$\mathcal{N}$-extended Chern-Simons Carrollian supergravities in $2+1$
spacetime dimensions

Farhad Ali∗, Lucrezia Ravera⋆

∗Department of Mathematics, Kohat University of Science and Technology,
26000 Kohat KPK, Pakistan.
⋆INFN, Sezione di Milano,
Via Celoria 16, I-20133 Milano, Italy.
<EMAIL_ADDRESS><EMAIL_ADDRESS>

In this work we present the ultra-relativistic $\mathcal{N}$-extended AdS
Chern-Simons supergravity theories in three spacetime dimensions invariant
under $\mathcal{N}$-extended AdS Carroll superalgebras. We first consider the
$(2,0)$ and $(1,1)$ cases; subsequently, we generalize our analysis to
$\mathcal{N}=(\mathcal{N},0)$, with $\mathcal{N}$ even, and to
$\mathcal{N}=(p,q)$, with $p,q>0$. The $\mathcal{N}$-extended AdS Carroll
superalgebras are obtained through the Carrollian (i.e., ultra-relativistic)
contraction applied to an $so(2)$ extension of
$\mathfrak{osp}(2|2)\otimes\mathfrak{sp}(2)$, to
$\mathfrak{osp}(2|1)\otimes\mathfrak{osp}(2,1)$, to an
$\mathfrak{so}(\mathcal{N})$ extension of
$\mathfrak{osp}(2|\mathcal{N})\otimes\mathfrak{sp}(2)$, and to the direct sum
of an $\mathfrak{so}(p)\oplus\mathfrak{so}(q)$ algebra and
$\mathfrak{osp}(2|p)\otimes\mathfrak{osp}(2,q)$, respectively. We also analyze
the flat limit ($\ell\rightarrow\infty$, being $\ell$ the length parameter) of
the aforementioned $\mathcal{N}$-extended Chern-Simons AdS Carroll
supergravities, in which we recover the ultra-relativistic
$\mathcal{N}$-extended (flat) Chern-Simons supergravity theories invariant
under $\mathcal{N}$-extended super-Carroll algebras. The flat limit is applied
at the level of the superalgebras, Chern-Simons actions, supersymmetry
transformation laws, and field equations.

##  1 Introduction

The study of spacetime symmetries has proved to be fundamental for analyzing
and understanding various physical models. For instance, think of Newtonian
gravity, Maxwell’s electromagnetism, special and general relativity, string
and supergravity theory. Most of these theories are based on relativistic
symmetries. On the other hand, during the years models with non-relativistic
symmetries have also been developed and analyzed, and and are still the
subject of in-depth studies.

In this context, Carroll symmetries [1, 2], arising when the velocity of light
is sent to zero (i.e., in the ultra-relativistic limit, $c\rightarrow 0$),
have attracted some interest over recent years. In fact, models with Carroll
symmetries occurred in the literature in the study of tachyon condensation
[3], warped conformal field theories [4], and tensionless strings [5, 6, 7, 8,
9]. Moreover, a study exploring the Carroll limit corresponding to M2- as well
as M3-branes propagating over $D=11$ supergravity backgrounds in M-theory has
been recently presented in [10].

Concerning gravity theories, models of Carrollian (i.e., ultra-relativistic)
gravity have been developed and analyzed in [11, 12, 13]. In particular, non-
and ultra-relativistic Chern-Simons (CS) type actions in $2+1$ dimensions were
constructed in [12], where the authors included a spin-$3$ field coupled to
gravity. The geometry of flat and Anti-de Sitter (AdS) Carroll spaces were
investigated, both in the bosonic as well as in the supersymmetric case, in
[14], were the authors focused on the symmetries of a particle moving in such
spaces. Afterwards, in [15], the AdS Carroll CS gravity theory was discussed
for the first time.111Here let us also mention that the geometric realization
of the (A)dS Carroll algebra corresponds to a null surface of (A)dS space (see
Refs. [16, 17]).

Non-relativistic symmetry groups play a remarkable role also in holography
[18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. More
specifically, in the work [24], connections among Carrollian physics,
holography of flat space, and the Bondi-Metzner-Sachs (BMS) algebra were
discovered and followed up in [25] (see also Refs. [26, 27] and [28]).
Besides, conformal extensions of the Carroll group were explored and related
to the BMS group in [29, 30], while in [31, 32, 33, 34] it was shewed the way
in which Carrollian structures and geometry emerge in the flat holography and
fluid/gravity correspondence framework.

Recently, in [35] the construction of the three-dimensional $\mathcal{N}=1$ CS
supergravity theory invariant under the so-called AdS Carroll superalgebra
(ultra-relativistic contraction of the $\mathcal{N}=1$ AdS superalgebra, see
[14]), together with the study of its flat limit, has been presented for the
first time. In the study done in [35], the method introduced in [36] was
adopted. In [36] a generalization of the standard Inönü-Wigner (IW)
contraction [37, 38] was presented, consisting in rescaling not only the
generators of a Lie (super)algebra but also the arbitrary constants appearing
in the components of the invariant tensor; the latter is the key ingredient
for formulating an action à la CS, that is invariant under the IW contracted
(super)algebra by construction. This procedure was further improved in [35] by
considering dimensionful generators from the very beginning, on the same lines
of [39]. As shown in the same paper, this allows to obtain, for instance, the
Poincaré limit from the $\mathfrak{osp}(2|2)\otimes\mathfrak{sp}(2)$ CS
supergravity action directly considering the flat limit
$\ell\rightarrow\infty$, being $\ell$ the length parameter related to the
cosmological constant.

Moving to higher $\mathcal{N}$, we have that the $\mathcal{N}=2$
supersymmetric extensions of the Poincaré and AdS algebras are not unique, and
can be subdivided into two inequivalent classes: The $(2,0)$ and the $(1,1)$
cases. Here we mention that, in the extension to the $(p,q)$ case, when either
$p$ or $q$ is greater than one some subtleties arise. Indeed, even though the
$(p,q)$ Poincaré superalgebra can be derived as an IW contraction of the
$(p,q)$ AdS superalgebra, the Poincaré limit applied at the level of the (CS)
action requires to enlarge the AdS superalgebra, considering, in particular, a
direct sum of an $\mathfrak{so}(p)\oplus\mathfrak{so}(q)$ algebra and the
$(p,q)$ AdS superalgebra $\mathfrak{osp}(2|p)\otimes\mathfrak{osp}(2,q)$ [40,
41, 42]; this is related to the fact that, as it was proven in [40], the semi-
direct extension of the $\mathfrak{so}(p)\oplus\mathfrak{so}(q)$ automorphism
algebra by the $(p,q)$ Poincaré superalgebra allows to produce a non-
degenerate invariant tensor which is used to construct a well-defined three-
dimensional CS $(p,q)$ Poincaré supergravity theory (in particular, when
either $p$ or $q$ is greater than one, it is not possible to obtain a non-
degenerate invariant tensor without considering this extension).

It is well assumed that a (super)gravity theory in three spacetime dimensions
can be described by a CS action as a gauge theory, providing a useful toy
model to approach higher-dimensional models [43, 44, 45, 46, 47, 48, 49, 50,
51]. In the last decades, diverse three-dimensional supergravity theories have
been studied, and, in this context, there has also been a growing interest to
extend AdS and Poincaré supergravity theories to other symmetries (see [39,
52] and references therein).

In the present work, we apply the method of [36] with the improvements of [35]
to develop in a systematic way the ultra-relativistic $\mathcal{N}$-extended
AdS CS supergravity theories in three (that is $2+1$) spacetime dimensions
invariant under $\mathcal{N}$-extended AdS Carroll superalgebras. In
particular, we will distinguish between the two $\mathcal{N}$-extended cases
$\mathcal{N}=(\mathcal{N},0)$ and $\mathcal{N}=(p,q)$, generalizing the
results presented at the algebraic level in [14] and also developing the
associated CS supergravity theories in three dimensions. More specifically, we
start by considering the $(2,0)$ and $(1,1)$ cases, and then generalize our
analysis to $\mathcal{N}=(\mathcal{N},0)$, with $\mathcal{N}$ even, and to
$\mathcal{N}=(p,q)$, that is $\mathcal{N}=p+q$, with $p,q>0$.222We concentrate
on the $\mathcal{N}=(\mathcal{N},0)$ case with $\mathcal{N}$ even in order to
reproduce a well-defined ultra-relativistic limit at the supersymmetric level
(see also [53] and references therein, which deals with similar situations but
in the non-relativistic limit $c\rightarrow\infty$). The
$\mathcal{N}$-extended AdS Carroll superalgebras are obtained through the
Carrollian (i.e., ultra-relativistic) contraction applied to an $so(2)$
extension of $\mathfrak{osp}(2|2)\otimes\mathfrak{sp}(2)$, to
$\mathfrak{osp}(2|1)\otimes\mathfrak{osp}(2,1)$, to an
$\mathfrak{so}(\mathcal{N})$ extension of
$\mathfrak{osp}(2|\mathcal{N})\otimes\mathfrak{sp}(2)$ (with $\mathcal{N}$
even), and to the direct sum of an $\mathfrak{so}(p)\oplus\mathfrak{so}(q)$
algebra and $\mathfrak{osp}(2|p)\otimes\mathfrak{osp}(2,q)$, respectively. Let
us mention that the $\mathcal{N}=(\mathcal{N},0)$ case (and thus also the
$\mathcal{N}=(2,0)$ one) will be more subtle, since it will require the
definitions of new supersymmetry generators in order to properly study the
Carroll limit, on the same lines of what was done in [14] (see also [53],
which deals with non-relativistic superalgebras, and references therein). The
ultra-relativistic $\mathcal{N}$-extended AdS Carroll supergravity actions are
constructed à la CS, by exploiting the non-vanishing components of the
corresponding invariant tensor. The aforementioned actions are all based on a
non-degenerate, invariant bilinear form (i.e., an invariant metric). The
results we have obtained in the present work were also open problems suggested
in Ref. [14], and represent $\mathcal{N}$-extended generalizations of [35].

