smalltableaux aainstitutetext: Bogoliubov Laboratory of Theoretical Physics,
JINR,
141980 Dubna, Moscow region, Russia bbinstitutetext: Moscow Institute of
Physics and Technology,
141700 Dolgoprudny, Moscow region, Russia

# Off-shell invariants of linearized $4D,\mathcal{N}=2$ supergravity in the
harmonic approach

Evgeny Ivanov a,b Nikita Zaigraev<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Using the harmonic superspace approach, we construct, at the linearized level,
$\mathcal{N}=2$ supersymmetric curvatures generalizing scalar curvature, Ricci
curvature and Weyl tensor. These supercurvatures are the building blocks of
various linearized $4D,\,\mathcal{N}=2$ Einstein supergravity invariants. The
supercurvatures involving the scalar and Ricci curvatures are analytic
harmonic ${\cal N}=2$ superfields, while the Weyl supertensor is a chiral
$\mathcal{N}=2$ superfield. As the basic distinguished feature of our
construction, all these objects are expressed through the fundamental analytic
gauge prepotentials $h^{++M},M=(\alpha\dot{\alpha},+\alpha,+\dot{\alpha},5).$
The related characteristic features are the heavy use of harmonic derivatives
and harmonic zero-curvature equations. On a number of instructive examples, we
describe the component reduction of the superfield invariants constructed.

##  1 Introduction

Among various generalizations of Einstein’s theory of gravity, supergravity
and higher-curvature gravity are of most attention for long. Supergravity
naturally unifies fields of different spins within a single theory and leads
to essential improvements of UV behavior (see, e.g., Freedman:2012zz ;
Deser:1977yyz ). Theories with higher curvatures also exhibit a number of non-
trivial quantum properties, e.g.: (a) modification of the Gilbert-Einstein
action by terms quadratic in curvature yields a renormalizable theory
Stelle:1976gc ; (b) the $R+R^{2}$ Starobinsky theory Starobinsky:1980te  is
ghost free Whitt:1984pd  and is exploited in models of cosmological inflation;
(c) the pure $R^{2}$ gravity (without $R$-term) is also ghost-free Alvarez-
Gaume:2015rwa . It is of significant interest to study the supergravity
theories involving higher-curvature terms in the bosonic limit Cecotti:1987sa
. The invariants of the higher-derivative supergravities find applications as
possible supergravity counterterms, are relevant to the black hole entropy
problem, have applications in cosmology and also provide information on the
relevant conformal anomalies. Supersymmetric generalizations of the terms
constructed from curvatures are also of interest on their own as they can shed
more light on the intrinsic algebraic and geometric structure of diverse off-
shell formulations of supergravity. An overview of applications of the higher-
order supergravity invariants and a detailed discussion of methods of their
construction can be found, e.g., in ref. Ozkan:2024euj .

The most efficient and convenient way to deal with supersymmetric theories is
the superspace approach Wess:1992cp ; 1001 ; Buchbinder:1998qv . In this
paper, we consider linearized higher-order invariants of $\mathcal{N}=2$
supergravity in the harmonic superspace Galperin:1984av . Among all the
covariant superspace approaches to $\mathcal{N}=2$ supergravity (for review,
see Kuzenko:2022ajd ) the formulation in harmonic superspace has a number of
advantages. The main merit of such a formulation is the manifest presence of
$\mathcal{N}=2$ supergravity prepotentials, which have a natural geometric
interpretation as vielbeins covariantizing the derivatives with respect to
harmonic variables. This way, the harmonic formulation of $\mathcal{N}=2$
supergravities non-trivially generalizes the geometric Ogievetsky-Sokatchev
approach to $\mathcal{N}=1$ supergravity Ogievetsky:1978mt .

The $\mathcal{N}=2$ supergravity unconstrained prepotentials appear as
analytic vielbeins in the covariant harmonic derivative which preserves the
notion of harmonic analyticity:

$\mathfrak{D}^{++}=\mathcal{D}^{++}+h^{++\alpha\dot{\alpha}}\partial_{\alpha\dot{\alpha}}+h^{++\alpha+}\partial^{-}_{\alpha}+h^{++\dot{\alpha}+}\partial^{-}_{\dot{\alpha}}+h^{++5}\partial_{5}.$
(1.1)

The prepotentials with a clear geometric origin remain “terra incognita” in
all other formulations of $\mathcal{N}=2$ supergravity111The basic difference
from the Mezincescu-type Mez  prepotentials is that the latter carry some
gauge invariances of non-geometric character which admit no a direct
generalization to the full nonlinear level (see, e.g., discussion in 18 )..
Meanwhile, the presence of prepotentials is a necessary ingredient for
studying the equations of motion and quantizing the theory in a manifestly
supersymmetric manner. Another striking feature of the harmonic superspace is
the possibility to construct actions that do not admit formulations in the
conventional superspaces.

Despite the active developments of superfield formulations of extended
supergravity theories for recent years, not too many studies were specially
devoted to the harmonic superspace formulation of $\mathcal{N}=2$ supergravity
and so many relevant questions remained opened. In this regard, it makes sense
to start by briefly listing the main works on the subject (for a comprehensive
review, see Ivanov:2022vwc ; 18 ).

$\mathcal{N}=2$ harmonic superspace was introduced in Galperin:1984av  as the
most adequate superspace for description of general theories with manifest
off-shell $\mathcal{N}=2$ supersymmetry. In Galperin:1984av , $\mathcal{N}=2$
Einstein supergravity analytic prepotentials and the corresponding gauge group
were discovered. The invariant nonlinear action for $\mathcal{N}=2$ Einstein
supergravity was built by Galperin, Ky and Sokatchev in Galperin:1987em . In
the accompanying work Galperin:1987ek  there was constructed a multiplet of
$\mathcal{N}=2$ conformal supergravity ($\mathcal{N}=2$ Weyl multiplet) in
harmonic superspace and various versions of $\mathcal{N}=2$ Einstein
supergravity were recovered through the appropriate superconformal harmonic
compensators. In particular, the “principal” version of $\mathcal{N}=2$
Einstein supergravity with an infinite number of auxiliary fields was
introduced. In this version, an off-shell hypermultiplet is used as a
superconformal compensator (in addition to the $\mathcal{N}=2$ vector
multiplet) and just this version permits the most general interaction
Lagrangians of hypermultiplet matter in the $\mathcal{N}=2$ supergravity
background. The harmonic superspace construction of general off-shell
hypermultiplet couplings in $\mathcal{N}=2$ supergravity background and their
relation to quaternionic-Kähler geometry were elaborated in a series of papers
Galperin:1992pj ; Ivanov:1998de ; Ivanov:1998ih ; Ivanov:1999vg . The
linearized action for $\mathcal{N}=2$ Einstein supergravity was constructed by
Zupnik Zupnik:1998td  and recently it was generalized to $\mathcal{N}=2$
higher spin supergravities in Buchbinder:2021ite . The interrelations between
$\mathcal{N}=2$ supercurrents and supergravity prepotentials were discussed in
Kuzenko:1999pi ; Butter:2010sc . More recently Butter:2015nza , there was
suggested a new covariant approach to the harmonic superspace as a symbiosis
of the latter with conformal $4D,{\cal N}=2$ superspace Butter:2011sr .
Nevertheless, many issues related to the harmonic superspace formulation of
$\mathcal{N}=2$ supergravity have never been addressed. In particular, this
concerns the construction of off-shell higher-derivative $\mathcal{N}=2$
supergravity invariants in the harmonic approach.

On the other hand, quite a few remarkable results in this area were achieved
in other superfield approaches to $\mathcal{N}=2$ supergravity. The
$\mathcal{N}=2$ curvature-squared off-shell invariants were constructed in
Butter:2013lta  and Kuzenko:2015jxa  (see also review Kuzenko:2017uqg ), based
upon the formulation of $\mathcal{N}=2$ conformal supergravity in $SU(2)$
superspace Kuzenko:2008ep  and $\mathcal{N}=2$ conformal superspace
Butter:2011sr . The $\mathcal{N}=2$ supersymmetrizations of $R^{4}$ invariants
were considered in Moura:2002ip ; deWit:2010za . To the best of our knowledge,
the higher-derivative $\mathcal{N}=2$ supergravity invariants beyond the
quartic order have never been constructed in the off-shell superfield
approaches.

Note that the construction of the on-shell $\mathcal{N}$-extended supergravity
invariants in harmonic superspace have also been extensively discussed in a
number of papers devoted to the on-shell extended supergravity counterterms.
For example, in Drummond:2003ex  some invariants of $4D,\mathcal{N}=8$
linearized supergravity were constructed; the on-shell harmonic superspace
linearized counterterms ($\partial^{2(\mathcal{N}-4)}R^{4}$) for
$\mathcal{N}\geq 4$ supergravities were presented in Bossard:2011tq . A short
summary of the results on such invariants can be found in a review
Ozkan:2024euj . For a recent discussion of these on-shell superinvariants at
the nonlinear level, see, e.g., Kallosh:2023pkr .

In this paper, we construct off-shell higher-derivative $\mathcal{N}=2$
supergravity invariants in harmonic superspace at the linearized level,
consistently applying the whole machinery of the harmonic approach. The main
distinction of this scheme from the more conventional superspace methods
Butter:2013lta ; Kuzenko:2015jxa ; Kuzenko:2017uqg ; Moura:2002ip ;
deWit:2010za  is that its basic building blocks are directly related to the
unconstrained analytical prepotentials underlying the structure of curved
$\mathcal{N}=2$ harmonic superspace and encapsulating the off-shell
supergravity multiplet in Wess-Zumino (WZ) gauge. This radically simplify
deducing the component forms of the invariants. Though the most difficult part
of the road to components remains solving the so called harmonic zero-
curvature equations, it proves to be rather straightforward in WZ gauge.

As we shall show, the basic objects in constructing $\mathcal{N}=2$ higher-
derivative supergravity invariants are linearized gauge-invariant
supercurvatures which have a surprisingly simple structure:

$\mathcal{F}^{++\alpha\dot{\alpha}}=(\mathcal{D}^{+})^{4}G^{--\alpha\dot{\alpha}}=-8i\theta^{+}_{\rho}\bar{\theta}^{+}_{\dot{\rho}}\left(\mathcal{R}^{(\alpha\rho)(\dot{\alpha}\dot{\rho})}-\frac{1}{8}\epsilon^{\alpha\rho}\epsilon^{\dot{\alpha}\dot{\rho}}R\right)+\dots,$
(1.2)
$\mathcal{F}^{++5}=(\mathcal{D}^{+})^{4}G^{--5}=\frac{i}{2}(\theta^{+})^{2}R-\frac{i}{2}(\bar{\theta}^{+})R+\dots.,$
(1.3)
$\mathcal{W}_{(\alpha\beta)}=(\bar{\mathcal{D}}^{+})^{2}\left(\mathcal{D}^{+}_{(\alpha}G^{---}_{\beta)}+\mathcal{D}^{-}_{(\alpha}G^{--+}_{\beta)}-\partial_{(\alpha}^{\dot{\rho}}G^{--}_{\beta)\dot{\rho}}\right)=32\theta^{-(\gamma}\theta^{+\delta)}\mathcal{R}_{(\alpha\beta\gamma\delta)}+\dots.$
(1.4)

The linearized $\mathcal{N}=2$ supercurvatures
$\mathcal{F}^{++\alpha\dot{\alpha}}$ and $\mathcal{F}^{++5}$ are analytic
superfields, while the linearized $\mathcal{N}=2$ Weyl tensor
$\mathcal{W}_{(\alpha\beta)}$ is a chiral superfield. All these
supercurvatures are constructed out of the superfield potentials
$G^{--\alpha\dot{\alpha}},G^{--5},G^{--\alpha\pm}$ related to the basic
analytic prepotentials via the well defined harmonic zero-curvature equations.
In WZ gauge, these equations can be directly solved, giving rise to the
explicit expressions for the $G$-potentials in terms of the $\mathcal{N}=2$
supergravity multiplet fields and their derivatives. In their component
expansion, the supercurvatures involve all possible irreducible linearized
gravity curvatures quadratic in derivatives: the scalar curvature $R$, the
irreducible part of Ricci tensor
$\mathcal{R}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}$ and the linearized Weyl
tensor $\mathcal{R}_{(\alpha\beta\gamma\delta)}$. So these supercurvatures can
be viewed as the building blocks of various linearized $\mathcal{N}=2$
supergravity invariants.

The paper is organized as follows. In section 2 we introduce notations and
shortly recall the structure of harmonic superspace. In section 3 we describe
prepotentials of $\mathcal{N}=2$ Einstein supergravity multiplet and the
corresponding gauge group. Next we perform the linearization procedure and
study in detail the component contents of the analytic prepotentials. In
section 4, the covariant $\mathcal{N}=2$ superfields $G^{\pm\pm M}$ are
introduced and the linearized $\mathcal{N}=2$ supersymmetric Einstein action
is constructed. In section 5, the linearized supercurvatures are defined and
all invariant quadratic actions in harmonic superspace are presented. In
section 6, we elaborate on the component reduction of the linearized
$\mathcal{N}=2$ Einstein supergravity, obtain the linearized supercurvatures
and the linearized quadratic invariants. In section 7 we discuss construction
of more general linearized $\mathcal{N}=2$ supergravity invariants. Section 8
is for conclusions and outlook. In particular, we briefly sketch the higher-
spin generalizations of the invariants constructed.

Three Appendices collect some technical details. Appendix A provides the
linearized gravity invariants in the spinor notations used throughout the
paper. In Appendix B we introduce the linearized spin $\frac{3}{2}$ invariants
which are of need, when performing the component reduction in the fermionic
sector. In Appendix C we present the solutions of the harmonic zero-curvature
equations, which, as was already mentioned, are a necessary part of passing to
the component Lagrangians.

##  2 Harmonic superspace

We proceed from $\mathcal{N}=2$ harmonic superspace in the analytic basis,
with an extra $x^{5}$ coordinate (for a pedagogical introduction, see 18
)222Here we use standard notations:
$x^{\alpha\dot{\alpha}}=x^{m}(\tilde{\sigma}_{m})^{\alpha\dot{\alpha}}$,
$\partial_{\alpha\dot{\alpha}}=\frac{1}{2}\sigma^{m}_{\alpha\dot{\alpha}}\partial_{m}$
(so
$\partial_{\alpha\dot{\alpha}}x^{\beta\dot{\beta}}=\delta^{\beta}_{\alpha}\delta^{\dot{\beta}}_{\dot{\alpha}}$,
$\partial_{\alpha\dot{\beta}}\partial^{\dot{\beta}\beta}=\frac{1}{4}\delta_{\alpha}^{\beta}\Box$),
$\hat{\alpha}=(\alpha,\dot{\alpha})$,
$\partial^{-}_{\hat{\alpha}}=\frac{\partial}{\partial\theta^{+\hat{\alpha}}}$,
$\partial^{+}_{\hat{\alpha}}=\frac{\partial}{\partial\theta^{-\hat{\alpha}}}$,
$(\psi\chi)=\psi^{\alpha}\chi_{\alpha}$,
$(\bar{\psi}\bar{\chi})=\bar{\psi}_{\dot{\alpha}}\chi^{\dot{\alpha}}$,
$(\theta^{\hat{+}})^{2}=\theta^{+\hat{\alpha}}\theta^{+}_{\hat{\alpha}}$ etc.:

$\left\\{x^{\alpha\dot{\alpha}},\theta^{+\hat{\alpha}},\theta^{-\hat{\alpha}},u^{\pm}_{i},x^{5}\right\\},\quad\hat{\alpha}=(\alpha,\dot{\alpha})\,.$
(2.1)

The harmonics $u^{\pm}_{i}$ are auxiliary coordinates that parametrize an
internal sphere $S^{2}\sim SU(2)/U(1)$ and satisfy the relation
$u^{+i}u^{-}_{i}=1$. The presence of harmonics $u^{\pm}_{i}$ in the coordinate
set inevitably leads to an infinite number of component fields in any
unconstrained harmonic superfield. The additional coordinate $x^{5}$ is needed
to introduce the interaction of $\mathcal{N}=2$ Maxwell multiplet with
hypermultiplet, as well as to describe massive hypermultiplet (with a mass
equal to central charge). In this paper, all harmonic superfields are assumed
to be $x^{5}$-independent. The coordinate $x^{5}$ will be merely used for the
purpose to simplify handling $\mathcal{N}=2$ supergravity prepotentials in the
linearized theory. In the analytic basis, rigid $\mathcal{N}=2$ supersymmetry
transformations act on the coordinates (2.1) as:

$\delta_{\epsilon}x^{\alpha\dot{\alpha}}=-4i\left(\epsilon^{-\alpha}\bar{\theta}^{+\dot{\alpha}}+\theta^{+\alpha}\bar{\epsilon}^{-\dot{\alpha}}\right),\quad\delta_{\epsilon}\theta^{\pm\hat{\alpha}}=\epsilon^{\pm\hat{\alpha}},\quad\delta_{\epsilon}u^{\pm}_{i}=0,\quad\delta_{\epsilon}x^{5}=2i\epsilon^{-\hat{\alpha}}\theta^{+}_{\hat{\alpha}}.$
(2.2)

Here $\epsilon^{\pm\hat{\alpha}}:=\epsilon^{\hat{\alpha}i}u^{\pm}_{i}$ are
relevant spinorial parameters. These transformations preserve an analytic
subspace of harmonic superspace:

$\zeta:=\left\\{x^{\alpha\dot{\alpha}},\theta^{+\hat{\alpha}},u^{\pm}_{i}\right\\}.$
(2.3)

This invariant subspace contains just half the original Grassmann variables.

An analogue of complex conjugation in harmonic superspace is the tilde
conjugation. On complex functions, it acts like the ordinary complex
conjugation. Its action on the harmonic superspace coordinates is as follows

$\widetilde{x^{\alpha\dot{\alpha}}}=x^{\alpha\dot{\alpha}},\quad\widetilde{\theta^{\pm}_{\alpha}}=\bar{\theta}^{\pm}_{\dot{\alpha}},\quad\widetilde{\bar{\theta}^{\pm}_{\dot{\alpha}}}=-\theta^{\pm}_{\alpha},\quad\widetilde{u^{\pm
i}}=-u_{i}^{\pm},\quad\widetilde{u^{\pm}_{i}}=u^{\pm
i},\quad\widetilde{x^{5}}=x^{5}.$ (2.4)

The spinor derivatives which are covariant with respect to $\mathcal{N}=2$
supersymmetry (2.2) are defined as:

$\begin{split}&\mathcal{D}^{+}_{\hat{\alpha}}:=\partial^{+}_{\hat{\alpha}},\quad\\\
&\mathcal{D}^{-}_{\alpha}:=-\partial^{-}_{\alpha}+4i\bar{\theta}^{-\dot{\alpha}}\partial_{\alpha\dot{\alpha}}-2i\theta^{-}_{\alpha}\partial_{5},\quad\\\
&\bar{\mathcal{D}}^{-}_{\dot{\alpha}}:=-\partial^{-}_{\dot{\alpha}}-4i\theta^{-\alpha}\partial_{\alpha\dot{\alpha}}+2i\bar{\theta}^{-}_{\dot{\alpha}}\partial_{5}.\end{split}$
(2.5)

The covariant harmonic derivatives are given by:

$\begin{split}&\mathcal{D}^{++}=\partial^{++}-4i\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}\partial_{\rho\dot{\rho}}+\theta^{+\hat{\rho}}\partial^{+}_{\hat{\rho}}+i\theta^{+\hat{\rho}}\theta^{+}_{\hat{\rho}}\partial_{5},\\\
&\mathcal{D}^{--}=\partial^{--}-4i\theta^{-\rho}\bar{\theta}^{-\dot{\rho}}\partial_{\rho\dot{\rho}}+\theta^{-\hat{\rho}}\partial^{-}_{\hat{\rho}}+i\theta^{-\hat{\rho}}\theta^{-}_{\hat{\rho}}\partial_{5},\\\
&\mathcal{D}^{0}=\partial^{0}+\theta^{+\hat{\rho}}\partial^{-}_{\hat{\rho}}-\theta^{-\hat{\rho}}\partial^{+}_{\hat{\rho}}.\end{split}$
(2.6)

Here we used the following notations for the partial harmonic derivatives:

$\partial^{++}=u^{+i}\frac{\partial}{\partial
u^{-i}},\qquad\partial^{--}=u^{-i}\frac{\partial}{\partial
u^{+i}},\qquad\partial^{0}=u^{+i}\frac{\partial}{\partial
u^{+i}}-u^{-i}\frac{\partial}{\partial u^{-i}}.$ (2.7)

They satisfy the relations:

$[\partial^{++},\partial^{--}]=\partial^{0},\qquad[\partial^{0},\partial^{\pm\pm}]=\pm
2\partial^{\pm\pm}.$ (2.8)

The covariant harmonic derivatives play an important role in the formulation
of $\mathcal{N}=2$ supergravity in harmonic superspace. They satisfy the
$SU(2)$-algebra relations similar to (2.8):

$[\mathcal{D}^{++},\mathcal{D}^{--}]=\mathcal{D}^{0},\qquad[\mathcal{D}^{0},\mathcal{D}^{\pm\pm}]=\pm
2\mathcal{D}^{\pm\pm}.$ (2.9)

The spinor covariant derivatives (2.5) obey the relations:

$[\mathcal{D}^{--},\mathcal{D}^{+}_{\hat{\alpha}}]=\mathcal{D}^{-}_{\hat{\alpha}},\qquad[\mathcal{D}^{++},\mathcal{D}^{-}_{\hat{\alpha}}]=\mathcal{D}^{+}_{\hat{\alpha}},\qquad[\mathcal{D}^{++},\mathcal{D}^{+}_{\hat{\alpha}}]=0,\qquad[\mathcal{D}^{--},\mathcal{D}^{-}_{\hat{\alpha}}]=0,$
(2.10a)
$\\{\mathcal{D}^{+}_{\alpha},\bar{\mathcal{D}}^{-}_{\dot{\alpha}}\\}=-4i\partial_{\alpha\dot{\alpha}},\;\;\\{\bar{\mathcal{D}}^{+}_{\dot{\alpha}},\mathcal{D}^{-}_{\alpha}\\}=4i\partial_{\alpha\dot{\alpha}},\;\;\\{\mathcal{D}^{+}_{\alpha},\mathcal{D}^{-}_{\beta}\\}=2i\epsilon_{\alpha\beta}\partial_{5},\;\;\\{\bar{\mathcal{D}}^{+}_{\dot{\alpha}},\bar{\mathcal{D}}^{-}_{\dot{\beta}}\\}=-2i\epsilon_{\dot{\alpha}\dot{\beta}}\partial_{5}.$
(2.10b)

Harmonic superspace can be looked upon as the most general $4D,\mathcal{N}=2$
superspace Kuzenko:1998xm ; Jain:2009aj , and it is indispensable for
formulation and study of classical and quantum $\mathcal{N}=2$ theories 18 .
The analytic superspace (2.3) plays the fundamental role as it underlies the
geometric structure of all $\mathcal{N}=2$ theories (see, e.g.,
Buchbinder:2022kzl  for application to the construction of the hypermultiplet
couplings to higher spins). In this paper, $\mathcal{N}=2$ harmonic
analyticity will be shown to be a necessary ingredient also in constructing
higher-derivative ${\cal N}=2$ supergravity invariants. In the next section,
as an instructive example, we shall expound how the analytic unconstrained
prepotentials of $\mathcal{N}=2$ Einstein supergravity naturally appear in the
harmonic superspace framework. After that, we shall proceed to the analysis of
the linearized theory.

