionic fields.

As a result, $\mathcal{N}=2$ spin $\mathbf{s}$ supermultiplet involves
$\mathbf{8(s^{2}+(s-1)^{2})}_{B}+\mathbf{8(s^{2}+(s-1)^{2})}_{F}$ off-shell
degrees of freedom555On shell there survive $4_{B}+4_{F}$ degrees of freedom
for each $\mathcal{N}=2$ spin $\mathbf{s}$.. In the simplest $s=2$ case one
reproduces the off-shell multiplet of the “minimal” $\mathcal{N}=2$ Einstein
supergravity.

The manifestly $\mathcal{N}=2$ supersymmetric and gauge invariant linearized
action has the universal form for any ${\bf s}$:

$\begin{split}S_{(s)}=(-1)^{s+1}\int d^{4}xd^{8}\theta
du\,\Big{\\{}&G^{++\alpha(s-1)\dot{\alpha}(s-1)}G^{--}_{\alpha(s-1)\dot{\alpha}(s-1)}\\\
&\qquad+4G^{++\alpha(s-2)\dot{\alpha}(s-2)}G^{--}_{\alpha(s-2)\dot{\alpha}(s-2)}\Big{\\}},\end{split}$
(19)

where we have introduced $\mathcal{N}=2$ supersymmetry-covariant fields
($\delta_{\epsilon}G^{++\dots}=0$)

$\begin{split}G^{++\alpha(s-1)\dot{\alpha}(s-1)}=h^{++\alpha(s-1)\dot{\alpha}(s-1)}+4i\big{[}&h^{++\alpha(s-1)(\dot{\alpha}(s-2)+}\bar{\theta}^{-\dot{\alpha}_{s-1})}\\\
&-h^{++\dot{\alpha}(s-1)(\alpha(s-2)+}\,\theta^{-\alpha_{s-1})}\big{]},\\\
G^{++\alpha(s-2)\dot{\alpha}(s-2)}=h^{++\alpha(s-2)\dot{\alpha}(s-2)}-2i\big{[}&h^{++\alpha(s-2)\alpha_{s-1})\dot{\alpha}(s-2)+}\theta^{-}_{\alpha_{s-1}}\\\
&+h^{++\alpha(s-2)(\dot{\alpha}(s-2)\dot{\alpha}_{(s-1)})+}\,\bar{\theta}^{-}_{\dot{\alpha}_{s-1}}\big{]},\end{split}$
(20)

and the negatively charged potentials are related to the basic ones by the
appropriate harmonic zero-curvature conditions:

$\begin{split}\mathcal{D}^{++}G^{--\alpha(s-1)\dot{\alpha}(s-1)}=\mathcal{D}^{--}G^{++\alpha(s-1)\dot{\alpha}(s-1)},\\\
\mathcal{D}^{++}G^{--\alpha(s-2)\dot{\alpha}(s-2)}=\mathcal{D}^{--}G^{++\alpha(s-2)\dot{\alpha}(s-2)}.\end{split}$
(21)

Using the explicit form of the WZ gauge (16) and eliminating the auxiliary
fields, one can verify that the action (19) at the component level is reduced
to the sum of the free Fronsdal actions for the spins $s$ and $s-1$, as well
as two Fang-Fronsdal actions for the spin $s-1/2$.

##  4 $\mathcal{N}=2$ supersymmetric interaction of higher spins with
hypermultiplet

Using the spin $\mathbf{s}$ analytic gauge potentials (13), one can define the
analytic differential operator:

$\begin{split}\hat{\mathcal{H}}^{++}_{(s)}:=&h^{++\alpha(s-1)\dot{\alpha}(s-1)}\partial^{s-1}_{\alpha(s-1)\dot{\alpha}(s-1)}+h^{++\alpha(s-1)\dot{\alpha}(s-2)+}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\partial^{-}_{\alpha}\\\
&+h^{++\alpha(s-2)\dot{\alpha}(s-1)+}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\partial^{-}_{\dot{\alpha}}+h^{++\alpha(s-2)\dot{\alpha}(s-2)}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\partial_{5}.\end{split}$
(22)

It is direct to check that this operator is invariant under $\mathcal{N}=2$
supersymmetry (1), (3), (14). For spin $s$, it contains $s-1$ derivative.

