Generalized cosmological term in $D=4$ supergravity from a new $AdS$-Lorentz
superalgebra

Diego M. Peñafiel∗, Lucrezia Ravera‡

∗Instituto de Física, Pontificia Universidad Católica de Valparaíso,
Casilla 4059, Valparaiso-Chile.
‡INFN, Sezione di Milano,
Via Celoria 16, I-20133 Milano-Italy.
<EMAIL_ADDRESS><EMAIL_ADDRESS>

A new supersymmetrization of the so-called $AdS$-Lorentz algebra is presented.
It involves two fermionic generators and is obtained by performing an abelian
semigroup expansion of the superalgebra $\mathfrak{osp}(4|1)$. The peculiar
properties of the aforesaid expansion method are then exploited to construct a
$D=4$ supergravity action involving a generalized supersymmetric cosmological
term in a geometric way, only from the curvatures of the novel superalgebra.
The action obtained with this procedure is a MacDowell-Mansouri like action.
Gauge invariance and supersymmetry of the action are also analyzed.

##  1 Introduction

As it is well known, a good candidate for describing dark energy is the
cosmological constant (see, for example, [1, 2]). Then, it becomes interesting
to analyze the ways in which cosmological constant terms can be introduced in
(super)gravity theories. In particular, the $D=4$ supergravity theory with a
cosmological term can be developed in a geometric formulation, where the
theory is construted from the curvatures of the superalgebra
$\mathfrak{osp}(4|1)$ and the resulting action is the so-called MacDowell-
Mansouri action [3].

On the other hand, as it was shown in [4, 5], the renormalized action for
$AdS$ gravity in four-dimensions, corresponding to the (bosonic) MacDowell-
Mansouri action, is on-shell equivalent to the square of the Weyl tensor
describing conformal gravity. Then, a further motivation in the construction
of (super)gravity MacDowell-Mansouri like actions lies in the fact that they
suggest a (super)conformal structure.111Let us mention, here, that the
development of more complicated supergravity theories, such as matter-couple
ones, can be carried on in a systematic way with the superconformal method
(see the book [6] for a review). Indeed, in this approach, which is based on
early work [7, 8, 9, 10] on the understanding of supergravity from the
superconformal algebra [11], the conformal symmetry is properly used as a tool
for constructing matter-coupled supergravity theories with local Poincaré
supersymmetry in such a way to gain insight into their structure.

Considering, instead, Minkowski spacetime, its symmetries are described by the
Poincaré algebra, which is generated by $\\{J_{ab},P_{a}\\}$, being $J_{ab}$
and $P_{a}$ the Lorentz and spacetime translations generators, respectively,
where, in particular,

${[P_{a},P_{b}]=0.}$ (1.1)

The symmetries of Minkowski spacetime can be generalized and extended from the
Poincaré to the Maxwell symmetries [12, 13, 14, 15, 16, 17, 18, 19, 20, 21]
(see also the more recent work [22] for an alternative way of closing Maxwell-
like algebras). The most general deformation is

$[P_{a},P_{b}]=Z_{ab},$ (1.2)

where $Z_{ab}$ ($Z_{ba}=-Z_{ab}$) transforms as a tensor under Lorentz
transformations and is associated with a constant electromagnetic background
field. Then, one can analyze the commutator of $Z_{ab}$ with $P_{a}$ and
iterate the procedure [17]. In this way, as it was shown in [23] (see also
[24]), the most general structure one obtains is the infinite-dimensional free
Lie algebra, called Maxwell∞ in the quoted references, generated by the
$P_{a}$’s.

The (“conventional”) Maxwell algebra $\mathcal{M}$ of [13, 14, 15, 17] (also
denoted in the literature as $\mathfrak{B}_{4}$) is generated by the set
$\\{J_{ab},P_{a},Z_{ab}\\}$ and its commutation relations read as follows:

$\begin{split}\left[P_{a},P_{b}\right]&=\Lambda Z_{ab},\\\
\left[J_{ab},J_{cd}\right]&=\eta_{bc}J_{ad}-\eta_{ac}J_{bd}-\eta_{bd}J_{ac}+\eta_{ad}J_{bc},\\\
\left[J_{ab},P_{c}\right]&=\eta_{bc}P_{a}-\eta_{ac}P_{b},\\\
\left[J_{ab},Z_{cd}\right]&=\eta_{bc}Z_{ad}-\eta_{ac}Z_{bd}-\eta_{bd}Z_{ac}+\eta_{ad}Z_{bc},\\\
\left[Z_{ab},Z_{cd}\right]&=0,\quad\left[Z_{ab},P_{c}\right]=0.\end{split}$
(1.3)

Here, the bosonic generators $Z_{ab}$ are tensorial abelian charges. We stress
that the Maxwell algebra is not unique at bosonic (and supersymmetric) level.
From the Lie algebra cohomology perspective, one can deform also the
commutation relations $\left[Z_{ab},Z_{cd}\right]=0$ and
$\left[Z_{ab},P_{c}\right]=0$ requiring the Jacobi identity to be obeyed in
order to end up with a consistent Lie algebra [17, 23, 24], and iterate the
construction to properly obtain all possible multiple commutators of the
$P_{a}$’s (namely, the so-called free Lie algebra generated by the $P_{a}$’s).

The constant $\Lambda$ appearing in (1.3) can be related to the cosmological
constant when $[\Lambda]=M^{2}$. Setting $\Lambda=e$, being $e$ the
electromagnetic coupling constant, we get the description of an enlarged
spacetime in the presence of a constant electromagnetic background field.
Indeed, in order to interpret the Maxwell algebra (and the corresponding
Maxwell group), a Maxwell group-invariant particle model on the extended
spacetime $(x^{\mu},\phi^{\mu\nu})$, with the translations of $\phi^{\mu\nu}$
generated by $Z_{\mu\nu}$ (that is $Z_{ab}$ in the components language), was
studied [16, 17, 18, 19, 20, 21]: The interaction term described by a Maxwell-
invariant $1$-form introduces new tensor degrees of freedom $f_{\mu\nu}$,
momenta conjugate to $\phi^{\mu\nu}$, that, in the equations of motion, play
the role of a background electromagnetic field which is constant on-shell and
leads to a closed, Maxwell-invariant $2$-form. Interestingly, in [25] the
authors presented an alternative way of introducing the generalized
cosmological constant term adopting the Maxwell algebra.

On the other hand, it was shown that the Maxwell symmetries can be deformed in
such a way to obtain the $\mathfrak{so}(D-1,2)\oplus\mathfrak{so}(D-1,1)$ or
$\mathfrak{so}(D,1)\oplus\mathfrak{so}(D-1,1)$ algebras [18, 20, 26, 27],
where the $Z_{ab}$ generators are non-abelian. Then, if spacetime symmetries
are considered as local symmetries, one can construct Chern-Simons gravity
actions in which dark energy can be interpreted as part of the metric of
spacetime.

Subsequently, in [28] it was shown that a generalized cosmological constant
term can also be introduced in a Born-Infeld like action constructed from the
curvatures of the so-called $AdS$-Lorentz algebra, $AdS-\mathcal{L}_{4}$ (also
known as $\mathfrak{so}(D-1,1)\oplus\mathfrak{so}(D-1,2)$, or Poincaré
semisimple extended algebra). The $AdS$-Lorentz algebra, which was first
introduced and described as a tensorial semi-simple enlargement of the
Poincaré algebra and as a Lie algebra deformation of the Maxwell algebra [15,
18, 20, 26, 27], can also be obtained by performing an abelian semigroup
expansion ($S$-expansion, for short) of the $AdS$ algebra, as it was shown in
[29]. The general theory of expansions of Lie (super)algebras, which allows to
derive new Lie (super)algebras and, correspondingly, new physical theories,
was first introduced in [30]. Subsequently, in [31] the authors formulated the
so-called $S$-expansion procedure, which is based on combining the
multiplication law of a semigroup $S$ with the structure constants of a Lie
(super)algebra $\mathfrak{g}$. The new Lie (super)algebra one ends up with is
called the $S$-expanded (super)algebra $\mathfrak{g}_{S}=S\times\mathfrak{g}$,
and the $S$-expansion method also provide in a simple way an invariant tensor
for it, in terms of an invariant tensor for the starting (super)algebra
$\mathfrak{g}$, which is particularly helpful in the (geometric) construction
of new (super)gravity theories. In fact, diverse (super)gravity theories have
been studied by exploiting the $S$-expansion method and its properties (see,
for instance, the relevant results presented in [32, 33, 34, 35, 36, 37, 38,
39, 40, 41, 42, 43, 44, 45, 46]).

The $S$-expansion method allows to obtain two separate types of Lie algebras,
denoted in the literature as $\mathfrak{B}_{m}$ (Maxwell algebras type) and
$AdS-\mathcal{L}_{m}$ (for a concise review see, for example, [22]). Both can
be related with each other by Inönü-Wigner contraction. The integer index
$m>2$ labels different representatives, where standard generators of the
Lorentz transformations $J_{ab}$ and translations $P_{a}$ become equipped with
another set (or sets) of new generators $Z_{ab}$ and $R_{a}$. The value $m-1$
might be used to indicate the total number of different generators
($J_{ab},P_{a},Z_{ab}=Z_{ab}^{(1)},R_{a},=R_{a}^{(1)},Z_{ab}^{(2)},R_{a}^{(2)},\ldots$).
In particular, $\mathfrak{B}_{3}$ and $AdS-\mathcal{L}_{3}$ correspond to the
Poincaré and to the $AdS$ algebras, respectively, while $\mathfrak{B}_{4}$
represents the Maxwell algebra $\mathcal{M}$ written in (1.3) (obtained by
including the $Z_{ab}$ generator).

Referring to superalgebras, the minimal supersymmetrization of the Maxwell
algebra ($s\mathcal{M}$) was introduced in [47], and it requires two fermionic
charges. Extensive studies and analyses of the minimal Maxwell superalgebra
and its generalizations have been preformed using expansion methods in [48,
49]. Let us mention that other relevant superalgebras containing two fermionic
charges were introduced and deeply analyzed in [50, 51, 52, 53, 54, 55]. In
particular, we are referring to the D’Auria-Fré superalgebra, introduced in
[50] and further analyzed in [52, 53], underlying the Free Differential
Algebra describing $D=11$ supergravity, the Green algebra [51], in the context
of superstring, and Maxwell-type superalgebras related to $D=4$ and $D=11$
supergravity [54, 55].

Concerning the minimal Maxwell superalgebra $s\mathcal{M}$, in [56] it was
shown that the $\mathcal{N}=1$, $D=4$ pure supergravity Lagrangian can be
obtained as a quadratic expression in the curvatures associated with
$s\mathcal{M}$. The action of [56] does not include the cosmological constant.

Let us specify, here, that in the recent paper [24] the authors gave the
structure of a free Lie superalgebra for studying extensions of the Poincaré
superalgebra, and that the contractions of the $\mathfrak{B}_{m}$ algebras we
have considered so far are subalgebras of the free Lie (Maxwell super)algebra,
which can be viewed as a universal structure comprising, by quotienting,
generalizations existing in the literature. In particular, as shown in [24],
the superalgebra $s\mathcal{M}$ arises as a particular finite-dimensional
quotient of the free Lie superalgebra in four spacetime dimensions generated
by odd supertranslations $Q_{\alpha}$.

On the other hand, in [39] the authors demonstrated that the $AdS$-Lorentz
superalgebra $sAdS-\mathcal{L}_{4}$ (minimal supersymmetrization of the
$AdS$-Lorentz algebra $AdS-\mathcal{L}_{4}$ of [28]) allows to construct in a
geometric way the supergravity containing a generalized supersymmetric
cosmological constant. The four-dimensional action of [39] is built only from
the curvatures of $sAdS-\mathcal{L}_{4}$ and corresponds to a MacDowell-
Mansouri like action. In the same paper, the authors also extended their
result introducing the so-called generalized minimal $AdS$-Lorentz
superalgebra, building a more general action.

The superalgebra $sAdS-\mathcal{L}_{4}$ of [39] presents the following
anticommutation relation:

$\left\\{Q_{\alpha},Q_{\beta}\right\\}=-\frac{1}{2}\left[\left(\gamma^{ab}C\right)_{\alpha\beta}Z_{ab}-2\left(\gamma^{a}C\right)_{\alpha\beta}P_{a}\right],$
(1.4)

being $Q_{\alpha}$ a four-components Majorana spinor charge. In
$(s)AdS-\mathcal{L}_{4}$, unlike the case of the (minimal supersymmetrization
of) the Maxwell algebra $(s)\mathcal{M}$, the new generators $Z_{ab}$ are non-
abelian and behave as Lorentz generators. Their presence implies the
introduction of a new bosonic field which modified the definition of the
curvatures when building a (super)gravity theory for the $AdS$-Lorentz
(super)algebra $(s)AdS-\mathcal{L}_{4}$. Furthermore, $sAdS-\mathcal{L}_{4}$
contains only one spinor charge, while, as shown in [47], the minimal
supersymmetrization of the Maxwell algebra requires two Majorana spinor
charges.