##  3 $\mathcal{N}=2$ Einstein supergravity multiplet in harmonic superspace

This section is devoted to a detailed discussion of $\mathcal{N}=2$ Einstein
supergravity multiplet and gauge freedom in both the full-fledged nonlinear
case and its linearized approximation. We will also define the auxiliary
fields redefinitions needed for performing the component reduction in section
6.

###  3.1 $\mathcal{N}=2$ supergravity multiplet: nonlinear set-up

The gauge group of $\mathcal{N}=2$ Einstein supergravity is constituted by the
analyticity-preserving transformations of harmonic superspace Galperin:1987ek
; Galperin:1987em ; 18  333They can be treated as an analytic gauging of rigid
$\mathcal{N}=2$ supersymmetry transformations (2.2).

$\delta_{\lambda}x^{\alpha\dot{\alpha}}=\lambda^{\alpha\dot{\alpha}}(\zeta),\quad\delta_{\lambda}\theta^{+\hat{\alpha}}=\lambda^{+\hat{\alpha}}(\zeta),\quad\delta_{\lambda}\theta^{-\hat{\alpha}}=\lambda^{-\hat{\alpha}}(\zeta,\theta^{-}),\quad\delta_{\lambda}u^{\pm}_{i}=0,\quad\delta_{\lambda}x^{5}=\lambda^{5}(\zeta).$
(3.1)

Note that the only parameters which are not analytic are
$\lambda^{-\hat{\alpha}}$. Another peculiarity is that the harmonic variables
retain inert under gauge transformations 444It is the gauge group of
$\mathcal{N}=2$ conformal supergravity that affects harmonics, see
Galperin:1987ek ; 18 ; Ivanov:2022vwc . . It is useful to introduce a multi-
index $M:=(\alpha\dot{\alpha},+\hat{\alpha},-\hat{\alpha},5)$ and denote the
harmonic superspace coordinates as
$z^{M}:=(x^{\alpha\dot{\alpha}},\theta^{+\hat{\alpha}},\theta^{-\hat{\alpha}},x^{5})$.
Then super-diffeomorphism transformations can be represented as the action of
the differential operator:

$\delta_{\lambda}z^{M}=[\hat{\Lambda},z^{M}],\qquad\hat{\Lambda}:=\lambda^{M}\partial_{M}.$
(3.2)

This is a very convenient notation, because, for instance, transformations of
the superspace derivatives
$\partial_{M}:=(\partial_{\alpha\dot{\alpha}},\partial^{-}_{\hat{\alpha}},\partial^{+}_{\hat{\alpha}},\partial_{5})$
can be succinctly written as

$\delta_{\lambda}\partial_{M}=[\hat{\Lambda},\partial_{M}].$ (3.3)

In particular, it implies that the harmonic derivative $\mathcal{D}^{++}$
ceases to be covariant555The rigid $\mathcal{N}=2$ supersymmetry
transformations can be defined as a subgroup of (3.1) distinguished by the
condition $\delta_{\epsilon}\mathcal{D}^{++}=0$. under supertranslations
(3.1):

$\begin{split}\delta_{\lambda}\mathcal{D}^{++}=[\hat{\Lambda},\mathcal{D}^{++}]=&-\mathcal{D}^{++}\lambda^{M}\partial_{M}\\\
&-4i\lambda^{+\rho}\bar{\theta}^{+\dot{\rho}}\partial_{\rho\dot{\rho}}-4i\theta^{+\rho}\bar{\lambda}^{+\dot{\rho}}\partial_{\rho\dot{\rho}}+\lambda^{+\hat{\rho}}\partial^{+}_{\hat{\rho}}+2(\lambda^{\hat{+}}\theta^{\hat{+}})\partial_{5}.\end{split}$
(3.4)

The unwanted terms can be canceled by introducing a set of gauge
prepotentials:

$\mathcal{D}^{++}\quad\to\quad\mathfrak{D}^{++}=\mathcal{D}^{++}+h^{++\alpha\dot{\alpha}}\partial_{\alpha\dot{\alpha}}+h^{++\hat{\alpha}+}\partial^{-}_{\hat{\alpha}}+h^{++\hat{\alpha}-}\partial^{+}_{\hat{\alpha}}+h^{++5}\partial_{5}$
(3.5)

with the gauge freedom:

$\begin{cases}\delta_{\lambda}h^{++\alpha\dot{\alpha}}=\mathfrak{D}^{++}\lambda^{\alpha\dot{\alpha}}+4i\lambda^{+\alpha}\bar{\theta}^{+\dot{\alpha}}+4i\theta^{+\alpha}\bar{\lambda}^{+\dot{\alpha}}-\hat{\Lambda}h^{++\alpha\dot{\alpha}},\\\
\delta_{\lambda}h^{++\hat{\alpha}+}=\mathfrak{D}^{++}\lambda^{+\hat{\alpha}}-\hat{\Lambda}h^{++\hat{\alpha}+},\\\
\delta_{\lambda}h^{++\hat{\alpha}-}=\mathfrak{D}^{++}\lambda^{-\hat{\alpha}}-\lambda^{+\hat{\alpha}}-\hat{\Lambda}h^{++\hat{\alpha}-},\\\
\delta_{\lambda}h^{++5}=\mathfrak{D}^{++}\lambda^{5}-2\lambda^{+\hat{\alpha}}\theta^{+}_{\hat{\alpha}}-\hat{\Lambda}h^{++5}.\end{cases}$
(3.6)

The modified derivative $\mathfrak{D}^{++}$ is covariant under (3.1), i.e. it
satisfies:

$\delta_{\lambda}\mathfrak{D}^{++}=[\hat{\Lambda},\mathfrak{D}^{++}]+\delta_{\lambda}h^{++M}\partial_{M}=0.$
(3.7)

Since the harmonic derivative $\mathcal{D}^{++}$ is real with respect to the
tilde conjugation, it is natural to require that $\mathfrak{D}^{++}$ obeys the
same property:

$\widetilde{\mathfrak{D}^{++}}=\mathfrak{D}^{++}\quad\Rightarrow\quad\begin{aligned}
&\widetilde{h^{++\alpha\dot{\alpha}}}=h^{++\alpha\dot{\alpha}},\\\
&\widetilde{h^{++5}}=h^{++5},\\\
&\widetilde{h^{++\alpha+}}=h^{++\dot{\alpha}+},\\\
&\widetilde{h^{++\dot{\alpha}+}}=-h^{++\alpha+}.\end{aligned}$ (3.8)

Note that the transformation law of $h^{++\hat{\alpha}-}$ in (3.6) allows to
impose the analytic gauge  $h^{++\hat{\alpha}-}=0$, which expresses
$\lambda^{-\hat{\alpha}}$ in terms of the analytic parameter
$\lambda^{+\hat{\alpha}}$:

$\mathfrak{D}^{++}\lambda^{-\hat{\alpha}}=\lambda^{+\hat{\alpha}}.$ (3.9)

In the analytic gauge, the differential operator $\mathfrak{D}^{++}$ becomes
analytic, i.e. it satisfies the condition
$[\mathcal{D}^{+}_{\hat{\alpha}},\mathfrak{D}^{++}]=0$. Hereafter, we always
stick to this gauge.

Notably, the gauge freedom (3.6) allows one to gauge away infinitely many
fields from the prepotentials and to end up with the off-shell $\mathcal{N}=2$
Einstein supergravity multiplet in the component expansion of $h^{++M}$. In
this WZ gauge we have666The numerical coefficients are chosen so as to have
the field transformation laws as simple as possible, see (3.19).:

$\begin{split}&h_{WZ}^{++\alpha\dot{\alpha}}=-4i\theta^{+\beta}\bar{\theta}^{+\dot{\beta}}\Phi_{\beta\dot{\beta}}^{\alpha\dot{\alpha}}+16(\bar{\theta}^{+})^{2}\theta^{+\beta}\psi_{\beta}^{\alpha\dot{\alpha}i}u^{-}_{i}-16(\theta^{+})^{2}\bar{\theta}^{+\dot{\beta}}\bar{\psi}_{\dot{\beta}}^{\alpha\dot{\alpha}i}u^{-}_{i}+(\theta^{+})^{4}V^{\alpha\dot{\alpha}(ij)}u^{-}_{i}u^{-}_{j}\\\
&h_{WZ}^{++5}=-4i\theta^{+\beta}\bar{\theta}^{+\dot{\beta}}C_{\beta\dot{\beta}}+8(\bar{\theta}^{+})^{2}\theta^{+\beta}\rho_{\beta}^{i}u^{-}_{i}-8(\theta^{+})^{2}\bar{\theta}^{+\dot{\beta}}\bar{\rho}_{\dot{\beta}}^{i}u^{-}_{i}+(\theta^{+})^{4}S^{(ij)}u^{-}_{i}u^{-}_{j}\\\
&h_{WZ}^{++\alpha+}=(\bar{\theta}^{+})^{2}\theta^{+}_{\beta}T^{(\alpha\beta)}+(\bar{\theta}^{+})^{2}\theta^{+\alpha}T+(\theta^{+})^{2}\bar{\theta}^{+}_{\dot{\beta}}P^{\alpha\dot{\beta}}+(\theta^{+})^{4}\chi^{\alpha
i}u^{-}_{i},\\\
&h_{WZ}^{++\dot{\alpha}+}=(\theta^{+})^{2}\bar{\theta}^{+}_{\dot{\beta}}\bar{T}^{(\dot{\alpha}\dot{\beta})}+(\theta^{+})^{2}\bar{\theta}^{+\dot{\alpha}}\bar{T}-(\bar{\theta}^{+})^{2}\theta^{+}_{\beta}\bar{P}^{\beta\dot{\alpha}}+(\theta^{+})^{4}\bar{\chi}^{\dot{\alpha}i}u^{-}_{i}.\end{split}$
(3.10)

The remaining fields are defined up to a residual gauge freedom. As usual, it
is specified by the requirement of preservation of WZ gauge, i.e.
$\delta^{res}_{\lambda}h^{++M}\sim h^{++M}_{WZ}$. As a result, the set of
residual parameters is obtained:

$\begin{cases}\lambda_{WZ}^{\alpha\dot{\alpha}}=a^{\alpha\dot{\alpha}}(x)-4i\theta^{+\alpha}\bar{\epsilon}^{\dot{\alpha}i}(x)u^{-}_{i}-4i\epsilon^{\alpha
i}(x)u^{-}_{i}\bar{\theta}^{+\dot{\alpha}},\\\
\lambda_{WZ}^{+\alpha}=\epsilon^{\alpha
i}(x)u^{+}_{i}+\theta^{+\beta}l_{(\beta}^{\;\;\alpha)}(x)+4i\theta^{+\beta}\bar{\theta}^{+\dot{\beta}}(\delta_{\beta}^{\rho}\delta_{\dot{\beta}}^{\dot{\rho}}+\Phi_{\beta\dot{\beta}}^{\rho\dot{\rho}})\partial_{\rho\dot{\rho}}\epsilon^{\alpha
i}(x)u^{-}_{i},\\\
\lambda_{WZ}^{+\dot{\alpha}}=\bar{\epsilon}^{\dot{\alpha}i}(x)u^{+}_{i}+\bar{\theta}^{+\dot{\beta}}l_{(\dot{\beta}}^{\;\;\dot{\alpha})}(x)+4i\theta^{+\beta}\bar{\theta}^{+\dot{\beta}}(\delta_{\beta}^{\rho}\delta_{\dot{\beta}}^{\dot{\rho}}+\Phi_{\beta\dot{\beta}}^{\rho\dot{\rho}})\partial_{\rho\dot{\rho}}\bar{\epsilon}^{\dot{\alpha}i}(x)u^{-}_{i},\\\
\lambda^{5}_{WZ}=c(x)+2i\epsilon^{\hat{\alpha}i}(x)u^{-}_{i}\theta^{+}_{\hat{\alpha}}.\end{cases}$
(3.11)

The realization of these transformations on the WZ fields directly follows
from (3.6). As an example, let us quote the transformations of the spin $2$,
$\frac{3}{2}$, $1$ gauge fields:

$\begin{split}\delta\Phi_{\beta\dot{\beta}}^{\alpha\dot{\alpha}}=&(\delta_{\beta}^{\rho}\delta_{\dot{\beta}}^{\dot{\rho}}+\Phi_{\beta\dot{\beta}}^{\rho\dot{\rho}})\partial_{\rho\dot{\rho}}a^{\alpha\dot{\alpha}}-a^{\rho\dot{\rho}}\partial_{\rho\dot{\rho}}\Phi_{\beta\dot{\beta}}^{\alpha\dot{\alpha}}-\left(l_{(\beta}^{\;\;\rho)}\delta^{\dot{\alpha}}_{\dot{\rho}}+\delta^{\alpha}_{\rho}\bar{l}_{(\dot{\beta}}^{\;\;\dot{\rho})}\right)\left(\delta_{\rho}^{\alpha}\delta_{\dot{\rho}}^{\dot{\alpha}}+\Phi_{\rho\dot{\rho}}^{\alpha\dot{\alpha}}\right)\\\
&-2i\epsilon_{\beta
i}\bar{\psi}^{\alpha\dot{\alpha}i}_{\dot{\beta}}-2i\bar{\epsilon}_{\dot{\beta}i}\psi_{\beta}^{\alpha\dot{\alpha}i},\end{split}$
(3.12a)
$\begin{split}\delta\psi^{\alpha\dot{\alpha}i}_{\beta}=&-\epsilon^{\dot{\alpha}\dot{\beta}}(\delta_{\beta}^{\rho}\delta_{\dot{\beta}}^{\dot{\rho}}+\Phi_{\beta\dot{\beta}}^{\rho\dot{\rho}})\partial_{\rho\dot{\rho}}\epsilon^{\alpha
i}+i\frac{1}{2}T^{\;\alpha)}_{(\beta}\bar{\epsilon}^{\dot{\alpha}i}-i\frac{1}{2}T\delta^{\alpha}_{\beta}\bar{\epsilon}^{\dot{\alpha}i}-i\frac{1}{2}\bar{P}^{\dot{\alpha}}_{\beta}\epsilon^{\alpha
i}+\frac{1}{6}\epsilon_{\beta j}V^{\alpha\dot{\alpha}(ij)}\\\
&+\psi^{\rho\dot{\rho}}_{\beta}\partial_{\rho\dot{\rho}}a^{\alpha\dot{\alpha}}-a^{\rho\dot{\rho}}\partial_{\rho\dot{\rho}}\psi^{\alpha\dot{\alpha}i}_{\beta}-l_{(\beta}^{\;\;\rho)}\psi_{\rho}^{\alpha\dot{\alpha}i},\end{split}$
(3.12b) $\delta
C_{\alpha\dot{\alpha}}=(\delta_{\alpha}^{\rho}\delta_{\dot{\alpha}}^{\dot{\rho}}+\Phi_{\alpha\dot{\alpha}}^{\rho\dot{\rho}})\partial_{\rho\dot{\rho}}c-a^{\rho\dot{\rho}}\partial_{\rho\dot{\rho}}C_{\alpha\dot{\alpha}}-(l^{\;\;\beta)}_{(\alpha}\delta^{\dot{\beta}}_{\dot{\alpha}}+\delta_{\alpha}^{\beta}l_{(\dot{\alpha}}^{\;\;\dot{\beta})})C_{\beta\dot{\beta}}-i\epsilon_{\beta
i}\bar{\rho}^{i}_{\dot{\beta}}-i\epsilon_{\dot{\beta}i}\rho^{i}_{\beta}.$
(3.12c)

These transformations make it possible to link the harmonic superspace
formulation to the component approach. For example, we see that the
combination
$\delta_{\beta}^{\alpha}\delta^{\dot{\alpha}}_{\dot{\beta}}+\Phi^{\alpha\dot{\alpha}}_{\beta\dot{\beta}}:=\frac{1}{2}\sigma^{\alpha\dot{\alpha}}_{m}\sigma^{a}_{\beta\dot{\beta}}e_{a}^{m}$
can be identified with the fierbein $e^{m}_{a}$, the parameter
$a^{\alpha\dot{\alpha}}$ stands for local translations and
$l^{(\alpha\beta)},\bar{l}^{(\dot{\alpha}\dot{\beta})}$ for local Lorentz
transformations, while $\epsilon^{\alpha i},\bar{\epsilon}^{\dot{\alpha}i}$
are the parameters of local supersymmetry. The metric tensor is then given by
$g^{mn}=e^{m}_{a}e^{n}_{b}\eta^{ab}$ and have the standard transformation law
$\delta
g^{mn}=\partial_{k}a^{m}g^{kn}+\partial_{k}a^{n}g^{mk}-a^{k}\partial_{k}g^{mn}$.
The fermionic field $\psi_{\beta}^{\alpha\dot{\alpha}i}$ describes the doublet
of the spin $\frac{3}{2}$ Rarita-Schwinger fields, $C_{\alpha\dot{\alpha}}$ is
the spin 1 gauge field. Other fields are auxiliary fields of the off-shell
$\mathcal{N}=2$ Einstein supergravity multiplet777Some auxiliary fields
require redefinition, see the explicit redefinition in the linearized limit
(3.26).:

$\begin{split}\text{Physical
sector}&:\qquad\Phi^{\alpha\dot{\alpha}}_{\beta\dot{\beta}},\psi^{\alpha\dot{\alpha}i}_{\beta},C_{\beta\dot{\beta}};\\\
\text{Auxiliary sector}&:\qquad
V^{\alpha\dot{\alpha}(ij)},\rho^{i}_{\beta},S^{(ij)},T,T^{(\alpha\beta)},P^{\alpha\dot{\beta}},\chi^{\alpha
i}.\end{split}$ (3.13)

Totally, the multiplet carries $\mathbf{40_{B}+40_{F}}$ off-shell degrees of
freedom. The dimensions of the involved fields can be directly determined from
that of the prepotentials in WZ gauge (3.10). The above component WZ expansion
corresponds to the “minimal” version of $\mathcal{N}=2$ supergravity obtained
from the conformal one through the compensation by the so called “nonlinear
multiplet” (in our formulation there are in fact two hidden compensators:
$\mathcal{N}=2$ vector multiplet and the nonlinear multiplet, see, e.g., 18 ;
Freedman:2012zz ; Lauria:2020rhc ). The off-shell component content of this
version of $\mathcal{N}=2$ Einstein supergravity was firstly found by Fradkin
and Vasiliev Fradkin:1979as ; Fradkin:1979cw  and, independently, by de Wit
and van Holten deWit:1979xpv  .

###  3.2 $\mathcal{N}=2$ supergravity multiplet: linearization

Linearizing gauge transformations (3.6) yields:

$\begin{cases}\delta_{\lambda}h^{++\alpha\dot{\alpha}}=\mathcal{D}^{++}\lambda^{\alpha\dot{\alpha}}+4i\lambda^{+\alpha}\bar{\theta}^{+\dot{\alpha}}+4i\theta^{+\alpha}\bar{\lambda}^{+\dot{\alpha}},\\\
\delta_{\lambda}h^{++\hat{\alpha}+}=\mathcal{D}^{++}\lambda^{+\hat{\alpha}},\\\
\delta_{\lambda}h^{++\hat{\alpha}-}=\mathcal{D}^{++}\lambda^{-\hat{\alpha}}-\lambda^{+\hat{\alpha}},\\\
\delta_{\lambda}h^{++5}=\mathcal{D}^{++}\lambda^{5}-2i\lambda^{+\hat{\alpha}}\theta^{+}_{\hat{\alpha}}.\end{cases}$
(3.14)

In the linearized case we also impose the analytic gauge
$h^{++\hat{\alpha}-}=0$, so
$\mathcal{D}^{++}\lambda^{-\hat{\alpha}}=\lambda^{+\hat{\alpha}}$. In this
gauge all prepotentials are analytic. The WZ gauge has the same form as in the
nonlinear theory, eq. (3.10).

Rigid supersymmetry transformations of the analytic prepotentials can be
obtained by requiring invariance of the covariant derivative
$\mathfrak{D}^{++}$ under the rigid $\mathcal{N}=2$ supersymmetry:

$\delta_{\epsilon}\mathfrak{D}^{++}=[\hat{\Lambda}_{\epsilon},\mathcal{D}^{++}]+\delta_{\epsilon}h^{++M}\partial_{M}=0.$
(3.15)

As a solution of (3.15), all parameters $\lambda_{\epsilon}^{M}$ span rigid
supersymmetry:

$\lambda_{\epsilon}^{\alpha\dot{\alpha}}=-4i\theta^{+\alpha}\bar{\epsilon}^{-\dot{\alpha}}-4i\epsilon^{-\alpha}\bar{\theta}^{+\dot{\alpha}},\quad\lambda_{\epsilon}^{+\alpha}=\epsilon^{+\alpha},\quad\lambda_{\epsilon}^{+\dot{\alpha}}=\bar{\epsilon}^{+\dot{\alpha}},\quad\lambda_{\epsilon}^{5}=2i\epsilon^{-\hat{\alpha}}\theta^{+}_{\hat{\alpha}}.$
(3.16)

Note that, in full agreement with (2.2),
$\delta_{\epsilon}z^{M}=\lambda_{\epsilon}^{M}$.

From (3.15) we find that the rigid supersymmetry transformations of the
prepotentials are given by:

$\begin{cases}\delta_{\epsilon}h^{++\alpha\dot{\alpha}}=-4ih^{++\alpha+}\bar{\epsilon}^{-\dot{\alpha}}-4i\epsilon^{-\alpha}h^{++\dot{\alpha}+}-\delta_{\epsilon}z^{M}\partial_{M}h^{++\alpha\dot{\alpha}},\\\
\delta_{\epsilon}h^{++\alpha+}=-\delta_{\epsilon}z^{M}\partial_{M}\delta_{\epsilon}h^{++\alpha+},\\\
\delta_{\epsilon}h^{++\dot{\alpha}+}=-\delta_{\epsilon}z^{M}\partial_{M}h^{++\dot{\alpha}+},\\\
\delta_{\epsilon}h^{++5}=2ih^{++\hat{\alpha}+}\epsilon^{-}_{\hat{\alpha}}-\delta_{\epsilon}z^{M}\partial_{M}h^{++5}.\end{cases}$
(3.17)

Note that the prepotentials $h^{++\alpha\dot{\alpha}}$ and $h^{++5}$ have non-
standard transformation laws. As usual, in WZ gauge, supersymmetry
transformations must be accompanied by compensating gauge transformations that
ensure the preservation of the gauge-fixed form of $h^{++M}_{WZ}$.