Subsequently, we study the flat limit ($\ell\rightarrow\infty$, being $\ell$
the length parameter) of the aforesaid $\mathcal{N}$-extended CS AdS Carroll
supergravities, in which we recover the ultra-relativistic
$\mathcal{N}$-extended (flat) CS supergravity theories invariant under
$\mathcal{N}$-extended super-Carroll algebras. The flat limit is applied at
the level of the superalgebras, CS actions, supersymmetry transformation laws,
and field equations.

The remain of the paper is organized as follows: In Section 2, we first
introduce a new $\mathcal{N}=(2,0)$ AdS Carroll superalgebra, which is
obtained as the ultra-relativistic contraction of an $\mathfrak{so}(2)$
extension of $\mathfrak{osp}(2|2)\otimes\mathfrak{sp}(2)$. Here, the
$\mathfrak{so}(2)$ extension is necessary in order to end up with an invariant
non-degenerate inner product in the ultra-relativistic limit, providing a
well-defined CS action. In fact, this allows us to subsequently develop the
three-dimensional CS supergravity action invariant under the $(2,0)$ AdS
Carroll superalgebra, which we call $(2,0)$ CS AdS Carroll supergravity in
$2+1$ dimensions. In Section 3, we repeat the same analysis for the $(1,1)$
case, ending up with the CS supergravity action invariant under the $(1,1)$
AdS Carroll superalgebra. Subsequently, we generalize our study to the cases
of $\mathcal{N}=(\mathcal{N},0)$, with $\mathcal{N}$ even, and
$\mathcal{N}=(p,q)$, with $p,q>0$, respectively in Sections 4 and 5. In
Section 6, we discuss the flat limit $\ell\rightarrow\infty$ of the
$\mathcal{N}$-extended CS AdS Carroll supergravities introduced in the
previous part of the work. Section 7 contains some final comments and remarks.

##  2 $(2,0)$ AdS Carroll supergravity in $2+1$ dimensions

In this section, we first introduce a new $\mathcal{N}=(2,0)$ AdS Carroll
superalgebra, which is obtained as the ultra-relativistic contraction of an
$\mathfrak{so}(2)$ extension of $\mathfrak{osp}(2|2)\otimes\mathfrak{sp}(2)$.
The $\mathfrak{so}(2)$ extension of
$\mathfrak{osp}(2|2)\otimes\mathfrak{sp}(2)$ is needed in order to end up with
an invariant non-degenerate inner product in the ultra-relativistic limit,
namely with a well-defined invariant tensor (this is reminescent of what was
done in [40] in the case of relativistic theories), in such a way to be able
to construct a well-defined CS action. Indeed, this allows us to subsequently
develop the three-dimensional CS supergravity action invariant under this
$\mathcal{N}=(2,0)$ AdS Carroll superalgebra, which we call $(2,0)$ CS AdS
Carroll supergravity.

Let us mention, here, that a $\mathcal{N}=(2,0)$ AdS Carroll superalgebra has
been first introduced in [14]. Nevertheless, due to the degeneracy of the
invariant tensor for that superalgebra, one could not construct a well-defined
CS action in that case. On the other hand, as we will see in the following,
our $\mathcal{N}=(2,0)$ AdS Carroll superalgebra will be different from the
one presented in [14], allowing, in particular, the formulation of a three-
dimensional CS action in the supergravity context.

###  2.1 $\mathcal{N}=(2,0)$ AdS Carroll superalgebra

An $\mathcal{N}=(2,0)$ supersymmetric extension of the AdS Carroll algebra was
obtained in [14] as the ultra-relativistic contraction of
$\mathfrak{osp}(2|2)\otimes\mathfrak{sp}(2)$, the latter being generated by
the set
$\\{\tilde{J}_{AB},\tilde{P}_{A},\tilde{Z}^{ij},\tilde{Q}^{i}_{\alpha}\\}$,
with $A,B,\ldots=0,1,2$, $\alpha=1,2$, and $i=1,2$, where $\tilde{J}_{AB}$ are
the Lorentz generators, $\tilde{P}_{A}$ represent the spacetime translations,
$\tilde{Z}^{ij}=\epsilon^{ij}\tilde{Z}$ are internal symmetry generators, and
$\tilde{Q}^{i}_{\alpha}$ are the supersymmetry generators ($2$-components
Majorana spinor charges). The (anti)commutation relations of
$\mathfrak{osp}(2|2)\otimes\mathfrak{sp}(2)$ read as follows:

$\begin{split}&\left[\tilde{J}_{AB},\tilde{J}_{CD}\right]=\eta_{BC}\tilde{J}_{AD}-\eta_{AC}\tilde{J}_{BD}-\eta_{BD}\tilde{J}_{AC}+\eta_{AD}\tilde{J}_{BC}\,,\\\
&\left[\tilde{J}_{AB},\tilde{P}_{C}\right]=\eta_{BC}\tilde{P}_{A}-\eta_{AC}\tilde{P}_{B}\,,\\\
&\left[\tilde{P}_{A},\tilde{P}_{B}\right]=\frac{1}{\ell^{2}}\tilde{J}_{AB}\,,\\\
&\left[\tilde{J}_{AB},\tilde{Q}^{i}_{\alpha}\right]=-\frac{1}{2}\left(\Gamma_{AB}\tilde{Q}^{i}\right)_{\alpha}\,,\quad\quad\left[\tilde{P}_{A},\tilde{Q}^{i}_{\alpha}\right]=-\frac{1}{2\ell}\left(\Gamma_{A}\tilde{Q}^{i}\right)_{\alpha}\,,\\\
&\left[\tilde{Z},\tilde{Q}^{i}_{\alpha}\right]=-\epsilon^{ij}\tilde{Q}^{j}_{\alpha}\,,\\\
&\left\\{\tilde{Q}^{i}_{\alpha},\tilde{Q}^{j}_{\beta}\right\\}=\delta^{ij}\left[-\frac{1}{2\ell}\left(\Gamma^{AB}C\right)_{\alpha\beta}\tilde{J}_{AB}+\left(\Gamma^{A}C\right)_{\alpha\beta}\tilde{P}_{A}\right]+\frac{1}{\ell}\epsilon^{ij}C_{\alpha\beta}\tilde{Z}\,,\end{split}$
(2.1)

where $\ell$ is a length parameter, $C$ is the charge conjugation matrix, and
$\Gamma_{A}$ and $\Gamma_{AB}$ represent the Dirac matrices in three
dimensions. The generators $\tilde{J}_{AB}$, $\tilde{P}_{A}$,
$\tilde{Z}^{ij}$, and $\tilde{Q}^{i}_{\alpha}$ have a dual description in
terms of $1$-form fields, $\tilde{\omega}^{AB}$ (spin connection),
$\tilde{V}^{A}$ (vielbein), $\tilde{z}_{ij}$ ($1$-form field dual to the
generator $\tilde{Z}^{ij}$), and $\tilde{\psi}_{i}^{\alpha}$ (gravitinos),
respectively.

Here, we consider the ultra-relativistic contraction of an $\mathfrak{so}(2)$
extension of $\mathfrak{osp}(2|2)\otimes\mathfrak{sp}(2)$, involving an extra
generator $\tilde{S}^{ij}=\epsilon^{ij}\tilde{S}$. This will allow the
formulation of a well-defined ultra-relativistic CS action, based on a non-
degenerate invariant tensor, which would not be possible by considering the
$\mathcal{N}=(2,0)$ superalgebra of [14]. In particular, we extend (2.1) by
adding the extra $\tilde{S}$ generator and we perform, on the same lines of
[40], the redefinition

$\tilde{T}\equiv\tilde{Z}-\ell\tilde{S}\,,$ (2.2)

to eliminate $\tilde{Z}$ in favour of $\tilde{T}$ (this redefinition is
particularly convenient for discussing the flat limit, see also [40]).
Consequently, we rewrite the (anti)commutation relations (2.1) as follows (we
consider dimensionful generators from the very beginning, on the same lines of
[39]):333We use the metric $\eta_{AB}$ with the signature $(-,+,+)$.

$\begin{split}&\left[\tilde{J}_{AB},\tilde{J}_{CD}\right]=\eta_{BC}\tilde{J}_{AD}-\eta_{AC}\tilde{J}_{BD}-\eta_{BD}\tilde{J}_{AC}+\eta_{AD}\tilde{J}_{BC}\,,\\\
&\left[\tilde{J}_{AB},\tilde{P}_{C}\right]=\eta_{BC}\tilde{P}_{A}-\eta_{AC}\tilde{P}_{B}\,,\\\
&\left[\tilde{P}_{A},\tilde{P}_{B}\right]=\frac{1}{\ell^{2}}\tilde{J}_{AB}\,,\\\
&\left[\tilde{J}_{AB},\tilde{Q}^{i}_{\alpha}\right]=-\frac{1}{2}\left(\Gamma_{AB}\tilde{Q}^{i}\right)_{\alpha}\,,\quad\quad\left[\tilde{P}_{A},\tilde{Q}^{i}_{\alpha}\right]=-\frac{1}{2\ell}\left(\Gamma_{A}\tilde{Q}^{i}\right)_{\alpha}\,,\\\
&\left[\tilde{T},\tilde{Q}^{i}_{\alpha}\right]=-\epsilon^{ij}\tilde{Q}^{j}_{\alpha}\,,\\\
&\left\\{\tilde{Q}^{i}_{\alpha},\tilde{Q}^{j}_{\beta}\right\\}=\delta^{ij}\left[-\frac{1}{2\ell}\left(\Gamma^{AB}C\right)_{\alpha\beta}\tilde{J}_{AB}+\left(\Gamma^{A}C\right)_{\alpha\beta}\tilde{P}_{A}\right]+\epsilon^{ij}C_{\alpha\beta}\left(\frac{1}{\ell}\tilde{T}+\tilde{S}\right)\,.\end{split}$
(2.3)

Note that, in the flat limit $\ell\rightarrow\infty$, $\tilde{S}$ becomes the
central element of the $\mathcal{N}=(2,0)$ Poincaré superalgebra extended with
the extra $so(2)$ generator $\tilde{T}$ (see [40]).