Actually, as we have already mentioned, in [39] the authors also presented the
generalized minimal $AdS$-Lorentz superalgebra, which involves two fermionic
charges. However, it also includes two extra bosonic generators
($\tilde{Z}_{ab}$ and $\tilde{Z}_{a}$), and an Inönü-Wigner contraction of the
generalized minimal $AdS$-Lorentz superalgebra provides the generalized
minimal Maxwell superalgebra $s\mathcal{M}_{4}$ [38], which involves an extra
bosonic generator $\tilde{Z}_{ab}$ with respect to the minimal super-Maxwell
algebra $s\mathcal{M}$ of [47]. Then, in order to end up with the Maxwell
algebra $\mathcal{M}$, one should further contract the bosonic subalgebra of
$s\mathcal{M}_{4}$ in such a way to remove the extra bosonic generator
$\tilde{Z}_{ab}$.

The $AdS$-Lorentz type (super)algebras also possesses the non-commutativity
$\left[P_{a},P_{b}\right]=Z_{ab}$, which is also present in the Maxwell
(super)symmetries. Nevertheless, unlike the MacDowell-Mansouri Lagrangians for
$\mathfrak{osp}(4|1)$ and for $sAdS-\mathcal{L}_{4}$, it was shown in [38]
that the supergravity action à la MacDowell-Mansouri based on the generalized
minimal Maxwell superalgebra $s\mathcal{M}_{4}$ does not reproduce the
supersymmetric cosmological constant term in the action. This is a direct
consequence of the $S$-expansion procedure. The result of [38] corresponds to
a generalization of the one previously presented in [56]. In both cases, the
super-Maxwell fields only appears in boundary terms of the resulting
Lagrangian.

Let us specify that, in general, the (super-)Maxwell (super)gravity theories
are not invariant under the corresponding Maxwell (super)algebra. On the other
hand, there has been a growing interest in the study of three-dimensional
Chern-Simons (super)gravity theories invariant under Maxwell type
(super)algebras [42, 44, 46, 45, 57, 58, 59, 60]. In particular, in [46] the
authors presented the construction of the $(2+1)$-dimensional Chern-Simons
supergravity theory invariant, by construction, under the minimal Maxwell
superalgebra (that is, under the super-Maxwell gauge transformation) in
absence of extra fields, obtaining a supergravity action without cosmological
constant term characterized by three coupling constants, and showed that the
Maxwell supergravity obtained appears as a vanishing cosmological constant
limit of a minimal $AdS$-Lorentz supergravity (the flat limit was applied at
the level of the superalgebra, Chern-Simons action, supersymmetry
transformation laws, and field equations).

Thus, introducing a generalized supersymmetric cosmological term through
super-Maxwell symmetries in the context of supergravity seems to be a hard
task.222Actually, this was done in [61], but in the formalism of the so-called
contrained BF theories (see [61] and references therein). Conversely, the
$AdS$-Lorentz type superalgebras seem to be better candidates for introducing
the cosmological term in supergravity, in the presence of bosonic generators
$Z_{ab}$. Let us also mention that, in [39], unlike the case of Maxwell-type
superalgebras, the bosonic fields associated to the Lorentz-like generators
$Z_{ab}$ appear not only in the boundary terms but also in the bulk Lagrangian
of the model.

With this motivation, the aim of the present paper it to construct a
MacDowell-Mansouri like action based on a new $AdS$-Lorentz type superalgebra
involving two fermionic charges and possessing a bosonic subalgebra that does
not contains any additional generator with respect to the Maxwell algebra
$\mathcal{M}$, in such a way that, when contracted, it directly reproduces
$\mathcal{M}$. In this sense, such a superalgebra could also be viewed as the
minimal supersymmetrization of a minimal Maxwell-like algebra (actually, of a
deformation of $\mathcal{M}$) in which the bosonic generator $Z_{ab}$ is non-
abelian (in particular,
$\left[Z_{ab},Z_{cd}\right]=\eta_{bc}Z_{ad}-\eta_{ac}Z_{bd}-\eta_{bd}Z_{ac}+\eta_{ad}Z_{bc}$,
see also [18, 20] for non-abelian deformations of Maxwell (super)algebras) and
where $\left[Z_{ab},P_{c}\right]=\eta_{bc}P_{a}-\eta_{ac}P_{b}$, even if, as
we will see, the generators $Z_{ab}$ in this case will not behave as Lorentz
generators when considering the supersymmetric extension and the corresponding
commutation relations with the fermionic charges. Thus, in this paper we first
present the aforesaid new supersymmetrization of $AdS-\mathcal{L}_{4}$ (we
will call it $\tilde{s}AdS-\mathcal{L}_{4}$, for short) as an abelian
semigroup expansion of $\mathfrak{osp}(4|1)$. Then, we show that
$\tilde{s}AdS-\mathcal{L}_{4}$ allows to construct in a geometric way a $D=4$
supergravity action containing a generalized supersymmetric cosmological term.
The action we end up with corresponds to a MacDowell-Mansouri like action
written in terms of the $\tilde{s}AdS-\mathcal{L}_{4}$ curvatures. Our result
is a new supersymmetric extension of [28] involving two fermionic generators.
In our model, the $\tilde{s}AdS-\mathcal{L}_{4}$ fields will appear not only
in the boundary terms, but also in the bulk Lagrangian (analogously to what
happened in [39] and differently from what happened in [56, 38]). Referring to
the recent works [62, 63, 64, 65], we conjecture that the presence of the
$\tilde{s}AdS-\mathcal{L}_{4}$ fields in the bulk and in the boundary would
allow to recover the supersymmetry invariance of a supergravity theory based
on the superalgebra $\tilde{s}AdS-\mathcal{L}_{4}$ in the presence of a non-
trivial boundary of spacetime in the so-called rheonomic (i.e., geometric)
approach.

The paper is organized as follows: In Section 2, we present the new
superalgebra $\tilde{s}AdS-\mathcal{L}_{4}$ as an $S$-expansion of
$\mathfrak{osp}(4|1)$. Subsequently, in Section 3 we construct in a geometric
way a $D=4$ supergravity model containing a generalized supersymmetric
cosmological term only from the curvatures of $\tilde{s}AdS-\mathcal{L}_{4}$;
the action obtained corresponds to a MacDowell-Mansouri like action. Then, we
analyze the supersymmetry invariance of the theory. Finally, Section 4
contains the conclusions and possible future developments. In the Appendix we
collect our conventions and some useful formulas.

##  2 $AdS$-Lorentz superalgebra $\tilde{s}AdS-\mathcal{L}_{4}$ as an
$S$-expansion of $\mathfrak{osp}(4|1)$

In the following, we apply the $S$-expansion method to $\mathfrak{osp}(4|1)$
by using a particular abelian semigroup and obtain a new supersymmetrization
of $AdS-\mathcal{L}_{4}$. We will name this novel superalgebra
$\tilde{s}AdS-\mathcal{L}_{4}$, for short.

Let us first recall that the generators
$\\{\tilde{J}_{ab},\tilde{P}_{a},\tilde{Q}_{\alpha}\\}$ (with $a=0,1,2,3$,
$\alpha=1,2,3,4$) of $\mathfrak{osp}(4|1)$ fulfill the (anti)commutation
relations:333Here and in the following we denote the quantities referring to
$\mathfrak{osp}(4|1)$ with a tilde symbol on the top.

$\begin{split}\left[\tilde{J}_{ab},\tilde{J}_{cd}\right]&=\eta_{bc}\tilde{J}_{ad}-\eta_{ac}\tilde{J}_{bd}-\eta_{bd}\tilde{J}_{ac}+\eta_{ad}\tilde{J}_{bc},\\\
\left[\tilde{J}_{ab},\tilde{P}_{c}\right]&=\eta_{bc}\tilde{P}_{a}-\eta_{ac}\tilde{P}_{b},\\\
\left[\tilde{P}_{a},\tilde{P}_{b}\right]&=\tilde{J}_{ab},\\\
\left[\tilde{J}_{ab},\tilde{Q}_{\alpha}\right]&=-\frac{1}{2}\left(\gamma_{ab}\tilde{Q}\right)_{\alpha},\\\
\left[\tilde{P}_{a},\tilde{Q}_{\alpha}\right]&=-\frac{1}{2}\left(\gamma_{a}\tilde{Q}\right)_{\alpha},\\\
\left\\{\tilde{Q}_{\alpha},\tilde{Q}_{\beta}\right\\}&=-\frac{1}{2}\left[\left(\gamma^{ab}C\right)_{\alpha\beta}\tilde{J}_{ab}-2\left(\gamma^{a}C\right)_{\alpha\beta}\tilde{P}_{a}\right],\end{split}$
(2.1)

where $\gamma_{ab}$, $\gamma_{a}$ are Dirac gamma matrices in four dimensions
and $C$ is the charge conjugation matrix; $\tilde{J}_{ab}$ are the Lorentz
generators, $\tilde{P}_{a}$ the spacetime translations generators, and
$\tilde{Q}_{\alpha}$ is a four-components Majorana spinor charge.

Then, let us consider, on the same lines of [39], the following decomposition
of the superalgebra $\mathfrak{g}=\mathfrak{osp}(4|1)$ in three subspaces
$V_{p}$, $p=0,1,2$:

$\mathfrak{g}=\mathfrak{osp}(4|1)=\mathfrak{so}(3,1)\oplus\frac{\mathfrak{osp}(4|1)}{\mathfrak{sp}(4)}\oplus\frac{\mathfrak{sp}(4)}{\mathfrak{so}(3,1)}=V_{0}\oplus
V_{1}\oplus V_{2},$ (2.2)

where $V_{0}=\\{\tilde{J}_{ab}\\}$, $V_{1}=\\{\tilde{Q}_{\alpha}\\}$, and
$V_{2}=\\{\tilde{P}_{a}\\}$. Consequently, we can write the subspace structure

$\begin{split}[V_{0},V_{0}]&\subset V_{0},\quad[V_{1},V_{1}]\subset
V_{0}\oplus V_{2}{,}\\\ [V_{0},V_{1}]&\subset V_{1},\quad[V_{1},V_{2}]\subset
V_{1}{,}\\\ [V_{0},V_{2}]&\subset V_{2},\quad[V_{2},V_{2}]\subset
V_{0}.\end{split}$ (2.3)

Now, let us consider the abelian semigroup
$\hat{S}=\\{\lambda_{0},\lambda_{1},\lambda_{2},\lambda_{3}\\}$ described by
the multiplication table given in Table 1.

| $\lambda_{0}$ | $\lambda_{1}$ | $\lambda_{2}$ | $\lambda_{3}$
---|---|---|---|---
$\lambda_{0}$ | $\lambda_{0}$ | $\lambda_{1}$ | $\lambda_{2}$ | $\lambda_{3}$
$\lambda_{1}$ | $\lambda_{1}$ | $\lambda_{2}$ | $\lambda_{3}$ | $\lambda_{2}$
$\lambda_{2}$ | $\lambda_{2}$ | $\lambda_{3}$ | $\lambda_{2}$ | $\lambda_{3}$
$\lambda_{3}$ | $\lambda_{3}$ | $\lambda_{2}$ | $\lambda_{3}$ | $\lambda_{2}$
Table 1: Multiplication table of the semigroup $\hat{S}$.