From (3.17) one can deduce supersymmetry transformations of the component
fields. For example, the gravity field transforms as:

$\delta_{\epsilon}\Phi^{\alpha\dot{\alpha}}_{\beta\dot{\beta}}=2i\left(\epsilon^{i}_{\beta}\bar{\psi}^{\alpha\dot{\alpha}}_{\dot{\beta}\;\;\;i}-\bar{\epsilon}^{i}_{\dot{\beta}}\psi^{\alpha\dot{\alpha}}_{\beta\;\;\;i}\right).$
(3.18)

The complete set of linearized gauge transformation laws for the component
fields can be deduced from (3.14) and the linearized version of the residual
gauge parameters (3.11):

$\delta_{\lambda}\Phi^{\alpha\dot{\alpha}}_{\beta\dot{\beta}}=\partial_{\beta\dot{\beta}}a^{\alpha\dot{\alpha}}-l^{\;\;\alpha)}_{(\beta}\delta^{\dot{\alpha}}_{\dot{\beta}}-\delta^{\alpha}_{\beta}l^{\;\;\dot{\alpha})}_{(\dot{\beta}},$
(3.19a)
$\delta_{\lambda}\psi_{\beta}^{\alpha\dot{\alpha}i}=-\partial^{\dot{\alpha}}_{\beta}\epsilon^{\alpha
i},$ (3.19b)
$\delta_{\lambda}C_{\alpha\dot{\alpha}}=\partial_{\alpha\dot{\alpha}}c,$
(3.19c)
$\delta_{\lambda}P^{\alpha\dot{\alpha}}=-2i\partial^{\dot{\alpha}\beta}l_{(\beta}^{\;\;\alpha)},$
(3.19d) $\delta_{\lambda}\chi^{\alpha i}=2\Box\epsilon^{\alpha i},$ (3.19e)
$\delta_{\lambda}\rho_{\alpha}^{i}=\partial_{\alpha\dot{\alpha}}\bar{\epsilon}^{\dot{\alpha}i}.$
(3.19f)

Other fields in WZ gauge are invariant under gauge transformations. The
transformations (3.19a), (3.19b) and (3.19c) are linearized gauge
transformations for the spin $2$, spin $\frac{3}{2}$ and spin $1$ fields,
respectively. The fields $P^{\alpha\dot{\alpha}},\chi^{\alpha i}$ and
$\rho_{\alpha}^{i}$ can be properly redefined with the help of the spin $2$
and spin $\frac{3}{2}$ gauge fields so as to ensure trivial gauge
transformation laws for them. Such redefinitions play an important role in the
component reduction, so we will explicitly present them here.

Spin 2 field has the following decomposition into the irreducible parts:

$\Phi^{\alpha\dot{\alpha}\beta\dot{\beta}}=\Phi^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}+\epsilon^{\alpha\beta}\Phi^{(\dot{\alpha}\dot{\beta})}+\epsilon^{\dot{\alpha}\dot{\beta}}\Phi^{(\alpha\beta)}+\epsilon^{\alpha\beta}\epsilon^{\dot{\alpha}\dot{\beta}}\Phi.$
(3.20)

Using local Lorentz transformations, one can fix the “symmetric” gauge:
$\Phi^{(\alpha\beta)}=0$, $\Phi^{(\dot{\alpha}\dot{\beta})}=0$, thus
expressing local Lorentz transformations in terms of local translations
$a^{\alpha\dot{\alpha}}$:

$l^{\;\;\alpha)}_{(\beta}=\frac{1}{2}\partial_{(\beta\dot{\beta}}a^{\alpha)\dot{\beta}},\qquad
l^{\;\;\dot{\alpha})}_{(\dot{\beta}}=\frac{1}{2}\partial_{\beta(\dot{\beta}}a^{\beta\dot{\alpha})}.$
(3.21)

In this gauge, the remaining components undergo the transformation laws:

$\delta\Phi^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}=\partial^{(\alpha(\dot{\alpha}}a^{\beta)\dot{\beta})},\qquad\delta\Phi=\frac{1}{4}\partial_{\beta\dot{\beta}}a^{\beta\dot{\beta}}.$
(3.22)

The field $P^{\alpha\dot{\alpha}}$ possesses the non-trivial gauge
transformation law (3.19d). Because of the relations (3.21), in the symmetric
gauge we have:

$\partial^{\dot{\alpha}\beta}l_{(\beta}^{\;\;\alpha)}=\frac{1}{16}\Box
a^{\alpha\dot{\alpha}}-\frac{1}{4}\partial^{\alpha}_{\dot{\beta}}\partial^{\dot{\alpha}}_{\beta}a^{\beta\dot{\beta}}=\frac{1}{8}\Box
a^{\alpha\dot{\alpha}}-\frac{1}{4}\partial^{\alpha\dot{\alpha}}\partial^{\beta\dot{\beta}}a_{\beta\dot{\beta}}.$
(3.23)

Then the variation (3.19d) can be canceled by adding to
$P^{\alpha\dot{\alpha}}$ the term $iB^{\alpha\dot{\alpha}}$, where:

$B^{\alpha\dot{\alpha}}=-\partial_{\beta\dot{\beta}}\Phi^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}+3\partial^{\alpha\dot{\alpha}}\Phi,\quad\delta
B^{\alpha\dot{\alpha}}=-2\partial^{\dot{\alpha}\beta}l_{(\beta}^{\;\;\alpha)}=-\frac{1}{4}\Box
a^{\alpha\dot{\alpha}}+\frac{1}{2}\partial^{\alpha\dot{\alpha}}\partial^{\beta\dot{\beta}}a_{\beta\dot{\beta}}.$
(3.24)

Note the important property,
$\partial_{\beta\dot{\beta}}B^{\beta\dot{\beta}}=-\frac{1}{4}R$, where $R$ is
the linearized scalar curvature defined in (A.5). It is invariant under the
linearized gauge transformations (3.22).

The spin $\frac{3}{2}$ field can also be decomposed into the irreducible
pieces:

$\psi^{\beta\alpha\dot{\alpha}i}=\psi^{(\beta\alpha)\dot{\alpha}i}+\epsilon^{\beta\alpha}\psi^{\dot{\alpha}i},\qquad\delta_{\lambda}\psi^{(\beta\alpha)\dot{\alpha}i}=-\partial^{(\beta\dot{\alpha}}\epsilon^{\alpha)i},\quad\delta_{\lambda}\psi^{\dot{\alpha}i}=-\frac{1}{2}\partial^{\dot{\alpha}}_{\alpha}\epsilon^{\alpha
i}.$ (3.25)

The fields $\chi^{\alpha i}$ and $\rho_{\alpha}^{i}$ have non-trivial gauge
transformation laws (3.19e) and (3.19f). So they need to be redefined in a
similar way.

Finally, the complete set of the field redefinitions required amounts to:

$P^{\alpha\dot{\alpha}}\to
P^{\alpha\dot{\alpha}}+iB^{\alpha\dot{\alpha}},\qquad\chi^{\alpha
i}\to\chi^{\alpha
i}+4\partial^{\beta}_{\dot{\beta}}\psi_{\beta}^{\alpha\dot{\beta}i},\qquad\rho^{i}_{\alpha}\to\rho^{i}_{\alpha}-2\bar{\psi}^{i}_{\alpha}.$
(3.26)

After these redefinitions, the fields $P^{\alpha\dot{\alpha}}$, $\chi^{\alpha
i}$ and $\rho_{\alpha}^{i}$ become invariant under gauge transformations and
so can be treated as genuine auxiliary fields.

##  4 $\mathcal{N}=2$ supergravity actions: nonlinear and linearized

This section is a review of the construction of the nonlinear $\mathcal{N}=2$
supergravity action and its linearized version. An important ingredient of the
construction is the non-analytic negatively charged gauge potentials related
to the fundamental analytic ones by the harmonic zero-curvature conditions.

###  4.1 Nonlinear theory

The action of $\mathcal{N}=2$ Einstein supergravity in harmonic superspace was
constructed by Galperin, Ky and Sokatchev Galperin:1987ek . Here we briefly
recall its main features. The action describes $\mathcal{N}=2$ abelian vector
multiplet in the $\mathcal{N}=2$ supergravity background888Integration over
Grassmann variables in defined as:
$d^{8}\theta=d^{4}\theta^{+}d^{4}\theta^{-}:=(\mathcal{D}^{-})^{4}(\mathcal{D}^{+})^{4}$,
$(\mathcal{D}^{\pm})^{4}:=\frac{1}{16}(\mathcal{D}^{+})^{2}(\bar{\mathcal{D}}^{+})^{2}$.
Integral over harmonic variables is defined as $\int du\,1=1$ and zero
otherwise.:

$S_{full}=-\int d^{4}xd^{8}\theta du\,E\,H^{++5}H^{--5}.$ (4.1)

Here $E$ is the $\mathcal{N}=2$ full harmonic superspace integration measure
with the transformation law:

$\delta_{\lambda}E=-\left(\partial_{\alpha\dot{\alpha}}\lambda^{\alpha\dot{\alpha}}-\partial^{-}_{\hat{\alpha}}\lambda^{+\hat{\alpha}}-\partial^{+}_{\hat{\alpha}}\lambda^{-\hat{\alpha}}\right)E.$
(4.2)

The measure $E$ can be constructed from supergravity prepotentials and it
satisfies some useful properties. We shall not present the whole construction
here, for details see Galperin:1987ek ; 18 ; Ivanov:2022vwc . It will be
important that it is expressed in terms of the potentials $H^{++M}$ and
$H^{--M}$ which covariantize the harmonic derivatives ${\cal D}^{\pm\pm}$

$\mathfrak{D}^{++}=\partial^{++}+H^{++\alpha\dot{\alpha}}\partial_{\alpha\dot{\alpha}}+H^{++\hat{\alpha}+}\partial^{-}_{\hat{\alpha}}+\theta^{+\hat{\alpha}}\partial^{+}_{\hat{\alpha}}+H^{++5}\partial_{5},$
(4.3a)
$\mathfrak{D}^{--}=\partial^{--}+H^{--\alpha\dot{\alpha}}\partial_{\alpha\dot{\alpha}}+H^{--\hat{\alpha}+}\partial^{-}_{\hat{\alpha}}+H^{--\hat{\alpha}-}\partial^{+}_{\hat{\alpha}}+H^{--5}\partial_{5}.$
(4.3b)

The non-analytic potentials $H^{--M}$ are uniquely determined in terms of
$H^{++M}$ as solutions of the harmonic zero-curvature condition:

$[\mathfrak{D}^{++},\mathfrak{D}^{--}]=D^{0}.$ (4.4)

One could also consider another nonlinear action:

$S_{alt}=\int d^{4}xd^{8}\theta
du\,E\,H^{++\alpha\dot{\alpha}}H^{--}_{\alpha\dot{\alpha}}.$ (4.5)

However, it vanishes due to the property (easily checkable in WZ gauge):

$(\mathcal{D}^{+})^{4}\left(EH^{--\alpha\dot{\alpha}}\right)=0.$ (4.6)

In this paper we limit our attention to studying invariants and their
component reductions in the linear approximations, so we shall not discuss any
higher-curvature generalization of (4.1) or (4.5).

###  4.2 Linearized action and covariant superfields

To construct the linearized $\mathcal{N}=2$ supergravity action with manifest
$\mathcal{N}=2$ supersymmetry it is natural to pass to the covariant
superfields, since the analytic prepotentials defined above have non-standard
transformations laws (3.17) under rigid ${\cal N}=2$ supersymmetry. The most
natural way to introduce $\mathcal{N}=2$ covariant superfields is to use the
basis of covariant derivatives. To this end, one needs to split the covariant
harmonic derivative into the flat and the supergravity parts:

$\mathfrak{D}^{++}=\mathcal{D}^{++}+\hat{\mathcal{H}}^{++},\qquad\hat{\mathcal{H}}^{++}:=h^{++M}\partial_{M}.$
(4.7)

These parts are independently invariant under rigid supersymmetry:
$\delta_{\epsilon}\mathcal{D}^{++}=0$,
$\delta_{\epsilon}\hat{\mathcal{H}}^{++}=0$. As the next step, one rewrites
the supergravity part $\hat{\mathcal{H}}^{++}$ in the basis of covariant
derivatives (2.5):

$\hat{\mathcal{H}}^{++}=G^{++\alpha\dot{\alpha}}\partial_{\alpha\dot{\alpha}}+G^{++\alpha+}\mathcal{D}^{-}_{\alpha}+G^{++\dot{\alpha}+}\bar{\mathcal{D}}^{-}_{\dot{\alpha}}+G^{++5}\partial_{5}.$
(4.8)

Due to the invariance of both $\hat{\mathcal{H}}^{++}$ and covariant
derivatives under rigid ${\cal N}=2$ supersymmetry, the new vielbeins
$G^{++M}$ are $\mathcal{N}=2$ superfields with the standard rigid
supersymmetry laws. However, most of the newly introduced superfields
$G^{++M}$ are non-analytic:

$\begin{split}&G^{++\alpha\dot{\alpha}}=h^{++\alpha\dot{\alpha}}+4ih^{++\alpha+}\bar{\theta}^{-\dot{\alpha}}+4i\theta^{-\alpha}h^{++\dot{\alpha}+},\\\
&G^{++5}\;\,\,=h^{++5}-2ih^{++\hat{\alpha}+}\theta^{-}_{\hat{\alpha}},\\\
&G^{++\hat{\alpha}+}=-h^{++\hat{\alpha}+}.\\\
&G^{++\hat{\alpha}-}=h^{++\hat{\alpha}-}=0.\end{split}$ (4.9)

The linearized gauge transformations of $\hat{\mathcal{H}}^{++}$ are given by:

$\delta_{\lambda}\hat{\mathcal{H}}^{++}=[\mathcal{D}^{++},\hat{\Lambda}],\qquad\hat{\Lambda}:=\lambda^{M}\partial_{M}=\Lambda^{M}\mathcal{D}_{M},$
(4.10)

where the parameters $\Lambda^{M}$ are defined as:

$\begin{split}&\Lambda^{\alpha\dot{\alpha}}=\lambda^{\alpha\dot{\alpha}}+4i\lambda^{+\alpha}\bar{\theta}^{-\dot{\alpha}}+4i\theta^{-\alpha}\lambda^{+\dot{\alpha}},\\\
&\Lambda^{5}\;\;\,=\lambda^{5}-2i\lambda^{+\hat{\alpha}}\theta^{-}_{\hat{\alpha}},\\\
&\Lambda^{+\hat{\alpha}}=-\lambda^{+\hat{\alpha}},\\\
&\Lambda^{-\hat{\alpha}}=\lambda^{-\hat{\alpha}},\qquad\mathcal{D}^{++}\Lambda^{-\hat{\alpha}}+\Lambda^{+\hat{\alpha}}=0.\end{split}$
(4.11)

The linearized gauge transformations of the covariant superfields $G^{++M}$
are then obtained in the form:

$\delta_{\lambda}G^{++\alpha\dot{\alpha}}=\mathcal{D}^{++}\Lambda^{\alpha\dot{\alpha}},\quad\delta_{\lambda}G^{++5}=\mathcal{D}^{++}\Lambda^{5},\quad\delta_{\lambda}G^{++\hat{\alpha}+}=\mathcal{D}^{++}\Lambda^{+\hat{\alpha}}.$
(4.12)

The linearized zero-curvature condition reads:

$[\mathcal{D}^{++},\hat{\mathcal{H}}^{--}]=[\mathcal{D}^{--},\hat{\mathcal{H}}^{++}].$
(4.13)

Here we have introduced the negatively charged differential operator
$\hat{\mathcal{H}}^{--}$:

$\displaystyle\mathfrak{D}^{--}:=\mathcal{D}^{--}+\hat{\mathcal{H}}^{--},$
(4.14)
$\displaystyle\hat{\mathcal{H}}^{--}:=G^{--\alpha\dot{\alpha}}\partial_{\alpha\dot{\alpha}}+G^{--\hat{\alpha}+}\mathcal{D}^{-}_{\hat{\alpha}}+G^{--\hat{\alpha}-}\mathcal{D}^{+}_{\hat{\alpha}}+G^{--5}\partial_{5}.$

By construction, it trivially transforms under rigid $\mathcal{N}=2$
supersymmetry, $\delta_{\epsilon}\hat{\mathcal{H}}^{--}=0$.

In terms of the covariant superfields the zero-curvature condition (4.13)
amounts to the set of equations:

$\begin{cases}\mathcal{D}^{++}G^{--\alpha\dot{\alpha}}=\mathcal{D}^{--}G^{++\alpha\dot{\alpha}},\\\
\mathcal{D}^{++}G^{--5}=\mathcal{D}^{--}G^{++5},\\\
\mathcal{D}^{++}G^{--\hat{\alpha}+}=\mathcal{D}^{--}G^{++\hat{\alpha}+},\\\
\mathcal{D}^{++}G^{--\hat{\alpha}-}+G^{--\hat{\alpha}+}=0.\end{cases}$ (4.15)

These are linear harmonic equations and they have a unique solution 18 . Some
constituents of this solution are explicitly quoted in Appendix C.

The zero-curvature condition (4.13) is invariant under the linearized gauge
transformations:

$\delta_{\lambda}\hat{\mathcal{H}}^{++}=[\mathcal{D}^{++},\hat{\Lambda}],\qquad\delta_{\lambda}\hat{\mathcal{H}}^{--}=[\mathcal{D}^{--},\hat{\Lambda}].$
(4.16)

Then the linearized gauge freedom of the potentials $G^{--M}$ is given by:

$\begin{split}\delta_{\lambda}G^{--\alpha\dot{\alpha}}=\mathcal{D}^{--}\Lambda^{\alpha\dot{\alpha}},\quad\delta_{\lambda}G^{--5}=\mathcal{D}^{--}\Lambda^{5},\quad\\\
\delta_{\lambda}G^{--\hat{\alpha}+}=\mathcal{D}^{--}\Lambda^{+\hat{\alpha}}+\Lambda^{-\hat{\alpha}},\quad\delta_{\lambda}G^{--\hat{\alpha}-}=\mathcal{D}^{--}\Lambda^{-\hat{\alpha}}.\end{split}$
(4.17)

Now it is not difficult to verify that the supersymmetric and gauge invariant
quadratic action has the form:

$S_{lin}=-\int d^{4}xd^{8}\theta
du\left[G^{++\alpha\dot{\alpha}}G^{--}_{\alpha\dot{\alpha}}+4G^{++5}G^{--5}\right].$
(4.18)

Indeed, the gauge transformation of this action is:

$\delta_{\lambda}S_{lin}=-2\int d^{4}xd^{8}\theta
du\left[G^{++\alpha\dot{\alpha}}\mathcal{D}^{--}\Lambda_{\alpha\dot{\alpha}}+4G^{++5}\mathcal{D}^{--}\Lambda^{5}\right].$
(4.19)

Integrating over non-analytic Grassmann variables
$d^{4}\theta^{-}:=(\mathcal{D}^{+})^{4}=\frac{1}{16}(\mathcal{D}^{+})^{2}(\bar{\mathcal{D}}^{+})^{2}$
yields:

$(\mathcal{D}^{+})^{4}\left(G^{++\alpha\dot{\alpha}}\mathcal{D}^{--}\Lambda_{\alpha\dot{\alpha}}\right)=16ih^{++\alpha+}\partial_{\alpha\dot{\alpha}}\lambda^{+\dot{\alpha}}-16ih^{++\dot{\alpha}+}\partial_{\alpha\dot{\alpha}}\lambda^{+\alpha},$
(4.20)
$(\mathcal{D}^{+})^{4}\left(G^{++5}\mathcal{D}^{--}\Lambda^{4}\right)=-4ih^{++\alpha+}\partial_{\alpha\dot{\alpha}}\lambda^{+\dot{\alpha}}+4ih^{++\dot{\alpha}+}\partial_{\alpha\dot{\alpha}}\lambda^{+\alpha}.$
(4.21)

We observe that these terms in (4.19) explicitly cancel each other.

In section 6 we shall perform a detailed analysis of the component structure
of this action and explicitly demonstrate that it indeed describes the
linearized off-shell $\mathcal{N}=2$ supergravity.

##  5 Linearized super-curvatures and quadratic invariants

In this section, using the covariant superfields $G^{\pm\pm M}$ introduced in
the previous section, we construct the linearized supercurvatures which
contain linearized gravity invariants $R$,
$\mathcal{R}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}$,
$\mathcal{R}_{(\alpha\beta\gamma\delta)}$,
$\bar{\mathcal{R}}_{(\dot{\alpha}\dot{\beta}\dot{\gamma}\dot{\delta})}$ (see
Appendix A for notations). We shall present all the superfield actions which
in components involve terms quadratic in the gravity curvatures.

Inspecting the gauge transformations of the negatively charged potentials
(4.17) and generalizing the method which was exploited for construction of the
linearized $\mathcal{N}=2$ supergravity action, we can construct two
manifestly supersymmetric analytic objects:

$\mathcal{F}^{++\alpha\dot{\alpha}}:=\left(\mathcal{D}^{+}\right)^{4}G^{--\alpha\dot{\alpha}},$
(5.1a) $\mathcal{F}^{++5}:=\left(\mathcal{D}^{+}\right)^{4}G^{--5}.$ (5.1b)

These quantities are invariant under the linearized gauge transformations
(4.17):

$\delta_{\lambda}\mathcal{F}^{++\alpha\dot{\alpha}}=(\mathcal{D}^{+})^{4}\delta
G^{--\alpha\dot{\alpha}}=(\mathcal{D}^{+})^{4}\mathcal{D}^{--}\Lambda^{\alpha\dot{\alpha}}=0,$
(5.2a) $\delta_{\lambda}\mathcal{F}^{++5}=(\mathcal{D}^{+})^{4}\delta
G^{--5}=(\mathcal{D}^{+})^{4}\mathcal{D}^{--}\Lambda^{5}=0.$ (5.2b)

It is then natural to treat these quantities as linearized $\mathcal{N}=2$
supergravity curvatures. Note that, as a consequence of the superfield
equations of motion of $\mathcal{N}=2$ linearized Einstein supergravity (4.18)
Buchbinder:2022vra , these supercurvatures are zero on shell of the latter
theory:

$\mathcal{F}^{++\alpha\dot{\alpha}}=0,\qquad\mathcal{F}^{++5}=0.$ (5.3)

Using these linearized supercurvatures, one can build two $\mathcal{N}=2$
linearized supergravity invariants

$I_{1}:=\int
d\zeta^{(-4)}\mathcal{F}^{++\alpha\dot{\alpha}}\mathcal{F}^{++}_{\alpha\dot{\alpha}}=\int
d^{4}xd^{8}\theta
du\;G^{--\alpha\dot{\alpha}}(\mathcal{D}^{+})^{4}G^{--}_{\alpha\dot{\alpha}},$
(5.4)

$I_{2}:=\int d\zeta^{(-4)}\mathcal{F}^{++5}\mathcal{F}^{++5}=\int
d^{4}xd^{8}\theta du\;G^{--5}(\mathcal{D}^{+})^{4}G^{--5}.$ (5.5)

In the second expressions the integration goes over the total harmonic
superspace. The invariants (5.4) and (5.5) have the same form as the invariant
constructed in Ivanov:2005qf  in six-dimensional renormalizable higher-
derivative gauge theory and so can be considered as its spin 2 generalization.
As we shall see in the next section, at the component level these invariants
provide $\mathcal{N}=2$ supergravity extensions of the linearized invariants
$\frac{1}{4}R^{mn}R_{mn}=16\mathcal{R}^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\mathcal{R}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}+R^{2}$
and $R^{2}$, respectively.