The non-vanishing components of an invariant tensor for the superalgebra
(2.3), which will be useful in the sequel, are given by

$\begin{split}&\langle\tilde{J}_{AB}\tilde{J}_{CD}\rangle=\alpha_{0}\left(\eta_{AD}\eta_{BC}-\eta_{AC}\eta_{BD}\right)\,,\\\
&\langle\tilde{J}_{AB}\tilde{P}_{C}\rangle=\alpha_{1}\epsilon_{ABC}\,,\\\
&\langle\tilde{P}_{A}\tilde{P}_{B}\rangle=\frac{\alpha_{0}}{\ell^{2}}\eta_{AB}\,,\\\
&\langle\tilde{T}\tilde{T}\rangle=-2\alpha_{0}\,,\\\
&\langle\tilde{T}\tilde{S}\rangle=2\alpha_{1}\,,\\\
&\langle\tilde{S}\tilde{S}\rangle=-\frac{2\alpha_{1}}{\ell}\,,\\\
&\langle\tilde{Q}^{i}_{\alpha}\tilde{Q}^{j}_{\beta}\rangle=2\left(\alpha_{1}-\frac{\alpha_{0}}{\ell}\right)C_{\alpha\beta}\delta^{ij}\,,\end{split}$
(2.4)

where $\epsilon_{ABC}$ is the Levi-Civita symbol in three dimensions and
$\alpha_{0}$ and $\alpha_{1}$ are arbitrary constants.

To take the Carrollian (i.e., ultra-relativistic) contraction of the
superalgebra (2.3), we decompose the indices $A,B,\ldots$ as

$A\rightarrow(0,a)\,,\quad a=1,2\,.$ (2.5)

This first step induces the following decomposition of the generators:

$\tilde{J}_{AB}\rightarrow\\{\tilde{J}_{ab},\tilde{J}_{a0}\equiv\tilde{K}_{a}\\}\,,\quad\tilde{P}_{A}\rightarrow\\{\tilde{P}_{a},\tilde{P}_{0}\equiv\tilde{H}\\}\,.$
(2.6)

We also have

$\Gamma_{AB}\rightarrow\\{\Gamma_{ab},\Gamma_{a0}\\}\,,\quad\Gamma_{A}\rightarrow\\{\Gamma_{a},\Gamma_{0}\\}\,.$
(2.7)

Furthermore, we define new supersymmetry charges by

$\tilde{Q}^{\pm}_{\alpha}=\frac{1}{\sqrt{2}}\left(\tilde{Q}^{1}_{\alpha}\pm\left(\Gamma_{0}\right)_{\alpha\beta}\tilde{Q}^{2}_{\beta}\right)\,{,}$
(2.8)

on the same lines of what was done in [54] (see also references therein) in
the study of supersymmetric non-relativistic models. Then, we rescale the
generators with a parameter $\sigma$ as follows:

$\tilde{H}\rightarrow\sigma H\,,\quad\tilde{K}_{a}\rightarrow\sigma
K_{a}\,,\quad\tilde{S}\rightarrow\sigma
S\,,\quad\tilde{Q}^{\pm}_{\alpha}\rightarrow\sqrt{\sigma}Q^{\pm}_{\alpha}\,.$
(2.9)

Taking the limit $\sigma\rightarrow\infty$444Let us highlight that
$\sigma\rightarrow\infty$ corresponds to the limit
$\frac{1}{c}\rightarrow\infty$ (where $c$ denotes the velocity of light), that
is $c\rightarrow 0$ (ultra-relativistic limit). and removing the tilde symbol
also on the generators that we have not rescaled, we end up with a new
$\mathcal{N}=(2,0)$ AdS Carroll superalgebra (differing from the one of [14],
due to the presence of the generator $S$), whose non-trivial (anti)commutation
relations read as follows:

$\begin{split}&\left[K_{a},J_{bc}\right]=\delta_{ab}K_{c}-\delta_{ac}K_{b}\,,\\\
&\left[J_{ab},P_{c}\right]=\delta_{bc}P_{a}-\delta_{ac}P_{b}\,,\\\
&\left[K_{a},P_{b}\right]=-\delta_{ab}H\,,\\\
&\left[P_{a},P_{b}\right]=\frac{1}{\ell^{2}}J_{ab}\,,\\\
&\left[P_{a},H\right]=\frac{1}{\ell^{2}}K_{a}\,,\\\
&\left[J_{ab},Q^{\pm}_{\alpha}\right]=-\frac{1}{2}\left(\Gamma_{ab}Q^{\pm}\right)_{\alpha}\,,\\\
&\left[P_{a},Q^{\pm}_{\alpha}\right]=-\frac{1}{2\ell}\left(\Gamma_{a}Q^{\mp}\right)_{\alpha}\,,\\\
&\left[T,Q^{+}_{\alpha}\right]=\left(\Gamma_{0}\right)_{\alpha\beta}Q^{+}_{\beta}\,,\\\
&\left[T,Q^{-}_{\alpha}\right]=-\left(\Gamma_{0}\right)_{\alpha\beta}Q^{-}_{\beta}\,,\\\
&\left\\{Q^{+}_{\alpha},Q^{+}_{\beta}\right\\}=\left(\Gamma^{0}C\right)_{\alpha\beta}\left(H+S\right)\,,\\\
&\left\\{Q^{+}_{\alpha},Q^{-}_{\beta}\right\\}=-\frac{1}{\ell}\left(\Gamma^{a0}C\right)_{\alpha\beta}K_{a}\,,\\\
&\left\\{Q^{-}_{\alpha},Q^{-}_{\beta}\right\\}=\left(\Gamma^{0}C\right)_{\alpha\beta}\left(H-S\right)\,.\end{split}$
(2.10)

We will now construct a CS supergravity action in three dimensions invariant
under the $\mathcal{N}=(2,0)$ AdS Carroll superalgebra (2.10): The $(2,0)$ AdS
Carroll CS supergravity action.

###  2.2 $(2,0)$ AdS Carroll supergravity action

The general form of a three-dimensional CS action is given by

$I_{CS}=\frac{k}{4\pi}\int_{\mathcal{M}}\Big{\langle}AdA+\frac{2}{3}A^{3}\Big{\rangle}=\frac{k}{4\pi}\int_{\mathcal{M}}\Big{\langle}AdA+\frac{1}{3}A\left[A,A\right]\Big{\rangle}\,,$
(2.11)

where $k=1/(4G)$ is the CS level of the theory,555For gravitational theories,
the CS level is related to the gravitational constant $G$. $A$ is the gauge
connection $1$-form, and $\langle\ldots\rangle$ denotes the invariant tensor.
The integral in (2.11) is over a three-dimensional manifold
$\mathcal{M}$.666In the sequel, we will omit the wedge product “$\wedge$”
between differential forms. The CS action (2.11) can also be rewritten as

$I_{CS}=\frac{k}{4\pi}\int_{\mathcal{M}}\Big{\langle}AF-\frac{1}{3}A^{3}\Big{\rangle}\,,$
(2.12)

in terms of the curvature $2$-form
$F=dA+A^{2}=dA+\frac{1}{2}\left[A,A\right]$.

In the case of the $\mathcal{N}=(2,0)$ AdS Carroll superalgebra (2.10), the
connection $1$-form reads777Here and in the following, for simplicity, we will
omit the spinor index $\alpha$.

$A=\frac{1}{2}\omega^{ab}J_{ab}+k^{a}K_{a}+V^{a}P_{a}+hH+tT+sS+\psi^{+}Q^{+}+\psi^{-}Q^{-}\,,$
(2.13)

where $\omega^{ab}$, $k^{a}$, $V^{a}$, $h$, $t$, $s$, $\psi^{+}$, and
$\psi^{-}$ are the $1$-form fields dual to the generators $J_{ab}$, $K_{a}$,
$P_{a}$, $H$, $T$, $S$, $Q^{+}$, and $Q^{-}$, respectively.