Then, consider the subset decomposition
$\hat{S}=\hat{S}_{0}\cup\hat{S}_{1}\cup\hat{S}_{2}$ with

$\hat{S}_{0}=\\{\lambda_{0},\lambda_{2}\\},\quad\hat{S}_{1}=\\{\lambda_{1},\lambda_{3}\\},\quad\hat{S}_{2}=\\{\lambda_{2}\\}.$
(2.4)

The decomposition given in (2.4) is resonant (see [31] for details), since it
satisfies

$\begin{split}\hat{S}_{0}\cdot\hat{S}_{0}&\subset\hat{S}_{0},\quad\hat{S}_{1}\cdot\hat{S}_{1}\subset\hat{S}_{0}\cap
S_{2}{,}\\\
\hat{S}_{0}\cdot\hat{S}_{1}&\subset\hat{S}_{1},\quad\hat{S}_{1}\cdot\hat{S}_{2}\subset\hat{S}_{1}{,}\\\
\hat{S}_{0}\cdot\hat{S}_{2}&\subset\hat{S}_{2},\quad\hat{S}_{2}\cdot\hat{S}_{2}\subset\hat{S}_{0},\end{split}$
(2.5)

which has the same form of (2.3). Then, according to Theorem IV.2 of [31], the
subalgebra

$\mathfrak{g}_{R}=W_{0}\oplus W_{1}\oplus W_{2},$ (2.6)

where

$\begin{split}&W_{0}=(\hat{S}_{0}\times
V_{0})=\\{\lambda_{0},\lambda_{2}\\}\times\\{\tilde{J}_{ab}\\}=\\{\lambda_{0}\tilde{J}_{ab},\lambda_{2}\tilde{J}_{ab}\\},\\\
&W_{1}=(\hat{S}_{1}\times
V_{1})=\\{\lambda_{1},\lambda_{3}\\}\times\\{\tilde{Q}_{\alpha}\\}=\\{\lambda_{1}\tilde{Q}_{\alpha},\lambda_{3}\tilde{Q}_{\alpha}\\},\\\
&W_{2}=(\hat{S}_{2}\times
V_{2})=\\{\lambda_{2}\\}\times\\{\tilde{P}_{a}\\}=\\{\lambda_{2}\tilde{P}_{a}\\},\end{split}$
(2.7)

is a resonant subalgebra of $\hat{S}\times\mathfrak{g}$. We can then perform
the following identification:

$J_{ab}=\lambda_{0}\tilde{J}_{ab},\quad Z_{ab}=\lambda_{2}\tilde{J}_{ab},\quad
P_{a}=\lambda_{2}\tilde{P}_{a},\quad
Q_{\alpha}=\lambda_{1}\tilde{Q}_{\alpha},\quad\Sigma_{\alpha}=\lambda_{3}\tilde{Q}_{\alpha},$
(2.8)

being $\\{J_{ab},P_{a},Z_{ab},Q_{\alpha},\Sigma_{\alpha}\\}$ the set of
generators of the new superalgebra obtained after a resonant
$\hat{S}$-expansion of $\mathfrak{osp}(4|1)$. This superalgebra actually
corresponds to a new supersymmetrization of the $AdS$-Lorentz algebra
$AdS-\mathcal{L}_{4}$ of [28]. Let us call it $\tilde{s}AdS-\mathcal{L}_{4}$.
Its (anti)commutation relations can be obtained by using the multiplication
rules of the semigroup $\hat{S}$ (see Table 1) together with the commutation
relations (2.1) of $\mathfrak{osp}(4|1)$; they read as follows:

$\begin{split}\left[J_{ab},J_{cd}\right]&=\eta_{bc}J_{ad}-\eta_{ac}J_{bd}-\eta_{bd}J_{ac}+\eta_{ad}J_{bc},\\\
\left[J_{ab},P_{c}\right]&=\eta_{bc}P_{a}-\eta_{ac}P_{b},\\\
\left[J_{ab},Z_{cd}\right]&=\eta_{bc}Z_{ad}-\eta_{ac}Z_{bd}-\eta_{bd}Z_{ac}+\eta_{ad}Z_{bc},\\\
\left[Z_{ab},Z_{cd}\right]&=\eta_{bc}Z_{ad}-\eta_{ac}Z_{bd}-\eta_{bd}Z_{ac}+\eta_{ad}Z_{bc},\\\
\left[Z_{ab},P_{c}\right]&=\eta_{bc}P_{a}-\eta_{ac}P_{b},\\\
\left[P_{a},P_{b}\right]&=Z_{ab},\\\
\left[J_{ab},Q_{\alpha}\right]&=-\frac{1}{2}\left(\gamma_{ab}Q\right)_{\alpha},\quad\left[J_{ab},\Sigma_{\alpha}\right]=-\frac{1}{2}\left(\gamma_{ab}\Sigma\right)_{\alpha},\\\
\left[Z_{ab},Q_{\alpha}\right]&=-\frac{1}{2}\left(\gamma_{ab}\Sigma\right)_{\alpha},\quad\left[Z_{ab},\Sigma_{\alpha}\right]=-\frac{1}{2}\left(\gamma_{ab}\Sigma\right)_{\alpha},\\\
\left[P_{a},Q_{\alpha}\right]&=-\frac{1}{2}\left(\gamma_{a}\Sigma\right)_{\alpha},\quad\left[P_{a},\Sigma_{\alpha}\right]=-\frac{1}{2}\left(\gamma_{a}\Sigma\right)_{\alpha},\\\
\left\\{Q_{\alpha},Q_{\beta}\right\\}&=-\frac{1}{2}\left[\left(\gamma^{ab}C\right)_{\alpha\beta}Z_{ab}-2\left(\gamma^{a}C\right)_{\alpha\beta}P_{a}\right],\\\
\left\\{Q_{\alpha},\Sigma_{\beta}\right\\}&=-\frac{1}{2}\left[\left(\gamma^{ab}C\right)_{\alpha\beta}Z_{ab}-2\left(\gamma^{a}C\right)_{\alpha\beta}P_{a}\right],\\\
\left\\{\Sigma_{\alpha},\Sigma_{\beta}\right\\}&=-\frac{1}{2}\left[\left(\gamma^{ab}C\right)_{\alpha\beta}Z_{ab}-2\left(\gamma^{a}C\right)_{\alpha\beta}P_{a}\right].\end{split}$
(2.9)

Observe that a new Majorana spinor charge $\Sigma_{\alpha}$ has been
introduced as a direct consequence of the $\hat{S}$-expansion procedure.

The new $AdS$-Lorentz superalgebra (2.9) contains the so-called
$AdS-\mathcal{L}_{4}$ algebra generated by $\\{J_{ab},P_{a},Z_{ab}\\}$ as a
bosonic subalgebra. The $AdS-\mathcal{L}_{4}$ algebra and its generalizations
have been largely studied and analyzed in [28]. In particular, it was proven
that $AdS-\mathcal{L}_{4}$ allows to include a generalized cosmological term
in a Born-Infeld gravity action. Furthermore, performing the rescaling

$J_{ab}\rightarrow J_{ab},\quad Z_{ab}\rightarrow\mu^{2}Z_{ab},\quad
P_{a}\rightarrow\mu P_{a}$ (2.10)

and taking the limit $\mu\rightarrow\infty$ (Inönü-Wigner contraction) in
$AdS-\mathcal{L}_{4}$, one obtains the minimal Maxwell algebra (1.3).

On the other hand, $AdS-\mathcal{L}_{4}$ is also a bosonic subalgebra of the
$AdS$-Lorentz superalgebra of [39]. In other words, the new $AdS$-Lorentz
superalgebra (2.9) and the $AdS$-Lorentz superalgebra of [39] share the same
bosonic subalgebra $AdS-\mathcal{L}_{4}$.

Nevertheless, as we have already mentioned in the Introduction, the
$AdS$-Lorentz superalgebra of [39] has just one fermionic generator.
Differently, $\tilde{s}AdS-\mathcal{L}_{4}$, given by (2.9), possesses two
fermionic charges $Q_{\alpha}$ and $\Sigma_{\alpha}$. Then, since it also
contains $AdS-\mathcal{L}_{4}$ as a bosonic subalgebra, in this sense (2.9) it
could also be viewed as the minimal supersymmetrization of a minimal Maxwell-
like algebra (that is, actually, a deformation of the Maxwell algebra
$\mathcal{M}$, see also [18]) in which the bosonic generator $Z_{ab}$ is non-
abelian
($\left[Z_{ab},Z_{cd}\right]=\eta_{bc}Z_{ad}-\eta_{ac}Z_{bd}-\eta_{bd}Z_{ac}+\eta_{ad}Z_{bc}$)
and where $\left[Z_{ab},P_{c}\right]=\eta_{bc}P_{a}-\eta_{ac}P_{b}$, even if
the generators $Z_{ab}$, in this case, do not behave as Lorentz generators
when considering the supersymmetric extension and the corresponding
commutation relations with the fermionic charges. In this context, we observe
that the behavior of the generators $Z_{ab}$ in $\tilde{s}AdS-\mathcal{L}_{4}$
is also different from the behavior of the $Z_{ab}$’s in (a contraction of)
the generalized minimal $AdS$-Lorentz superalgebra of [39].

##  3 Generalized supersymmetric cosmological term in $D=4$ from
$\tilde{s}AdS-\mathcal{L}_{4}$

In order to construct an action based on $\tilde{s}AdS-\mathcal{L}_{4}$ we
start, on the same lines of [38, 39], from the following $1$-form connection:

$A=A^{A}T_{A}=\frac{1}{2}\omega^{ab}J_{ab}+\frac{1}{\ell}V^{a}P_{a}+\frac{1}{\ell}k^{ab}Z_{ab}+\frac{1}{\sqrt{\ell}}\psi^{\alpha}Q_{\alpha}+\frac{1}{\sqrt{\ell}}\xi^{\alpha}\Sigma_{\alpha},$
(3.1)

where the $1$-form gauge fields are given by

$\omega^{ab}=\lambda_{0}\tilde{\omega}^{ab},\quad
V^{a}=\lambda_{2}\tilde{V}^{a},\quad
k^{ab}=\lambda_{2}\tilde{\omega}^{ab},\quad\psi^{\alpha}=\lambda_{1}\tilde{\psi}^{\alpha},\quad\xi^{\alpha}=\lambda_{3}\tilde{\psi}^{\alpha},$
(3.2)

in terms of the components of the $\mathfrak{osp}(4|1)$ connection

$\tilde{A}=\frac{1}{2}\tilde{\omega}^{ab}\tilde{J}_{ab}+\frac{1}{\ell}\tilde{V}^{a}\tilde{P}_{a}+\frac{1}{\sqrt{\ell}}\tilde{\psi}^{\alpha}\tilde{Q}_{\alpha}.$
(3.3)

Note that, in order to properly interpret the gauge fields, it is necessary to
introduce a length scale $\ell$. The $1$-forms
$\omega^{ab},V^{a},k^{ab},\psi^{\alpha},\xi^{\alpha}$ are the spin connection,
the vielbein, a bosonic $1$-form field, the gravitino field, and an extra
spinor $1$-form field, respectively ($\psi^{\alpha}$ and $\xi^{\alpha}$ are
both Majorana spinors).

The $2$-form curvature $F=dA+A\wedge A$ associated with the connection (3.1)
reads

$F=F^{A}T_{A}=\frac{1}{2}R^{ab}J_{ab}+\frac{1}{\ell}R^{a}P_{a}+\frac{1}{2}F^{ab}k_{ab}+\frac{1}{\sqrt{\ell}}\rho^{\alpha}Q_{\alpha}+\frac{1}{\sqrt{\ell}}\Xi^{\alpha}\Sigma_{\alpha},$
(3.4)

where, in particular, we have444Here as well as in the sequel, for simplicity,
we omit the spinor index $\alpha$, and the wedge product $\wedge$ between
differential forms is understood.

$\displaystyle R^{ab}$
$\displaystyle=d\omega^{ab}+\omega^{a}_{\;c}\omega^{cb},$ (3.5a)
$\displaystyle R^{a}$
$\displaystyle=dV^{a}+\omega^{a}_{\;b}V^{b}+k^{a}_{\;b}V^{b}-\frac{1}{2}\bar{\psi}\gamma^{a}\psi-\bar{\psi}\gamma^{a}\xi-\frac{1}{2}\bar{\xi}\gamma^{a}\xi$
$\displaystyle=DV^{a}+k^{a}_{\;b}V^{b}-\frac{1}{2}\bar{\psi}\gamma^{a}\psi-\bar{\psi}\gamma^{a}\xi-\frac{1}{2}\bar{\xi}\gamma^{a}\xi,$
(3.5b) $\displaystyle F^{ab}$
$\displaystyle=dk^{ab}+2\omega^{a}_{\;c}k^{cb}+k^{a}_{\;c}k^{cb}+\frac{1}{\ell^{2}}V^{a}V^{b}+\frac{1}{2\ell}\bar{\psi}\gamma^{ab}\psi+\frac{1}{\ell}\bar{\psi}\gamma^{ab}\xi+\frac{1}{2\ell}\bar{\xi}\gamma^{ab}\xi$
$\displaystyle=Dk^{ab}+k^{a}_{\;c}k^{cb}+\frac{1}{\ell^{2}}V^{a}V^{b}+\frac{1}{2\ell}\bar{\psi}\gamma^{ab}\psi+\frac{1}{\ell}\bar{\psi}\gamma^{ab}\xi+\frac{1}{2\ell}\bar{\xi}\gamma^{ab}\xi{,}$
(3.5c) $\displaystyle\rho$
$\displaystyle=d\psi+\frac{1}{4}\omega^{ab}\gamma_{ab}\psi=D\psi,$ (3.5d)
$\displaystyle\Xi$
$\displaystyle=d\xi+\frac{1}{4}\omega^{ab}\gamma_{ab}\xi+\frac{1}{2\ell}V^{a}\gamma_{a}\psi+\frac{1}{2\ell}V^{a}\gamma_{a}\xi+\frac{1}{4}k^{ab}\gamma_{ab}\psi+\frac{1}{4}k^{ab}\gamma_{ab}\xi$
$\displaystyle=D\xi+\frac{1}{2\ell}V^{a}\gamma_{a}\psi+\frac{1}{2\ell}V^{a}\gamma_{a}\xi+\frac{1}{4}k^{ab}\gamma_{ab}\psi+\frac{1}{4}k^{ab}\gamma_{ab}\xi,$
(3.5e)

being $D=d+\omega$ the Lorentz covariant derivative. Setting $F=0$ one
retrieves the Maurer-Cartan equations for the superalgebra
$\tilde{s}AdS-\mathcal{L}_{4}$.