It is also possible to construct one more quadratic invariant, with a more
complicated structure. First, we define the object:

$\mathcal{H}^{++}_{\alpha\beta}:=\mathcal{D}^{-}_{\alpha}G^{+++}_{\beta}-\partial_{\alpha}^{\dot{\rho}}G^{++}_{\beta\dot{\rho}}.$
(5.6)

Since it is constructed out of the covariant superfields, it is manifestly
supersymmetric. Under the linearized gauge transformation it behaves as:

$\delta_{\lambda}\mathcal{H}^{++}_{\alpha\beta}=\mathcal{D}^{++}\Lambda_{\alpha\beta},\qquad\Lambda_{\alpha\beta}:=\mathcal{D}^{-}_{\alpha}\Lambda^{+}_{\beta}-\partial_{\alpha}^{\dot{\rho}}\Lambda_{\beta\dot{\rho}}.$
(5.7)

The newly introduced object is half-analytic :

$\bar{\mathcal{D}}^{+}_{\dot{\alpha}}\mathcal{H}^{++}_{\alpha\beta}=0,\qquad\bar{\mathcal{D}}^{+}_{\dot{\alpha}}\Lambda_{\alpha\beta}=0.$
(5.8)

This property fixes the relative coefficient in (5.6). The superfield
$\mathcal{H}^{++}_{\alpha\beta}$ can be decomposed into the irreducible parts
as
$\mathcal{H}^{++}_{\alpha\beta}=\mathcal{H}^{++}_{(\alpha\beta)}+\epsilon_{\alpha\beta}\mathcal{H}^{++}$.
Using the zero-curvature equation, one can define its negatively charged
counterpart $\mathcal{H}^{--}_{\alpha\beta}$:

$\mathcal{D}^{++}\mathcal{H}^{--}_{\alpha\beta}=\mathcal{D}^{--}\mathcal{H}^{++}_{\alpha\beta}.$
(5.9)

The superfield $\mathcal{H}^{--}_{\alpha\beta}$ has the following linearized
gauge transformation law:

$\delta_{\lambda}\mathcal{H}^{--}_{\alpha\beta}=\mathcal{D}^{--}\Lambda_{\alpha\beta}.$
(5.10)

Using the zero-curvature equations (4.15) one can explicitly express
$\mathcal{H}^{--}_{\alpha\beta}$ in terms of the negatively charged potentials
$G^{--M}$:

$\mathcal{H}^{--}_{\alpha\beta}=\mathcal{D}^{+}_{\alpha}G^{---}_{\beta}+\mathcal{D}^{-}_{\alpha}G^{--+}_{\beta}-\partial_{\alpha}^{\dot{\rho}}G^{--}_{\beta\dot{\rho}}.$
(5.11)

With the help of the irreducible parts of $\mathcal{H}^{++}_{\alpha\beta}$ and
$\mathcal{H}^{--}_{\alpha\beta}$ one can construct two invariant actions:

$I_{3}=\int d^{4}xd^{8}\theta
du\;\mathcal{H}^{++(\alpha\beta)}\mathcal{H}^{--}_{(\alpha\beta)},$ (5.12)
$I_{4}=\int d^{4}xd^{8}\theta du\;\mathcal{H}^{++}\mathcal{H}^{--}.$ (5.13)

The invariance of $I_{3}$ and $I_{4}$ directly follows from the half-
analyticity condition $\eqref{eq: half-analiticity}$. For example, the
transformation of $I_{3}$ is given by:

$\begin{split}\delta_{\lambda}I_{3}&=2\int d^{4}xd^{8}\theta
du\;\mathcal{H}^{++(\alpha\beta)}\mathcal{D}^{--}\Lambda_{(\alpha\beta)}\\\
&=\frac{1}{8}\int
d^{4}xd^{4}\theta^{+}(\mathcal{D}^{+})^{2}\mathcal{H}^{++(\alpha\beta)}(\bar{\mathcal{D}}^{+})^{2}\mathcal{D}^{--}\Lambda_{(\alpha\beta)}.\end{split}$
(5.14)

Then we can use the half-analyticity property (5.8),

$(\bar{\mathcal{D}}^{+})^{2}\mathcal{D}^{--}\Lambda_{(\alpha\beta)}=-\bar{\mathcal{D}}^{+}_{\dot{\alpha}}\bar{\mathcal{D}}^{-\dot{\alpha}}\Lambda_{(\alpha\beta)}=\bar{\mathcal{D}}^{-\dot{\alpha}}\bar{\mathcal{D}}^{+}_{\dot{\alpha}}\Lambda_{(\alpha\beta)}=0.$
(5.15)

The invariants $I_{3}$ and $I_{4}$ have the same structure as the action of
$\mathcal{N}=2$ vector multiplet 18 . The gauge invariance of the latter
action was ensured by the analyticity of both the prepotential $V^{++}$ and
the gauge parameter $\lambda$. In checking the invariance of $I_{3}$ and
$I_{4}$ we used the weaker condition of half-analyticity.

There is yet another interesting way to represent these invariants. Let us
define the gauge invariant superfield:

$\mathcal{W}_{\alpha\beta}:=(\bar{\mathcal{D}}^{+})^{2}\mathcal{H}^{--}_{\alpha\beta},\qquad\qquad\delta_{\lambda}\mathcal{W}_{\alpha\beta}=0.$
(5.16)

As we shall show in the next section, the tensor $\mathcal{W}_{(\alpha\beta)}$
in components involves the linearized Weyl tensor
$\mathcal{R}_{(\alpha\beta\gamma\delta)}$, so it is natural to dub
$\mathcal{W}_{(\alpha\beta)}$ as the linearized $\mathcal{N}=2$ super-Weyl
tensor. Antisymmetric part $\mathcal{W}$ does not involve gravitational
curvatures and so (5.13) cannot lead to any gravitation quadratic invariant,
as will be shown below.

The tensor (5.16) satisfies some useful properties:

$\mathcal{D}^{++}\mathcal{W}_{\alpha\beta}=0,\qquad\mathcal{D}^{--}\mathcal{W}_{\alpha\beta}=0,\qquad\bar{\mathcal{D}}^{+}_{\dot{\alpha}}\mathcal{W}_{\alpha\beta}=0,\qquad\bar{\mathcal{D}}^{-}_{\dot{\alpha}}\mathcal{W}_{\alpha\beta}=0.$
(5.17)

The last property can be checked in the following way:

$\mathcal{D}^{++}\bar{\mathcal{D}}^{-}_{\dot{\alpha}}\mathcal{W}_{\alpha\beta}=\bar{\mathcal{D}}^{+}_{\dot{\alpha}}\mathcal{W}_{\alpha\beta}+\bar{\mathcal{D}}^{-}_{\dot{\alpha}}\mathcal{D}^{++}\mathcal{W}_{\alpha\beta}=0\quad\Rightarrow\quad\bar{\mathcal{D}}^{-}_{\dot{\alpha}}\mathcal{W}_{\alpha\beta}=0.$
(5.18)

Here we used the proposition that a negatively charged harmonic object
annihilated by the harmonic derivative $\mathcal{D}^{++}$ is zero. So we
conclude, that $\mathcal{W}_{\alpha\beta}$ is covariantly harmonic-independent
chiral tensor.

Integrals over Grassmann variables can be represented as
$d^{8}\theta:=d^{4}\theta
d^{4}\bar{\theta}=\frac{1}{16}d^{4}\theta(\bar{\mathcal{D}}^{+})^{2}(\bar{\mathcal{D}}^{-})^{2}$,
so due to the half-analyticity condition for $\mathcal{H}^{++(\alpha\beta)}$,
the invariant $I_{3}$ can be rewritten as:

$I_{3}=\frac{1}{16}\int d^{4}xd^{4}\theta
du\;(\bar{\mathcal{D}}^{-})^{2}\mathcal{H}^{++(\alpha\beta)}\mathcal{W}_{(\alpha\beta)}.$
(5.19)

Using the relation
$\bar{\mathcal{D}}^{-\dot{\alpha}}=[\mathcal{D}^{--},\bar{\mathcal{D}}^{+\dot{\alpha}}]$
and the properties of the linearized $\mathcal{N}=2$ super-Weyl tensor (5.17),
we obtain:

$I_{3}=\frac{1}{16}\int d^{4}xd^{4}\theta
du\;\bar{\mathcal{D}}^{-}_{\dot{\alpha}}\mathcal{D}^{+\dot{\alpha}}\mathcal{D}^{--}\mathcal{H}^{++(\alpha\beta)}\mathcal{W}_{(\alpha\beta)}.$
(5.20)

Finally, using the zero-curvature condition (5.9) and integrating by parts, we
find:

$I_{3}=-\frac{1}{16}\int
d^{4}xd^{4}\theta\;\mathcal{W}^{(\alpha\beta)}\mathcal{W}_{(\alpha\beta)}.$
(5.21)

Here we have also used the covariant harmonic-independence of
$\mathcal{W}_{(\alpha\beta)}$. So we end up with the integral over
$\mathcal{N}=2$ chiral superspace.

Using the same reasonings, the invariant $I_{4}$ can also be represented in
the chiral form:

$I_{4}=-\frac{1}{16}\int d^{4}xd^{4}\theta\;\mathcal{W}^{2}.$ (5.22)

In such a form, the invariants $I_{3}$ and $I_{4}$ are manifestly gauge
invariant.

Quite analogously, one can introduce the conjugated superfields
$\bar{\mathcal{H}}^{++}_{(\dot{\alpha}\dot{\beta})}$,
$\bar{\mathcal{H}}^{--}_{(\dot{\alpha}\dot{\beta})}$,
$\bar{\mathcal{W}}_{(\dot{\alpha}\dot{\beta})}$ and construct the conjugated
invariants.

##  6 Component contents of the linearized invariants

Given the component expansion of the prepotentials in WZ gauge (3.10), it is
straightforward to deduce the component expansion of the invariants
constructed. Technically, the most involved part of the component reduction
procedure is solving the harmonic zero-curvature equations (4.15). The
solutions of these equations are collected in Appendix C. Using these results,
in this section we present component expansions of all invariants constructed
earlier. In some cases, if necessary, we present the full component structure,
while otherwise quote only the relevant terms in the gravity sector.

###  6.1 Einstein supergravity

We start with the component reduction of the linearized $\mathcal{N}=2$
supergravity action (4.18). It is convenient to consider the bosonic and
fermionic parts of the action separately.

After substituting the solutions of the zero-curvature equations and
integrating over Grassmann and harmonic co-ordinates, we obtain the linearized
off-shell $\mathcal{N}=2$ Einstein supergravity action in the bosonic sector
as:

$\begin{split}S^{(bos)}_{lin}=8\int
d^{4}x\;\Bigr{[}&\Phi^{\alpha\dot{\alpha}\beta\dot{\beta}}\mathcal{G}_{\alpha\dot{\alpha}\beta\dot{\beta}}+4C^{\rho\dot{\rho}}\left(\partial_{\rho\dot{\rho}}\partial^{\beta\dot{\beta}}C_{\beta\dot{\beta}}-\frac{1}{2}\Box
C_{\rho\dot{\rho}}\right)\\\
&+4iT^{(\alpha\beta)}\partial_{(\alpha}^{\dot{\beta}}C_{\dot{\beta}\beta)}-4i\bar{T}^{(\dot{\alpha}\dot{\beta})}\partial_{(\dot{\alpha}}^{\beta}C_{\beta\dot{\beta})}\\\
&+\frac{1}{2}T^{(\alpha\beta)}T_{(\alpha\beta)}+\frac{1}{2}T^{(\dot{\alpha}\dot{\beta})}T_{(\dot{\alpha}\dot{\beta})}-2T\bar{T}\\\
&-\frac{1}{2}P^{\alpha\dot{\alpha}}\bar{P}_{\alpha\dot{\alpha}}-\frac{3}{4}P^{\alpha\dot{\alpha}}P_{\alpha\dot{\alpha}}-\frac{3}{4}\bar{P}^{\alpha\dot{\alpha}}\bar{P}_{\alpha\dot{\alpha}}\\\
&-\frac{1}{72}V^{\alpha\dot{\alpha}(ij)}V_{\alpha\dot{\alpha}(ij)}-\frac{1}{18}S^{ij}S_{ij}\Bigr{]}.\end{split}$
(6.1)

It will be useful to decompose the electromagnetic tensor into the irreducible
parts:

$\mathcal{F}_{\rho\dot{\rho}\beta\dot{\beta}}=\partial_{\rho\dot{\rho}}C_{\beta\dot{\beta}}-\partial_{\beta\dot{\beta}}C_{\rho\dot{\rho}}=\epsilon_{\rho\beta}\partial_{(\dot{\rho}}^{\sigma}C_{\sigma\dot{\beta})}+\epsilon_{\dot{\rho}\dot{\beta}}\partial_{(\rho}^{\dot{\sigma}}C_{\dot{\sigma}\beta)}:=\epsilon_{\rho\beta}\bar{\mathcal{F}}_{(\dot{\rho}\dot{\beta})}+\epsilon_{\dot{\rho}\dot{\beta}}\mathcal{F}_{(\rho\beta)}.$
(6.2)

The irreducible spin-tensors $\mathcal{F}_{(\rho\beta)}$ and
$\bar{\mathcal{F}}_{(\dot{\rho}\dot{\beta})}$ are gauge invariant and they
will be used for finding out the component structure of invariants.

For the Maxwell part of the action (6.1) we find (up to integration by parts):

$\begin{split}32C^{\rho\dot{\rho}}\left(\partial_{\rho\dot{\rho}}\partial^{\beta\dot{\beta}}C_{\beta\dot{\beta}}-\frac{1}{2}\Box
C_{\rho\dot{\rho}}\right)&=16\left(\partial^{\beta\dot{\beta}}C^{\rho\dot{\rho}}-\partial^{\rho\dot{\rho}}C^{\beta\dot{\beta}}\right)\left(\partial_{\beta\dot{\beta}}C_{\rho\dot{\rho}}-\partial_{\rho\dot{\rho}}C_{\beta\dot{\beta}}\right)\\\
&=-32\bar{\mathcal{F}}^{(\dot{\beta}\dot{\rho})}\bar{\mathcal{F}}_{(\dot{\beta}\dot{\rho})}-32\bar{\mathcal{F}}^{(\beta\rho)}\bar{\mathcal{F}}_{(\beta\rho)}.\end{split}$
(6.3)

This Maxwell action has the incorrect sign. This is because, within the
compensating procedure of deriving the Einstein supergravity action from the
conformal supergravity one, the above Maxwell field comes from the vector
superconformal compensator action with a wrong sign. However, the situation
can be remedied by taking into account the tensorial auxiliary fields. The
equations of motion for them are algebraic:

$\begin{split}T_{(\alpha\beta)}=-4i\mathcal{F}_{(\alpha\beta)},\quad&\bar{T}_{(\dot{\alpha}\dot{\beta})}=4i\mathcal{F}_{(\dot{\alpha}\dot{\beta})},\\\
T=\bar{T}=0,\quad
P_{\alpha\dot{\alpha}}=\bar{P}_{\alpha\dot{\alpha}}=0,\quad&V^{\alpha\dot{\alpha}(ij)}=0,\quad
S^{(ij)}=0\,\end{split}$ (6.4)

and so these auxiliary fields can be eliminated from the action. As a result
of this elimination, we arrive at the on-shell massless Pauli-Fiertz and
Maxwell actions with the correct signs:

$S^{(bos)}_{lin|on-shell}=8\int
d^{4}x\left(\Phi^{\alpha\dot{\alpha}\beta\dot{\beta}}\mathcal{G}_{\alpha\dot{\alpha}\beta\dot{\beta}}+4\bar{\mathcal{F}}^{(\dot{\beta}\dot{\rho})}\bar{\mathcal{F}}_{(\dot{\beta}\dot{\rho})}+4\bar{\mathcal{F}}^{(\beta\rho)}\bar{\mathcal{F}}_{(\beta\rho)}\right),$
(6.5)

thereby confirming the status of $C_{\alpha\dot{\alpha}}$ as the physical
“graviphoton” field.

In the fermionic sector we are left with the following off-shell action999Note
that, in the process of calculating the component action, many unexpected
cancelations occur.:

$\begin{split}S_{lin}^{(ferm)}=32i\int
d^{4}x\Bigr{(}&\psi^{\beta\alpha\dot{\alpha}i}\bar{\mathcal{G}}_{\beta\alpha\dot{\alpha}i}+2\rho^{\beta
i}\partial_{\beta}^{\dot{\beta}}\bar{\rho}_{\dot{\beta}i}+\frac{1}{16}\rho^{\alpha
i}\chi_{\alpha i}+c.c.\Bigr{)}.\end{split}$ (6.6)

Here $\bar{\mathcal{G}}_{\beta\alpha\dot{\alpha}i}$ is a complex spin
$\frac{3}{2}$ gauge field strength (see eqs. (B.3)). The auxiliary fields
$\chi^{\alpha i}$ and $\rho^{\alpha i}$ have the algebraic equations of motion
and are equal to zero on shell. So on shell we end up with the free action of
the $SU(2)$ doublet of the spin $\frac{3}{2}$ Rarita-Schwinger gauge fields
$\psi^{\beta\alpha\dot{\alpha}i}=\psi^{(\beta\alpha)\dot{\alpha}i}+\epsilon^{\alpha\beta}\psi^{\dot{\alpha}i}$.

Thus we have come to the component off-shell action of the linearized
$\mathcal{N}=2$ supergravity and confirmed that on shell it is reduced to the
sum of the free spin $2$ action, the action of the doublet of spins
$\frac{3}{2}$ and the spin $1$ action.

###  6.2 $R^{mn}R_{mn}$ invariant

Using the relations of Appendix C, one can straightforwardly deduce the
component contents of the linearized supercurvatures. The supercurvature
(5.1a) in WZ gauge has the following component expansion:

$\begin{split}\mathcal{F}^{++\alpha\dot{\alpha}}=&-8i\theta^{+}_{\rho}\bar{\theta}^{+}_{\dot{\rho}}\left(\mathcal{R}^{(\alpha\rho)(\dot{\alpha}\dot{\rho})}-\frac{1}{8}\epsilon^{\alpha\rho}\epsilon^{\dot{\alpha}\dot{\rho}}R+i\epsilon^{(\dot{\alpha}\dot{\rho}}\partial^{\dot{\beta})\rho}P^{\alpha}_{\dot{\beta}}-i\epsilon^{(\alpha\rho}\partial^{\beta)\dot{\rho}}\bar{P}^{\dot{\alpha}}_{\beta}\right)\\\
&+4(\bar{\theta}^{+})^{2}\left(\partial^{\dot{\alpha}}_{\rho}T^{(\alpha\rho)}+\partial^{\alpha\dot{\alpha}}T\right)+4(\theta^{+})^{2}\left(\partial^{\alpha}_{\dot{\rho}}\bar{T}^{(\dot{\alpha}\dot{\rho})}+\partial^{\alpha\dot{\alpha}}\bar{T}\right)\\\
&+\frac{1}{3}V^{\alpha\dot{\alpha}(ij)}u^{+}_{i}u^{+}_{j}+\frac{4}{3}i\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}\partial_{\rho\dot{\rho}}V^{\alpha\dot{\alpha}(ij)}u^{+}_{i}u^{-}_{j}-\frac{2}{3}(\theta^{+})^{4}\Box
V^{\alpha\dot{\alpha}(ij)}u^{-}_{i}u^{-}_{j}\\\
&-16i\bar{\theta}^{+}_{\dot{\rho}}\mathcal{G}^{\dot{\rho}\dot{\alpha}\alpha
i}u^{+}_{i}-16i\theta^{+}_{\rho}\bar{\mathcal{G}}^{\rho\alpha\dot{\alpha}i}u^{+}_{i}\\\
&+32(\bar{\theta}^{+})^{2}\theta^{+\rho}\partial_{\rho\dot{\rho}}\mathcal{G}^{\dot{\rho}\dot{\alpha}\alpha
i}u^{-}_{i}-32(\theta^{+})^{2}\bar{\theta}^{+\dot{\rho}}\partial_{\rho\dot{\rho}}\bar{\mathcal{G}}^{\rho\alpha\dot{\alpha}i}u^{-}_{i}\\\
&+2i\theta^{+\alpha}\bar{\chi}^{\dot{\alpha}i}u^{+}_{i}-2i\bar{\theta}^{+\dot{\alpha}}\chi^{\alpha
i}u^{+}_{i}-4(\bar{\theta}^{+})^{2}\theta^{+\rho}\partial_{\rho}^{\dot{\alpha}}\chi^{\alpha
i}u^{-}_{i}-4(\theta^{+})^{2}\bar{\theta}^{+\dot{\rho}}\partial_{\dot{\rho}}^{\alpha}\bar{\chi}^{\dot{\alpha}i}u^{-}_{i}.\end{split}$
(6.7)

From this expression we conclude that the analytic superfield
$\mathcal{F}^{++\alpha\dot{\alpha}}$ is $\mathcal{N}=2$ generalization of the
linearized Einstein tensor (A.11).