The corresponding curvature $2$-form $F$ is

$F=\frac{1}{2}\mathcal{R}^{ab}J_{ab}+\mathcal{K}^{a}K_{a}+R^{a}P_{a}+\mathcal{H}H+\mathcal{T}T+\mathcal{S}S+\nabla\psi^{+}Q^{+}+\nabla\psi^{-}Q^{-}\,,$
(2.14)

with

$\begin{split}\mathcal{R}^{ab}&=d\omega^{ab}+\frac{1}{\ell^{2}}V^{a}V^{b}=R^{ab}+\frac{1}{\ell^{2}}V^{a}V^{b}\,,\\\
\mathcal{K}^{a}&=dk^{a}+\omega^{a}_{\phantom{a}b}k^{b}+\frac{1}{\ell^{2}}V^{a}h+\frac{1}{\ell}\bar{\psi}^{+}\Gamma^{a0}\psi^{-}=\mathfrak{K}^{a}+\frac{1}{\ell^{2}}V^{a}h+\frac{1}{\ell}\bar{\psi}^{+}\Gamma^{a0}\psi^{-}\,,\\\
R^{a}&=dV^{a}+\omega^{a}_{\phantom{a}b}V^{b}\,,\\\
\mathcal{H}&=dh+V^{a}k_{a}-\frac{1}{2}\bar{\psi}^{+}\Gamma^{0}\psi^{+}-\frac{1}{2}\bar{\psi}^{-}\Gamma^{0}\psi^{-}=\mathfrak{H}-\frac{1}{2}\bar{\psi}^{+}\Gamma^{0}\psi^{+}-\frac{1}{2}\bar{\psi}^{-}\Gamma^{0}\psi^{-}\,,\\\
\mathcal{T}&=dt\,,\\\
\mathcal{S}&=ds-\frac{1}{2}\bar{\psi}^{+}\Gamma^{0}\psi^{+}+\frac{1}{2}\bar{\psi}^{-}\Gamma^{0}\psi^{-}\,,\\\
\nabla\psi^{+}&=d\psi^{+}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\psi^{+}+\frac{1}{2\ell}V^{a}\Gamma_{a}\psi^{-}-t\Gamma_{0}\psi^{+}\,,\\\
\nabla\psi^{-}&=d\psi^{-}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\psi^{-}+\frac{1}{2\ell}V^{a}\Gamma_{a}\psi^{+}+t\Gamma_{0}\psi^{-}\,.\end{split}$
(2.15)

Now, in order to formulate a CS action of the form (2.11) invariant under the
$\mathcal{N}=(2,0)$ super-AdS Carroll group, we make use of the connection
$1$-form given in (2.13) and of the corresponding non-vanishing components of
the invariant tensor.

Concerning, in particular, the invariant tensor, we now apply the method of
[36], meaning that we rescale not only the generators but also the
coefficients appearing in the invariant tensor before applying the, in the
present case, ultra-relativistic contraction, in order to end up with a non-
trivial invariant tensor for the contracted (super)algebra on which the CS
theory will be based. Specifically, we consider the non-vanishing components
of the invariant tensor for the $\mathfrak{so}(2)$ extension of
$\mathfrak{osp}(2|2)\otimes\mathfrak{sp}(2)$ (see (2.3)) given in (2.4), we
decompose the indices as in (2.5) and consider the new supersymmetry charges
(2.8), and then we rescale not only the generators in compliance with (2.9)
but also the coefficients appearing in (2.4) as follows:

$\alpha_{0}\rightarrow\alpha_{0}\,,\quad\alpha_{1}\rightarrow\sigma\alpha_{1}\,.$
(2.16)

Taking the limit $\sigma\rightarrow\infty$, we end up with the following non-
vanishing components of an invariant tensor for the $\mathcal{N}=(2,0)$ AdS
Carroll superalgebra:

$\begin{split}&\langle
J_{ab}J_{cd}\rangle=\alpha_{0}\left(\delta_{ad}\delta_{bc}-\delta_{ac}\delta_{bd}\right)\,,\\\
&\langle J_{ab}H\rangle=\alpha_{1}\epsilon_{ab}\,,\\\ &\langle
K_{a}P_{b}\rangle=-\alpha_{1}\epsilon_{ab}\,,\\\ &\langle
P_{a}P_{b}\rangle=\frac{\alpha_{0}}{\ell^{2}}\delta_{ab}\,,\\\ &\langle
TT\rangle=-2\alpha_{0}\,,\\\ &\langle TS\rangle=2\alpha_{1}\,,\\\ &\langle
Q^{+}_{\alpha}Q^{+}_{\beta}\rangle=\langle
Q^{-}_{\alpha}Q^{-}_{\beta}\rangle=2\alpha_{1}C_{\alpha\beta}\,.\end{split}$
(2.17)

This bilinear form is non-degenerate if $\alpha_{1}\neq 0$.

After that, using the connection $1$-form in (2.13) and the non-vanishing
components of the invariant tensor given in (2.17) in the general expression
(2.11) for a three-dimensional CS action, we can finally write the $(2,0)$ AdS
Carroll CS supergravity action in three spacetime dimensions invariant under
(2.10), which reads as follows:

$\begin{split}I^{(2,0)}_{CS}&=\frac{k}{4\pi}\int_{\mathcal{M}}\Bigg{\\{}\frac{\alpha_{0}}{2}\left(\omega^{a}_{\phantom{a}b}R^{b}_{\phantom{b}a}+\frac{2}{\ell^{2}}V^{a}R_{a}-4tdt\right)+\alpha_{1}\bigg{(}\epsilon_{ab}R^{ab}h-2\epsilon_{ab}\mathfrak{K}^{a}V^{b}+\frac{1}{\ell^{2}}\epsilon_{ab}V^{a}V^{b}h\\\
&+4tds+2\bar{\psi}^{+}\nabla\psi^{+}+2\bar{\psi}^{-}\nabla\psi^{-}\bigg{)}-d\left(\frac{\alpha_{1}}{2}\epsilon_{ab}\omega^{ab}h-\alpha_{1}\epsilon_{ab}k^{a}V^{b}-2\alpha_{1}ts\right)\Bigg{\\}}\,,\end{split}$
(2.18)

written in terms of the curvatures appearing in (2.15). We can see that in
(2.18) we have two different sectors, one proportional to $\alpha_{0}$ and the
other proportional to $\alpha_{1}$. Observe that the term proportional to
$\alpha_{0}$ corresponds to the exotic Lagrangian involving the Lorentz
contribution, a torsional piece, and a contribution from the $1$-form field
$t$, while it does not contain any contribution from the $1$-form fields
$\psi^{+}$ and $\psi^{-}$. Let us mention that the CS action (2.18) can also
be rewritten up to boundary terms as

$\begin{split}I^{(2,0)}_{CS}&=\frac{k}{4\pi}\int_{\mathcal{M}}\Bigg{\\{}\frac{\alpha_{0}}{2}\left(\omega^{a}_{\phantom{a}b}R^{b}_{\phantom{b}a}+\frac{2}{\ell^{2}}V^{a}R_{a}-4tdt\right)+\alpha_{1}\bigg{(}\epsilon_{ab}R^{ab}h-2\epsilon_{ab}\mathfrak{K}^{a}V^{b}+\frac{1}{\ell^{2}}\epsilon_{ab}V^{a}V^{b}h\\\
&+4tds+2\bar{\psi}^{+}\nabla\psi^{+}+2\bar{\psi}^{-}\nabla\psi^{-}\bigg{)}\Bigg{\\}}\,.\end{split}$
(2.19)

The CS action (2.18), characterized by two coupling constants $\alpha_{0}$ and
$\alpha_{1}$, is invariant by construction under the $\mathcal{N}=(2,0)$ AdS
Carroll superalgebra (2.10). In particular, the local gauge transformations
$\delta_{\lambda}A=d\lambda+\left[A,\lambda\right]$ with gauge parameter

$\lambda=\frac{1}{2}\lambda^{ab}J_{ab}+\kappa^{a}K_{a}+\lambda^{a}P_{a}+\tau
H+\varphi T+\varsigma S+\varepsilon^{+}Q^{+}+\varepsilon^{-}Q^{-}$ (2.20)

are given by

$\begin{split}\delta\omega^{ab}&=d\lambda^{ab}+\frac{2}{\ell^{2}}V^{[a}\lambda^{b]}\,,\\\
\delta
k^{a}&=d\kappa^{a}-\lambda^{a}_{\phantom{a}b}k^{b}+\omega^{a}_{\phantom{a}b}\kappa^{b}-\frac{1}{\ell^{2}}\lambda^{a}h+\frac{1}{\ell^{2}}V^{a}\tau-\frac{1}{\ell}\bar{\varepsilon}^{+}\Gamma^{a0}\psi^{-}-\frac{1}{\ell}\bar{\varepsilon}^{-}\Gamma^{a0}\psi^{+}\,,\\\
\delta
V^{a}&=d\lambda^{a}-\lambda^{a}_{\phantom{a}b}V^{b}+\omega^{a}_{\phantom{a}b}\lambda^{b}\,,\\\
\delta
h&=d\tau-\lambda^{a}k_{a}+V^{a}\kappa_{a}+\bar{\varepsilon}^{+}\Gamma^{0}\psi^{+}+\bar{\varepsilon}^{-}\Gamma^{0}\psi^{-}\,,\\\
\delta t&=d\varphi\,,\\\ \delta
s&=d\varsigma+\bar{\varepsilon}^{+}\Gamma^{0}\psi^{+}-\bar{\varepsilon}^{-}\Gamma^{0}\psi^{-}\,,\\\
\delta\psi^{+}&=d\varepsilon^{+}-\frac{1}{4}\lambda^{ab}\Gamma_{ab}\psi^{+}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{+}-\frac{1}{2\ell}\lambda^{a}\Gamma_{a}\psi^{-}+\frac{1}{2\ell}V^{a}\Gamma_{a}\varepsilon^{-}+\varphi\Gamma_{0}\psi^{+}-t\Gamma_{0}\varepsilon^{+}\,,\\\
\delta\psi^{-}&=d\varepsilon^{-}-\frac{1}{4}\lambda^{ab}\Gamma_{ab}\psi^{-}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{-}-\frac{1}{2\ell}\lambda^{a}\Gamma_{a}\psi^{+}+\frac{1}{2\ell}V^{a}\Gamma_{a}\varepsilon^{+}-\varphi\Gamma_{0}\psi^{-}+t\Gamma_{0}\varepsilon^{-}\,.\end{split}$
(2.21)