The most general cubic hypermultiplet coupling to spin $\mathbf{s}$ higher-
spin consistent with analyticity and $\mathcal{N}=2$ supersymmetry has the
form:

$S_{free}+S_{cubic,s}=-\frac{1}{2}\int
d\zeta^{(-4)}\,q^{+a}\left(\mathcal{D}^{++}+\kappa_{s}\hat{\mathcal{H}}^{++}_{(s)}(J)^{P(s)}\right)q^{+}_{a}.$
(23)

Here we used the notation $P(s):=\frac{1-(-1)^{s}}{2}$ and $\kappa_{s}$ is the
spin $\mathbf{s}$ coupling constant. The generator $J$ acts as
$J\tilde{q}^{+}:=i\tilde{q}^{+}$, $Jq^{+}:=-iq^{+}$. The reason for
introducing this generator is that without it the vertices for odd spins prove
to be identically zero666This is an analogue of the well-known fact that it is
impossible to construct a minimal interaction of spin 1 (and all odd spins)
with a real scalar field. Generator $J$ indicates that the fields $q^{+}$ and
$\tilde{q}^{+}$ ( $f^{i}$ and $\bar{f}^{i}$ at the component level) have the
opposite charges.. So for odd spins this interaction explicitly breaks
$SU(2)_{PG}$.

Under the action of gauge transformations (15), operator
$\hat{\mathcal{H}}^{++}_{(s)}$ transforms as:

$\delta_{\lambda}\hat{\mathcal{H}}^{++}_{(s)}=[\mathcal{D}^{++},\hat{\Lambda}_{(s)}],$
(24)

where

$\begin{split}\hat{\Lambda}_{(s)}:=&\lambda^{\alpha(s-1)\dot{\alpha}(s-1)}\partial^{s-1}_{\alpha(s-1)\dot{\alpha}(s-1)}+\lambda^{\alpha(s-1)\dot{\alpha}(s-2)+}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\partial^{-}_{\alpha}\\\
&+\lambda^{\alpha(s-2)\dot{\alpha}(s-1)+}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\partial^{-}_{\dot{\alpha}}+\lambda^{\alpha(s-2)\dot{\alpha}(s-2)}\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\partial_{5}.\end{split}$
(25)

One can check that the action (23) is invariant to the leading order in
$\kappa_{s}$, if the gauge transformation of the hypermultiplet is:

$\delta^{(s)}_{\lambda}q^{+}_{a}=-\frac{\kappa_{s}}{2}\left\\{\hat{\Lambda}_{(s)}^{\alpha(s-2)\dot{\alpha}(s-2)}+\frac{1}{2}\Omega_{(s)}^{\alpha(s-2)\dot{\alpha}(s-2)},\partial^{s-2}_{\alpha(s-2)\dot{\alpha}(s-2)}\right\\}J^{P(s)}q^{+}_{a}.$
(26)

Here
$\hat{\Lambda}_{(s)}^{\alpha(s-2)\dot{\alpha}(s-2)}=\lambda^{\alpha(s-2)\dot{\alpha}(s-2)M}\partial_{M}$,
$\Omega_{(s)}^{\alpha(s-2)\dot{\alpha}(s-2)}=(-1)^{P(M)}\partial_{M}\lambda^{\alpha(s-2)\dot{\alpha}(s-2)M}$,
where $M:=(\alpha\dot{\alpha},\alpha+,\dot{\alpha}+)$, and $P(M)$ is defined
as $P(\alpha\dot{\alpha})=0,P(\alpha+)=P(\dot{\alpha}+)=1$.