After integration over Grassmann coordinates and harmonics, the linearized
invariant (5.4) yields the following component Lagrangian:

$\begin{split}I_{1}=-16\int
d^{4}x\;\Bigr{[}&\mathcal{R}^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\mathcal{R}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}+\frac{1}{16}R^{2}\\\
&+i\mathcal{R}^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\partial_{\alpha\dot{\alpha}}P_{\beta\dot{\beta}}-16i\mathcal{R}^{(\alpha\beta)(\dot{\alpha}\dot{\beta})}\partial_{\alpha\dot{\alpha}}\bar{P}_{\beta\dot{\beta}}\\\
&+\frac{3i}{8}\partial_{\beta\dot{\beta}}P^{\beta\dot{\beta}}R-\frac{3i}{8}\partial_{\beta\dot{\beta}}\bar{P}^{\beta\dot{\beta}}R\\\
&-5P^{\alpha\dot{\alpha}}\Box
P_{\alpha\dot{\alpha}}-5\bar{P}^{\alpha\dot{\alpha}}\Box\bar{P}_{\alpha\dot{\alpha}}+4P^{\alpha\dot{\beta}}\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\bar{P}^{\beta\dot{\alpha}}\\\
&-2T^{(\alpha\beta)}\partial^{\dot{\alpha}}_{\alpha}\partial^{\dot{\beta}}_{\beta}\bar{T}_{(\dot{\alpha}\dot{\beta})}+T\Box
T+\frac{1}{8}\frac{1}{27}V^{\alpha\dot{\alpha}(ij)}\Box
V_{\alpha\dot{\alpha}(ij)}\\\ &+16i\mathcal{G}^{\dot{\rho}\dot{\alpha}\alpha
i}\partial_{\dot{\rho}}^{\rho}\bar{\mathcal{G}}_{\rho\alpha\dot{\alpha}i}\\\
&+2i\partial_{\rho\dot{\rho}}\mathcal{G}^{\dot{\rho}\dot{\alpha}\rho
i}\bar{\chi}_{\dot{\alpha}i}-2i\partial_{\rho\dot{\rho}}\mathcal{G}^{\rho\alpha\dot{\rho}i}\chi_{\alpha
i}-\frac{i}{2}\bar{\chi}^{\dot{\alpha}i}\partial_{\alpha\dot{\alpha}}\chi^{\alpha}_{i}\Bigr{]}.\end{split}$
(6.8)

From this component structure we see that $I_{1}$ is just $\mathcal{N}=2$
superextension of the square of the linearized Ricci curvature. The auxiliary
fields in $I_{1}$ have non-algebraic equations of motions and so are
dynamical, in contract to those in the linearized $\mathcal{N}=2$ Einstein
supergravity101010This property is typical for super-invariants with higher
derivatives, see, e.g., Ivanov:2005qf ..

###  6.3 $R^{2}$ invariant

The component structure of the second linearized supercurvature (5.1b) is
worked out in a similar way:

$\begin{split}\mathcal{F}^{++5}=&\frac{i}{2}(\theta^{+})^{2}\left(R-4i\partial_{\beta\dot{\beta}}P^{\beta\dot{\beta}}\right)-\frac{i}{2}(\bar{\theta}^{+})\left(R+i\partial_{\beta\dot{\beta}}\bar{P}^{\beta\dot{\beta}}\right)\\\
&+8i\theta^{+}_{\alpha}\bar{\theta}^{+\dot{\alpha}}\left(\partial^{\alpha\dot{\beta}}\bar{\mathcal{F}}_{(\dot{\alpha}\dot{\beta})}+\frac{i}{2}\partial_{\dot{\alpha}\beta}T^{(\alpha\beta)}\right)-8i\bar{\theta}^{+}_{\dot{\alpha}}\theta^{+\alpha}\left(\partial^{\dot{\alpha}\beta}\mathcal{F}_{(\alpha\beta)}-\frac{i}{2}\partial_{\alpha\dot{\beta}}\bar{T}^{(\dot{\alpha}\dot{\beta})}\right)\\\
&+\frac{1}{3}S^{(ij)}u^{+}_{i}u^{+}_{j}+\frac{4}{3}i\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}\partial_{\rho\dot{\rho}}S^{(ij)}u^{+}_{i}u^{-}_{j}-\frac{2}{3}(\theta^{+})^{4}\Box
S^{(ij)}u^{-}_{i}u^{-}_{j}\\\ &+4i\theta^{+}_{\alpha}\mathcal{R}^{\alpha
i}u^{+}_{i}-4i\bar{\theta}^{+}_{\dot{\alpha}}\bar{\mathcal{R}}^{\dot{\alpha}i}u^{+}_{i}+8(\theta^{+})^{2}\bar{\theta}^{+\dot{\rho}}\partial_{\dot{\rho}\alpha}\mathcal{R}^{\alpha
i}u^{-}_{i}+8(\bar{\theta}^{+})^{2}\theta^{+\rho}\partial_{\rho\dot{\alpha}}\bar{\mathcal{R}}^{\dot{\alpha}i}u^{-}_{i}\\\
&+8i\theta^{+\sigma}\partial_{\sigma}^{\dot{\beta}}\bar{\rho}_{\dot{\beta}}^{i}u^{+}_{i}+8i\bar{\theta}^{+\dot{\sigma}}\partial_{\dot{\sigma}}^{\beta}\rho_{\beta}^{i}u^{+}_{i}-4(\bar{\theta}^{+})^{2}\theta^{+\beta}\Box\rho_{\beta}^{i}u^{-}_{i}+4(\theta^{+})^{2}\bar{\theta}^{+\dot{\beta}}\Box\bar{\rho}^{i}_{\dot{\beta}}u^{-}_{i}\\\
&-i\theta^{+\alpha}\chi^{i}_{\alpha}u^{+}_{i}+i\bar{\theta}^{+\dot{\alpha}}\bar{\chi}^{i}_{\dot{\alpha}}u^{+}_{i}-2(\theta^{+})^{2}\bar{\theta}^{+\dot{\rho}}\partial_{\dot{\rho}}^{\alpha}\chi^{i}_{\alpha}u^{-}_{i}-2(\bar{\theta}^{+})^{2}\theta^{+\rho}\partial_{\rho}^{\dot{\alpha}}\bar{\chi}_{\dot{\alpha}}^{i}u^{-}_{i}.\end{split}$
(6.9)

Then the component content of the invariant (5.5) is given by the expression:

$\begin{split}I_{2}=\int
d^{4}x\;\Bigr{[}&\frac{1}{4}R^{2}-iR\partial_{\beta\dot{\beta}}\bar{P}^{\beta\dot{\beta}}+iR\partial_{\beta\dot{\beta}}P^{\beta\dot{\beta}}+4(\partial_{\beta\dot{\beta}}P^{\beta\dot{\beta}})(\partial_{\rho\dot{\rho}}\bar{P}^{\rho\dot{\rho}})\\\
&+\left(2\bar{\mathcal{F}}^{(\dot{\alpha}\dot{\beta})}+i\bar{T}^{(\dot{\alpha}\dot{\beta})}\right)\Box\left(2\bar{\mathcal{F}}_{(\dot{\alpha}\dot{\beta})}+i\bar{T}_{(\dot{\alpha}\dot{\beta})}\right)\\\
&+\left(2\mathcal{F}^{(\alpha\beta)}-iT^{(\alpha\beta)}\right)\Box\left(2\mathcal{F}_{(\alpha\beta)}-iT_{(\alpha\beta)}\right)\\\
&-8\left(2\bar{\mathcal{F}}^{(\dot{\alpha}\dot{\beta})}+i\bar{T}^{(\dot{\alpha}\dot{\beta})}\right)\partial_{\alpha\dot{\alpha}}\partial_{\beta\dot{\beta}}\left(2\mathcal{F}^{(\alpha\beta)}-iT^{(\alpha\beta)}\right)+\dots\Bigr{]}.\end{split}$
(6.10)

This is $\mathcal{N}=2$ generalization of the $R^{2}$ invariant.

###  6.4 Weyl-squared invariant

Though the component reduction of the linearized $\mathcal{N}=2$ Weyl tensor
is technically more involved, it can be performed quite analogously to the
previous supercurvatures. We confine our consideration merely to the lowest
term in the supercurvature expansion and to the gravity sector. As a result,
we obtain:

$\mathcal{W}_{(\alpha\beta)}=32\theta^{-(\gamma}\theta^{+\delta)}\mathcal{R}_{(\alpha\beta\gamma\delta)}+\dots,$
(6.11) $\mathcal{W}=\theta^{-}\theta^{+}\cdot 0+\dots.$ (6.12)

From these formulas we observe that $\mathcal{W}$ does not contribute to the
pure gravity sector. The terms denoted by ellipses also contain at least 3
derivatives of the spin 2 field. So for the invariant $I_{3}$ defined in
(5.12) and (5.21) we have the following expression in the gravity sector:

$I_{3}\sim\int
d^{4}x\;\mathcal{R}^{(\alpha\beta\gamma\delta)}\mathcal{R}_{(\alpha\beta\gamma\delta)}.$
(6.13)

The conclusion is that the invariant $I_{3}$ is an irreducible part of the
linearized $\mathcal{N}=2$ Weyl-squared super-invariant. It is worth to note
that, using $I_{3}$, its complex conjugate and the invariants $I_{1},I_{2}$,
one can also construct a supersymmetric generalization of the square of
Riemann tensor.

To summarize, we have the complete set of the linearized off-shell
$\mathcal{N}=2$ supergravity invariants in terms of analytic prepotentials. In
components, they start with the squares of gravity curvatures: $I_{1}\sim
R^{mn}R_{mn}$, $I_{2}\sim R^{2}$ and
$I_{3}\sim\mathcal{R}^{(\alpha\beta\gamma\delta)}\mathcal{R}_{(\alpha\beta\gamma\delta)}$.

##  7 Linearized $\mathcal{N}=2$ supergravity invariants beyond the quadratic
order

In harmonic superspace one can construct invariant actions as integrals over
both analytic and full harmonic superspaces:

$I_{ASS}=\int d\zeta^{(-4)}\,\mathcal{L}^{(+4)},\qquad I_{HSS}=\int
d^{4}xd^{8}\theta du\;\mathcal{L}.$ (7.1)

Using the gauge-invariant supercurvatures
$\mathcal{F}^{++\alpha\dot{\alpha}},\mathcal{F}^{++5},\mathcal{W}_{(\alpha\beta)},\bar{\mathcal{W}}_{(\dot{\alpha}\dot{\beta})}$
defined in (5.1a), (5.1b), (5.16) and covariant derivatives (2.5), (2.6), one
can construct a family of such invariant actions. Being aware of the component
structure of supercurvatures, one can generate linearized higher-order
$\mathcal{N}=2$ supergravity invariants and deduce their component structure
in the gravity sector.

As examples, we present some interesting invariants:

$\bullet$ $R^{4}$ invariant

$\int d^{4}xd^{8}\theta
du\;\left(\mathcal{D}^{--}\mathcal{F}^{++5}\right)^{4}\sim\int
d^{4}x\;R^{4}+\dots\,.$ (7.2)

$\bullet$ $\Box R\Box R$ invariant:

$\int d^{4}xd^{8}\theta
du\;(\mathcal{D}^{-})^{2}\mathcal{F}^{++5}(\bar{\mathcal{D}}^{-})^{2}\mathcal{F}^{++5}\sim\int
d^{4}x\;\Box R\Box R\,.$ (7.3)

$\bullet$ $R^{2}\Box R$ invariant

$\int d^{4}xd^{8}\theta
du\;\left(\mathcal{D}^{--}\mathcal{F}^{++5}\right)^{2}(\mathcal{D}^{-})^{2}\mathcal{F}^{++5}\sim\int
d^{4}x\;R^{2}\Box R\,.$ (7.4)

$\bullet$ $R^{4+n}$ invariant

$\int d^{4}xd^{8}\theta
du\;\left(\mathcal{D}^{--}\mathcal{F}^{++5}\right)^{4}\left[(\mathcal{D}^{-})^{2}\mathcal{F}^{++5}\right]^{n}\sim\int
d^{4}x\;R^{4+n}+\dots\,.$ (7.5)

$\bullet$ $G^{4}$ invariant

$\int d^{4}xd^{8}\theta
du\;\left(\mathcal{D}^{--}\mathcal{F}^{++\alpha_{1}\dot{\alpha}_{1}}\right)\left(\mathcal{D}^{--}\mathcal{F}^{++\alpha_{2}\dot{\alpha}_{2}}\right)\left(\mathcal{D}^{--}\mathcal{F}^{++}_{\alpha_{1}\dot{\alpha}_{2}}\right)\left(\mathcal{D}^{--}\mathcal{F}^{++}_{\alpha_{2}\dot{\alpha}_{1}}\right)\\\
\sim\int
d^{4}x\;\mathcal{G}^{\alpha_{1}\beta_{1}\dot{\alpha}_{1}\dot{\beta}_{1}}\mathcal{G}^{\alpha_{2}\beta_{2}\dot{\alpha}_{2}\dot{\beta}_{2}}\mathcal{G}_{\beta_{2}\alpha_{1}\dot{\beta}_{1}\dot{\alpha}_{2}}\mathcal{G}_{\beta_{1}\alpha_{2}\dot{\beta}_{2}\dot{\alpha}_{1}}+\dots\,.$
(7.6)

$\bullet$ Bel-Robinson tensor squared

Bel-Robinson tensor in the spinor notations is defined as (see, e.g.
Ferrara:1977mv ):

$\mathcal{R}_{(\alpha\beta\gamma\delta)}\bar{\mathcal{R}}_{(\dot{\alpha}\dot{\beta}\dot{\gamma}\dot{\delta})}.$
(7.7)

Inspecting the component structure of supercurvatures, one immediately
observes that Bel-Robinson tensor is contained in the superfield

$\mathcal{W}_{(\alpha\beta)}\bar{\mathcal{W}}_{(\dot{\alpha}\dot{\beta})}\sim\theta^{+\gamma}\theta^{-\delta}\bar{\theta}^{+\dot{\gamma}}\bar{\theta}^{-\dot{\delta}}\mathcal{R}_{(\alpha\beta\gamma\delta)}\bar{\mathcal{R}}_{(\dot{\alpha}\dot{\beta}\dot{\gamma}\dot{\delta})}.$
(7.8)

Then one can construct $\mathcal{N}=2$ extension of the Bel-Robinson tensor-
squared invariant:

$\int d^{4}xd^{8}\theta
du\;\mathcal{W}^{(\alpha\beta)}\mathcal{W}_{(\alpha\beta)}\bar{\mathcal{W}}^{(\dot{\alpha}\dot{\beta})}\bar{\mathcal{W}}_{(\dot{\alpha}\dot{\beta})}\sim\int
d^{4}x\;\mathcal{R}^{(\alpha\beta\gamma\delta)}\mathcal{R}_{(\alpha\beta\gamma\delta)}\bar{\mathcal{R}}^{(\dot{\alpha}\dot{\beta}\dot{\gamma}\dot{\delta})}\bar{\mathcal{R}}_{(\dot{\alpha}\dot{\beta}\dot{\gamma}\dot{\delta})}\,.$
(7.9)

$\bullet$ $R^{3}$ invariants and two-loop finiteness of supergravity

It is known that ${\cal N}=2$ supergravity is finite to two loops. This means
the absence of the on-shell cubic supergravity invariants. It is interesting
to analyze this issue using our off-shell approach. On dimensional grounds,
$[R^{3}]=6$, so such invariants can be constructed as an integral over the
full harmonic superspace Lagrangian with the dimension $[\mathcal{L}]=2$ or as
an integral over the analytic superspace with $[\mathcal{L}^{(+4)}]=4$. The
supercurvatures have dimensions $[\mathcal{F}^{++\alpha\dot{\alpha}}]=1$,
$[\mathcal{F}^{++5}]=1$ and $[\mathcal{W}_{(\alpha\beta)}]=1$.

The only cubic invariant which does not vanish on shell is the product of
three Riemann tensors. The product of three Riemann tensors in the spinor
notation corresponds to the invariant

$\mathcal{R}_{\alpha\beta}^{\;\;\;\;\gamma\delta}\mathcal{R}_{\gamma\delta}^{\;\;\;\;\rho\kappa}\mathcal{R}_{\rho\kappa}^{\;\;\;\;\alpha\beta}.$
(7.10)

The $\mathcal{N}=2$ supersymmetric invariant extending such a product can
naturally be constructed from three $\mathcal{N}=2$ super-Weyl tensors
$\mathcal{W}_{(\alpha\beta)}$. Since $\mathcal{W}_{(\alpha\beta)}$ is not
analytic superfield, one needs to integrate over full harmonic superspace or
chiral superspace. Using the component structure, it is easy to prove that the
invariants containing (7.10) after passing to components, cannot be
constructed in such a way. Indeed, though a superfield containing (7.10) can
be constructed from the $\mathcal{N}=2$ Weyl tensor,

$\mathcal{D}^{+\delta}\mathcal{W}_{\alpha}^{\;\;\beta}\mathcal{W}_{\beta}^{\;\;\gamma}\mathcal{D}^{-}_{\delta}\mathcal{W}_{\gamma}^{\;\;\alpha}\sim(\theta^{+})^{2}(\theta^{-})^{2}\mathcal{R}_{\alpha\beta}^{\;\;\;\;\gamma\delta}\mathcal{R}_{\gamma\delta}^{\;\;\;\;\rho\kappa}\mathcal{R}_{\rho\kappa}^{\;\;\;\;\alpha\beta}+\dots\,,$
(7.11)

this superfield is not chiral and so does not contain the necessary invariant
as a coefficient of the highest $(\theta^{+})^{4}(\theta^{-})^{4}$ monomial.

This reasoning is similar to the $\mathcal{N}=1$ superspace arguments about
absence of such an invariant in $\mathcal{N}=1$ supergravity Ferrara:1977mv .
So we conclude, that it is not possible to $\mathcal{N}=2$ supersymmetrize the
invariant (7.10).

##  8 Conclusions and outlook

In this paper, using the prepotential formulation of $\mathcal{N}=2$
supergravity in harmonic superspace, we constructed the off-shell linearized
$\mathcal{N}=2$ supercurvatures

$\mathcal{F}^{++\alpha\dot{\alpha}}(\zeta),\qquad\mathcal{F}^{++5}(\zeta),\qquad\mathcal{W}_{(\alpha\beta)}(x,\theta,u),\qquad\bar{\mathcal{W}}_{(\dot{\alpha}\dot{\beta})}(x,\bar{\theta},u)\,,$
(8.1)

which contain, respectively, the linearized gravity curvatures:

$\mathcal{G}_{\alpha\beta\dot{\alpha}\dot{\beta}},\qquad
R,\qquad\mathcal{R}_{(\alpha\beta\gamma\delta)},\qquad\bar{\mathcal{R}}_{(\dot{\alpha}\dot{\beta}\dot{\gamma}\dot{\delta})}.$
(8.2)

From these supercurvatures we constructed all linearized quadratic invariants
and presented some examples of the linearized invariants beyond the quadratic
order. All the constructed objects are expressed in terms of unconstrained
analytical prepotentials, which is the main feature of our approach in
contrast to the previous superspace studies Butter:2013lta ; Kuzenko:2015jxa ;
Kuzenko:2017uqg ; Moura:2002ip ; deWit:2010za . This allows us to
systematically derive the component form of the invariants by solving the
harmonic zero-curvature equations in WZ gauge. It is interesting to note that
all the linearized higher-derivative supergravity invariants constructed have
an obvious tensor structure and are manifestly invariant with respect to the
gauge transformations. It is the basic difference of these invariants from the
action of $\mathcal{N}=2$ Einstein supergravity (4.18) which is gauge
invariant up to total derivatives and so represents a Chern-Simons-like
action.

These results raise a number of interesting and natural issues which we hope
to address in the near future.

$\bullet$ Nonlinear invariants in harmonic superspace

The most direct task is to construct nonlinear generalizations of the
supercurvatures and invariants introduced here. Since there exists no
analytical integration measure in this version of $\mathcal{N}=2$ Einstein
supergravity 18 , one can expect that the corresponding nonlinear quadratic
invariants can be written in the full harmonic superspace as:

$\int d^{4}xd^{8}\theta
du\;E\,H^{--\alpha\dot{\alpha}}(\mathfrak{D}^{+})^{4}H^{--}_{\alpha\dot{\alpha}},\qquad\int
d^{4}xd^{8}\theta du\;E\,H^{--5}(\mathfrak{D}^{+})^{4}H^{--5}.$ (8.3)

Checking the invariance of such expressions implies making use of the
properties of the integration measure $E$ and integrations by parts. We hope
to achieve some progress toward this goal elsewhere.

It is interesting to establish the connection of such nonlinear invariants
with those found in other approaches Butter:2013lta ; Kuzenko:2015jxa ;
Kuzenko:2017uqg . Despite the fact that the invariants constructed
supersymmetrize the same gravitational invariants, it is problematic to
compare the results obtained at the superfield level, since in our approach
the invariants are constructed in terms of analytic prepotentials, while the
standard approach uses supercurvatures and supertorsions. This issue is
discussed in more detail in the next item.

$\bullet$ Differential geometry of curved $\mathcal{N}=2$ harmonic superspace

The natural way to construct higher-derivative $\mathcal{N}=2$ supergravity
invariants is to use the supertorsions and supercurvatures appearing in the
algebra of covariant derivatives Howe:1981gz . Such a method was adopted in
Butter:2013lta ; Kuzenko:2015jxa . The covariant supergeometry of
$\mathcal{N}=2$ supergravity in harmonic superspace was considered in
Galperin:1987em ; Delamotte:1987yn . In the harmonic superspace approach all
the basic supergravity constraints are automatically solved in terms of
analytic prepotentials. In a separate publication, we shall present such
nonlinear invariants constructed out of the covariant torsions and curvatures,
starting from the full harmonic superspace algebra of the covariant
derivatives.

$\bullet$ $\mathcal{N}=2$ Gauss-Bonnet theorem in harmonic superspace

In the pure gravity, there are two topological invariants quadratic in the
Riemann tensor: Pontryagin and Gauss-Bonnet invariants. Gauss-Bonnet invariant
is given as the integral of Euler density:

$I_{GB}=\int d^{4}x\sqrt{g}\left(R^{mnkr}R_{mnkr}-4R^{mn}R_{mn}+R^{2}\right).$
(8.4)

Such invariants are topological in the sense that $\delta I_{GB}/\delta
g_{mn}=0$.

$\mathcal{N}=1$ Gauss-Bonnet invariant was constructed in Ferrara:1977mv  and
its ${\cal N}=2$ cousin in Butter:2013lta  in the framework of conventional
$\mathcal{N}=2$ superspace. It would be interesting to explicitly construct
$\mathcal{N}=2$ generalization of these invariants in harmonic superspace and
to explicitly check that such invariants are topological, i.e. satisfy the
relation

$\delta I=\int d^{4}xd^{8}\theta duE\;\frac{\delta{I}}{\delta h^{++M}}\delta
h^{++M}=0.$ (8.5)

In Butter:2013lta , the topological nature of $\mathcal{N}=2$ GB-term has been
established only at the component level.

$\bullet$ $\mathcal{N}=2$ conformal supergravity invariants

$\mathcal{N}=2$ conformal supergravity formulated in harmonic superspace
Galperin:1987ek  involves an additional analytic prepotential $h^{(+4)}$, so
the structure of the linearized Weyl-squared invariant will be modified as
compared to $\mathcal{N}=2$ Einstein supergravity. It is quite simple to
extend the construction of section 5 to the conformal case. We shall present
this result and its higher-spin generalizations elsewhere.