Restricting ourselves to supersymmetry, we have

$\begin{split}\delta\omega^{ab}&=0\,,\\\ \delta
k^{a}&=-\frac{1}{\ell}\bar{\varepsilon}^{+}\Gamma^{a0}\psi^{-}-\frac{1}{\ell}\bar{\varepsilon}^{-}\Gamma^{a0}\psi^{+}\,,\\\
\delta V^{a}&=0\,,\\\ \delta
h&=\bar{\varepsilon}^{+}\Gamma^{0}\psi^{+}+\bar{\varepsilon}^{-}\Gamma^{0}\psi^{-}\,,\\\
\delta t&=0\,,\\\ \delta
s&=\bar{\varepsilon}^{+}\Gamma^{0}\psi^{+}-\bar{\varepsilon}^{-}\Gamma^{0}\psi^{-}\,,\\\
\delta\psi^{+}&=d\varepsilon^{+}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{+}+\frac{1}{2\ell}V^{a}\Gamma_{a}\varepsilon^{-}-t\Gamma_{0}\varepsilon^{+}\,,\\\
\delta\psi^{-}&=d\varepsilon^{-}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{-}+\frac{1}{2\ell}V^{a}\Gamma_{a}\varepsilon^{+}+t\Gamma_{0}\varepsilon^{-}\,.\end{split}$
(2.22)

The equations of motion obtained from the variation of the action (2.18) with
respect to the fields $\omega^{ab}$, $k^{a}$, $V^{a}$, $h$, $t$, $s$,
$\psi^{+}$, and $\psi^{-}$ are, respectively,

$\begin{split}\delta\omega^{ab}&:\quad\alpha_{0}\mathcal{R}^{ab}+\alpha_{1}\epsilon^{ab}\mathcal{H}=0\,,\\\
\delta k^{a}&:\quad\alpha_{1}R^{a}=0\,,\\\ \delta
V^{a}&:\quad\frac{\alpha_{0}}{\ell^{2}}R^{a}+2\alpha_{1}\epsilon_{ab}\mathcal{K}^{b}=0\,,\\\
\delta h&:\quad\alpha_{1}\mathcal{R}^{ab}=0\,,\\\ \delta
t&:\quad-\alpha_{0}\mathcal{T}+\alpha_{1}\mathcal{S}=0\,,\\\ \delta
s&:\quad\alpha_{1}\mathcal{T}=0\,,\\\
\delta\psi^{+}&:\quad\alpha_{1}\nabla\psi^{+}=0\,,\\\
\delta\psi^{-}&:\quad\alpha_{1}\nabla\psi^{-}=0\,,\end{split}$ (2.23)

up to boundary terms, and we can see that when $\alpha_{1}\neq 0$ they reduce
to the vanishing of the $(2,0)$ super-AdS Carroll curvature $2$-forms, namely

$\mathcal{R}^{ab}=0\,,\quad\mathcal{K}^{a}=0\,,\quad
R^{a}=0\,,\quad\mathcal{H}=0\,,\quad\mathcal{T}=0\,,\quad\mathcal{S}=0\,,\quad\nabla\psi^{+}=0\,,\quad\nabla\psi^{-}=0\,.$
(2.24)

Here, we can also observe that $\alpha_{1}\neq 0$ is a sufficient condition to
recover (2.24), meaning that one could consistently set $\alpha_{0}=0$, which
corresponds to the vanishing of the exotic term in the CS action (2.18).

##  3 $(1,1)$ AdS Carroll supergravity in $2+1$ dimensions

In this section, we repeat the analysis done in Section 2 in the $(1,1)$ case.
To this aim, we first review the derivation of the $\mathcal{N}=(1,1)$ AdS
Carroll superalgebra introduced in [14], which is obtained as the ultra-
relativistic contraction of $\mathfrak{osp}(2|1)\otimes\mathfrak{osp}(2,1)$.
Then, we write the non-vanishing components of the invariant tensor of the
$\mathcal{N}=(1,1)$ AdS Carroll superalgebra (obtained as the Carrollian
contraction of the non-vanishing components of the invariant tensor for
$\mathfrak{osp}(2|1)\otimes\mathfrak{osp}(2,1)$). This allows us to construct
a three-dimensional CS supergravity action invariant under the
$\mathcal{N}=(1,1)$ AdS Carroll superalgebra, which we call the $(1,1)$ AdS
Carroll CS supergravity action.

###  3.1 Review of the $\mathcal{N}=(1,1)$ AdS Carroll superalgebra

Let us briefly review the derivation of the $\mathcal{N}=(1,1)$ AdS Carroll
superalgebra of [14] as the Carrollian contraction of
$\mathfrak{osp}(2|1)\otimes\mathfrak{osp}(2,1)$.

The superalgebra $\mathfrak{osp}(2|1)\otimes\mathfrak{osp}(2,1)$ is generated
by the set
$\\{\tilde{J}_{AB},\tilde{P}_{A},\tilde{Q}^{+}_{\alpha},\tilde{Q}^{-}_{\alpha}\\}$
obeying the following (anti)commutation relations:

$\begin{split}&\left[\tilde{J}_{AB},\tilde{J}_{CD}\right]=\eta_{BC}\tilde{J}_{AD}-\eta_{AC}\tilde{J}_{BD}-\eta_{BD}\tilde{J}_{AC}+\eta_{AD}\tilde{J}_{BC}\,,\\\
&\left[\tilde{J}_{AB},\tilde{P}_{C}\right]=\eta_{BC}\tilde{P}_{A}-\eta_{AC}\tilde{P}_{B}\,,\\\
&\left[\tilde{P}_{A},\tilde{P}_{B}\right]=\frac{1}{\ell^{2}}\tilde{J}_{AB}\,,\\\
&\left[\tilde{J}_{AB},\tilde{Q}^{\pm}_{\alpha}\right]=-\frac{1}{2}\left(\Gamma_{AB}\tilde{Q}^{\pm}\right)_{\alpha}\,,\quad\quad\left[\tilde{P}_{A},\tilde{Q}^{\pm}_{\alpha}\right]=\mp\frac{1}{2\ell}\left(\Gamma_{A}\tilde{Q}^{\pm}\right)_{\alpha}\,,\\\
&\left\\{\tilde{Q}^{+}_{\alpha},\tilde{Q}^{+}_{\beta}\right\\}=-\frac{1}{2\ell}\left(\Gamma^{AB}C\right)_{\alpha\beta}\tilde{J}_{AB}+\left(\Gamma^{A}C\right)_{\alpha\beta}\tilde{P}_{A}\,,\\\
&\left\\{\tilde{Q}^{-}_{\alpha},\tilde{Q}^{-}_{\beta}\right\\}=\frac{1}{2\ell}\left(\Gamma^{AB}C\right)_{\alpha\beta}\tilde{J}_{AB}+\left(\Gamma^{A}C\right)_{\alpha\beta}\tilde{P}_{A}\,.\end{split}$
(3.1)

Note that by taking the flat limit $\ell\rightarrow\infty$ of (3.1) one
recovers the $\mathcal{N}=(1,1)$ Poincaré superalgebra.

The non-vanishing components of an invariant tensor for the superalgebra (3.1)
are

$\begin{split}&\langle\tilde{J}_{AB}\tilde{J}_{CD}\rangle=\alpha_{0}\left(\eta_{AD}\eta_{BC}-\eta_{AC}\eta_{BD}\right)\,,\\\
&\langle\tilde{J}_{AB}\tilde{P}_{C}\rangle=\alpha_{1}\epsilon_{ABC}\,,\\\
&\langle\tilde{P}_{A}\tilde{P}_{B}\rangle=\frac{\alpha_{0}}{\ell^{2}}\eta_{AB}\,,\\\
&\langle\tilde{Q}^{+}_{\alpha}\tilde{Q}^{+}_{\beta}\rangle=2\left(\alpha_{1}-\frac{\alpha_{0}}{\ell}\right)C_{\alpha\beta}\,,\\\
&\langle\tilde{Q}^{-}_{\alpha}\tilde{Q}^{-}_{\beta}\rangle=2\left(\alpha_{1}+\frac{\alpha_{0}}{\ell}\right)C_{\alpha\beta}\,,\end{split}$
(3.2)

being $\alpha_{0}$ and $\alpha_{1}$ arbitrary independent constants.

Now, to take the Carrollian contraction of the superalgebra (3.1), we
decompose the indices $A,B,\ldots=0,1,2$ as in (2.5), which induces the
decomposition (2.6), together with (2.7). Then, we rescale the generators with
a parameter $\sigma$ as

$\tilde{H}\rightarrow\sigma H\,,\quad\tilde{K}_{a}\rightarrow\sigma
K_{a}\,,\quad\tilde{Q}^{\pm}_{\alpha}\rightarrow\sqrt{\sigma}Q^{\pm}_{\alpha}\,.$
(3.3)

Subsequently, taking the limit $\sigma\rightarrow\infty$ (and removing the
tilde symbol also on the generators that we have not rescaled), we end up with
the $\mathcal{N}=(1,1)$ AdS Carroll superalgebra introduced in Ref. [14],
whose (anti)commutation relations read

$\begin{split}&\left[K_{a},J_{bc}\right]=\delta_{ab}K_{c}-\delta_{ac}K_{b}\,,\\\
&\left[J_{ab},P_{c}\right]=\delta_{bc}P_{a}-\delta_{ac}P_{b}\,,\\\
&\left[K_{a},P_{b}\right]=-\delta_{ab}H\,,\\\
&\left[P_{a},P_{b}\right]=\frac{1}{\ell^{2}}J_{ab}\,,\\\
&\left[P_{a},H\right]=\frac{1}{\ell^{2}}K_{a}\,,\\\
&\left[J_{ab},Q^{\pm}_{\alpha}\right]=-\frac{1}{2}\left(\Gamma_{ab}Q^{\pm}\right)_{\alpha}\,,\quad\quad\left[P_{a},Q^{\pm}_{\alpha}\right]=\mp\frac{1}{2\ell}\left(\Gamma_{a}Q^{\pm}\right)_{\alpha}\,,\\\
&\left\\{Q^{+}_{\alpha},Q^{+}_{\beta}\right\\}=-\frac{1}{\ell}\left(\Gamma^{a0}C\right)_{\alpha\beta}K_{a}+\left(\Gamma^{0}C\right)_{\alpha\beta}H\,,\\\
&\left\\{Q^{-}_{\alpha},Q^{-}_{\beta}\right\\}=\frac{1}{\ell}\left(\Gamma^{a0}C\right)_{\alpha\beta}K_{a}+\left(\Gamma^{0}C\right)_{\alpha\beta}H\,.\end{split}$
(3.4)

In the sequel we will construct a CS action in three-dimension invariant under
the $\mathcal{N}=(1,1)$ AdS Carroll superalgebra (3.4).