The cubic vertices constructed are invariant only to the leading order. The
variation of the cubic part of the action (23) under (26) in the order
$\sim\kappa_{s}^{2}$ includes terms which are quadratic in spinor derivatives
and so cannot be compensated by cubic vertices of such a type. The only
exception is the case of $s=2$, i.e. $\mathcal{N}=2$ Einstein supergravity. In
this case, by a non-Abelian deformation of the prepotential gauge
transformation,

$\delta_{\lambda}\hat{\mathcal{H}}^{++}_{(s=2)}=[\mathcal{D}^{++},\hat{\Lambda}_{(s=2)}]\quad\to\quad\delta^{full}_{\lambda}\hat{\mathcal{H}}^{++}_{(s=2)}=[\mathcal{D}^{++}+\kappa_{2}\hat{\mathcal{H}}^{++}_{(s=2)},\hat{\Lambda}_{(s=2)}],$
(27)

the complete invariance of the action (23) can be ensured.

Perhaps for $s\geq 3$ a non-linear gauge invariance can be achieved by
introducing a new type of cubic vertices and a new type of gauge superfield
potentials. However, the meaning of such potentials, their gauge freedom and
their field contents are unclear at present and require further study. It is
not unlikely that they are nontrivially related to $\mathcal{N}=2$ gauge
potentials of the half-integer higher spins, which still remain to be
constructed. By introducing an infinite tower of $\mathcal{N}=2$ higher-spin
prepotentials (with both integer and half-integers spins), one could expect
the complete non-Abelian invariance. It is interesting to figure out possible
parallels of these assumptions with the relevant discussions in [10].

Note that, for a special choice of the transformation parameters777Such a set
of parameters is a $\mathcal{N}=2$ supersymmetric generalization of the
Killing tensor field (see, e.g., [10, 11]) and can be interpreted as a
“$\mathcal{N}=2$ Killing supertensor”., such that
$[\mathcal{D}^{++},\hat{\Lambda}_{(s)}]=0$, we obtain rigid “higher-spin”
supersymmetry transformations of the free hypermultiplet action with $s-1$
derivatives888In the $s=2$ case these transformations correspond to rigid
$\mathcal{N}=2$ supersymmetry.. Thus the interactions constructed have a
Noether origin and so can be viewed as gauging of the “higher-spin” rigid
supersymmetries of the free hypermultiplet through introducing the appropriate
analytic gauge potentials. Also it is possible to reconstruct both gauge
transformations and supersymmetry transformations of the $\mathcal{N}=2$
higher-spin potentials. All these reasonings will be helpful while
constructing $\mathcal{N}=2$ superconformal higher-spin gauge potentials in
the next section.

##  5 $\mathcal{N}=2$ superconformal higher-spin multiplets

$\mathcal{N}=2$ superconformal transformations non-trivially act on harmonics
[8] (in contrast to $\mathcal{N}=2$ supersymmetry, see (1)). So we extend the
set of indices $M$ by harmonic indices
$\Rightarrow\;M:=(\alpha\dot{\alpha},\alpha+,\dot{\alpha}+,++)$, $P(++)=0$.
Free off-shell action of massless hypermultiplet (9) is invariant under
$\mathcal{N}=2$ superconformal transformations (here it is useful to employ a
passive form of transformations):

$\delta_{sc}q^{+}_{a}=-\hat{\Lambda}q^{+}_{a}-\frac{1}{2}\Omega
q^{+}_{a},\qquad\hat{\Lambda}=\lambda^{M}\partial_{M},\quad\Omega=(-1)^{M}\partial_{M}\lambda^{M}$
(28)

with special rigid parameters:

$\begin{cases}\lambda_{sc}^{\alpha\dot{\alpha}}=&a^{\alpha\dot{\alpha}}-4i\left(\epsilon^{\alpha
i}\bar{\theta}^{+\dot{\alpha}}+\theta^{+\alpha}\bar{\epsilon}^{\dot{\alpha}i}\right)u^{-}_{i}\par+x^{\dot{\alpha}\rho}k_{\rho\dot{\rho}}x^{\dot{\rho}\alpha}+bx^{\alpha\dot{\alpha}}\\\
&-4i\theta^{+\alpha}\bar{\theta}^{+\dot{\alpha}}\lambda^{(ij)}u^{-}_{i}u^{-}_{j}-4i\left(x^{\alpha\dot{\rho}}\eta_{\dot{\rho}}^{i}\bar{\theta}^{+\dot{\alpha}}+\theta^{+\alpha}\eta^{i}_{\rho}x^{\rho\dot{\alpha}}\right)u^{-}_{i},\\\
\lambda^{\alpha+}_{sc}=&\epsilon^{\alpha
i}u^{+}_{i}+\frac{1}{2}\theta^{+\alpha}(b+i\gamma)+x^{\alpha\dot{\beta}}k_{\beta\dot{\beta}}\theta^{+\beta}+x^{\alpha\dot{\alpha}}\eta^{i}_{\dot{\alpha}}u^{+}_{i}\\\
&+\theta^{+\alpha}\left(\lambda^{(ij)}u^{+}_{i}u^{-}_{j}+4i\theta^{+\rho}\eta^{i}_{\rho}u^{-}_{i}\right),\\\
\bar{\lambda}^{\dot{\alpha}+}_{sc}=&\epsilon^{\dot{\alpha}i}u^{+}_{i}+\frac{1}{2}\bar{\theta}^{+\dot{\alpha}}(b-i\gamma)+x^{\dot{\alpha}\beta}k_{\beta\dot{\beta}}\bar{\theta}^{+\dot{\beta}}+x^{\alpha\dot{\alpha}}\eta^{i}_{\alpha}u^{+}_{i}\\\
&+\bar{\theta}^{+\dot{\alpha}}\left(\lambda^{(ij)}u^{+}_{i}u^{-}_{j}-4i\bar{\theta}^{+\dot{\rho}}\eta^{i}_{\dot{\rho}}u^{-}_{i}\right),\\\
\lambda^{++}_{sc}=&\lambda^{ij}u^{+}_{i}u^{+}_{j}+4i\theta^{+\alpha}\bar{\theta}^{+\dot{\alpha}}k_{\alpha\dot{\alpha}}+4i\left(\theta^{+\alpha}\eta^{i}_{\alpha}+\eta^{i}_{\dot{\alpha}}\bar{\theta}^{+\dot{\alpha}}\right)u^{+}_{i}.\end{cases}$
(29)

These parameters satisfy the system of equations
$[\mathcal{D}^{++},\hat{\Lambda}]=\lambda^{++}\mathcal{D}^{0}$.

Following the discussion in the end of previous section, one can determine
$\mathcal{N}=2$ higher-spin superconformal gauge potentials by requiring
superconformal invariance of general cubic vertex999An alternative approach to
determining the structure of $\mathcal{N}$-extended superconformal higher-spin
gauge prepotentials (of Mezincescu type) uses the method of supercurrent
multiplets and starts from the Fayet-Sohnius on shell hypermultiplets [12].
Our approach, like in the non-conformal case [1] \- [3], uses the analytic
gauge vielbein-type potentials covariantizing the ${\cal D}^{++}$ derivative,
and the off-shell $q^{+}$ hypermultiplets as most general ${\cal N}=2$ matter
multiplets. An interesting task is to learn how $\mathcal{N}=2$ Kuzenko-
Raptakis prepotentials emerge from our analytic potentials in a special
harmonic-independent gauge.. The analyticity and superconformal invariance
taken together imply the following general form of cubic interaction:

$S_{free}+S_{sc-cubic,s}=-\frac{1}{2}\int
d\zeta^{(-4)}\,q^{+a}\left(\mathcal{D}^{++}+\hat{\mathbb{H}}^{++}_{(s)}(J)^{P(s)}\right)q^{+}_{a},$
(30)

where we introduced the analytic differential operator of degree $s-1$ with
the odd number of superspace derivatives

$\hat{\mathbb{H}}^{++}_{(s)}:=h^{++M_{1}\dots
M_{s-1}}\partial_{M_{s-1}}\dots\partial_{M_{1}}+h^{++M_{1}\dots
M_{s-3}}\partial_{M_{s-3}}\dots\partial_{M_{1}}+\dots.$ (31)

Various parts of the operator can be viewed as corresponding to the spins
$\mathbf{s},\mathbf{s-2},\dots$. This operator generalizes the non-conformal
operator $\hat{\mathcal{H}}^{++}_{(s)}$ given in (22) in three aspects. First
of all, it contains all possible types of indices since the derivative
$\partial_{\alpha\dot{\alpha}}$ is non-trivially transformed under the
superconformal group (29). Secondly, because of the presence of a weight
factor in (28), terms with fewer derivatives are necessarily present. Thirdly,
no the irreducibility conditions with respect to the Lorentz indices are
imposed in advance on the potentials, as distinct from a symmetrization in the
non-conformal case (13).