The problem of constructing nonlinear $\mathcal{N}=2$ Weyl-squared action in
harmonic superspace remains a challenge.

$\bullet$ Other versions of $\mathcal{N}=2$ supergravity

It is well known that various types of $\mathcal{N}=2$ Einstein supergravity
can be constructed from the conformal $\mathcal{N}=2$ Weyl multiplet using
superconformal compensators 18 ; Ivanov:2022vwc ; Freedman:2012zz . The
$\mathcal{N}=2$ Maxwell multiplet must necessarily be present as one of the
compensators. The second compensator may be the hypermultiplet, the nonlinear
multiplet, the tensor or central-charged multiplets. These compensators lead
to different versions of supergravity. The “principal” version (when one of
the compensators is the hypermultiplet) contains an infinite number of off-
shell components, other versions contain $\mathbf{40_{B}}+\mathbf{40_{F}}$
off-shell degrees of freedom. In this article, we dealt with the version
corresponding to the nonlinear compensating multiplet. It is natural to
consider the construction of invariants also in other versions of
supergravity.

In this regard, we note the work Butter:2010sc  in which various linearized
versions of $\mathcal{N}=2$ Einstein supergravity were considered. It is
instructive to reproduce such versions within the harmonic superspace
framework. Such formulations can lead to the alternative versions of ${\cal
N}=2$ higher spins Buchbinder:2021ite  and can clarify their relationships
with the off-shell superconformal multiplets of $\mathcal{N}=2$ higher spins
recently considered in ref. Buchbinder:2024pjm .

It is also of interest to construct the linearized versions of $\mathcal{N}=2$
AdS supergravities, which were investigated in ref. Butter:2011ym . The
harmonic formulation of such theories is so far unknown.

$\bullet$ $\mathcal{N}=2$ higher-spin generalizations

Generalizing the invariants constructed to $\mathcal{N}=2$ higher-spin off-
shell supermultiplets goes straightforwardly. For example, in terms of the
covariant superfields $G^{\pm\pm\alpha(s-1)\dot{\alpha}(s-1)}$ and
$G^{\pm\pm\alpha(s-2)\dot{\alpha}(s-2)}$ introduced in Buchbinder:2021ite , we
can immediately derive the following generalizations of the invariants $I_{1}$
and $I_{2}$:

$\begin{split}&I_{1}^{(s)}=\int
d\zeta^{(-4)}\mathcal{F}^{++\alpha(s-1)\dot{\alpha}(s-1)}\mathcal{F}^{++}_{\alpha(s-1)\dot{\alpha}(s-1)},\\\
&\mathcal{F}^{++\alpha(s-1)\dot{\alpha}(s-1)}:=\left(\mathcal{D}^{+}\right)^{4}G^{--\alpha(s-1)\dot{\alpha}(s-1)},\end{split}$
(8.6) $\begin{split}&I_{2}^{(s)}=\int
d\zeta^{(-4)}\mathcal{F}^{++\alpha(s-2)\dot{\alpha}(s-2)}\mathcal{F}^{++}_{\alpha(s-2)\dot{\alpha}(s-2)},\\\
&\mathcal{F}^{++\alpha(s-2)\dot{\alpha}(s-2)}:=\left(\mathcal{D}^{+}\right)^{4}G^{--\alpha(s-2)\dot{\alpha}(s-2)}.\end{split}$
(8.7)

These invariants are the higher-spin generalizations of the $R^{2}$ and
$R^{mn}R_{mn}$ invariants of $\mathcal{N}=2$ supergravity. In a similar way,
one can generalize other invariants to $\mathcal{N}=2$ higher spins. In the
bosonic case, the higher-spin invariants of this kind were discussed in
Joung:2012qy .

$\bullet$ Induced actions

It is known that the higher-derivative gravity and higher-spin invariants can
be recovered as induced actions by starting from couplings of the higher-spin
gauge fields to quantum matter fields and then integrating out the matter
fields from the generating functional (see, e.g., BS ). The hypermultiplet
$q^{+}$ is the most general $\mathcal{N}=2$ matter multiplet. Using cubic
interacting vertices of hypermultiplet with gauge higher-spin superfields
Buchbinder:2022kzl , one can consider an effective action of the form:

$\int Dq^{+}e^{-\frac{1}{2}\int
d\zeta^{(-4)}q^{+a}\left(\mathcal{D}^{++}+\kappa\hat{\mathcal{H}}^{++}\right)q^{+}_{a}}$
(8.8)

(see Buchbinder:2022kzl  for details of the notation). Doing this functional
integral must yield some higher-derivative invariants of the higher-spin
${\cal N}=2$ gauge multiplets. In particular, it would be instructive to
reproduce in such a way the invariants introduced at the linearized level in
the present paper.

$\bullet$ $6D,\;\mathcal{N}=(1,0)$ supergravity invariants

Various higher-derivative invariants of $6D,\mathcal{N}=(1,0)$ supergravity
have been actively studied in the superfield approach (see, e.g.,
Butter:2016qkx ; Butter:2017jqu ; Novak:2017wqc ; Butter:2018wss ). $6D$
supergravity in harmonic superspace was constructed by Sokatchev in
Sokatchev:1988aa . However, the higher-derivative invariants of
$\mathcal{N}=(1,0)$ off-shell supergravity were never tackled in detail by the
harmonic superspace methods111111In ref. BHS  the harmonic approach was
applied for analysis of UV divergences of $6D$ supergravities.. It is worth
noting that the construction of such invariants in the formulations of ${\cal
N}=2$ supergravity without analytic prepotentials requires utilizing new
action principles based on the use of closed covariant super-forms
Butter:2016qkx . It is of interest to learn whether such invariants admit a
reformulation in harmonic superspace. Recall once more that the latter
provides the universal methods of setting up superfield invariants, based upon
the analytic gauge prepotentials and the harmonic zero-curvature equations.

###### Acknowledgements.

Work of N.Z. was partially supported by the grant 22-1-1-42-2 of the
Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”.

##  Appendix A Linearized gravity invariants in spinor notations

We use the standard definition of the linearized gravity field
$g_{mn}=\eta_{mn}+h_{mn}$, $h:=\eta^{mn}h_{mn}$. The linearizations of Riemann
curvature tensor, Ricci curvature and scalar curvature are given by:

$R_{krmn}=2\partial_{[r}\partial_{[m}h_{n]k]}=\frac{1}{2}\left(\partial_{m}\partial_{r}h_{nk}-\partial_{m}\partial_{k}h_{nr}-\partial_{n}\partial_{r}h_{mk}+\partial_{n}\partial_{k}h_{mr}\right),$
(A.1a)
$R_{mn}=\frac{1}{2}\left(\partial^{r}\partial_{m}h_{nr}+\partial^{r}\partial_{n}h_{mr}-\Box
h_{mn}-\partial_{m}\partial_{n}h\right),$ (A.1b)
$R=\partial^{m}\partial^{n}h_{mn}-\Box h.$ (A.1c)

All curvatures are invariant under the linearized gauge transformations
$\delta h_{mn}=\partial_{m}a_{n}+\partial_{n}a_{m}$.

The spin 2 field $\Phi^{\alpha\dot{\alpha}}_{\beta\dot{\beta}}$ from WZ gauge
(3.10) is related to the fierbein as
$\delta_{\beta}^{\alpha}\delta^{\dot{\alpha}}_{\dot{\beta}}+\Phi^{\alpha\dot{\alpha}}_{\beta\dot{\beta}}:=\frac{1}{2}\sigma^{\alpha\dot{\alpha}}_{m}\sigma^{a}_{\beta\dot{\beta}}e_{a}^{m}$.
The fierbein expansion near the flat Minkowski background is given by:
$e^{m}_{a}=\delta^{m}_{a}+\varepsilon_{a}^{m}$, so that
$h_{mn}=\varepsilon_{mn}+\varepsilon_{nm}$. After fixing the local Lorentz
freedom, in the symmetric gauge $\varepsilon_{mn}=\varepsilon_{(mn)}$ we have
:

$\begin{split}h_{mn}=2\varepsilon_{mn}&=\sigma_{m}^{\alpha\dot{\alpha}}\sigma_{n}^{\beta\dot{\beta}}\left(\Phi_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}+\epsilon_{\alpha\beta}\epsilon_{\dot{\alpha}\dot{\beta}}\Phi\right)\\\
&=\sigma_{m}^{\alpha\dot{\alpha}}\sigma_{n}^{\beta\dot{\beta}}\Phi_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}+2\eta_{mn}\Phi.\end{split}$
(A.2)

From this relation it follows that $h=8\Phi$.

One can rewrite121212Here we use standard notations, see e.g. 18 :
$\sigma_{[mn]}^{(\alpha\beta)}=\frac{i}{2}\left((\sigma_{m})^{\alpha}_{\;\dot{\alpha}}\sigma_{n}^{\dot{\alpha}\beta}-(\sigma_{n})^{\alpha}_{\;\dot{\alpha}}\sigma_{m}^{\dot{\alpha}\beta}\right),\qquad\tilde{\sigma}_{[mn]}^{(\dot{\alpha}\dot{\beta})}=\frac{i}{2}\left(\sigma_{m}^{\alpha\dot{\alpha}}(\sigma_{n})_{\alpha}^{\;\,\dot{\beta}}-\sigma_{n}^{\alpha\dot{\alpha}}(\sigma_{m})_{\alpha}^{\;\,\dot{\beta}}\right).$
the curvatures (A.1) in terms of the irreducible spin $2$ fields
$\Phi_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}$ and $\Phi$:

$R_{krmn}=\sigma_{[kr]}^{(\alpha\beta)}\sigma_{[mn]}^{(\gamma\delta)}\left(-\frac{1}{2}\mathcal{R}_{(\alpha\beta\gamma\delta)}-\frac{1}{48}\epsilon_{\alpha\gamma}\epsilon_{\beta\delta}R\right)-\frac{1}{4}\sigma_{[kr]}^{(\alpha\beta)}\tilde{\sigma}_{[mn]}^{(\dot{\alpha}\dot{\beta})}\mathcal{R}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}+c.c.,$
(A.3)
$\begin{split}R_{mn}=\sigma_{m}^{\alpha\dot{\alpha}}\sigma_{n}^{\beta\dot{\beta}}\Big{(}\mathcal{R}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}+\frac{1}{8}\epsilon_{\alpha\beta}\epsilon_{\dot{\alpha}\dot{\beta}}R\Big{)},\end{split}$
(A.4)
$R=4\partial^{\alpha\dot{\alpha}}\partial^{\beta\dot{\beta}}\Phi_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}-6\Box\Phi.$
(A.5)

Here we used the following notations for the irreducible parts of the
linearized curvatures:

$\mathcal{R}_{(\alpha\beta\gamma\delta)}:=\partial_{(\alpha}^{\dot{\alpha}}\partial_{\beta}^{\dot{\beta}}\Phi_{\gamma\delta)(\dot{\alpha}\dot{\beta})},\qquad\bar{\mathcal{R}}_{(\dot{\alpha}\dot{\beta}\dot{\gamma}\dot{\delta})}:=\partial^{\alpha}_{(\dot{\alpha}}\partial^{\beta}_{\dot{\beta}}\Phi_{\dot{\gamma}\dot{\delta})(\alpha\beta)},$
(A.6)
$\mathcal{R}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}:=2\partial_{(\alpha(\dot{\alpha}}\partial^{\rho\dot{\rho}}\Phi_{(\beta)\rho)(\dot{\beta})\dot{\rho})}-\frac{1}{2}\Box\Phi_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}-2\partial_{(\alpha(\dot{\alpha}}\partial_{\beta)\dot{\beta})}\Phi.$
(A.7)

Note that the linearized invariants (A.6) are constructed from the conformal
graviton and so can be identified with the self-dual and anti-self-dual parts
of the linearized Weyl tensor. For completeness, we give explicit relations
for the irreducible decomposition of the Weyl tensor:

$W_{krmn}:=R_{krmn}-\left(g_{k[m}R_{n]r}-g_{r[m}R_{n]k}\right)+\frac{1}{3}R\,g_{k[m}g_{n]r},$
(A.8)
$W_{krmn}=-\frac{1}{2}\sigma_{[kr]}^{(\alpha\beta)}\sigma_{[mn]}^{(\gamma\delta)}\mathcal{R}_{(\alpha\beta\gamma\delta)}-\frac{1}{2}\tilde{\sigma}_{[kr]}^{(\dot{\alpha}\dot{\beta})}\tilde{\sigma}_{[mn]}^{(\dot{\gamma}\dot{\delta})}\bar{\mathcal{R}}_{(\dot{\alpha}\dot{\beta}\dot{\gamma}\dot{\delta})}.$
(A.9)

The linearized Bianchi identities read:

$\partial^{\beta\dot{\beta}}\mathcal{R}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}=\frac{1}{8}\partial_{\alpha\dot{\alpha}}R,$
(A.10)

so the linearized Einstein tensor has the form:

$\mathcal{G}_{\alpha\beta\dot{\alpha}\dot{\beta}}=\mathcal{R}_{(\alpha\beta)(\dot{\alpha}\dot{\beta})}-\frac{1}{8}\epsilon_{\alpha\beta}\epsilon_{\dot{\alpha}\dot{\beta}}R,\qquad\partial^{\beta\dot{\beta}}\mathcal{G}_{\alpha\beta\dot{\alpha}\dot{\beta}}=0.$
(A.11)

Using the linearized Einstein tensor one can represent the linearized
Einstein-Gilbert action as:

$S_{2}\sim\int
d^{4}x\;\Phi^{\alpha\beta\dot{\alpha}\dot{\beta}}\mathcal{G}_{\alpha\beta\dot{\alpha}\dot{\beta}}.$
(A.12)

We make use of the above notations for curvatures, when writing solutions of
the harmonic zero-curvature equations in the gravity sector, and also in the
component reduction of $\mathcal{N}=2$ invariants. In such a formulation, the
component invariants are explicitly invariant with respect to the linearized
gauge transformations.

##  Appendix B Spin $\frac{3}{2}$ linearized curvatures

In this Appendix, we present the expressions for the linearized curvatures of
the Rarita-Schwinger spin $\frac{3}{2}$ gauge field. Such curvatures appear in
solutions of the zero curvature equations in the fermionic spin $\frac{3}{2}$
sector and in the component representation of $\mathcal{N}=2$ supergravity
invariants.

In accord with eq. (3.25), the irreducible parts of the spin $\frac{3}{2}$
Rarita-Schwinger field undergo the following linearized gauge transformations:

$\psi^{\beta\alpha\dot{\alpha}i}=\psi^{(\beta\alpha)\dot{\alpha}i}+\epsilon^{\beta\alpha}\psi^{\dot{\alpha}i},\qquad\delta_{\lambda}\psi^{(\beta\alpha)\dot{\alpha}i}=-\partial^{(\beta\dot{\alpha}}\epsilon^{\alpha)i},\quad\delta_{\lambda}\psi^{\dot{\alpha}i}=-\frac{1}{2}\partial^{\dot{\alpha}}_{\alpha}\epsilon^{\alpha
i}$ (B.1a) and, for the complex-conjugated field,
$\bar{\psi}^{\dot{\beta}\dot{\alpha}\alpha
i}=\bar{\psi}^{(\dot{\beta}\dot{\alpha})\dot{\alpha}i}+\epsilon^{\dot{\beta}\dot{\alpha}}\bar{\psi}^{\alpha
i},\qquad\delta_{\lambda}\bar{\psi}^{(\dot{\beta}\dot{\alpha})\alpha
i}=-\partial^{(\dot{\beta}\alpha}\bar{\epsilon}^{\dot{\alpha})i},\quad\delta_{\lambda}\bar{\psi}^{\alpha
i}=-\frac{1}{2}\partial^{\alpha}_{\dot{\alpha}}\epsilon^{\dot{\alpha}i}.$
(B.1b)

From these irreducible parts one can construct one-derivative irreducible
linearized gauge invariants:

$\mathcal{R}^{i}_{(\dot{\alpha}\dot{\beta})\alpha}:=\partial_{(\dot{\alpha}}^{\beta}\psi^{i}_{(\alpha\beta)\dot{\beta})}+\partial_{(\dot{\alpha}\alpha}\psi^{i}_{\dot{\beta})},\qquad\mathcal{R}^{i}_{\alpha}:=\partial^{\beta\dot{\beta}}\psi^{i}_{(\alpha\beta)\dot{\beta}}-3\partial_{\alpha}^{\dot{\beta}}\psi^{i}_{\dot{\beta}},$
(B.2a)
$\bar{\mathcal{R}}^{i}_{(\alpha\beta)\dot{\alpha}}:=\partial_{(\alpha}^{\dot{\beta}}\bar{\psi}^{i}_{(\dot{\alpha}\dot{\beta})\beta)}+\partial_{(\alpha\dot{\alpha}}\bar{\psi}^{i}_{\beta)},\qquad\bar{\mathcal{R}}^{i}_{\dot{\alpha}}:=\partial^{\beta\dot{\beta}}\bar{\psi}^{i}_{(\dot{\alpha}\dot{\beta})\beta}-3\partial_{\dot{\alpha}}^{\beta}\bar{\psi}^{i}_{\beta}.$
(B.2b)

The Einstein-like spin $\frac{3}{2}$ tensors have the form given below and
satisfy the linearized Bianchi identities:

$\mathcal{G}^{i}_{\dot{\alpha}\dot{\beta}\alpha}:=\mathcal{R}^{i}_{(\dot{\alpha}\dot{\beta})\alpha}+\frac{1}{2}\epsilon_{\dot{\alpha}\dot{\beta}}\mathcal{R}^{i}_{\alpha},\qquad\partial^{\alpha\dot{\alpha}}\mathcal{G}^{i}_{\dot{\alpha}\dot{\beta}\alpha}=0,$
(B.3a)
$\bar{\mathcal{G}}^{i}_{\alpha\beta\dot{\alpha}}:=\bar{\mathcal{R}}^{i}_{(\alpha\beta)\dot{\alpha}}+\frac{1}{2}\epsilon_{\alpha\beta}\bar{\mathcal{R}}^{i}_{\dot{\alpha}},\qquad\partial^{\alpha\dot{\alpha}}\bar{\mathcal{G}}^{i}_{\alpha\beta\dot{\alpha}}=0.$
(B.3b)

Using these linearized tensors, one can construct the gauge-invariant spin
$\frac{3}{2}$ action as:

$S_{\frac{3}{2}}\sim\int d^{4}x\;i\bar{\psi}^{\dot{\alpha}\dot{\beta}\alpha
i}\bar{\mathcal{G}}_{\dot{\alpha}\dot{\beta}\alpha i}.$ (B.4)

##  Appendix C Solutions of harmonic zero-curvature equations

To perform the component reduction of the supergravity invariants, one needs
solutions of four linear harmonic equations (4.15):

$\displaystyle\mathcal{D}^{++}G^{--\alpha\dot{\alpha}}=\mathcal{D}^{--}G^{++\alpha\dot{\alpha}},$
(C.1) $\displaystyle\mathcal{D}^{++}G^{--5}=\mathcal{D}^{--}G^{++5},$
$\displaystyle\mathcal{D}^{++}G^{--\hat{\alpha}+}=\mathcal{D}^{--}G^{++\hat{\alpha}+}$
$\displaystyle\mathcal{D}^{++}G^{--\hat{\alpha}+}+G^{--\hat{\alpha}+}=0.$

Because of linearity, one can solve these equations for each field in WZ gauge
(3.10) separately.

Note the important property

$\delta\left[(\mathcal{D}^{+})^{4}G^{--\alpha\dot{\alpha}}\right]=0,\qquad\delta\left[(\mathcal{D}^{+})^{4}G^{--5}\right]=0.$
(C.2)

So all $(\theta^{-})^{4}$ components of $G^{--\alpha\dot{\alpha}}$ and
$G^{--5}$ are invariant under gauge transformations (4.17). This useful
observation makes it possible to verify the correctness of the solutions
obtained. In order to explicitly identify gauge invariant terms in the
solutions of zero-curvature equations, we shall use the notation of Appendices
A and B.

In this Appendix, we explicitly quote solutions of the zero-curvature
equations necessary for the component reduction of the $\mathcal{N}=2$
supergravity curvatures and invariants defined earlier. We are working with
the redefined auxiliary fields $P_{\alpha\dot{\alpha}}$, $\rho_{\beta}^{i}$
and $\chi^{i}_{\beta}$, see eqs. (3.26).