Considering the Bianchi identity $\nabla F=0$ ($\nabla=d+[A,\cdot]$), we
obtain:

$\displaystyle DR^{ab}$ $\displaystyle=0,$ (3.6a) $\displaystyle DR^{a}$
$\displaystyle=R^{a}_{\;b}V^{b}+F^{a}_{\;b}V^{b}-k^{a}_{\;b}R^{b}+\bar{\psi}\gamma^{a}\rho+\bar{\psi}\gamma^{a}\Xi+\bar{\xi}\gamma^{a}\rho+\bar{\xi}\gamma^{a}\Xi,$
(3.6b) $\displaystyle DF^{ab}$
$\displaystyle=2R^{a}_{\;c}k^{cb}+2F^{a}_{\;c}k^{cb}+\frac{2}{\ell^{2}}R^{a}V^{b}-\frac{1}{\ell}\bar{\psi}\gamma^{ab}\rho-\frac{1}{\ell}\bar{\psi}\gamma^{ab}\Xi-\frac{1}{\ell}{\bar{\xi}}\gamma^{ab}\rho-\frac{1}{\ell}\bar{\xi}\gamma^{ab}\Xi,$
(3.6c) $\displaystyle D\rho$ $\displaystyle=\frac{1}{4}R^{ab}\gamma_{ab}\psi,$
(3.6d) $\displaystyle D\Xi$
$\displaystyle=\frac{1}{4}R^{ab}\gamma_{ab}\xi+\frac{1}{2\ell}R^{a}\gamma_{a}\psi-\frac{1}{2\ell}V^{a}\gamma_{a}\rho+\frac{1}{2\ell}R^{a}\gamma_{a}\xi-\frac{1}{2\ell}V^{a}\gamma_{a}\Xi$
$\displaystyle+\frac{1}{4}F^{ab}\gamma_{ab}\psi-\frac{1}{4}k^{ab}\gamma_{ab}\rho+\frac{1}{4}F^{ab}\gamma_{ab}\xi-\frac{1}{4}k^{ab}\gamma_{ab}\Xi.$
(3.6e)

On the other hand, for a clearer understanding of the procedure we are going
to follow, let us recall that the $2$-form curvature
$\tilde{F}=d\tilde{A}+\tilde{A}\wedge\tilde{A}$ associated with the
$\mathfrak{osp}(4|1)$ connection (3.3) is

$\tilde{F}=\frac{1}{2}\tilde{R}^{ab}\tilde{J}_{ab}+\frac{1}{\ell}\tilde{R}^{a}\tilde{P}_{a}+\frac{1}{\sqrt{\ell}}\tilde{\rho}^{\alpha}\tilde{Q}_{\alpha},$
(3.7)

where, as it is well known, we have

$\displaystyle\tilde{R}^{ab}$
$\displaystyle=d\tilde{\omega}^{ab}+\tilde{\omega}^{a}_{\;c}\tilde{\omega}^{cb}+\frac{1}{\ell^{2}}\tilde{V}^{a}\tilde{V}^{b}+\frac{1}{2\ell}\bar{\tilde{\psi}}\gamma^{ab}\tilde{\psi},$
(3.8a) $\displaystyle\tilde{R}^{a}$
$\displaystyle=d\tilde{V}^{a}+\tilde{\omega}^{a}_{\;b}\tilde{V}^{b}-\frac{1}{2}\bar{\tilde{\psi}}\gamma^{a}\tilde{\psi}=\tilde{D}\tilde{V}^{a}-\frac{1}{2}\bar{\tilde{\psi}}\gamma^{a}\tilde{\psi},$
(3.8b) $\displaystyle\tilde{\rho}$
$\displaystyle=d\tilde{\psi}+\frac{1}{4}\tilde{\omega}^{ab}\gamma_{ab}\tilde{\psi}+\frac{1}{2\ell}\tilde{V}^{a}\gamma_{a}\tilde{\psi}=\tilde{D}\tilde{\psi}+\frac{1}{2\ell}\tilde{V}^{a}\gamma_{a}\tilde{\psi},$
(3.8c)

being $\tilde{D}=d+\tilde{\omega}$. Now, as recalled in [38, 39], the general
form of the MacDowell-Mansouri action [3] constructed with the
$\mathfrak{osp}(4|1)$ $2$-form curvature $\tilde{F}$ is

$S=2\int\langle\tilde{F}\wedge\tilde{F}\rangle=2\int\tilde{F}^{A}\wedge\tilde{F}^{B}\langle\tilde{T}_{A}\tilde{T}_{B}\rangle,$
(3.9)

with the following choice of the invariant tensor:

$\langle\tilde{T}_{A}\tilde{T}_{B}\rangle=\left\\{\begin{aligned}
\langle\tilde{J}_{ab}\tilde{J}_{cd}\rangle=&\epsilon_{abcd},\\\
\langle\tilde{Q}_{\alpha}\tilde{Q}_{\beta}\rangle=&2\left(\gamma_{5}\right)_{\alpha\beta}.\end{aligned}\right.$
(3.10)

Observe that if one chooses the whole
$\langle\tilde{T}_{A}\tilde{T}_{B}\rangle$ as an invariant tensor (which
satisfies the Bianchi identities) for the $OSp(4|1)$ supergroup, then the
action (3.9) is a topological invariant and gives no equations of motion.
Nevertheless, with the choice (3.10) of the invariant tensor (which breaks the
$OSp(4|1)$ supergroup to its Lorentz subgroup), (3.9) becomes a dynamical
action that corresponds to the MacDowell-Mansouri action for the
$\mathfrak{osp}(4|1)$ superalgebra [3, 66]. Writing the explicit form of the
action (3.9) with the choice (3.10) and omitting the boundary terms, the
result is the $\mathcal{N}=1$ supergravity action in four dimensions, given by
the Einstein-Hilbert and Rarita-Schwinger terms plus the usual supersymmetric
cosmological terms; the aforementioned action is not invariant under the
$\mathfrak{osp}(4|1)$ gauge transformations. However, the invariance of the
action under supersymmetry transformation can be obtained by modifying the
supersymmetry transformation of the spin connection $\tilde{\omega}^{ab}$
[67].

Now, in order to construct a MacDowell-Mansouri like action for
$\tilde{s}AdS-\mathcal{L}_{4}$, we consider the $\hat{S}$-expansion of
$\langle\tilde{T}_{A}\tilde{T}_{B}\rangle$ and the $2$-form curvature $F$ in
(3.4). In particular, the action for $\tilde{s}AdS-\mathcal{L}_{4}$ can be
written as

$S=2\int F^{A}\wedge F^{B}\langle T_{A}T_{B}\rangle,$ (3.11)

where $\langle T_{A}T_{B}\rangle$ can be obtained from the components of the
invariant tensor written in (3.10), using Theorem VII.1 of [31]. One can then
show that the non-vanishing components of $\langle T_{A}T_{B}\rangle$ are

$\begin{split}&\langle
J_{ab}J_{cd}\rangle=C_{0}\langle\tilde{J}_{ab}\tilde{J}_{cd}\rangle=C_{0}\epsilon_{abcd},\\\
&\langle
J_{ab}Z_{cd}\rangle=C_{2}\langle\tilde{J}_{ab}\tilde{J}_{cd}\rangle=C_{2}\epsilon_{abcd},\\\
&\langle
Z_{ab}Z_{cd}\rangle=C_{2}\langle\tilde{J}_{ab}\tilde{J}_{cd}\rangle=C_{2}\epsilon_{abcd},\\\
&\langle
Q_{\alpha}Q_{\beta}\rangle=C_{2}\langle\tilde{Q}_{\alpha}\tilde{Q}_{\beta}\rangle=2C_{2}\left(\gamma_{5}\right)_{\alpha\beta},\\\
&\langle
Q_{\alpha}\Sigma_{\beta}\rangle=C_{2}\langle\tilde{Q}_{\alpha}\tilde{Q}_{\beta}\rangle=2C_{2}\left(\gamma_{5}\right)_{\alpha\beta},\\\
&\langle\Sigma_{\alpha}\Sigma_{\beta}\rangle=C_{2}\langle\tilde{Q}_{\alpha}\tilde{Q}_{\beta}\rangle=2C_{2}\left(\gamma_{5}\right)_{\alpha\beta},\end{split}$
(3.12)

where $C_{0}$ and $C_{2}$ are (dimensionless) independent constants.

Thus, considering (3.12) and the $2$-form curvature (3.4), we can write an
action of the form (3.11) as

$S=\int\left(\frac{C_{0}}{2}\epsilon_{abcd}R^{ab}R^{cd}+C_{2}\epsilon_{abcd}R^{ab}F^{cd}+\frac{C_{2}}{2}\epsilon_{abcd}F^{ab}F^{cd}+\frac{4C_{2}}{\ell}\bar{\rho}\gamma_{5}\rho+\frac{8C_{2}}{\ell}\bar{\rho}\gamma_{5}\Xi+\frac{4C_{2}}{\ell}\bar{\Xi}\gamma_{5}\Xi\right).$
(3.13)

Then, using the formulas collected in the Appendix and the Bianchi identities
originated from the superalgebra $\tilde{s}AdS-\mathcal{L}_{4}$, one can show
that the MacDowell-Mansouri like action (3.13) for
$\tilde{s}AdS-\mathcal{L}_{4}$ can be written explicitly as

$\begin{split}S&=\int\Bigg{\\{}\frac{C_{0}}{2}\epsilon_{abcd}R^{ab}R^{cd}+C_{2}\Bigg{(}\epsilon_{abcd}R^{ab}\mathcal{F}^{cd}+\frac{1}{2}\epsilon_{abcd}\mathcal{F}^{ab}\mathcal{F}^{cd}\Bigg{)}\\\
&+\frac{C_{2}}{\ell^{2}}\Bigg{(}{\epsilon_{abcd}}R^{ab}V^{c}V^{d}+4\bar{\psi}V^{a}\gamma_{a}\gamma_{5}\rho+4\bar{\psi}V^{a}\gamma_{a}\gamma_{5}\sigma+4\bar{\xi}V^{a}\gamma_{a}\gamma_{5}\sigma+4\bar{\xi}V^{a}\gamma_{a}\gamma_{5}\rho\Bigg{)}\\\
&+\frac{C_{2}}{\ell^{2}}\epsilon_{abcd}\Bigg{(}\mathcal{F}^{ab}V^{c}V^{d}+\frac{2}{\ell}\bar{\psi}\gamma^{ab}\xi
V^{c}V^{d}+\frac{1}{\ell}\bar{\xi}\gamma^{ab}\xi
V^{c}V^{d}+\frac{1}{\ell}\bar{\psi}\gamma^{ab}\psi
V^{c}V^{d}+\frac{1}{2\ell^{2}}V^{a}V^{b}V^{c}V^{d}\Bigg{)}\\\
&+\frac{C_{2}}{\ell}d\Bigg{(}4\bar{\psi}\gamma_{5}\rho+4\bar{\psi}\gamma_{5}\sigma+4\bar{\xi}\gamma_{5}\rho+4\bar{\xi}\gamma_{5}\sigma\Bigg{)}\Bigg{\\}},\end{split}$
(3.14)

where we have also isolated the boundary terms and defined

$\displaystyle\mathcal{F}^{ab}$ $\displaystyle=Dk^{ab}+k^{a}_{\;c}k^{cb},$
(3.15a) $\displaystyle\sigma$
$\displaystyle=D\xi+\frac{1}{4}k^{ab}\gamma_{ab}\psi+\frac{1}{4}k^{ab}\gamma_{ab}\xi.$
(3.15b)

Note that we have separated the action (3.14) in different pieces, in such a
way to make their physical meaning manifest: The piece proportional to $C_{0}$
corresponds to the Gauss-Bonnet term; the second term is an Euler invariant
and can be seen as a Gauss-Bonnet like term (it does not contribute to the
dynamics of the theory and can be written as a boundary term) that involves
the new $\tilde{s}AdS-\mathcal{L}_{4}$ field $k^{ab}$; the third piece
contains the Einstein-Hilbert and Rarita-Schwinger Lagrangian, describing pure
supergravity, plus three additional terms involving the new spinor $1$-form
field $\xi$ and the bosonic field $k^{ab}$; the fourth term corresponds to a
generalized supersymmetric cosmological term which contains, besides the usual
supersymmetric cosmological term, also two additional terms involving the
$\tilde{s}AdS-\mathcal{L}_{4}$ spinor field $\xi$ and one additional term
involving $k^{ab}$; the last term is a boundary term.