From now on, to avoid identical terms, we assume that the indices of the
fields are ordered according to the rule
$M=(\alpha\dot{\alpha},\alpha+,\dot{\alpha}+,++)$, i.e. $M_{1}\geq
M_{2}\dots\geq M_{s-1}$ and prepotentials satisfy the permutation condition
$h^{++M_{1}\dots M_{n}M_{k}\dots M_{r}}=(-1)^{P(M_{k})P(M_{n})}h^{++M_{1}\dots
M_{k}M_{n}\dots M_{r}}$.

For lucidity, we give the explicit form of the superconformal transformation
law of higher-spin prepotentials, which ensures the superconformal invariance
of the cubic vertex:

$\delta_{sc}\hat{\mathbb{H}}^{++}_{(s)}=[\hat{\mathbb{H}}^{++}_{(s)},\hat{\Lambda}]+\frac{1}{2}[\hat{\mathbb{H}}^{++}_{(s)},\Omega].$
(32)

Here we assume the appropriate integration by parts which brings the right
side of (32) to the form (31). All the peculiarities listed above are in
conjunction with this formula.

The gauge freedom of action (30) is realized by the transformations:

$\delta^{(s,k)}_{\lambda}q^{+}_{a}=-\frac{\kappa_{s}}{2}\left\\{\hat{\Lambda}^{M_{1}\dots
M_{k-2}}+\frac{1}{2}\Omega^{M_{1}\dots
M_{k-2}},\partial_{M_{k-2}}\dots\partial_{M_{1}}\right\\}_{AGB}(J)^{P(s)}q^{+}_{a},$
(33a)
$\delta^{(s,k)}_{\lambda}\hat{\mathbb{H}}^{++}_{(s)}=\frac{1}{2}\left[\mathcal{D}^{++},\left\\{\hat{\Lambda}^{M_{1}\dots
M_{k-2}},\partial_{M_{k-2}}\dots\partial_{M_{1}}\right\\}_{AGB}\right],$ (33b)

where in (33b) we performed the appropriate integration by parts. We also
define operators $\hat{\Lambda}^{M_{1}\dots M_{k-2}}:=\lambda^{M_{1}\dots
M_{k-2}N}\partial_{N}$, $\Omega^{M_{1}\dots
M_{k-2}}:=(-1)^{P(N)}\partial_{N}\lambda^{NM_{1}\dots M_{k-2}}$. The $k$-th
parameter takes the values $s,s-2.\dots$. The different values of $k$
correspond to the gauge freedom for different spin contributions appearing in
the operator (31). Anti-graduate bracket is defined as
$\\{F_{1},F_{2}\\}_{AGB}:=[F_{1},F_{2}]$ for fermionic objects,
$\\{B_{1},B_{2}\\}_{AGB}:=\\{B_{1},B_{2}\\}$ for bosonic and
$\\{B,F\\}_{AGB}:=\\{B,F\\}$ for bosonic and fermionic ones.

If we require
$q^{+a}\delta^{(s,k)}_{\lambda}\hat{\mathbb{H}}^{++}_{(s)}(J)^{P(s)}q^{+}_{a}=0$,
we obtain the conditions on “superconformal higher-spin” rigid symmetries of
the free hypermultiplet (in the $s=2$ case these symmetry transformations are
just $\mathcal{N}=2$ superconformal transformations). The corresponding
parameters $\lambda^{\dots}$ which solve this equation can be treated as
$\mathcal{N}=2$ spin $s$ “superconformal Killing supertensor”.