Gravity sector:

$G_{(\Phi)}^{++\alpha\dot{\alpha}}=-4i\theta^{+\beta}\bar{\theta}^{+\dot{\beta}}\Phi_{\beta\dot{\beta}}^{\alpha\dot{\alpha}}+4(\theta^{+})^{2}\bar{\theta}^{+\dot{\beta}}\bar{\theta}^{-\dot{\alpha}}B_{\dot{\beta}}^{\alpha}-4(\bar{\theta}^{+})^{2}\theta^{+\beta}\theta^{-\alpha}B_{\beta}^{\dot{\alpha}},$
(C.3a)
$\begin{split}G^{--\alpha\dot{\alpha}}_{(\Phi)}=&-4i\theta^{-\beta}\bar{\theta}^{-\dot{\beta}}\Phi^{\alpha\dot{\alpha}}_{\beta\dot{\beta}}+8(\theta^{-})^{2}\bar{\theta}^{-\dot{\rho}}\bar{\theta}^{+\dot{\beta}}\left(\partial^{\beta}_{(\dot{\rho}}\Phi^{\alpha\dot{\alpha}}_{\beta\dot{\beta})}-\frac{1}{2}\delta^{\dot{\alpha}}_{\dot{\rho}}B^{\alpha}_{\dot{\beta}}\right)\\\
&-8(\bar{\theta}^{-})^{2}\theta^{-\rho}\theta^{+\beta}\left(\partial_{(\rho}^{\dot{\beta}}\Phi^{\alpha\dot{\alpha}}_{\dot{\beta}\beta)}-\frac{1}{2}\delta^{\alpha}_{\rho}B_{\beta}^{\dot{\alpha}}\right)\\\
&-8i(\theta^{-})^{4}\theta^{+}_{\rho}\bar{\theta}^{+}_{\dot{\rho}}\left(\mathcal{R}^{(\alpha\rho)(\dot{\alpha}\dot{\rho})}-\frac{1}{8}\epsilon^{\alpha\rho}\epsilon^{\dot{\alpha}\dot{\rho}}R\right),\end{split}$
(C.3b)
$G_{(\Phi)}^{++5}=2(\theta^{+})^{2}\bar{\theta}^{+\dot{\beta}}\theta^{-\beta}B_{\beta\dot{\beta}}+2(\bar{\theta}^{+})^{2}\theta^{+\beta}\bar{\theta}^{-\dot{\beta}}B_{\beta\dot{\beta}},$
(C.3c)
$\begin{split}G_{(\Phi)}^{--5}=&-2(\theta^{-})^{2}\bar{\theta}^{-\dot{\beta}}\theta^{+\beta}B_{\beta\dot{\beta}}-\frac{i}{2}(\theta^{-})^{4}(\theta^{+})^{2}R\\\
&-2(\bar{\theta}^{-})^{2}\theta^{-\beta}\bar{\theta}^{+\dot{\beta}}B_{\beta\dot{\beta}}+\frac{i}{2}(\theta^{-})^{4}(\bar{\theta}^{+})^{2}R,\end{split}$
(C.3d)
$G^{++\alpha+}_{(\Phi)}=-i(\theta^{+})^{2}\bar{\theta}^{+}_{\dot{\beta}}B^{\alpha\dot{\beta}},$
(C.3e)
$\begin{split}G^{--\alpha+}_{(\Phi)}=&-\frac{i}{2}(\theta^{-})^{2}\bar{\theta}^{+}_{\dot{\beta}}B^{\alpha\dot{\beta}}-i(\theta^{+}\theta^{-})\bar{\theta}^{-}_{\dot{\beta}}B^{\alpha\dot{\beta}}\\\
&+(\theta^{+})^{2}(\bar{\theta}^{-})^{2}\theta^{-\rho}\partial_{\rho\dot{\beta}}B^{\alpha\dot{\beta}}-(\theta^{-})^{2}(\bar{\theta}^{+}\bar{\theta}^{-})\theta^{+\rho}\partial_{\rho\dot{\beta}}B^{\alpha\dot{\beta}}\\\
&+2(\theta^{-})^{2}\theta^{+\rho}\bar{\theta}^{+(\dot{\rho}}\theta^{-\dot{\beta})}\partial_{\rho\dot{\rho}}B^{\alpha}_{\dot{\beta}}-\frac{i}{2}(\theta^{-})^{4}(\theta^{+})^{2}\bar{\theta}^{+\dot{\beta}}\Box
B^{\alpha}_{\dot{\beta}},\end{split}$ (C.3f)
$G^{--\alpha-}_{(\Phi)}=-\frac{i}{2}(\theta^{-})^{2}\bar{\theta}^{-}_{\dot{\beta}}B^{\alpha\dot{\beta}}+(\theta^{-})^{4}\theta^{+\beta}\partial_{\beta\dot{\beta}}B^{\alpha\dot{\beta}}.$
(C.3g)

Maxwell sector:

$G^{++5}_{(C)}=-4i\theta^{+\beta}\bar{\theta}^{+\dot{\beta}}C_{\beta\dot{\beta}}\,,$
(C.4a)
$\begin{split}G^{--5}_{(C)}=&-4i\theta^{-\beta}\bar{\theta}^{-\dot{\beta}}C_{\beta\dot{\beta}}\\\
&+8(\theta^{-})^{2}\bar{\theta}^{-(\dot{\rho}}\bar{\theta}^{+\dot{\beta})}\bar{\mathcal{F}}_{(\dot{\rho}\dot{\beta})}-8(\bar{\theta}^{-})^{2}\theta^{-(\rho}\theta^{+\beta)}\mathcal{F}_{(\rho\beta)}\\\
&-8i(\theta^{-})^{4}\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}\left(\partial_{\rho}^{\dot{\beta}}\bar{\mathcal{F}}_{(\dot{\rho}\dot{\beta})}+\partial_{\dot{\rho}}^{\beta}\mathcal{F}_{(\rho\beta)}\right).\end{split}$
(C.4b)

Auxiliary field $T^{(\alpha\beta)}$:

$G_{T^{(\alpha\beta)}}^{++\alpha\dot{\alpha}}=4i(\bar{\theta}^{+})^{2}\theta^{+}_{\beta}\bar{\theta}^{-\dot{\alpha}}T^{(\alpha\beta)}-4i(\theta^{+})^{2}\bar{\theta}^{+}_{\dot{\beta}}\theta^{-\alpha}\bar{T}^{(\dot{\alpha}\dot{\beta})},$
(C.5a)
$\begin{split}G_{T^{(\alpha\beta)}}^{--\alpha\dot{\alpha}}=&-4i(\bar{\theta}^{-})^{2}\theta^{-}_{\beta}\bar{\theta}^{+\dot{\alpha}}T^{(\alpha\beta)}+4(\theta^{-})^{4}(\bar{\theta}^{+})^{2}\partial^{\dot{\alpha}}_{\rho}T^{(\alpha\rho)}\\\
&+4i(\theta^{-})^{2}\bar{\theta}^{-}_{\dot{\beta}}\theta^{+\alpha}\bar{T}^{(\dot{\alpha}\dot{\beta})}+4(\theta^{-})^{4}(\theta^{+})^{2}\partial^{\alpha}_{\dot{\rho}}\bar{T}^{(\dot{\alpha}\dot{\rho})},\end{split}$
(C.5b)

$G_{T^{(\alpha\beta)}}^{++5}=-2i(\bar{\theta}^{+})^{2}\theta^{+}_{\alpha}\theta^{-}_{\beta}T^{(\alpha\beta)}-2i(\theta^{+})^{2}\bar{\theta}^{+}_{\dot{\alpha}}\bar{\theta}^{-}_{\dot{\beta}}\bar{T}^{(\dot{\alpha}\dot{\beta})},$
(C.5c)
$\begin{split}G_{T^{(\alpha\beta)}}^{--5}=&-2i(\bar{\theta}^{-})^{2}\theta^{+}_{\alpha}\theta^{-}_{\beta}T^{(\alpha\beta)}-4(\theta^{-})^{4}\theta^{+}_{\alpha}\bar{\theta}^{+\dot{\sigma}}\partial_{\beta\dot{\sigma}}T^{(\alpha\beta)}\\\
&-2i(\theta^{-})^{2}\bar{\theta}^{+}_{\dot{\alpha}}\bar{\theta}^{-}_{\dot{\beta}}\bar{T}^{(\dot{\alpha}\dot{\beta})}-4(\theta^{-})^{4}\bar{\theta}^{+}_{\dot{\alpha}}\theta^{+\sigma}\partial_{\sigma\dot{\beta}}\bar{T}^{(\dot{\alpha}\dot{\beta})},\end{split}$
(C.5d)

$G_{T^{(\alpha\beta)}}^{++\alpha+}=-(\bar{\theta}^{+})^{2}\theta^{+}_{\beta}T^{(\alpha\beta)},$
(C.5e)
$\begin{split}G_{T^{(\alpha\beta)}}^{--\alpha+}=&-\frac{1}{2}(\bar{\theta}^{-})^{2}\theta^{+}_{\beta}T^{(\alpha\beta)}-(\bar{\theta}^{+}\bar{\theta}^{-})\theta^{-}_{\beta}T^{(\alpha\beta)}\\\
&+i(\bar{\theta}^{-})^{2}(\theta^{+}\theta^{-})\bar{\theta}^{+\dot{\rho}}\partial_{\dot{\rho}\beta}T^{(\alpha\beta)}-2i(\bar{\theta}^{-})^{2}\theta^{+(\beta}\theta^{-\rho)}\bar{\theta}^{+\dot{\rho}}\partial_{\rho\dot{\rho}}T^{(\alpha}_{\beta)}\\\
&-i(\bar{\theta}^{+})^{2}(\theta^{-})^{2}\bar{\theta}^{-\dot{\rho}}\partial_{\dot{\rho}\beta}T^{(\alpha\beta)}+\frac{1}{2}(\theta^{-})^{4}(\bar{\theta}^{+})^{2}\theta^{+}_{\beta}\Box
T^{(\alpha\beta)},\end{split}$ (C.5f)
$G_{T^{(\alpha\beta)}}^{--\alpha-}=\frac{1}{2}(\bar{\theta}^{-})^{2}\theta^{-}_{\beta}T^{(\alpha\beta)}-i(\theta^{-})^{4}\bar{\theta}^{+\dot{\rho}}\partial_{\dot{\rho}\beta}T^{(\alpha\beta)}.$
(C.5g)

Auxiliary field $T$:

$G^{++\alpha\dot{\alpha}}_{(T)}=4i(\bar{\theta}^{+})^{2}\theta^{+\alpha}\bar{\theta}^{-\dot{\alpha}}T-4i(\theta^{+})^{2}\bar{\theta}^{+\dot{\alpha}}\theta^{-\alpha}\bar{T}\,,$
(C.6a)
$\begin{split}G^{--\alpha\dot{\alpha}}_{(T)}=&-4i(\bar{\theta}^{-})^{2}\theta^{-\alpha}\bar{\theta}^{+\dot{\alpha}}T+4(\theta^{-})^{4}(\bar{\theta}^{+})^{2}\partial^{\alpha\dot{\alpha}}T\\\
&+4i(\theta^{-})^{2}\bar{\theta}^{-\dot{\alpha}}\theta^{+\alpha}\bar{T}+4(\theta^{-})^{4}(\theta^{+})^{2}\partial^{\alpha\dot{\alpha}}\bar{T},\end{split}$
(C.6b)
$G^{++5}_{(T)}=-2i(\bar{\theta}^{+})^{2}(\theta^{+}\theta^{-})T+2i(\theta^{+})^{2}(\bar{\theta}^{+}\bar{\theta}^{-})\bar{T},$
(C.6c)
$G^{--5}_{(T)}=-2i(\theta^{-})^{2}(\bar{\theta}^{+}\bar{\theta}^{-})T+2i(\bar{\theta}^{-})^{2}(\theta^{+}\theta^{-})\bar{T},$
(C.6d) $G^{++\alpha+}_{(T)}=-(\bar{\theta}^{+})^{2}\theta^{+\alpha}T,$ (C.6e)
$\begin{split}G^{--\alpha+}_{(T)}=&-\frac{1}{2}(\bar{\theta}^{-})^{2}\theta^{+\alpha}T-(\bar{\theta}^{+}\bar{\theta}^{-})\theta^{-\alpha}T\\\
&-i(\bar{\theta}^{+})^{2}(\theta^{-})^{2}\bar{\theta}^{-\dot{\rho}}\partial_{\dot{\rho}}^{\alpha}T+i(\bar{\theta}^{-})^{2}(\theta^{+}\theta^{-})\bar{\theta}^{+\dot{\rho}}\partial_{\dot{\rho}}^{\alpha}T\\\
&-2i(\bar{\theta}^{-})^{2}\theta^{+(\alpha}\theta^{-\rho)}\bar{\theta}^{+\dot{\rho}}\partial_{\rho\dot{\rho}}T+\frac{1}{4}(\theta^{-})^{4}(\bar{\theta}^{+})^{2}\theta^{-\alpha}\Box
T,\end{split}$ (C.6f)
$G^{--\alpha-}_{(T)}=\frac{1}{2}(\bar{\theta}^{-})^{2}\theta^{-\alpha}T-\frac{i}{2}(\theta^{-})^{4}\bar{\theta}^{+\dot{\rho}}\partial_{\dot{\rho}}^{\alpha}T.$
(C.6g)

Auxiliary field $P^{\alpha\dot{\alpha}}$:

$G^{++\alpha\dot{\alpha}}_{(P)}=4i(\theta^{+})^{2}\bar{\theta}^{+}_{\dot{\beta}}\bar{\theta}^{-\dot{\alpha}}P^{\alpha\dot{\beta}}+4i(\bar{\theta}^{+})^{2}\theta^{+}_{\beta}\theta^{-\alpha}\bar{P}^{\beta\dot{\alpha}},$
(C.7a)
$\begin{split}G^{--\alpha\dot{\alpha}}_{(P)}=&2i(\bar{\theta}^{-})^{2}(\theta^{+}\theta^{-})P^{\alpha\dot{\alpha}}+4i(\theta^{-})^{2}\bar{\theta}^{+}_{(\dot{\beta}}\bar{\theta}^{-\dot{\alpha})}P^{\alpha\dot{\beta}}-8(\theta^{-})^{4}\theta^{+\rho}\bar{\theta}^{+}_{(\dot{\beta}}\partial^{\dot{\alpha})}_{\rho}P^{\alpha\dot{\beta}}\\\
&-2i(\theta^{-})^{2}(\bar{\theta}^{+}\bar{\theta}^{-})\bar{P}^{\alpha\dot{\alpha}}+4i(\bar{\theta}^{-})^{2}\theta^{+}_{(\beta}\theta^{-\alpha)}\bar{P}^{\beta\dot{\alpha}}-8(\theta^{-})^{4}\bar{\theta}^{+\dot{\rho}}\theta^{+}_{(\beta}\partial^{\alpha)}_{\dot{\rho}}\bar{P}^{\beta\dot{\alpha}},\end{split}$
(C.7b)
$G^{++5}_{(P)}=2i(\theta^{+})^{2}\theta^{-}_{\beta}\bar{\theta}^{+}_{\dot{\beta}}P^{\beta\dot{\beta}}+2i(\bar{\theta}^{+})^{2}\theta^{+}_{\beta}\bar{\theta}^{-}_{\dot{\beta}}\bar{P}^{\beta\dot{\beta}},$
(C.7c)
$\begin{split}G^{--5}_{(P)}=&-2i(\theta^{-})^{2}\theta^{+}_{\beta}\bar{\theta}^{-}_{\dot{\beta}}P^{\beta\dot{\beta}}+2(\theta^{-})^{4}(\theta^{+})^{2}\partial_{\beta\dot{\beta}}P^{\beta\dot{\beta}}\\\
&-2i(\bar{\theta}^{-})^{2}\theta^{-}_{\beta}\bar{\theta}^{+}_{\dot{\beta}}\bar{P}^{\beta\dot{\beta}}+2(\theta^{-})^{4}(\bar{\theta}^{+})^{2}\partial_{\beta\dot{\beta}}\bar{P}^{\beta\dot{\beta}},\end{split}$
(C.7d)
$G^{++\alpha+}_{(P)}=-(\theta^{+})^{2}\bar{\theta}^{+}_{\dot{\beta}}P^{\alpha\dot{\beta}},$
(C.7e)
$\begin{split}G^{--\alpha+}_{(P)}=&-\frac{1}{2}(\theta^{-})^{2}\bar{\theta}^{+}_{\dot{\beta}}P^{\alpha\dot{\beta}}-(\theta^{+}\theta^{-})\bar{\theta}^{-}_{\dot{\beta}}P^{\alpha\dot{\beta}}\\\
&-i(\theta^{+})^{2}(\bar{\theta}^{-})^{2}\theta^{-\rho}\partial_{\rho\dot{\beta}}P^{\alpha\dot{\beta}}+i(\theta^{-})^{2}(\bar{\theta}^{+}\bar{\theta}^{-})\theta^{+\rho}\partial_{\rho\dot{\beta}}P^{\alpha\dot{\beta}}\\\
&-2i(\theta^{-})^{2}\theta^{+\rho}\bar{\theta}^{+(\dot{\rho}}\theta^{-\dot{\beta})}\partial_{\rho\dot{\rho}}P^{\alpha}_{\dot{\beta}}-\frac{1}{2}(\theta^{-})^{4}(\theta^{+})^{2}\bar{\theta}^{+\dot{\beta}}\Box
P^{\alpha}_{\dot{\beta}},\end{split}$ (C.7f)
$G^{--\alpha-}_{(P)}=-\frac{1}{2}(\theta^{-})^{2}\bar{\theta}^{-}_{\dot{\beta}}P^{\alpha\dot{\beta}}-i(\theta^{-})^{4}\theta^{+\beta}\partial_{\beta\dot{\beta}}P^{\alpha\dot{\beta}}.$
(C.7g)

Auxiliary field $V^{\alpha\dot{\alpha}(ij)}$:

$G^{++\alpha\dot{\alpha}}_{(V)}=(\theta^{+})^{4}V^{\alpha\dot{\alpha}(ij)}u^{-}_{i}u^{-}_{j}\,,$
(C.8a)
$\begin{split}G^{--\alpha\dot{\alpha}}_{(V)}=&\frac{1}{3}(\theta^{-})^{2}(\bar{\theta}^{+})^{2}V^{\alpha\dot{\alpha}(ij)}u^{-}_{i}u^{-}_{j}+\frac{1}{3}(\bar{\theta}^{-})^{2}(\theta^{+})^{2}V^{\alpha\dot{\alpha}(ij)}u^{-}_{i}u^{-}_{j}+\frac{4}{3}(\theta^{+}\theta^{-})(\bar{\theta}^{+}\bar{\theta}^{-})V^{\alpha\dot{\alpha}(ij)}u^{-}_{i}u^{-}_{j}\\\
&-\frac{2}{3}(\theta^{-})^{2}(\bar{\theta}^{+}\bar{\theta}^{-})V^{\alpha\dot{\alpha}(ij)}u^{+}_{i}u^{-}_{j}-\frac{2}{3}(\bar{\theta}^{-})^{2}(\theta^{+}\theta^{-})V^{\alpha\dot{\alpha}(ij)}u^{+}_{i}u^{-}_{j}\\\
&+\frac{4}{3}i(\theta^{+})^{2}(\bar{\theta}^{-})^{2}\theta^{-\rho}\bar{\theta}^{+\dot{\rho}}\partial_{\rho\dot{\rho}}V^{\alpha\dot{\alpha}(ij)}u^{-}_{i}u^{-}_{j}+\frac{4}{3}i(\bar{\theta}^{+})^{2}(\theta^{-})^{2}\theta^{+\rho}\bar{\theta}^{-\dot{\rho}}\partial_{\rho\dot{\rho}}V^{\alpha\dot{\alpha}(ij)}u^{-}_{i}u^{-}_{j}\\\
&+\frac{1}{3}(\theta^{-})^{4}V^{\alpha\dot{\alpha}(ij)}u^{+}_{i}u^{+}_{j}+\frac{4}{3}i(\theta^{-})^{4}\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}\partial_{\rho\dot{\rho}}V^{\alpha\dot{\alpha}(ij)}u^{+}_{i}u^{-}_{j}-\frac{2}{3}(\theta^{-})^{4}(\theta^{+})^{4}\Box
V^{\alpha\dot{\alpha}(ij)}u^{-}_{i}u^{-}_{j}.\end{split}$ (C.8b)

Auxiliary field $S^{(ij)}$:

$G^{++5}_{(S)}=(\theta^{+})^{4}S^{(ij)}u^{-}_{i}u^{-}_{j},$ (C.9a)
$\begin{split}G^{--5}_{(S)}=&\frac{1}{3}(\theta^{-})^{2}(\bar{\theta}^{+})^{2}S^{(ij)}u^{-}_{i}u^{-}_{j}+\frac{1}{3}(\bar{\theta}^{-})^{2}(\theta^{+})^{2}S^{(ij)}u^{-}_{i}u^{-}_{j}+\frac{4}{3}(\theta^{+}\theta^{-})(\bar{\theta}^{+}\bar{\theta}^{-})S^{(ij)}u^{-}_{i}u^{-}_{j}\\\
&-\frac{2}{3}(\theta^{-})^{2}(\bar{\theta}^{+}\bar{\theta}^{-})S^{(ij)}u^{+}_{i}u^{-}_{j}-\frac{2}{3}(\bar{\theta}^{-})^{2}(\theta^{+}\theta^{-})S^{(ij)}u^{+}_{i}u^{-}_{j}\\\
&+\frac{4}{3}i(\theta^{+})^{2}(\bar{\theta}^{-})^{2}\theta^{-\rho}\bar{\theta}^{+\dot{\rho}}\partial_{\rho\dot{\rho}}S^{(ij)}u^{-}_{i}u^{-}_{j}+\frac{4}{3}i(\bar{\theta}^{+})^{2}(\theta^{-})^{2}\theta^{+\rho}\bar{\theta}^{-\dot{\rho}}\partial_{\rho\dot{\rho}}S^{(ij)}u^{-}_{i}u^{-}_{j}\\\
&+\frac{1}{3}(\theta^{-})^{4}S^{ij}u^{+}_{i}u^{+}_{j}+\frac{4}{3}i(\theta^{-})^{4}\theta^{+\rho}\bar{\theta}^{+\dot{\rho}}\partial_{\rho\dot{\rho}}S^{(ij)}u^{+}_{i}u^{-}_{j}-\frac{2}{3}(\theta^{-})^{4}(\theta^{+})^{4}\Box
S^{(ij)}u^{-}_{i}u^{-}_{j}.\end{split}$ (C.9b)

Rarita-Schwinger field
$\psi^{\alpha\beta\dot{\alpha}i}=\psi^{(\alpha\beta)\dot{\alpha}i}+\epsilon^{\alpha\beta}\psi^{\dot{\alpha}i}$:

$\begin{split}G_{(\psi)}^{++\alpha\dot{\alpha}}=&16(\bar{\theta}^{+})^{2}\theta^{+\beta}\psi_{\beta}^{\alpha\dot{\alpha}i}u^{-}_{i}-16(\theta^{+})^{2}\bar{\theta}^{+\dot{\beta}}\bar{\psi}_{\dot{\beta}}^{\alpha\dot{\alpha}i}u^{-}_{i}\\\
&-16i(\theta^{+})^{4}\bar{\theta}^{-\dot{\beta}}\partial_{\dot{\beta}}^{\beta}\psi^{\alpha\dot{\alpha}i}_{\beta}u^{-}_{i}-16i(\theta^{+})^{4}\theta^{-\beta}\partial_{\beta}^{\dot{\beta}}\bar{\psi}^{\alpha\dot{\alpha}i}_{\dot{\beta}}u^{-}_{i},\end{split}$
(C.10a) $\begin{split}G_{(\psi)}^{--\alpha\dot{\alpha}}=&\quad
16(\bar{\theta}^{+}\bar{\theta}^{-})\theta^{-\beta}\psi_{\beta}^{\alpha\dot{\alpha}i}u^{-}_{i}+8(\bar{\theta}^{-})^{2}\theta^{+\beta}\psi_{\beta}^{\alpha\dot{\alpha}i}u^{-}_{i}-8(\bar{\theta}^{-})^{2}\theta^{-\beta}\psi_{\beta}^{\alpha\dot{\alpha}i}u^{+}_{i}\\\
&-16i(\bar{\theta}^{-})^{2}(\theta^{+}\theta^{-})\bar{\theta}^{+\dot{\sigma}}\left(\partial_{\dot{\sigma}}^{\beta}\psi_{\beta}^{\alpha\dot{\alpha}i}-\delta^{\dot{\alpha}}_{\dot{\sigma}}\partial^{\beta}_{\dot{\beta}}\psi_{\beta}^{\alpha\dot{\beta}i}\right)u^{-}_{i}\\\
&+8i(\bar{\theta}^{+})^{2}(\theta^{-})^{2}\bar{\theta}^{-\dot{\sigma}}\left(2\partial_{\dot{\sigma}}^{\beta}\psi_{\beta}^{\alpha\dot{\alpha}i}-\delta^{\dot{\alpha}}_{\dot{\sigma}}\partial^{\beta}_{\dot{\beta}}\psi_{\beta}^{\alpha\dot{\beta}i}\right)u^{-}_{i}\\\
&-32i(\bar{\theta}^{-})^{2}\theta^{-(\beta}\theta^{+\sigma)}\bar{\theta}^{+\dot{\sigma}}\partial_{\sigma\dot{\sigma}}\psi_{\beta}^{\alpha\dot{\alpha}i}u^{-}_{i}\\\
&-16i(\theta^{-})^{4}\bar{\theta}^{+}_{\dot{\sigma}}\mathcal{G}^{\dot{\sigma}\dot{\alpha}\alpha
i}u^{+}_{i}+32(\theta^{-})^{4}(\bar{\theta}^{+})^{2}\theta^{+\rho}\partial_{\rho\dot{\rho}}\mathcal{G}^{\dot{\rho}\dot{\alpha}\alpha
i}u^{-}_{i}\\\
&-16(\theta^{+}\theta^{-})\bar{\theta}^{-\dot{\beta}}\bar{\psi}^{\alpha\dot{\alpha}i}_{\dot{\beta}}u^{-}_{i}-8(\theta^{-})^{2}\bar{\theta}^{+\dot{\beta}}\bar{\psi}^{\alpha\dot{\alpha}i}_{\dot{\beta}}u^{-}_{i}+8(\theta^{-})^{2}\bar{\theta}^{-\dot{\beta}}\bar{\psi}^{\alpha\dot{\alpha}i}_{\dot{\beta}}u^{+}_{i}\\\
&-16i(\theta^{-})^{2}(\bar{\theta}^{+}\bar{\theta}^{-})\theta^{+\sigma}\left(\partial_{\sigma}^{\dot{\beta}}\bar{\psi}^{\alpha\dot{\alpha}i}_{\dot{\beta}}-\delta_{\sigma}^{\alpha}\partial_{\beta}^{\dot{\beta}}\bar{\psi}_{\dot{\beta}}^{\beta\dot{\alpha}i}\right)u^{-}_{i}\\\
&+8i(\theta^{+})^{2}(\bar{\theta}^{-})^{2}\theta^{-\sigma}\left(2\partial_{\sigma}^{\dot{\beta}}\bar{\psi}^{\alpha\dot{\alpha}i}_{\dot{\beta}}-\delta^{\alpha}_{\sigma}\partial^{\dot{\beta}}_{\beta}\bar{\psi}^{\beta\dot{\alpha}i}_{\dot{\beta}}\right)u^{-}_{i}\\\
&+32i(\theta^{-})^{2}\bar{\theta}^{-(\dot{\beta}}\bar{\theta}^{+\dot{\sigma})}\theta^{+\sigma}\partial_{\sigma\dot{\sigma}}\bar{\psi}^{\alpha\dot{\alpha}i}_{\dot{\beta}}u^{-}_{i}\\\
&-16i(\theta^{-})^{4}\theta^{+}_{\sigma}\bar{\mathcal{G}}^{\sigma\alpha\dot{\alpha}i}u^{+}_{i}-32(\theta^{-})^{4}(\theta^{+})^{2}\bar{\theta}^{+\dot{\rho}}\partial_{\rho\dot{\rho}}\bar{\mathcal{G}}^{\rho\alpha\dot{\alpha}i}u^{-}_{i},\end{split}$
(C.10b)

$\begin{split}G^{++5}_{(\psi)}=&-16(\theta^{+})^{2}\bar{\theta}^{+\dot{\beta}}\psi^{i}_{\dot{\beta}}u^{-}_{i}+8i(\theta^{+})^{4}\theta^{-}_{\alpha}\partial^{\beta}_{\dot{\beta}}\psi_{\beta}^{\alpha\dot{\beta}i}u^{-}_{i}\\\
&+16(\bar{\theta}^{+})^{2}\theta^{+\beta}\bar{\psi}^{i}_{\beta}u^{-}_{i}-8i(\theta^{+})^{4}\bar{\theta}^{-}_{\dot{\alpha}}\partial_{\beta}^{\dot{\beta}}\bar{\psi}_{\dot{\beta}}^{\beta\dot{\alpha}i}u^{-}_{i},\end{split}$
(C.10c)
$\begin{split}G^{--5}_{(\psi)}=&-16(\theta^{+}\theta^{-})\bar{\theta}^{-\dot{\beta}}\psi^{i}_{\dot{\beta}}u^{-}_{i}-8(\theta^{-})^{2}\bar{\theta}^{+\dot{\beta}}\psi^{i}_{\dot{\beta}}u^{-}_{i}+8(\theta^{-})^{2}\bar{\theta}^{-\dot{\beta}}\psi_{\dot{\beta}}^{i}u^{+}_{i}\\\
&-8i(\theta^{-})^{2}(\bar{\theta}^{+}\bar{\theta}^{-})\theta^{+}_{\alpha}\mathcal{R}^{\alpha
i}u^{-}_{i}-4i(\theta^{+})^{2}(\bar{\theta}^{-})^{2}\theta^{-}_{\alpha}(\mathcal{R}^{\alpha
i}+2\partial^{\alpha\dot{\beta}}\psi^{-}_{\dot{\beta}})u^{-}_{i}\\\
&+4i(\theta^{-})^{4}\theta^{+}_{\alpha}\mathcal{R}^{\alpha
i}u^{+}_{i}+8(\theta^{-})^{4}(\theta^{+})^{2}\bar{\theta}^{+\dot{\rho}}\partial_{\dot{\rho}\alpha}\mathcal{R}^{\alpha
i}u^{-}_{i}+\\\
&+16(\bar{\theta}^{+}\bar{\theta}^{-})\theta^{-\beta}\bar{\psi}^{i}_{\beta}u^{-}_{i}+8(\bar{\theta}^{-})^{2}\theta^{+\beta}\bar{\psi}^{i}_{\beta}u^{-}_{i}-8(\bar{\theta}^{-})^{2}\theta^{-\beta}\bar{\psi}^{i}_{\beta}u^{+}_{i}\\\
&-8i(\bar{\theta}^{-})^{2}(\theta^{+}\theta^{-})\bar{\theta}^{+}_{\dot{\alpha}}\bar{\mathcal{R}}^{\dot{\alpha}i}u^{-}_{i}-4i(\bar{\theta}^{+})^{2}(\theta^{-})^{2}\bar{\theta}^{-}_{\dot{\alpha}}(\bar{\mathcal{R}}^{\dot{\alpha}i}+2\partial^{\dot{\alpha}\beta}\bar{\psi}_{\beta}^{i})u^{-}_{i}\\\
&-4i(\theta^{-})^{4}\bar{\theta}^{+}_{\dot{\alpha}}\bar{\mathcal{R}}^{\dot{\alpha}i}u^{+}_{i}+8(\theta^{-})^{4}(\bar{\theta}^{+})^{2}\theta^{+\rho}\partial_{\rho\dot{\alpha}}\bar{\mathcal{R}}^{\dot{\alpha}i}u^{-}_{i}.\end{split}$
(C.10d)

Auxiliary fermionic field $\rho_{\beta}^{i}$:

$G_{(\rho)}^{++5}=8(\bar{\theta}^{+})^{2}\theta^{+\beta}\rho^{i}_{\beta}u^{-}_{i}-8(\theta^{+})^{2}\bar{\theta}^{+\dot{\beta}}\bar{\rho}^{i}_{\dot{\beta}}u^{-}_{i},$
(C.11a) $\begin{split}G^{--5}_{(\rho)}=&\quad
8(\bar{\theta}^{+}\bar{\theta}^{-})\theta^{-\beta}\rho^{i}_{\beta}u^{-}_{i}+4(\bar{\theta}^{-})^{2}\theta^{+\beta}\rho_{\beta}^{i}u^{-}_{i}-4(\bar{\theta}^{-})^{2}\theta^{-\beta}\rho_{\beta}^{i}u^{+}_{i}\\\
&-8i(\bar{\theta}^{-})^{2}(\theta^{+}\theta^{-})\bar{\theta}^{+\dot{\sigma}}\partial_{\dot{\sigma}}^{\beta}\rho_{\beta}^{i}u^{-}_{i}+8i(\bar{\theta}^{+})^{2}(\theta^{-})^{2}\bar{\theta}^{-\dot{\sigma}}\partial_{\dot{\sigma}}^{\beta}\rho_{\beta}^{i}u^{-}_{i}\\\
&-16i(\bar{\theta}^{-})^{2}\theta^{-(\beta}\theta^{+\sigma)}\bar{\theta}^{+\dot{\sigma}}\partial_{\sigma\dot{\sigma}}\rho_{\beta}^{i}u^{-}_{i}\\\
&+8i(\theta^{-})^{4}\bar{\theta}^{+\dot{\sigma}}\partial^{\beta}_{\dot{\sigma}}\rho_{\beta}^{i}u^{+}_{i}-4(\theta^{-})^{4}(\bar{\theta}^{+})^{2}\theta^{+\beta}\Box\rho_{\beta}^{i}u^{-}_{i}\\\
&-8(\theta^{+}\theta^{-})\bar{\theta}^{-\dot{\beta}}\bar{\rho}^{i}_{\dot{\beta}}u^{-}_{i}-4(\theta^{-})^{2}\bar{\theta}^{+\dot{\beta}}\bar{\rho}^{i}_{\dot{\beta}}u^{-}_{i}+4(\theta^{-})^{2}\bar{\theta}^{-\dot{\beta}}\bar{\rho}^{i}_{\dot{\beta}}u^{+}_{i}\\\
&-8i(\theta^{-})^{2}(\bar{\theta}^{+}\bar{\theta}^{-})\theta^{+\sigma}\partial_{\sigma}^{\dot{\beta}}\bar{\rho}_{\dot{\beta}}^{i}u^{-}_{i}+8i(\theta^{+})^{2}(\bar{\theta}^{-})^{2}\theta^{-\sigma}\partial_{\sigma}^{\dot{\beta}}\bar{\rho}^{i}_{\dot{\beta}}u^{-}_{i}\\\
&+16i(\theta^{-})^{2}\bar{\theta}^{-(\dot{\beta}}\bar{\theta}^{+\dot{\sigma})}\theta^{+\sigma}\partial_{\sigma\dot{\sigma}}\bar{\rho}^{i}_{\dot{\beta}}u^{-}_{i}\\\
&+8i(\theta^{-})^{4}\theta^{+\sigma}\partial_{\sigma}^{\dot{\beta}}\bar{\rho}_{\dot{\beta}}^{i}u^{+}_{i}+4(\theta^{-})^{4}(\theta^{+})^{2}\bar{\theta}^{+\dot{\beta}}\Box\bar{\rho}^{i}_{\dot{\beta}}u^{-}_{i}.\end{split}$
(C.11b)

Auxiliary fermionic field $\chi_{\beta}^{i}$:

$G^{++\alpha\dot{\alpha}}_{(\chi)}=-4i(\theta^{+})^{4}\bar{\theta}^{-\dot{\alpha}}\chi^{\alpha
i}u^{-}_{i}+4i(\theta^{+})^{4}\theta^{-\alpha}\bar{\chi}^{\dot{\alpha}i}u^{-}_{i},$
(C.12a)
$\begin{split}G^{--\alpha\dot{\alpha}}_{(\chi)}=&\quad\;4i(\bar{\theta}^{-})^{2}(\theta^{+}\theta^{-})\bar{\theta}^{+\dot{\alpha}}\chi^{\alpha
i}u^{-}_{i}-2i(\theta^{-})^{2}(\bar{\theta}^{+})^{2}\bar{\theta}^{-\dot{\alpha}}\chi^{\alpha
i}u^{-}_{i}\\\ &-2i(\theta^{-})^{4}\bar{\theta}^{+\dot{\alpha}}\chi^{\alpha
i}u^{+}_{i}-4(\theta^{-})^{4}(\bar{\theta}^{+})^{2}\theta^{+\rho}\partial_{\rho}^{\dot{\alpha}}\chi^{\alpha
i}u^{-}_{i}\\\
&-4i(\theta^{-})^{2}(\bar{\theta}^{+}\bar{\theta}^{-})\theta^{+\alpha}\bar{\chi}^{\dot{\alpha}i}u^{-}_{i}+2i(\bar{\theta}^{-})^{2}(\theta^{+})^{2}\theta^{-\alpha}\bar{\chi}^{\dot{\alpha}i}u^{-}_{i}\\\
&+2i(\theta^{-})^{4}\theta^{+\alpha}\bar{\chi}^{\dot{\alpha}i}u^{+}_{i}-4(\theta^{-})^{4}(\theta^{+})^{2}\bar{\theta}^{+\dot{\rho}}\partial_{\dot{\rho}}^{\alpha}\bar{\chi}^{\dot{\alpha}i}u^{-}_{i},\end{split}$
(C.12b)

$G^{++5}_{(\chi)}=-2i(\theta^{+})^{4}\theta^{-\alpha}\chi_{\alpha}^{i}u^{-}_{i}+2i(\theta^{+})^{4}\bar{\theta}^{-\dot{\alpha}}\bar{\chi}_{\dot{\alpha}}^{i}u^{-}_{i},$
(C.12c) $\begin{split}G^{--5}_{(\chi)}=&\quad
2i(\theta^{-})^{2}(\bar{\theta}^{+}\bar{\theta}^{-})\theta^{+\alpha}\chi^{i}_{\alpha}u^{-}_{i}-i(\theta^{+})^{2}(\bar{\theta}^{-})^{2}\theta^{-\alpha}\chi^{i}_{\alpha}u^{-}_{i}\\\
&-i(\theta^{-})^{4}\theta^{+\alpha}\chi^{i}_{\alpha}u^{+}_{i}-2(\theta^{-})^{4}(\theta^{+})^{2}\bar{\theta}^{+\dot{\rho}}\partial^{\alpha}_{\dot{\rho}}\chi^{i}_{\alpha}u^{-}_{i}+\\\
&-2i(\bar{\theta}^{-})^{2}(\theta^{+}\theta^{-})\bar{\theta}^{+\dot{\alpha}}\bar{\chi}^{i}_{\dot{\alpha}}u^{-}_{i}+i(\bar{\theta}^{+})^{2}(\theta^{-})^{2}\theta^{-\dot{\alpha}}\bar{\chi}^{i}_{\dot{\alpha}}u^{-}_{i}\\\
&+i(\theta^{-})^{4}\bar{\theta}^{+\dot{\alpha}}\bar{\chi}^{i}_{\dot{\alpha}}u^{+}_{i}-2(\theta^{-})^{4}(\bar{\theta}^{+})^{2}\theta^{+\rho}\partial_{\rho}^{\dot{\alpha}}\bar{\chi}^{i}_{\dot{\alpha}}u^{-}_{i}.\end{split}$
(C.12d)

## References

  * (1) D. Z. Freedman and A. Van Proeyen, Supergravity, Cambridge Univ. Press, 2012.
  * (2) S. Deser, J. H. Kay and K. S. Stelle, Renormalizability Properties of Supergravity, Phys. Rev. Lett. 38 (1977) 527 [arXiv:1506.03757 [hep-th]].
  * (3) K. S. Stelle, Renormalization of Higher Derivative Quantum Gravity, Phys. Rev. D 16 (1977) 953-969.
  * (4) A. A. Starobinsky, A New Type of Isotropic Cosmological Models Without Singularity, Phys. Lett. B 91 (1980) 99-102.
  * (5) B. Whitt, Fourth Order Gravity as General Relativity Plus Matter, Phys. Lett. B 145 (1984) 176-178.
  * (6) L. Alvarez-Gaume, A. Kehagias, C. Kounnas, D. Lüst and A. Riotto, Aspects of Quadratic Gravity, Fortsch. Phys. 64 (2016) no.2-3, 176-189 [arXiv:1505.07657 [hep-th]].
  * (7) S. Cecotti, Higher derivative supergravity is equivalent to standard supergravity coupled to matter, Phys. Lett. B 190 (1987) 86-92.
  * (8) M. Ozkan, Y. Pang and E. Sezgin, Higher Derivative Supergravities in Diverse Dimensions, [arXiv:2401.08945 [hep-th]].
  * (9) J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press, 1983.
  * (10) S. J. Gates, M. T. Grisaru, M. Roček, W. Siegel, Superspace, Benjamin/Cummings, Reading, Mass., 1983.
  * (11) I. L. Buchbinder and S. M. Kuzenko, Ideas and methods of supersymmetry and supergravity: Or a walk through superspace, Taylor and Francis, 1998.
  * (12) A. Galperin, E. Ivanov, S. Kalitzin, V. Ogievetsky and E. Sokatchev, Unconstrained $\mathcal{N}=2$ Matter, Yang-Mills and Supergravity Theories in Harmonic Superspace, Class. Quant. Grav. 1 (1984) 469-498 [erratum: Class. Quant. Grav. 2 (1985), 127].
  * (13) S. M. Kuzenko, E. S. N. Raptakis and G. Tartaglino-Mazzucchelli, Covariant superspace approaches to ${\cal N}=2$ supergravity, [arXiv:2211.11162 [hep-th]].
  * (14) V. Ogievetsky and E. Sokatchev, Structure of Supergravity Group, Phys. Lett. B 79 (1978) 222.
  * (15) L. Mezincescu, On the superfield formulation of $O(2)$ supersymmetry, Dubna preprint JINR-P2-12572 (June, 1979).
  * (16) A. S. Galperin, E. A. Ivanov, V. I. Ogievetsky, E. S. Sokatchev, Harmonic superspace, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2001, 306 p.
  * (17) E. Ivanov, ${\cal N}=2\,$ Supergravities in Harmonic Superspace, [arXiv:2212.07925 [hep-th]].
  * (18) A. S. Galperin, N. A. Ky and E. Sokatchev, $N=2$ Supergravity in Superspace: Solution to the Constraints, Class. Quant. Grav. 4 (1987) 1235.
  * (19) A. S. Galperin, E. A. Ivanov, V. I. Ogievetsky and E. Sokatchev, $\mathcal{N}=2$ Supergravity in Superspace: Different Versions and Matter Couplings, Class. Quant. Grav. 4 (1987) 1255.
  * (20) A. Galperin, E. Ivanov and O. Ogievetsky, Harmonic space and quaternionic manifolds, Annals Phys. 230 (1994) 201-249 [arXiv:hep-th/9212155 [hep-th]].
  * (21) E. Ivanov and G. Valent, Quaternionic Taub - NUT from the harmonic space approach, Phys. Lett. B 445 (1998) 60-68 [arXiv:hep-th/9809108 [hep-th]].
  * (22) E. Ivanov and G. Valent, Harmonic space construction of the quaternionic Taub - NUT metric, Class. Quant. Grav. 16 (1999) 1039-1056 [arXiv:hep-th/9810005 [hep-th]].
  * (23) E. Ivanov and G. Valent, Quaternionic metrics from harmonic superspace: Lagrangian approach and quotient construction, Nucl. Phys. B 576 (2000) 543-577 [arXiv:hep-th/0001165 [hep-th]].
  * (24) B. M. Zupnik, Background harmonic superfields in $\mathcal{N}=2$ supergravity, Theor. Math. Phys. 116 (1998) 964-977 [arXiv:hep-th/9803202 [hep-th]].
  * (25) I. Buchbinder, E. Ivanov and N. Zaigraev, Unconstrained off-shell superfield formulation of 4D, $\mathcal{N}$ = 2 supersymmetric higher spins, JHEP 12 (2021) 016 [arXiv:2109.07639 [hep-th]].
  * (26) S. M. Kuzenko and S. Theisen, Correlation functions of conserved currents in $\mathcal{N}=2$ superconformal theory, Class. Quant. Grav. 17 (2000) 665-696 [arXiv:hep-th/9907107 [hep-th]].
  * (27) D. Butter and S. M. Kuzenko, $\mathcal{N}=2$ supergravity and supercurrents, JHEP 12 (2010) 080 [arXiv:1011.0339 [hep-th]].
  * (28) D. Butter, On conformal supergravity and harmonic superspace, JHEP 03 (2016) 107 [arXiv:1508.07718 [hep-th]].
  * (29) D. Butter, $\mathcal{N}=2$ Conformal Superspace in Four Dimensions, JHEP 10 (2011) 030 [arXiv:1103.5914 [hep-th]].
  * (30) D. Butter, B. de Wit, S. M. Kuzenko and I. Lodato, New higher-derivative invariants in $\mathcal{N}=2$ supergravity and the Gauss-Bonnet term, JHEP 12 (2013) 062 [arXiv:1307.6546 [hep-th]].
  * (31) S. M. Kuzenko and J. Novak, On curvature squared terms in $\mathcal{N}=2$ supergravity, Phys. Rev. D 92 (2015) no.8, 085033 [arXiv:1507.04922 [hep-th]].
  * (32) S. M. Kuzenko and J. Novak, Higher derivative couplings in supergravity, Phys. Part. Nucl. Lett. 14 (2017) no.2, 271-276.
  * (33) S. M. Kuzenko, U. Lindstrom, M. Rocek and G. Tartaglino-Mazzucchelli, 4D $\mathcal{N}=2$ Supergravity and Projective Superspace, JHEP 09 (2008) 051 [arXiv:0805.4683 [hep-th]].
  * (34) F. Moura, Four-dimensional ’old minimal’ $\mathcal{N}=2$ supersymmetrization of $R^{4}$, JHEP 07 (2003) 057 [arXiv:hep-th/0212271 [hep-th]].
  * (35) B. de Wit, S. Katmadas and M. van Zalk, New supersymmetric higher-derivative couplings: Full $\mathcal{N}=2$ superspace does not count!, JHEP 01 (2011) 007 [arXiv:1010.2150 [hep-th]].
  * (36) J. M. Drummond, P. J. Heslop, P. S. Howe and S. F. Kerstan, Integral invariants in $\mathcal{N}=4$ SYM and the effective action for coincident D-branes, JHEP 08 (2003) 016 [arXiv:hep-th/0305202 [hep-th]].
  * (37) G. Bossard, P. S. Howe, K. S. Stelle and P. Vanhove, The vanishing volume of D=4 superspace, Class. Quant. Grav. 28 (2011) 215005 [arXiv:1105.6087 [hep-th]].
  * (38) R. Kallosh, Superinvariants Below Critical Loop Order, [arXiv:2312.06794 [hep-th]].
  * (39) S. M. Kuzenko, Projective superspace as a double punctured harmonic superspace, Int. J. Mod. Phys. A 14 (1999) 1737-1758 [arXiv:hep-th/9806147 [hep-th]].
  * (40) D. Jain and W. Siegel, Deriving Projective Hyperspace from Harmonic, Phys. Rev. D 80 (2009) 045024 [arXiv:0903.3588 [hep-th]].