Thus, we have shown that the MacDowell-Mansouri like action constructed
applying the properties of the $S$-expansion procedure in the case of a
resonant $\hat{S}$-expansion (see Table 1 for the multiplication table of the
semigroup $\hat{S}$) of $\mathfrak{osp}(4|1)$ describes a supergravity model
with a generalized supersymmetric cosmological term. In other words, we have
introduced in alternative way the supersymmetric cosmological term in a
supergravity model, building a (deformed) $D=4$ supergravity action from the
new $AdS$-Lorentz superalgebra $\tilde{s}AdS-\mathcal{L}_{4}$. Our result
corresponds to a new supersymmetric extension of [28] involving two fermionic
generators.

Let us observe that, if we consider $k^{ab}=0$ and $\xi=0$ in the action
(3.14), we obtain the MacDowell-Mansouri action for the supergroup $OSp(4|1)$.
On the other hand, setting only $\xi=0$ in (3.14), we obtain the action for
the $AdS$-Lorentz superalgebra found in [39].

Notice that if we omit the boundary terms in (3.14), we get:

$\begin{split}S&=\int\Bigg{\\{}\frac{C_{2}}{\ell^{2}}\Bigg{(}{\epsilon_{abcd}}R^{ab}V^{c}V^{d}+4\bar{\psi}V^{a}\gamma_{a}\gamma_{5}\rho+4\bar{\psi}V^{a}\gamma_{a}\gamma_{5}\sigma+4\bar{\xi}V^{a}\gamma_{a}\gamma_{5}\sigma+4\bar{\xi}V^{a}\gamma_{a}\gamma_{5}\rho\Bigg{)}\\\
&+\frac{C_{2}}{\ell^{2}}\epsilon_{abcd}\Bigg{(}\mathcal{F}^{ab}V^{c}V^{d}+\frac{2}{\ell}\bar{\psi}\gamma^{ab}\xi
V^{c}V^{d}+\frac{1}{\ell}\bar{\xi}\gamma^{ab}\xi
V^{c}V^{d}+\frac{1}{\ell}\bar{\psi}\gamma^{ab}\psi
V^{c}V^{d}+\frac{1}{2\ell^{2}}V^{a}V^{b}V^{c}V^{d}\Bigg{)}\Bigg{\\}}.\end{split}$
(3.16)

Then, if we consider $k^{ab}=0$ and $\xi=0$ in the action (3.16), we are left
with the Einstein-Hilbert and Rarita-Schwinger Lagrangian plus the usual
supersymmetric cosmological term.

Let us now compute the variation of the Lagrangian with respect to the
different $\tilde{s}AdS-\mathcal{L}_{4}$ $1$-form fields in order to obtain
the field equations. One can prove that, computing the variation of the
Lagrangian with respect to $\omega^{ab}$ and imposing
$\delta_{\omega}\mathcal{L}=0$, we get the following field equation (modulo
boundary terms) for the $\tilde{s}AdS-\mathcal{L}_{4}$ supertorsion:

$\epsilon_{abcd}R^{c}V^{d}=0.$ (3.17)

From the variation of the Lagrangian with respect to $k^{ab}$, we obtain the
same equation. On the other hand, computing the variation of the Lagrangian
with respect to $V^{a}$ and imposing $\delta_{V}\mathcal{L}=0$, we get

$2\epsilon_{abcd}\left(R^{ab}+F^{ab}\right)V^{c}+4\left(\bar{\psi}+\bar{\xi}\right)\gamma_{d}\gamma_{5}\rho+4\bar{\xi}\gamma_{d}\gamma_{5}\left(\rho+\Xi\right)=0,$
(3.18)

and from the variation of the Lagrangian with respect to $\psi$, imposing
$\delta_{\psi}\mathcal{L}=0$, we obtain (modulo boundary terms):

$8V^{a}\gamma_{a}\gamma_{5}\left(\rho+\Xi\right)+4\gamma_{a}\gamma_{5}\left(\psi+\xi\right)R^{a}=0.$
(3.19)

The variation of the Lagrangian with respect to the spinor $1$-form field
$\xi$ leads to the same equation. Note that the field equations (3.17),
(3.18), and (3.19) are similar to those of $\mathfrak{osp}(4|1)$ supergravity,
the only differences being related to the presence of the new fields $k^{ab}$
and $\xi$.

Interestingly, we observe that one can define a new bosonic field

$\hat{\omega}^{ab}\equiv\omega^{ab}+k^{ab},$ (3.20)

which can be interpreted as an extension of the Riemannian connection
$\omega^{ab}_{\mu}dx^{\mu}$ to a non-Riemannian one with torsion, together
with the covariant derivative

$\hat{D}\equiv d+\hat{\omega},$ (3.21)

and a new spinor $1$-form field

$\hat{\psi}\equiv\psi+\xi$ (3.22)

implying a redefinition of the gravitino $1$-form field. Then, exploiting the
new definitions, the equations of motion (3.17), (3.18), and (3.19) become,
respectively:

$\displaystyle\epsilon_{abcd}\hat{R}^{c}V^{d}=0,$ (3.23a) $\displaystyle
2\epsilon_{abcd}\hat{R}^{ab}V^{c}+4\bar{\hat{\psi}}\gamma_{d}\gamma_{5}\hat{\rho}=0,$
(3.23b) $\displaystyle
8V^{a}\gamma_{a}\gamma_{5}\hat{\rho}+4\gamma_{a}\gamma_{5}\hat{\psi}\hat{R}^{a}=0,$
(3.23c)

where we have also defined

$\displaystyle\hat{R}^{ab}$ $\displaystyle\equiv
d\hat{\omega}^{ab}+\hat{\omega}^{a}_{\;c}\hat{\omega}^{cb}+\frac{1}{2\ell}\bar{\hat{\psi}}\gamma^{ab}\hat{\psi},$
(3.24a) $\displaystyle\hat{R}^{a}$
$\displaystyle\equiv\hat{D}V^{a}-\frac{1}{2}\bar{\hat{\psi}}\gamma_{a}\hat{\psi},$
(3.24b) $\displaystyle\hat{\rho}$
$\displaystyle\equiv\hat{D}\hat{\psi}+\frac{1}{2\ell}V^{a}\gamma_{a}\hat{\psi}.$
(3.24c)

These new curvatures have the same form of the $\mathfrak{osp}(4|1)$ ones, and
the fields $\\{\hat{\omega}^{ab},V^{a},\hat{\psi}\\}$ fulfill equations of
motion that have the same form of those of the $\mathfrak{osp}(4|1)$
supergravity theory. In fact, let us also observe that, at the price of
introducing the $1$-form fields $k^{ab}$ and $\xi$ (and the corresponding
Maurer-Cartan equations), the $\mathfrak{osp}(4|1)$ superalgebra can be mapped
into $\tilde{s}AdS-\mathcal{L}_{4}$, whereby the spin connection and the
gravitino are respectively identified with the Lorentz connection and
gravitino of a $D=4$ Minkowski spacetime with vanishing Lorentz curvature and
vanishing gravitino super field-strength, albeit with a modification of the
supertorsion, the latter being non-vanishing in both cases. This point will be
further analyzed in a future work.

Then, exploiting the definitions (3.20), (3.22), (3.24a), and (3.24c), the
action (3.16) can be rewritten as follows:

${S=\int\frac{C_{2}}{\ell^{2}}\Bigg{(}\epsilon_{abcd}\hat{R}^{ab}V^{c}V^{d}+4\bar{\hat{\psi}}V^{a}\gamma_{a}\gamma_{5}\hat{\rho}+\frac{1}{\ell}\epsilon_{abcd}\bar{\hat{\psi}}\gamma^{ab}\hat{\psi}V^{c}V^{d}+\frac{1}{2\ell^{2}}\epsilon_{abcd}V^{a}V^{b}V^{c}V^{d}\Bigg{)}.}$
(3.25)

The action (3.25) contains the Einstein-Hilbert and Rarita-Schwinger
Lagrangian, plus the supersymmetric cosmological term, for the new connection
$\hat{\omega}^{ab}$ defined in (3.20) and the new gravitino $1$-form field
$\hat{\psi}$ defined in (3.22), which can be seen as a shifted connection and
as a shifted gravitino, respectively. In this sense, our result may also be
considered as the specific extension to a non-Riemannian framework determined
by the structure of the $\tilde{s}AdS-\mathcal{L}_{4}$ algebra. Indeed, in
this context, the antisymmetry $k^{ab}=-k^{ba}$ implies that we are dealing
with an Einstein-Cartan geometry with non-metricity tensor equal to zero,
because in (3.20) we have $\hat{\omega}^{(ab)}=0$, while a symmetric part of
$\hat{\omega}^{ab}$ would have defined a non-vanishing non-metricity tensor
(see, for example, [68] for details).

On the other hand, let us also observe that an Inönü-Wigner contraction of the
action (3.16) leads us to the $D=4$ pure supergravity action

${S=\int\frac{C_{2}}{\ell^{2}}\Bigg{(}\epsilon_{abcd}R^{ab}V^{c}V^{d}+4\bar{\psi}V^{a}\gamma_{a}\gamma_{5}\rho\Bigg{)}.}$
(3.26)

In fact, performing the rescaling

${\omega^{ab}\rightarrow\omega^{ab},\quad
V^{a}\rightarrow\mu^{2}V^{a},\quad\psi\rightarrow\mu\psi,\quad
k^{ab}\rightarrow\mu^{4}k^{ab},\quad\xi\rightarrow\mu^{3}\xi}$ (3.27)

in (3.16) and dividing the action by $\mu^{4}$, the four-dimensional pure
supergravity action (3.26) is retrieved by taking the limit $\mu\rightarrow
0$.

Finally, allowing also the rescaling

${\ell\rightarrow\mu^{2}\ell}$ (3.28)

on the length scale $\ell$ in the action (3.16), together with the rescaling
(3.27), dividing the action by $\mu^{4}$, and taking the limit $\mu\rightarrow
0$, the $D=4$ supergravity action with supersymmetric cosmological term is
retrieved, namely

${S=\int\frac{C_{2}}{\ell^{2}}\Bigg{(}\epsilon_{abcd}R^{ab}V^{c}V^{d}+4\bar{\psi}V^{a}\gamma_{a}\gamma_{5}\rho+\frac{1}{\ell}\epsilon_{abcd}\bar{\psi}\gamma^{ab}\psi
V^{c}V^{d}+\frac{1}{2\ell^{2}}\epsilon_{abcd}V^{a}V^{b}V^{c}V^{d}\Bigg{)}.}$
(3.29)

###  3.1 $\tilde{s}AdS-\mathcal{L}_{4}$ gauge transformations and
supersymmetry invariance

The gauge transformation of the connection $A$ in (3.1) is

$\delta_{\varrho}A=D\varrho=d\varrho+[A,\varrho],$ (3.30)

where $\rho$ is the $\tilde{s}AdS-\mathcal{L}_{4}$ gauge parameter

$\varrho=\frac{1}{2}\varrho^{ab}J_{ab}+\frac{1}{2}\kappa^{ab}Z_{ab}+\frac{1}{\ell}\varrho^{a}P_{a}+\frac{1}{\sqrt{\ell}}\epsilon^{\alpha}Q_{\alpha}+\frac{1}{\sqrt{\ell}}\varepsilon^{\alpha}\Sigma_{\alpha}.$
(3.31)

Then, using

$\delta\left(A^{A}T_{A}\right)=d\varrho+\left[A^{B}T_{B},\varrho^{C}T_{C}\right],$
(3.32)

we obtain that the $\tilde{s}AdS-\mathcal{L}_{4}$ gauge transformations are
given by:

$\displaystyle\delta\omega^{ab}$ $\displaystyle=D\varrho^{ab},$ (3.33a)
$\displaystyle\delta k^{ab}$
$\displaystyle=D\kappa^{ab}-2\varrho^{a}_{\;c}k^{cb}+2k^{a}_{\;c}\kappa^{cb}+\frac{2}{\ell^{2}}V^{a}\varrho^{b}-\frac{1}{\ell}\bar{\epsilon}\gamma^{ab}\psi-\frac{1}{\ell}\bar{\epsilon}\gamma^{ab}\xi-\frac{1}{\ell}\bar{\varepsilon}\gamma^{ab}\psi-\frac{1}{\ell}\bar{\varepsilon}\gamma^{ab}\xi,$
(3.33b) $\displaystyle\delta V^{a}$
$\displaystyle=D\varrho^{a}-\varrho^{a}_{\;b}V^{b}+k^{a}_{\;b}\varrho^{b}-\kappa^{a}_{\;b}V^{b}+\bar{\epsilon}\gamma^{a}\psi+\bar{\epsilon}\gamma^{a}\xi+\bar{\varepsilon}\gamma^{a}\psi+\bar{\varepsilon}\gamma^{a}\xi,$
(3.33c) $\displaystyle\delta\psi$
$\displaystyle=d\epsilon+\frac{1}{4}\omega^{ab}\gamma_{ab}\epsilon-\frac{1}{4}\varrho^{ab}\gamma_{ab}\psi,$
(3.33d) $\displaystyle\delta\xi$
$\displaystyle=d\varepsilon+\frac{1}{4}\omega^{ab}\gamma_{ab}\varepsilon-\frac{1}{4}\varrho^{ab}\gamma_{ab}\xi-\frac{1}{2\ell}\varrho^{a}\gamma_{a}\psi+\frac{1}{2\ell}V^{a}\gamma_{a}\epsilon-\frac{1}{2\ell}\varrho^{a}\gamma_{a}\xi+\frac{1}{2\ell}V^{a}\gamma_{a}\varepsilon$
$\displaystyle+\frac{1}{4}k^{ab}\gamma_{ab}\epsilon-\frac{1}{4}\kappa^{ab}\gamma_{ab}\psi+\frac{1}{4}k^{ab}\gamma_{ab}\varepsilon-\frac{1}{4}\kappa^{ab}\gamma_{ab}\xi.$
(3.33e)

Analogously, from the gauge variation of the curvature $F$,

$\delta_{\varrho}F=\left[F,\varrho\right],$ (3.34)

we can write the following gauge transformations:

$\displaystyle\delta R^{ab}$ $\displaystyle=2R^{a}_{\;c}\varrho^{cb},$ (3.35a)
$\displaystyle\delta F^{ab}$
$\displaystyle=2R^{a}_{\;c}\kappa^{cb}-2\varrho^{a}_{\;c}F^{cb}+2F^{a}_{\;c}\kappa^{cb}+\frac{2}{\ell^{2}}R^{a}\varrho^{b}-\frac{1}{\ell}\bar{\epsilon}\gamma^{ab}\rho-\frac{1}{\ell}\bar{\epsilon}\gamma^{ab}\Xi-\frac{1}{\ell}\bar{\varepsilon}\gamma^{ab}\rho-\frac{1}{\ell}\bar{\varepsilon}\gamma^{ab}\Xi,$
(3.35b) $\displaystyle\delta R^{a}$
$\displaystyle=R^{a}_{\;b}\varrho^{b}-\varrho^{a}_{\;b}R^{b}+F^{a}_{\;b}\varrho^{b}-\kappa^{a}_{\;b}R^{b}+\bar{\epsilon}\gamma^{a}\rho+\bar{\epsilon}\gamma^{a}\Xi+\bar{\varepsilon}\gamma^{a}\rho+\bar{\varepsilon}\gamma^{a}\Xi,$
(3.35c) $\displaystyle\delta\rho$
$\displaystyle=\frac{1}{4}R^{ab}\gamma_{ab}\epsilon-\frac{1}{4}\varrho^{ab}\gamma_{ab}\rho,$
(3.35d) $\displaystyle\delta\Xi$
$\displaystyle=\frac{1}{4}R^{ab}\gamma_{ab}\varepsilon-\frac{1}{4}\varrho^{ab}\gamma_{ab}\Xi-\frac{1}{2\ell}\varrho^{a}\gamma_{a}\rho+\frac{1}{2\ell}R^{a}\gamma_{a}\epsilon-\frac{1}{2\ell}\varrho^{a}\gamma_{a}\Xi+\frac{1}{2\ell}R^{a}\gamma_{a}\varepsilon$
$\displaystyle+\frac{1}{4}F^{ab}\gamma_{ab}\epsilon-\frac{1}{4}\kappa^{ab}\gamma_{ab}\rho+\frac{1}{4}F^{ab}\gamma_{ab}\varepsilon-\frac{1}{4}\kappa^{ab}\gamma_{ab}\Xi.$
(3.35e)

Although the action (3.14) is built from $\tilde{s}AdS-\mathcal{L}_{4}$, one
can prove that it is not invariant under the $\tilde{s}AdS-\mathcal{L}_{4}$
gauge transformations. Furthermore, if we consider the variation of (3.14)
under gauge supersymmetry, we find:

$\delta_{\text{susy}}S=-\frac{4C_{2}}{\ell^{2}}\int
R^{a}\left(\bar{\rho}+\bar{\Xi}\right)\gamma_{a}\gamma_{5}\epsilon.$ (3.36)

Thus, as in the $\mathfrak{osp}(4|1)$ and super-Poincaré cases, the action is
invariant under gauge supersymmetry imposing the supertorsion constraint

$R^{a}=0.$ (3.37)

However, this causes $\omega^{ab}$ to be expressed in terms of the other
fields $V^{a},k^{ab},\psi,\xi$ (second order formalism).

Alternatively, on the same lines of [38, 39], one can recover the
supersymmetry invariance of the action in the first order formalism by
modifying the supersymmetry transformation of the spin connection
$\omega^{ab}$. Indeed, if we consider the variation of the action under an
arbitrary $\delta\omega^{ab}$, we find

$\delta_{\omega}S=\frac{2C_{2}}{\ell^{2}}\int\epsilon_{abcd}R^{a}V^{b}\delta\omega^{cd},$
(3.38)

which vanishes for arbitrary $\delta\omega^{ab}$ if $R^{a}=0$. Then, it is
possible to modify $\delta\omega^{ab}$ with the addition of an extra piece to
the gauge transformation in such a way that the variation of the action reads

$\delta S=-\frac{4C_{2}}{\ell^{2}}\int
R^{a}\left[\left(\bar{\rho}+\bar{\Xi}\right)\gamma_{a}\gamma_{5}\epsilon-\frac{1}{2}\epsilon_{abcd}V^{b}\delta_{\text{extra}}\omega^{cd}\right],$
(3.39)

the supersymmetry invariance being fulfilled when

$\delta_{\text{extra}}\omega^{ab}=2\epsilon^{abcd}\left(\bar{\zeta}_{ec}\gamma_{d}\epsilon+\bar{\zeta}_{de}\gamma_{c}\gamma_{5}\epsilon-\bar{\zeta}_{cd}\gamma_{e}\gamma_{5}\epsilon\right)V^{e},$
(3.40)

with $\bar{\rho}+\bar{\Xi}=\bar{\zeta}_{ab}V^{a}V^{b}$. Thus, we can conclude
that the action in the first order formalism is invariant under the
supersymmetry transformations555Let us mention that, actually, one could
retrieve supersymmetry invariance also by modifying the whole
$\delta\left(\omega^{ab}+k^{ab}\right)$, obtaining that the action is
invariant under
$\delta\left(\omega^{ab}+k^{ab}\right)=2\epsilon^{abcd}\left(\bar{\zeta}_{ec}\gamma_{d}\epsilon+\bar{\zeta}_{de}\gamma_{c}\gamma_{5}\epsilon-\bar{\zeta}_{cd}\gamma_{e}\gamma_{5}\epsilon\right)V^{e}-\frac{1}{\ell}\bar{\epsilon}\gamma^{ab}\psi-\frac{1}{\ell}\bar{\epsilon}\gamma^{ab}\xi$,
together with the other transformations (3.41c) and (3.41e). The
transformations we have written in (3.41a), (3.41d), and (3.41b) correspond to
a particular choice for $\delta\omega^{ab}$ and $\delta k^{ab}$ fulfilling
$\delta\left(\omega^{ab}+k^{ab}\right)=2\epsilon^{abcd}\left(\bar{\zeta}_{ec}\gamma_{d}\epsilon+\bar{\zeta}_{de}\gamma_{c}\gamma_{5}\epsilon-\bar{\zeta}_{cd}\gamma_{e}\gamma_{5}\epsilon\right)V^{e}-\frac{1}{\ell}\bar{\epsilon}\gamma^{ab}\psi-\frac{1}{\ell}\bar{\epsilon}\gamma^{ab}\xi$.

$\displaystyle\delta\omega^{ab}$
$\displaystyle=2\epsilon^{abcd}\left(\bar{\zeta}_{ec}\gamma_{d}\epsilon+\bar{\zeta}_{de}\gamma_{c}\gamma_{5}\epsilon-\bar{\zeta}_{cd}\gamma_{e}\gamma_{5}\epsilon\right)V^{e},$
(3.41a) $\displaystyle\delta k^{ab}$
$\displaystyle=-\frac{1}{\ell}\bar{\epsilon}\gamma^{ab}\psi-\frac{1}{\ell}\bar{\epsilon}\gamma^{ab}\xi,$
(3.41b) $\displaystyle\delta V^{a}$
$\displaystyle=\bar{\epsilon}\gamma^{a}\psi+\bar{\epsilon}\gamma^{a}\xi,$
(3.41c) $\displaystyle\delta\psi$
$\displaystyle=d\epsilon+\frac{1}{4}\omega^{ab}\gamma_{ab}\epsilon=D\epsilon,$
(3.41d) $\displaystyle\delta\xi$
$\displaystyle=\frac{1}{2\ell}V^{a}\gamma_{a}\epsilon+\frac{1}{4}k^{ab}\gamma_{ab}\epsilon.$
(3.41e)

Observe that supersymmetry is not a gauge symmetry of the action, and the
supersymmetry transformations do not close off-shell, while the
$\tilde{s}AdS-\mathcal{L}_{4}$ gauge transformations close off-shell by
construction.

Finally, note that there is also another kind of supersymmetry (i.e., a
supersymmetry-like symmetry), related to the fermionic generator
$\Sigma_{\alpha}$. The new supersymmetry-like transformations read as follows:

$\displaystyle\delta\omega^{ab}$ $\displaystyle=0,$ (3.42a)
$\displaystyle\delta k^{ab}$
$\displaystyle=-\frac{1}{\ell}\bar{\varepsilon}\gamma^{ab}\psi-\frac{1}{\ell}\bar{\varepsilon}\gamma^{ab}\xi,$
(3.42b) $\displaystyle\delta V^{a}$
$\displaystyle=\bar{\varepsilon}\gamma^{a}\psi+\bar{\varepsilon}\gamma^{a}\xi,$
(3.42c) $\displaystyle\delta\psi$ $\displaystyle=0,$ (3.42d)
$\displaystyle\delta\xi$
$\displaystyle=d\varepsilon+\frac{1}{4}\omega^{ab}\gamma_{ab}\varepsilon+\frac{1}{2\ell}V^{a}\gamma_{a}\varepsilon+\frac{1}{4}k^{ab}\gamma_{ab}\varepsilon=D\varepsilon+\frac{1}{2\ell}V^{a}\gamma_{a}\varepsilon+\frac{1}{4}k^{ab}\gamma_{ab}{\varepsilon}.$
(3.42e)

If we now consider the variation of the action under the new supersymmetry-
like transformations, we get:

$\delta_{\text{susy-like}}S=-\frac{4C_{2}}{\ell^{2}}\int
R^{a}\left(\bar{\rho}+\bar{\Xi}\right)\gamma_{a}\gamma_{5}\varepsilon,$ (3.43)

which has the same form of (3.36), the only difference relying in the
parameter $\varepsilon$. Then, one can repeat the procedure described above,
obtaining that the action in the first order formalism is invariant under the
following new supersymmetry-like transformations:666Again, one could retrieve
the invariance also by modifying the whole
$\delta\left(\omega^{ab}+k^{ab}\right)$, obtaining
$\delta\left(\omega^{ab}+k^{ab}\right)=2\epsilon^{abcd}\left(\bar{\zeta}_{ec}\gamma_{d}\epsilon+\bar{\zeta}_{de}\gamma_{c}\gamma_{5}\epsilon-\bar{\zeta}_{cd}\gamma_{e}\gamma_{5}\epsilon\right)V^{e}-\frac{1}{\ell}\bar{\varepsilon}\gamma^{ab}\psi-\frac{1}{\ell}\bar{\varepsilon}\gamma^{ab}\xi$,
together with (3.44c), (3.44d), and (3.44e).