###  3.2 $(1,1)$ AdS Carroll supergravity action

We will now construct a three-dimensional CS supergravity action invariant
under the $\mathcal{N}=(1,1)$ AdS Carroll superalgebra (3.4), which we call
the $(1,1)$ AdS Carroll CS supergravity action.

To this aim, we introduce the connection $1$-form $A$ associated with (3.4),
that is

$A=\frac{1}{2}\omega^{ab}J_{ab}+k^{a}K_{a}+V^{a}P_{a}+hH+\psi^{+}Q^{+}+\psi^{-}Q^{-}\,,$
(3.5)

being $\omega^{ab}$, $k^{a}$, $V^{a}$, $h$, $\psi^{+}$, and $\psi^{-}$ the
$1$-form fields respectively dual to the generators $J_{ab}$, $K_{a}$,
$P_{a}$, $H$, $Q^{+}$, and $Q^{-}$ obeying the (anti)commutation relations
given in (3.4), and the related curvature $2$-form $F$, which reads

$F=\frac{1}{2}\mathcal{R}^{ab}J_{ab}+\mathcal{K}^{a}K_{a}+R^{a}P_{a}+\mathcal{H}H+\nabla\psi^{+}Q^{+}+\nabla\psi^{-}Q^{-}\,,$
(3.6)

with

$\begin{split}\mathcal{R}^{ab}&=d\omega^{ab}+\frac{1}{\ell^{2}}V^{a}V^{b}=R^{ab}+\frac{1}{\ell^{2}}V^{a}V^{b}\,,\\\
\mathcal{K}^{a}&=dk^{a}+\omega^{a}_{\phantom{a}b}k^{b}+\frac{1}{\ell^{2}}V^{a}h+\frac{1}{2\ell}\bar{\psi}^{+}\Gamma^{a0}\psi^{+}-\frac{1}{2\ell}\bar{\psi}^{-}\Gamma^{a0}\psi^{-}=\mathfrak{K}^{a}+\frac{1}{2\ell}\bar{\psi}^{+}\Gamma^{a0}\psi^{+}-\frac{1}{2\ell}\bar{\psi}^{-}\Gamma^{a0}\psi^{-}\,,\\\
R^{a}&=dV^{a}+\omega^{a}_{\phantom{a}b}V^{b}\,,\\\
\mathcal{H}&=dh+V^{a}k_{a}-\frac{1}{2}\bar{\psi}^{+}\Gamma^{0}\psi^{+}-\frac{1}{2}\bar{\psi}^{-}\Gamma^{0}\psi^{-}=\mathfrak{H}-\frac{1}{2}\bar{\psi}^{+}\Gamma^{0}\psi^{+}-\frac{1}{2}\bar{\psi}^{-}\Gamma^{0}\psi^{-}\,,\\\
\nabla\psi^{+}&=d\psi^{+}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\psi^{+}+\frac{1}{2\ell}V^{a}\Gamma_{a}\psi^{+}\,,\\\
\nabla\psi^{-}&=d\psi^{-}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\psi^{-}-\frac{1}{2\ell}V^{a}\Gamma_{a}\psi^{-}\,.\end{split}$
(3.7)

Now, we move to the explicit construction of a CS action invariant under the
$\mathcal{N}=(1,1)$ super-AdS Carroll group, on the same lines of what we have
previously done in Section 2 for $\mathcal{N}=(2,0)$. Thus, we consider the
non-vanishing components of the (relativistic) invariant tensor given in
(3.2), we decompose the indices as in (2.5), and we rescale not only the
generators in compliance with (3.3) but also the coefficients appearing in
(3.2) as in (2.16). Then, taking the ultra-relativistic limit
$\sigma\rightarrow\infty$, we get the following non-vanishing components of an
invariant tensor for the $\mathcal{N}=(1,1)$ AdS Carroll superalgebra:

$\begin{split}&\langle
J_{ab}J_{cd}\rangle=\alpha_{0}\left(\delta_{ad}\delta_{bc}-\delta_{ac}\delta_{bd}\right)\,,\\\
&\langle J_{ab}H\rangle=\alpha_{1}\epsilon_{ab}\,,\\\ &\langle
K_{a}P_{b}\rangle=-\alpha_{1}\epsilon_{ab}\,,\\\ &\langle
P_{a}P_{b}\rangle=\frac{\alpha_{0}}{\ell^{2}}\delta_{ab}\,,\\\ &\langle
Q^{+}_{\alpha}Q^{+}_{\beta}\rangle=\langle
Q^{-}_{\alpha}Q^{-}_{\beta}\rangle=2\alpha_{1}C_{\alpha\beta}\,.\end{split}$
(3.8)

This invariant tensor is non-degenerate if $\alpha_{1}\neq 0$.

Substituting the connection $1$-form in (3.5) and the non-vanishing components
of the invariant tensor (3.8) in the general expression (2.11) for a three-
dimensional CS action, we end up with the $(1,1)$ AdS Carroll CS supergravity
action in $2+1$ spacetime dimensions, that is

$\begin{split}I^{(1,1)}_{CS}&=\frac{k}{4\pi}\int_{\mathcal{M}}\Bigg{\\{}\frac{\alpha_{0}}{2}\left(\omega^{a}_{\phantom{a}b}R^{b}_{\phantom{b}a}+\frac{2}{\ell^{2}}V^{a}R_{a}\right)+\alpha_{1}\bigg{(}\epsilon_{ab}R^{ab}h-2\epsilon_{ab}\mathfrak{K}^{a}V^{b}+\frac{1}{\ell^{2}}\epsilon_{ab}V^{a}V^{b}h\\\
&+2\bar{\psi}^{+}\nabla\psi^{+}+2\bar{\psi}^{-}\nabla\psi^{-}\bigg{)}-d\left(\frac{\alpha_{1}}{2}\epsilon_{ab}\omega^{ab}h-\alpha_{1}\epsilon_{ab}k^{a}V^{b}\right)\Bigg{\\}}\,,\end{split}$
(3.9)

which can also be rewritten omitting boundary terms as follows:

$\begin{split}I^{(1,1)}_{CS}&=\frac{k}{4\pi}\int_{\mathcal{M}}\Bigg{\\{}\frac{\alpha_{0}}{2}\left(\omega^{a}_{\phantom{a}b}R^{b}_{\phantom{b}a}+\frac{2}{\ell^{2}}V^{a}R_{a}\right)+\alpha_{1}\bigg{(}\epsilon_{ab}R^{ab}h-2\epsilon_{ab}\mathfrak{K}^{a}V^{b}+\frac{1}{\ell^{2}}\epsilon_{ab}V^{a}V^{b}h\\\
&+2\bar{\psi}^{+}\nabla\psi^{+}+2\bar{\psi}^{-}\nabla\psi^{-}\bigg{)}\Bigg{\\}}\,.\end{split}$
(3.10)

The action (3.9) has been written in terms of the curvatures appearing in
(3.7), it involves two different sectors, respectively proportional to
$\alpha_{0}$ (which corresponds to the exotic Lagrangian) and to $\alpha_{1}$,
and it is invariant by construction under the $\mathcal{N}=(1,1)$ AdS Carroll
superalgebra (3.4). The local gauge transformations
$\delta_{\lambda}A=d\lambda+\left[A,\lambda\right]$ with gauge parameter

$\lambda=\frac{1}{2}\lambda^{ab}J_{ab}+\kappa^{a}K_{a}+\lambda^{a}P_{a}+\tau
H+\varepsilon^{+}Q^{+}+\varepsilon^{-}Q^{-}$ (3.11)

are

$\begin{split}\delta\omega^{ab}&=d\lambda^{ab}+\frac{2}{\ell^{2}}V^{[a}\lambda^{b]}\,,\\\
\delta
k^{a}&=d\kappa^{a}-\lambda^{a}_{\phantom{a}b}k^{b}+\omega^{a}_{\phantom{a}b}\kappa^{b}-\frac{1}{\ell^{2}}\lambda^{a}h+\frac{1}{\ell^{2}}V^{a}\tau-\frac{1}{\ell}\bar{\varepsilon}^{+}\Gamma^{a0}\psi^{+}+\frac{1}{\ell}\bar{\varepsilon}^{-}\Gamma^{a0}\psi^{-}\,,\\\
\delta
V^{a}&=d\lambda^{a}-\lambda^{a}_{\phantom{a}b}V^{b}+\omega^{a}_{\phantom{a}b}\lambda^{b}\,,\\\
\delta
h&=d\tau-\lambda^{a}k_{a}+V^{a}\kappa_{a}+\bar{\varepsilon}^{+}\Gamma^{0}\psi^{+}+\bar{\varepsilon}^{-}\Gamma^{0}\psi^{-}\,,\\\
\delta\psi^{+}&=d\varepsilon^{+}-\frac{1}{4}\lambda^{ab}\Gamma_{ab}\psi^{+}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{+}-\frac{1}{2\ell}\lambda^{a}\Gamma_{a}\psi^{+}+\frac{1}{2\ell}V^{a}\Gamma_{a}\varepsilon^{+}\,,\\\
\delta\psi^{-}&=d\varepsilon^{-}-\frac{1}{4}\lambda^{ab}\Gamma_{ab}\psi^{-}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{-}+\frac{1}{2\ell}\lambda^{a}\Gamma_{a}\psi^{-}-\frac{1}{2\ell}V^{a}\Gamma_{a}\varepsilon^{-}\,,\end{split}$
(3.12)

and, restricting ourselves to supersymmetry, we are left with the following
transformation rules:

$\begin{split}\delta\omega^{ab}&=0\,,\\\ \delta
k^{a}&=-\frac{1}{\ell}\bar{\varepsilon}^{+}\Gamma^{a0}\psi^{+}+\frac{1}{\ell}\bar{\varepsilon}^{-}\Gamma^{a0}\psi^{-}\,,\\\
\delta V^{a}&=0\,,\\\ \delta
h&=\bar{\varepsilon}^{+}\Gamma^{0}\psi^{+}+\bar{\varepsilon}^{-}\Gamma^{0}\psi^{-}\,,\\\
\delta\psi^{+}&=d\varepsilon^{+}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{+}+\frac{1}{2\ell}V^{a}\Gamma_{a}\varepsilon^{+}\,,\\\
\delta\psi^{-}&=d\varepsilon^{-}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{-}-\frac{1}{2\ell}V^{a}\Gamma_{a}\varepsilon^{-}\,.\end{split}$
(3.13)

The equations of motion obtained from the variation of the action (3.9) with
respect to the $1$-form fields $\omega^{ab}$, $k^{a}$, $V^{a}$, $h$,
$\psi^{+}$, and $\psi^{-}$ are

$\begin{split}\delta\omega^{ab}&:\quad\alpha_{0}\mathcal{R}^{ab}+\alpha_{1}\epsilon^{ab}\mathcal{H}=0\,,\\\
\delta k^{a}&:\quad\alpha_{1}R^{a}=0\,,\\\ \delta
V^{a}&:\quad\frac{\alpha_{0}}{\ell^{2}}R^{a}+2\alpha_{1}\epsilon_{ab}\mathcal{K}^{b}=0\,,\\\
\delta h&:\quad\alpha_{1}\mathcal{R}^{ab}=0\,,\\\
\delta\psi^{+}&:\quad\alpha_{1}\nabla\psi^{+}=0\,,\\\
\delta\psi^{-}&:\quad\alpha_{1}\nabla\psi^{-}=0\,,\end{split}$ (3.14)

respectively; for $\alpha_{1}\neq 0$, they reduce to the vanishing of the
$(1,0)$ super-AdS Carroll curvature $2$-forms, namely

$\mathcal{R}^{ab}=0\,,\quad\mathcal{K}^{a}=0\,,\quad
R^{a}=0\,,\quad\mathcal{H}=0\,,\quad\nabla\psi^{+}=0\,,\quad\nabla\psi^{-}=0\,.$
(3.15)

Analogously to what happened in the $(2,0)$ case discussed in Section 2, we
can see that $\alpha_{1}\neq 0$ is a sufficient condition to recover (3.15),
which means that one could consistently set $\alpha_{0}=0$, making the exotic
term in the CS action (3.9) disappear.

##  4 $(\mathcal{N},0)$ AdS Carroll supergravity theories in $2+1$ dimensions

Now, we generalize our analysis to the $(\mathcal{N},0)$ case, with
$\mathcal{N}$ even. First, we present the derivation of the
$\mathcal{N}=(\mathcal{N},0)$ AdS Carroll superalgebra as the Carrollian
contraction of an $\mathfrak{so}(\mathcal{N})$ extension of
$\mathfrak{osp}(2|\mathcal{N})\otimes\mathfrak{sp}(2)$. This also provides us
with a non-degenerate invariant tensor in the ultra-relativistic limit. Then,
we can subsequently formulate a well-defined three-dimensional CS supergravity
action invariant under the aforesaid $\mathcal{N}=(\mathcal{N},0)$ AdS Carroll
superalgebra.

###  4.1 $\mathcal{N}=(\mathcal{N},0)$ AdS Carroll superalgebra

Let us first take the direct sum of
$\mathfrak{osp}(2|\mathcal{N})\otimes\mathfrak{sp}(2)$ and an
$\mathfrak{so}(\mathcal{N})$ algebra (we consider $\mathcal{N}$ even), that is
reminiscent of what was done in Ref. [40]. In this case, the non-trivial
(anti)commutation relations are

$\begin{split}&\left[\tilde{J}_{AB},\tilde{J}_{CD}\right]=\eta_{BC}\tilde{J}_{AD}-\eta_{AC}\tilde{J}_{BD}-\eta_{BD}\tilde{J}_{AC}+\eta_{AD}\tilde{J}_{BC}\,,\\\
&\left[\tilde{J}_{AB},\tilde{P}_{C}\right]=\eta_{BC}\tilde{P}_{A}-\eta_{AC}\tilde{P}_{B}\,,\quad\quad\left[\tilde{P}_{A},\tilde{P}_{B}\right]=\frac{1}{\ell^{2}}\tilde{J}_{AB}\,,\\\
&\left[\tilde{Z}^{ij},\tilde{Z}^{kl}\right]=\delta^{jk}\tilde{Z}^{il}-\delta^{ik}\tilde{Z}^{jl}-\delta^{jl}\tilde{Z}^{ik}+\delta^{il}\tilde{Z}^{jk}\,,\\\
&\left[\tilde{S}^{ij},\tilde{S}^{kl}\right]=-\frac{1}{\ell}\left(\delta^{jk}\tilde{S}^{il}-\delta^{ik}\tilde{S}^{jl}-\delta^{jl}\tilde{S}^{ik}+\delta^{il}\tilde{S}^{jk}\right)\,,\\\
&\left[\tilde{J}_{AB},\tilde{Q}^{i}_{\alpha}\right]=-\frac{1}{2}\left(\Gamma_{AB}\tilde{Q}^{i}\right)_{\alpha}\,,\quad\quad\left[\tilde{P}_{A},\tilde{Q}^{i}_{\alpha}\right]=-\frac{1}{2\ell}\left(\Gamma_{A}\tilde{Q}^{i}\right)_{\alpha}\,,\\\
&\left[\tilde{Z}^{ij},\tilde{Q}^{k}_{\alpha}\right]=\delta^{jk}\tilde{Q}^{i}_{\alpha}-\delta^{ik}\tilde{Q}^{j}_{\alpha}\,,\\\
&\left\\{\tilde{Q}^{i}_{\alpha},\tilde{Q}^{j}_{\beta}\right\\}=\delta^{ij}\left[-\frac{1}{2\ell}\left(\Gamma^{AB}C\right)_{\alpha\beta}\tilde{J}_{AB}+\left(\Gamma^{A}C\right)_{\alpha\beta}\tilde{P}_{A}\right]+\frac{1}{\ell}C_{\alpha\beta}\tilde{Z}^{ij}\,,\end{split}$
(4.1)

with $A,B,\ldots=0,1,2$, $i,j,\ldots=1,\ldots,\mathcal{N}$ (where we have
considered $\mathcal{N}=2x$, $x=1,\ldots,\frac{\mathcal{N}}{2}$), and where
$\tilde{Z}^{ij}=-\tilde{Z}^{ji}$, $\tilde{S}^{ij}=-\tilde{S}^{ij}$. Then, we
do the following redefinition (on the same lines of [40]):

$\tilde{T}^{ij}\equiv\tilde{Z}^{ij}-\ell\tilde{S}^{ij}\,,$ (4.2)

which is a generalization of the one performed in Section 2. Thus, we can now
rewrite the (anti)commutation relations (4.1) as