Using the gauge freedom (33b), one can impose WZ gauge101010For the $s=3$ case
we have derived this result directly, by exploring the gauge freedom (33b).
The WZ gauge for general $s$ was conjectured as a natural generalization of
the spin $3$ WZ gauge.:

$\begin{split}&h_{WZ}^{++\alpha(s-1)\dot{\alpha}(s-1)}=-4i\theta^{+}_{\rho}\bar{\theta}^{+}_{\dot{\rho}}\Phi^{(\rho\alpha(s-1))(\dot{\rho}\dot{\alpha}(s-1))}-(\bar{\theta}^{+})^{2}\theta^{+}_{\rho}\psi^{(\rho\alpha(s-1))\dot{\alpha}(s-1)i}u_{i}^{-}\\\
&\;\;\;\;\qquad\qquad\qquad\quad-(\theta^{+})^{2}\bar{\theta}^{+}_{\dot{\rho}}\bar{\psi}^{\alpha(s-1)(\dot{\alpha}(s-1)\dot{\rho})i}u_{i}^{-}+(\theta^{+})^{2}(\bar{\theta}^{+})^{2}V^{\alpha(s-1)\dot{\alpha}(s-1)ij}u^{-}_{i}u^{-}_{j}\,,\\\
&h_{WZ}^{++\alpha(s-1)\dot{\alpha}(s-2)+}=(\theta^{+})^{2}\bar{\theta}^{+}_{\dot{\nu}}P^{\alpha(s-1)(\dot{\alpha}(s-2)\dot{\nu})}+(\bar{\theta}^{+})^{2}\theta^{+}_{\nu}T^{(\alpha(s-1)\nu)\dot{\alpha}(s-2)}\\\
&\qquad\qquad\qquad\qquad\qquad\quad\qquad\qquad\qquad\quad\;\;\,+(\theta^{+})^{4}\chi^{\alpha(s-1)\dot{\alpha}(s-2)i}u_{i}^{-}\,,\\\
&h_{WZ}^{++\alpha(s-2)\dot{\alpha}(s-1)+}=\widetilde{h_{WZ}^{++\alpha(s-1)\dot{\alpha}(s-2)+}}\,,\\\
&h_{WZ}^{(+4)\alpha(s-2)\dot{\alpha}(s-2)}=(\theta^{+})^{2}(\bar{\theta}^{+})^{2}D^{\alpha(s-2)\dot{\alpha}(s-2)}\,.\end{split}$
(34)

From (33b) it also follows, that all other potentials are purely gauge degrees
of freedom. In the gauge that we have fixed, they are set to zero. In
principle, it is possible to initially fix the gauge in which these fields are
zero, but in this case the superconformal group will be implemented non-
linearly. This situation is typical for the conformal theory of higher spins,
see, e.g., [13].

The $\mathcal{N}=2$ superconformal higher-spin multiplet displays a number of
new features in comparison to the non-conformal case. First of all, there are
no auxiliary fields in the multiplet for $s\geq 3$ – all fields prove to be
the gauge fields. Secondly, the physical degrees of freedom for each spin in
the multiplet are described by one irreducible field and their gauge
transformation laws have a slightly different form111111Some fields require
proper redefinitions to avoid contributions of other fields in the relevant
transformation laws:

$\bullet$ Bosonic fields $\Phi^{\alpha(s)\dot{\alpha}(s)}$,
$V^{\alpha(s-1)\dot{\alpha}(s-1)ij}$, $P^{(\alpha(s-1)\dot{\alpha}(s-1))}$,
$D^{\alpha(s-2)\dot{\alpha}(s-2)}$ are conformal Fradkin-Tseytlin fields [14]
with gauge freedom of the form:

$\delta\Phi^{\alpha(s)\dot{\alpha}(s)}\sim\partial^{(\alpha(\dot{\alpha}}a^{\alpha(s-1))\dot{\alpha}(s-1))},\quad\delta
V^{\alpha(s-1)\dot{\alpha}(s-1)ij}\sim\partial^{(\alpha(\dot{\alpha}}v^{\alpha(s-2))\dot{\alpha}(s-2))(ij)},\dots.$
(35)

$\bullet$ Complex field $T^{\alpha(s)\dot{\alpha}(s-2)}$ have the gauge
freedom:

$\delta
T^{\alpha(s)\dot{\alpha}(s-2)}\sim\partial^{(\alpha(\dot{\alpha}}t^{\alpha(s-1))\dot{\alpha}(s-3))}.$
(36)

This is more general type of conformal field, see, e.g., [15]. In the simplest
non-trivial $s=3$ case this field is equivalent to the traceless hook gauge
field $T^{a[bc]}$ with the algebraic symmetries corresponding to the simplest
hook Young diagram $\tiny\yng(2,1)$, see, e.g., [16].