$\displaystyle\delta\omega^{ab}$
$\displaystyle=2\epsilon^{abcd}\left(\bar{\zeta}_{ec}\gamma_{d}\epsilon+\bar{\zeta}_{de}\gamma_{c}\gamma_{5}\epsilon-\bar{\zeta}_{cd}\gamma_{e}\gamma_{5}\epsilon\right)V^{e},$
(3.44a) $\displaystyle\delta k^{ab}$
$\displaystyle=-\frac{1}{\ell}\bar{\varepsilon}\gamma^{ab}\psi-\frac{1}{\ell}\bar{\varepsilon}\gamma^{ab}\xi,$
(3.44b) $\displaystyle\delta V^{a}$
$\displaystyle=\bar{\varepsilon}\gamma^{a}\psi+\bar{\varepsilon}\gamma^{a}\xi,$
(3.44c) $\displaystyle\delta\psi$ $\displaystyle=0,$ (3.44d)
$\displaystyle\delta\xi$
$\displaystyle=d\varepsilon+\frac{1}{4}\omega^{ab}\gamma_{ab}\varepsilon+\frac{1}{2\ell}V^{a}\gamma_{a}\varepsilon+\frac{1}{4}k^{ab}\gamma_{ab}\varepsilon=D\varepsilon+\frac{1}{2\ell}V^{a}\gamma_{a}\varepsilon+\frac{1}{4}k^{ab}\gamma_{ab}{\varepsilon},$
(3.44e)

being $\bar{\rho}+\bar{\Xi}=\bar{\zeta}_{ab}V^{a}V^{b}$. Also this
supersymmetry-like symmetry is not a gauge symmetry of the action. Further
investigation of this symmetry in the context of supergravity theories are
currently under analysis (work in progress).

##  4 Discussion

The Maxwell and $AdS$-Lorentz algebras of all types, together with their
supersymmetric extensions, have found interesting applications in
(super)gravity, although the new generators (and the consequent introduced
modifications) still require a clearer physical interpretation, particularly
concerning the presence of extra fermionic generators in the supersymmetric
cases (for some progress achieved in this context, see [55]).

In this paper, driven by the fact that from $AdS$-Lorentz type (super)algebras
one can introduce the cosmological term in (super)gravity in the presence of
an extra bosonic generator $Z_{ab}$ [28, 39], we have presented a new
supersymmetrization of the $AdS$-Lorentz algebra $AdS-\mathcal{L}_{4}$ of
[28]. Compared to previous literature, the novel superalgebra contains two
fermionic generators rather than one and, interestingly, it allows for non-
abelian charges in the underlying Maxwell algebra (see also [18, 20] for non-
abelian deformations of Maxwell (super)algebras). Specifically, our new
$AdS$-Lorentz superalgebra (that we called $\tilde{s}AdS-\mathcal{L}_{4}$)
possesses, by construction, a bosonic subalgebra (that is
$AdS-\mathcal{L}_{4}$) that does not contains any additional generator with
respect to the Maxwell algebra $\mathcal{M}$, recalled in (1.3), in such a way
that, when contracted, it directly reproduces the Maxwell algebra
$\mathcal{M}$. In this sense, $\tilde{s}AdS-\mathcal{L}_{4}$ could also be
viewed as the minimal supersymmetrization of a minimal Maxwell-like algebra in
which the bosonic generator $Z_{ab}$ is non-abelian
($\left[Z_{ab},Z_{cd}\right]=\eta_{bc}Z_{ad}-\eta_{ac}Z_{bd}-\eta_{bd}Z_{ac}+\eta_{ad}Z_{bc}$)
and where $\left[Z_{ab},P_{c}\right]=\eta_{bc}P_{a}-\eta_{ac}P_{b}$, even if
the generators $Z_{ab}$, in this case, do not behave as Lorentz generators
when considering the supersymmetric extension and the corresponding
commutation relations with the fermionic charges.

In particular, we have obtained $\tilde{s}AdS-\mathcal{L}_{4}$ as an
$S$-expansion of the superalgebra $\mathfrak{osp}(4|1)$, using the semigroup
described by Table 1, and, exploiting some peculiar and useful properties of
the abelian semigroup expansion method (on the same lines of [38, 39]), we
have shown that it allows to construct in a geometric way a $D=4$ supergravity
model involving a generalized supersymmetric cosmological term. The action we
have obtained with this procedure corresponds to a MacDowell-Mansouri like
action. Our result is a new supersymmetric extension of [28] involving two
fermionic generators. Moreover, we have shown that the final action (omitting
the boundary terms) can be rewritten as the Einstein-Hilbert and Rarita-
Schwinger action, plus the supersymmetric cosmological term, for a new
connection $\hat{\omega}^{ab}=\omega^{ab}+k^{ab}$ and gravitino
$\hat{\psi}=\psi+\xi$ (that is, a shifted connection and gravitino,
respectively). Then, the antisymmetry $k^{ab}=-k^{ba}$ implies that we are
dealing with an Einstein-Cartan geometry (determined by the structure of the
$\tilde{s}AdS-\mathcal{L}_{4}$ algebra and involving a redefinition of the
gravitino $1$-form field, since we are working in superspace) with vanishing
non-metricity.

Interestingly, in our model the bosonic $1$-form field $k^{ab}$ (associated to
$Z_{ab}$) and the spinor $1$-form field $\xi$ (associated to the fermionic
charge $\Sigma$) appear not only in the boundary terms but also in the bulk
Lagrangian (analogously to what happened in [39] and differently from what
happened in [56, 38]). In particular, the presence of the fields $k^{ab}$ and
$\xi$ in the boundary could be useful in the context of the $AdS$/CFT duality
(see [69, 70, 71, 72, 73, 74] and references therein). Interestingly, as shown
in [4], the introduction of a topological boundary in a bosonic action in
four-dimensions is equivalent to the holographic renormalization procedure
(for a review of the holographic renormalization, see, for example, [75]) in
the $AdS$/CFT context. Then, we conjecture that the presence of the fields
$k^{ab}$ and $\xi$ in the boundary would allow to regularize the supergravity
action in the holographic renormalization context (some work is in progress on
this point). On the other hand, at the purely supergravity level, it was
shown, adopting the so-called rheonomic (geometric) approach, that the
supersymmetry invariance of different supergravity actions in the presence of
a non-trivial boundary of spacetime can be recovered by adding appropriate
boundary terms, reproducing MacDowell-Mansouri like actions [62, 63, 64, 65].
Then, one could investigate the possibility of obtaining the action (3.14)
from the rheonomic approach adopted in [62, 63, 64, 65] in the presence of a
non-trivial boundary of spacetime, where we conjecture that the presence of
the fields $k^{ab}$ and $\xi$ in the boundary would allow to recover the
supersymmetry invariance of the action. These analyses could also shed some
light on the physical interpretation of the new generators and of the induced
modifications in the theory appearing when (Maxwell and) $AdS$-Lorentz type
algebras are considered.

It would also be interesting to carry on an analysis in $2+1$ dimensions (in
the context of Chern-Simons theories), on the same lines of [46], considering
the restriction to three dimensions of $\tilde{s}AdS-\mathcal{L}_{4}$.

On the other hand, a possible future development could consist in analyzing
$\mathcal{N}$-extended versions of $\tilde{s}AdS-\mathcal{L}_{4}$ and
constructing $\mathcal{N}$-extended supergravity models (also higher-
dimensional and matter-coupled ones) from the aforementioned
$\mathcal{N}$-extended superalgebras. In this context, the $S$-expansion
procedure could play an important role. In fact, as one can see from the
results we have obtained in this paper, the $S$-expansion procedure is not
only a useful mathematical method to derive new Lie (super)algebras but also a
powerful tool in order to construct, within a geometric formulation, a
(super)gravity action for an $S$-expanded (super)algebra. This can give rise
to many new extensions and generalizations of the results obtained here and in
the literature, as well as to a clearer understanding of the algebraic and
physical relations among different theories.

##  5 Acknowledgments

D.M.P. acknowledges DI-VRIEA for financial support through Proyecto
Postdoctorado 2018 VRIEAPUCV. The authors wish to thank P. Concha and E.
Rodríguez for the illuminating discussions.

This is a preprint of the article published in Eur. Phys. J. C 78 (2018)
no.11, 945. The final authenticated version is available online at:
https://doi.org/10.1140/epjc/s10052-018-6421-9.

##  6 Conventions and useful formulas

For the Minkowski metric we adopt the convention $\eta_{ab}\equiv(-1,1,1,1)$.
The Dirac gamma matrices in four spacetime dimensions are defined through
$\\{{\gamma_{a},\gamma_{b}}\\}=-2\eta_{ab}$ and obey the following relations:

$\begin{split}&[\gamma_{a},\gamma_{b}]=2\gamma_{ab},\;\gamma_{5}=-\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3},\;\gamma^{2}_{5}=-1,\;\\{{\gamma_{5},\gamma_{a}}\\}=0,\;[\gamma_{5},\gamma_{ab}]=0,\;\gamma_{ab}\gamma_{5}=-\frac{1}{2}\epsilon_{abcd}\gamma^{cd},\\\
&\gamma_{a}\gamma_{b}=\gamma_{ab}-\eta_{ab},\;\gamma^{ab}\gamma_{cd}=\epsilon^{ab}_{\;\;\;cd}\gamma_{5}-4\delta^{[a}_{\;\;[c}\gamma^{b]}_{\;\;d]}-2\delta^{ab}_{cd},\;\gamma^{ab}\gamma^{c}=2\gamma^{[a}\delta^{b]}_{c}-\epsilon^{abcd}\gamma_{5}\gamma_{d},\\\
&\gamma^{c}\gamma^{ab}=-2\gamma^{[a}\delta^{b]}_{c}-\epsilon^{abcd}\gamma_{5}\gamma_{d},\;\gamma_{m}\gamma^{ab}\gamma^{m}=0,\;\gamma_{ab}\gamma_{m}\gamma^{ab}=0,\;\gamma_{ab}\gamma_{cd}\gamma^{ab}=4\gamma_{cd},\;\gamma_{m}\gamma^{a}\gamma^{m}=-2\gamma^{a}.\end{split}$
(6.1)

Furthermore, we have

$(C\gamma_{a})^{T}=C\gamma_{a},\quad(C\gamma_{ab})^{T}=C\gamma_{ab},\quad(C\gamma_{5})^{T}=-C\gamma_{5},\quad(C\gamma_{5}\gamma_{a})^{T}=-C\gamma_{5}\gamma_{a},$
(6.2)

where $C$ ($C^{T}=-C$) is the charge conjugation matrix. For a generic
Majorana spinor $p$-form $\eta$ ($\bar{\eta}=\eta^{T}C$) and a generic
Majorana spinor $q$-form $\chi$ ($\bar{\chi}=\chi^{T}C$), the following
identities hold:

$\bar{\eta}\chi=(-1)^{pq}\bar{\chi}\eta,\quad\bar{\eta}S\chi=-(-1)^{pq}\bar{\chi}S\eta,\quad\bar{\eta}A\chi=(-1)^{pq}\bar{\chi}A\eta,$
(6.3)

where $S$ and $A$ are symmetric and a antisymmetric matrices, respectively.
Finally, in $\mathcal{N}=1$, $D=4$ we can write the following relevant Fierz
identities:

$\begin{split}&\psi\bar{\psi}=\frac{1}{2}\gamma_{a}\bar{\psi}\gamma^{a}\psi-\frac{1}{8}\gamma_{ab}\bar{\psi}\gamma^{ab}\psi,\\\
&\gamma_{a}\psi\bar{\psi}\gamma^{a}\psi=0,\\\
&\gamma_{ab}\psi\bar{\psi}\gamma^{ab}\psi=0,\\\
&\gamma_{ab}\psi\bar{\psi}\gamma^{a}\psi=\psi\bar{\psi}\gamma_{b}\psi.\end{split}$
(6.4)

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# A Monotone Function Given by a Low-Depth Decision Tree that is not an
Approximate Junta

Daniel M. Kane

##  1 Introduction

In [3], O’Donnell and Servedio show that any monotone function given by a
depth-$d$ decision tree can be learned to constant accuracy from random
samples in $\textrm{poly}(n,2^{d})$ time. The impact of this result is
somewhat lessened by an apparent lack of interesting monotone functions given
by low-depth decision trees. In particular, it was independently suggested by
Elad Verbin and by Rocco Servedio and Li-Yang Tan, that all such functions
might be approximated by functions on few variables (see [2], page 10).

######  Conjecture 1.

For every $\epsilon>0$ and every monotone function
$f:\\{0,1\\}^{n}\rightarrow\\{0,1\\}$ given by a depth-$d$ decision tree,
there is a $k$-junta, $g$, for $k=\textrm{poly}_{\epsilon}(d)$ so that $f$ and
$g$ agree on all but an $\epsilon$-fraction of inputs.

In this note, we disprove the above conjecture, and in particular provide an
example of a monotone low-degree function that is not well approximated by any
small junta. In particular we prove:

######  Theorem 2.

There exists a constant $\epsilon>0$ so that for every positive integer $d$,
there exists a $k=\exp(\Omega(\sqrt{d}))$ and a monotone function
$f:\\{0,1\\}^{n}\rightarrow\\{0,1\\}$ given by a depth-$d$ decision tree, so
that for every $k$-junta $g$, $f$ and $g$ disagree on at least an
$\epsilon$-fraction of inputs.

In fact it is known that the bound on $k$ in Theorem 2 is tight up to the
constant in the exponent. In particular, it is shown in [3] that any monotone
function given by a depth-$d$ decision tree has total influence
$I(f)=O(\sqrt{d})$. We combine this with the main result of [1], which says
that any boolean function $f$ can be $\epsilon$-approximated by a $k$-junta
for $k=\exp(O(I(f)/\epsilon)).$ Combining these results we find that:

######  Observation.

If $f$ is a monotone function given by a depth-$d$ decision tree, and if
$\epsilon>0$, then there is a $k$-junta $g$ that agrees with $f$ on all but an
$\epsilon$ fraction of inputs for $k=\exp(O(\sqrt{d}/\epsilon))$.

The function we construct to show Theorem 2 will combine ideas from two
previous constructions, the monotone addressing function and Talagrand’s
function.

The monotone addressing function is defined by

$f(x_{1},\ldots,x_{d-1},y_{0},\ldots,y_{2^{d-1}-1})=\begin{cases}1&\textrm{ if
}\sum x_{i}>\left\lfloor(d-1)/2\right\rfloor\\\ y_{x_{0}\ldots
x_{d-1}}&\textrm{ if }\sum x_{i}=\left\lfloor(d-1)/2\right\rfloor\\\
0&\textrm{ if }\sum x_{i}<\left\lfloor(d-1)/2\right\rfloor\end{cases}.$

This is an example of a monotone function given by a depth-$d$ decision tree
that depends on exponentially many variables, and thus provides us with a good
starting point. The monotone addressing function fails to provide a counter-
example to Conjecture 1 though since it agrees with the majority function
except on a set of measure $O(1/\sqrt{d})$.

Given the bound on the total sensitivity of a low-depth monotone function, we
know that any $f$ satisfying the conditions of Theorem 2 must not only have
near the maximum possible total influence for a low-depth monotone function,
but also must not be approximable by a function with much lower total
influence. Because of this restriction, our construction will look somewhat
similar to a construction of Talagrand in [4]. In particular, Talagrand
constructs a monotone function $f$ on $\\{0,1\\}^{d}$ so that on a constant
fraction of inputs, $f$ has sensitivity (i.e. the number of coordinates such
that changing the input at that coordinate would change the output of $f$)
$\Omega(\sqrt{d})$. Since, as is easily seen, the average sensitivity over all
inputs is equal to the total influence, this is as large as possible. On the
other hand, this condition tells us that $f$ retains large average sensitivity
even after ignoring any $\epsilon$-fraction of inputs for sufficiently small
constant $\epsilon$. Talagrand’s function fails to provide a counter-example
to Conjecture 1 on its own, because it is already a $d$-junta.

##  2 The Construction

In order to define the function $f$ with the properties specified by Theorem
2, we first introduce some background notation. We let $d,t$ and $m$ be
integers with $t=\Theta(\sqrt{d})$ and $m=\Theta(2^{t})$. We furthermore
assume that $2^{-t}m$ is sufficiently small given the value of $t/\sqrt{d}$.
We let $\mathcal{S}=(S_{1},\ldots,S_{m})$ be a random sequence of sets, where
the $S_{i}$ are chosen independently and uniformly from the set of subsets of
$\\{1,2,\ldots,d-1\\}$ of size exactly $t$. Given this $\mathcal{S}$, we
define the function $T_{\mathcal{S}}$ on $\\{0,1\\}^{d-1}$ as follows:

$T_{\mathcal{S}}(x_{1},\ldots,x_{d-1})=\\{1\leq i\leq m:x_{j}=1\textrm{ for
all }j\in S_{i}\\}.$

We will hereafter abbreviate $T$ by suppressing the explicit dependence on
$\mathcal{S}$, and abbreviate $(x_{1},\ldots,x_{d-1})$ by $x$.

We finally define $f$ as

$f_{\mathcal{S}}(x_{1},\ldots,x_{d-1},y_{1},\ldots,y_{m})=\begin{cases}1&\textrm{if
}|T(x)|\geq 2\\\ 0&\textrm{if }|T(x)|=0\\\ y_{i}&\textrm{if
}T(x)=\\{i\\}\end{cases}.$

Again, we will often suppress the dependence of $f$ on $\mathcal{S}$. It is
clear that $f$ is monotone. Furthermore, $f$ is given by a depth-$d$ decision
tree, since after fixing the values of the $x_{i}$, the value of $f$ depends
on at most one more coordinate. In the next Section, we show that $f$ cannot
be approximated by any $k$-junta for small $k$.

Note that Talagrand’s function is given (for appropriately chosen
$\mathcal{S}$) by

$G(x_{1},\ldots,x_{d-1})=\begin{cases}1&\textrm{if }|T(x)|\geq 1\\\
0&\textrm{if }|T(x)|=0\end{cases}.$

##  3 Approximation Bounds

Theorem 2 will follow from the following Proposition:

######  Proposition 3.

There exists an $\epsilon>0$ so that for $f_{\mathcal{S}}$ defined as above,
with constant probability over the choice of $\mathcal{S}$, $f$ is not
$\epsilon$-approximated by any $k$-junta for $k=o(2^{t})$.

A key step in our proof will be to show that with constant probability $f$
actually depends on one of the $y_{i}$.

######  Lemma 4.

With $T$ as above,

$\textrm{Pr}_{\mathcal{S},x}(|T_{\mathcal{S}}(x)|=1)=\Omega(1).$

###### Proof.

We will show the further claim that

$\mathbb{E}\left[|T_{\mathcal{S}}(x)|(2-|T_{\mathcal{S}}(x)|)\right]=\Omega(1).$
(1)

Since the term in the expectation is positive only if $|T|=1$, this will
complete our proof. We note that

$\displaystyle\mathbb{E}\left[|T_{\mathcal{S}}(x)|\right]$
$\displaystyle=\sum_{i=1}^{m}\textrm{Pr}(i\in T_{\mathcal{S}}(x))$
$\displaystyle=\sum_{i=1}^{m}\textrm{Pr}(x_{j}=1\textrm{ for all }j\in S_{i})$
$\displaystyle=m2^{-t}.$

On the other hand, we have that

$\displaystyle\mathbb{E}\left[|T_{\mathcal{S}}(x)|(|T_{\mathcal{S}}(x)|-1)\right]$
$\displaystyle=\sum_{i\neq j}\textrm{Pr}(i,j\in T_{\mathcal{S}}(x))$
$\displaystyle=\sum_{i\neq j}\textrm{Pr}(i\in
T_{\mathcal{S}}(x))\textrm{Pr}(j\in T_{\mathcal{S}}(x)|i\in
T_{\mathcal{S}}(x))$ $\displaystyle=\sum_{i\neq
j}2^{-t}\textrm{Pr}(x_{\ell}=1\textrm{ for all }\ell\in
S_{j}|x_{\ell}=1\textrm{ for all }\ell\in S_{i}).$

To compute this conditional probability we let
$S_{j}=\\{a_{1},\ldots,a_{t}\\}$ where the $a_{i}$ are picked randomly from
$\\{1,2,\ldots,d-1\\}$ without replacement. We compute it as the product

$\prod_{k=1}^{t}\textrm{Pr}(x_{a_{k}}=1|x_{a_{1}}=\ldots=x_{a_{k-1}}=1\textrm{
and }x_{\ell}=1\textrm{ for all }\ell\in S_{i}).$

These probabilities are approximated by first fixing the values of $S_{i}$ and
$a_{1},\ldots,a_{k-1}$. After additionally fixing the value of $a_{k}$, the
probability in question becomes $1$ if $a_{k}\in S_{i}$ and $1/2$ otherwise.
Thus the probability that $x_{a_{r}}=1$ is

$(1+\textrm{Pr}(a_{r}\in
S_{i}))/2=\left(1+\frac{|S_{i}\backslash\\{a_{1},\ldots,a_{r-1}\\}|}{d-r}\right)/2=1/2+O(t/d).$

Hence the probability that $j\in T_{\mathcal{S}}(x)$ given that $i\in
T_{\mathcal{S}}(x)$ is

$(1/2+O(t/d))^{t}=2^{-t}\exp(O(t^{2}/d)).$

Therefore, we have that

$\mathbb{E}\left[|T_{\mathcal{S}}(x)|(|T_{\mathcal{S}}(x)|-1)\right]=\sum_{i\neq
j}2^{-2t}\exp(O(t^{2}/d))\leq(2^{-t}m)^{2}\exp(O(t^{2}/d)).$

Therefore, we have that

$\displaystyle\mathbb{E}\left[|T_{\mathcal{S}}(x)|(2-|T_{\mathcal{S}}(x)|)\right]$
$\displaystyle=\mathbb{E}\left[|T_{\mathcal{S}}(x)|\right]-\mathbb{E}\left[|T_{\mathcal{S}}(x)|(|T_{\mathcal{S}}(x)|-1)\right]$
$\displaystyle\geq(2^{-t}m)-(2^{-t}m)^{2}\exp(O(t^{2}/d))$
$\displaystyle=(2^{-t}m)\left(1-(2^{-t}m)\exp(O(t^{2}/d))\right).$

As long as $2^{-t}m$ is bounded below by a constant and above by
$\exp(-O(t^{2}/d))/2$, this is $\Omega(1)$. ∎

We are now ready to prove Proposition 3. By Lemma 4, we note that with
constant probability over $\mathcal{S}$, that
$\textrm{Pr}_{x}(|T(x)|=1)=\Omega(1)$. For such $\mathcal{S}$, we claim that
$f$ has the desired property. In particular we claim the following:

######  Lemma 5.

If $f$ is as above and $g$ is a $k$-junta, then

$\textrm{Pr}(f(x,y)\neq
g(x,y))\geq\frac{\textrm{Pr}_{x}(|T(x)|=1)-k2^{-t}}{2}.$

###### Proof.

This follows from the simple observation that if, after fixing the value of
$x$, we have that $T=\\{i\\}$ where $g$ does not depend on $y_{i}$, then
$\textrm{Pr}_{y}(f(x,y)\neq g(x,y))=1/2$. This is because after further
conditioning on the values of all $y_{j}$ for $j\neq i$, $g$ becomes a
constant function (by assumption) and $f$ takes the values $0$ and $1$ each
with probability $1/2$. Therefore we have that

Pr $\displaystyle(f(x,y)\neq g(x,y))$
$\displaystyle\geq\frac{\textrm{Pr}(T(x)=\\{i\\}\textrm{ for some }i,\textrm{
and }g\textrm{ does not depend on }y_{i})}{2}$
$\displaystyle=\frac{\textrm{Pr}(|T(x)|=1)-\textrm{Pr}(T(x)=\\{i\\}\textrm{
for some }i,\textrm{ and }g\textrm{ depends on }y_{i})}{2}$
$\displaystyle=\frac{\textrm{Pr}(|T(x)|=1)-\sum_{i:g\textrm{ depends on
}y_{i}}\textrm{Pr}(T(x)=\\{i\\})}{2}$
$\displaystyle\geq\frac{\textrm{Pr}(|T(x)|=1)-\sum_{i:g\textrm{ depends on
}y_{i}}\textrm{Pr}(i\in T(x))}{2}$
$\displaystyle=\frac{\textrm{Pr}(|T(x)|=1)-\sum_{i:g\textrm{ depends on
}y_{i}}2^{-t}}{2}$
$\displaystyle\geq\frac{\textrm{Pr}_{x}(|T(x)|=1)-k2^{-t}}{2}.$

∎

Proposition 3 and Theorem 2 now follow immediately.

## Acknowledgements

I would like to thank Ryan O’Donnell for making me aware of this problem, and
for his help with finding appropriate references for this paper. This research
was done with the support of an NSF postdoctoral fellowship.

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