$\bullet$ Fields $\psi^{\alpha(s)\dot{\alpha}(s-1)i}$ and
$\chi^{\alpha(s-1)\dot{\alpha}(s-2)i}$ are conformal fermionic gauge fields:

$\delta\psi^{\alpha(s)\dot{\alpha}(s-1)i}\sim\partial^{(\alpha(\dot{\alpha}}\xi^{\alpha(s-1))\dot{\alpha}(s-2))i},\quad\delta\chi^{\alpha(s-1)\dot{\alpha}(s-2)i}\sim\partial^{\alpha\dot{\alpha}}\zeta^{\alpha(s-2))\dot{\alpha}(s-3))i}.$
(37)

Finally, the $\mathcal{N}=2$ superconformal spin $\mathbf{s}$ multiplet is
encompassed by $\mathbf{8(2s-1)_{B}+8(2s-1)_{F}}$ off-shell degrees of
freedom. The simplest $s=1,2$ multiplets (in contrast to $s\geq 3$) contain
auxiliary fields and comprise $\mathcal{N}=2$ Maxwell and conformal
supergravity (Weyl) gauge multiplets.

The action (30) with gauge transformations (33) is consistent in the first
order in $\kappa_{s}$. In the conformal case one can construct a fully
consistent coupling of an infinite tower of higher-spin potentials to the
hypermultiplet by applying to a non-abelian deformation of gauge freedom.
Also, the action constructed in this way can be consistently lifted to an
arbitrary background of $\mathcal{N}=2$ superconformal gravity. More detailed
discussion of these issues will be presented in [4].

##  6 Conclusion

In conclusion, we have found that the principle of Grassmann harmonic
analyticity plays a key role both in the off-shell formulation of
$\mathcal{N}=2$ non-conformal multiplets of higher spins and $\mathcal{N}=2$
superconformal higher-spins. All the fundamental gauge potentials are analytic
superfields, which, being combined with $\mathcal{N}=2$ supersymmetry and
conformal supersymmetry , significantly limits the possible form of the
interaction Lagrangians, as the example of a cubic vertex with a
hypermultiplet demonstrates. This opens up a wide range of new tasks. Let us
mention some:

  * •

Massive $\mathcal{N}=2$ higher spins

Surprisingly, the problem of constructing supersymmetric massive theories of
higher spins is more difficult compared to massless theories [17]. The off-
shell $4D,\mathcal{N}=1$ superfield theory of massive higher spins has been
elaborated quite recently [18]. The formulations of such theories look rather
cumbersome. One would expect that $\mathcal{N}=2$ theory of massive higher
spins in HSS would have more elegant formulation.

  * •

Induced actions

The vertices (23) and (30) can be used to construct effective theories after
calculation of the functional integral over hypermultiplets (see, e.g., [19]
for a general discussion of higher-spin effective actions). This would provide
a way of building consistent theories of (superconformal) $\mathcal{N}=2$
higher spins.

  * •

Component reduction

The non-conformal vertex (23) is completely self-consistent for $s=2$. The
infinite tower of $\mathcal{N}=2$ superconformal higher spins interacting with
the hypermultiplet is also completely self-consistent, as discussed above. The
constructed vertices are cubic in superfields. An interesting task is the
component reduction of such theories and exploration of the mechanism of
eliminating the hypermultiplet auxiliary fields, leading to nonlinearities at
the component level (see recent [20] for the related discussion).

Acknowledgements

The authors thank S.M. Kuzenko and E.S.N. Raptakis for useful correspondence.
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”.

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