$\begin{split}&\left[\tilde{J}_{AB},\tilde{J}_{CD}\right]=\eta_{BC}\tilde{J}_{AD}-\eta_{AC}\tilde{J}_{BD}-\eta_{BD}\tilde{J}_{AC}+\eta_{AD}\tilde{J}_{BC}\,,\\\
&\left[\tilde{J}_{AB},\tilde{P}_{C}\right]=\eta_{BC}\tilde{P}_{A}-\eta_{AC}\tilde{P}_{B}\,,\\\
&\left[\tilde{P}_{A},\tilde{P}_{B}\right]=\frac{1}{\ell^{2}}\tilde{J}_{AB}\,,\\\
&\left[\tilde{T}^{ij},\tilde{T}^{kl}\right]=\delta^{jk}\tilde{T}^{il}-\delta^{ik}\tilde{T}^{jl}-\delta^{jl}\tilde{T}^{ik}+\delta^{il}\tilde{T}^{jk}\,,\\\
&\left[\tilde{T}^{ij},\tilde{S}^{kl}\right]=\delta^{jk}\tilde{S}^{il}-\delta^{ik}\tilde{S}^{jl}-\delta^{jl}\tilde{S}^{ik}+\delta^{il}\tilde{S}^{jk}\,,\\\
&\left[\tilde{S}^{ij},\tilde{S}^{kl}\right]=-\frac{1}{\ell}\left(\delta^{jk}\tilde{S}^{il}-\delta^{ik}\tilde{S}^{jl}-\delta^{jl}\tilde{S}^{ik}+\delta^{il}\tilde{S}^{jk}\right)\,,\\\
&\left[\tilde{J}_{AB},\tilde{Q}^{i}_{\alpha}\right]=-\frac{1}{2}\left(\Gamma_{AB}\tilde{Q}^{i}\right)_{\alpha}\,,\\\
&\left[\tilde{P}_{A},\tilde{Q}^{i}_{\alpha}\right]=-\frac{1}{2\ell}\left(\Gamma_{A}\tilde{Q}^{i}\right)_{\alpha}\,,\\\
&\left[\tilde{T}^{ij},\tilde{Q}^{k}_{\alpha}\right]=\delta^{jk}\tilde{Q}^{i}_{\alpha}-\delta^{ik}\tilde{Q}^{j}_{\alpha}\,,\\\
&\left\\{\tilde{Q}^{i}_{\alpha},\tilde{Q}^{j}_{\beta}\right\\}=\delta^{ij}\left[-\frac{1}{2\ell}\left(\Gamma^{AB}C\right)_{\alpha\beta}\tilde{J}_{AB}+\left(\Gamma^{A}C\right)_{\alpha\beta}\tilde{P}_{A}\right]+C_{\alpha\beta}\left(\frac{1}{\ell}\tilde{T}^{ij}+\tilde{S}^{ij}\right)\,.\end{split}$
(4.3)

Observe that, taking the limit $\ell\rightarrow\infty$ of (4.3), we get the
$\mathcal{N}=(\mathcal{N},0)$ Poincaré superalgebra involving a semi-direct
$\mathfrak{so}(\mathcal{N})$ extension (with $\mathcal{N}$ even).

The non-vanishing components of an invariant tensor for (4.3), which will be
useful in the sequel, are given by

$\begin{split}&\langle\tilde{J}_{AB}\tilde{J}_{CD}\rangle=\alpha_{0}\left(\eta_{AD}\eta_{BC}-\eta_{AC}\eta_{BD}\right)\,,\\\
&\langle\tilde{J}_{AB}\tilde{P}_{C}\rangle=\alpha_{1}\epsilon_{ABC}\,,\\\
&\langle\tilde{P}_{A}\tilde{P}_{B}\rangle=\frac{\alpha_{0}}{\ell^{2}}\eta_{AB}\,,\\\
&\langle\tilde{T}^{ij}\tilde{T}^{kl}\rangle=2\alpha_{0}\left(\delta^{il}\delta^{kj}-\delta^{ik}\delta^{lj}\right)\,,\\\
&\langle\tilde{T}^{ij}\tilde{S}^{kl}\rangle=-2\alpha_{1}\left(\delta^{il}\delta^{kj}-\delta^{ik}\delta^{lj}\right)\,,\\\
&\langle\tilde{S}^{ij}\tilde{S}^{kl}\rangle=\frac{2\alpha_{1}}{\ell}\left(\delta^{il}\delta^{kj}-\delta^{ik}\delta^{lj}\right)\,,\\\
&\langle\tilde{Q}^{i}_{\alpha}\tilde{Q}^{j}_{\beta}\rangle=2\left(\alpha_{1}-\frac{\alpha_{0}}{\ell}\right)C_{\alpha\beta}\delta^{ij}\,,\end{split}$
(4.4)

being $\alpha_{0}$ and $\alpha_{1}$ arbitrary independent constants.

In order to take the ultra-relativistic contraction of (4.3), we decompose the
indices $A,B,\ldots=0,1,2$ as in (2.5), which induces the decomposition (2.6),
together with the gamma matrices decomposition (2.7).

Moreover, we define, on the same lines of what was done in [53] (see also
references therein) in the case of non-relativistic theories, new
supersymmetry charges by

$\tilde{Q}^{\pm\,\lambda}_{\alpha}=\frac{1}{\sqrt{2}}\left(\tilde{Q}^{\lambda}_{\alpha}\pm\left(\Gamma_{0}\right)_{\alpha\beta}\tilde{Q}^{x+\lambda}_{\beta}\right)\,,$
(4.5)

where in (4.5) we consider $\lambda,\mu,\ldots=1,\ldots,x$ (these new indices
must not be confused with the spinor ones $\alpha,\beta,\ldots=1,2$),
generalizing to the $\mathcal{N}=(\mathcal{N},0)$ case, with $\mathcal{N}$
even, what we have previously done in Section 2 (see, in particular, (2.8)).
We remind that $x=1,\ldots,\frac{\mathcal{N}}{2}$.

This also reflects on the generators $\tilde{T}^{ij}$ and $\tilde{S}^{ij}$,
which are now respectively described by

$\begin{split}&\tilde{T}^{\lambda\mu}\,,\quad\tilde{T}^{\prime\lambda\mu}\equiv\tilde{T}^{\lambda+x\
\mu+x}\,,\quad\tilde{U}^{\lambda\mu}\equiv\tilde{T}^{x+\lambda\
\mu}\,,\quad\tilde{U}^{\prime\lambda\mu}\equiv\tilde{T}^{\lambda\ x+\mu}\,,\\\
&\tilde{S}^{\lambda\mu}\,,\quad\tilde{S}^{\prime\lambda\mu}\equiv\tilde{S}^{\lambda+x\
\mu+x}\,,\quad\tilde{V}^{\lambda\mu}\equiv\tilde{S}^{x+\lambda\
\mu}\,,\quad\tilde{V}^{\prime\lambda\mu}\equiv\tilde{S}^{\lambda\
x+\mu}\,,\end{split}$ (4.6)

which satisfy the symmetry properties

$\begin{split}&\tilde{T}^{\lambda\mu}=-\tilde{T}^{\mu\lambda}\,,\quad\tilde{T}^{\prime\lambda\mu}=-\tilde{T}^{\prime\mu\lambda}\,,\quad\tilde{U}^{\lambda\mu}=-\tilde{U}^{\prime\mu\lambda}\,,\\\
&\tilde{S}^{\lambda\mu}=-\tilde{S}^{\mu\lambda}\,,\quad\tilde{S}^{\prime\lambda\mu}=-\tilde{S}^{\prime\mu\lambda}\,,\quad\tilde{V}^{\lambda\mu}=-\tilde{V}^{\prime\mu\lambda}\,.\end{split}$
(4.7)

In particular, using (4.5), (4.6), and (4.7), together with the decomposition
(2.6) and (2.7), and defining

$\begin{split}&\tilde{X}^{[\lambda\mu]}\equiv\frac{1}{2}\left(\tilde{T}^{\lambda\mu}+\tilde{T}^{\prime\lambda\mu}\right)\,,\quad\tilde{X}^{\prime[\lambda\mu]}\equiv\frac{1}{2}\left(\tilde{T}^{\lambda\mu}-\tilde{T}^{\prime\lambda\mu}\right)\,,\\\
&\tilde{Y}^{[\lambda\mu]}\equiv\frac{1}{2}\left(\tilde{S}^{\lambda\mu}+\tilde{S}^{\prime\lambda\mu}\right)\,,\quad\tilde{Y}^{\prime[\lambda\mu]}\equiv\frac{1}{2}\left(\tilde{S}^{\lambda\mu}-\tilde{S}^{\prime\lambda\mu}\right)\,,\end{split}$
(4.8)

the (anti)commutation relations in (4.3) become

$\begin{split}\\{\tilde{Q}^{\pm\,\lambda}_{\alpha},\tilde{Q}^{\pm\,\mu}_{\beta}\\}&=\delta^{\lambda\mu}\left[-\frac{1}{2\ell}\left(\Gamma^{ab}C\right)_{\alpha\beta}\tilde{J}_{ab}+\left(\Gamma^{0}C\right)_{\alpha\beta}\tilde{H}\right]+C_{\alpha\beta}\left(\frac{1}{\ell}\tilde{X}^{[\lambda\mu]}+\tilde{Y}^{[\lambda\mu]}\right)\\\
&\mp\left(\Gamma^{0}C\right)_{\alpha\beta}\left(\frac{1}{\ell}\tilde{U}^{(\lambda\mu)}+\tilde{V}^{(\lambda\mu)}\right)\,,\\\
\\{\tilde{Q}^{+\,\lambda}_{\alpha},\tilde{Q}^{-\,\mu}_{\beta}\\}&=\delta^{\lambda\mu}\left[-\frac{1}{\ell}\left(\Gamma^{a0}C\right)_{\alpha\beta}\tilde{K}_{a}+\left(\Gamma^{a}C\right)_{\alpha\beta}\tilde{P}_{a}\right]+C_{\alpha\beta}\left(\frac{1}{\ell}\tilde{X}^{\prime[\lambda\mu]}+\tilde{Y}^{\prime[\lambda\mu]}\right)\\\
&-\left(\Gamma^{0}C\right)_{\alpha\beta}\left(\frac{1}{\ell}\tilde{U}^{[\lambda\mu]}+\tilde{V}^{[\lambda\mu]}\right)\,,\end{split}$
(4.9)

where

$\begin{split}&\tilde{U}^{(\lambda\mu)}=\frac{1}{2}\left(\tilde{U}^{\lambda\mu}+\tilde{U}^{\mu\lambda}\right)\,,\quad\tilde{U}^{[\lambda\mu]}=\frac{1}{2}\left(\tilde{U}^{\lambda\mu}-\tilde{U}^{\mu\lambda}\right)\,,\\\
&\tilde{V}^{(\lambda\mu)}=\frac{1}{2}\left(\tilde{V}^{\lambda\mu}+\tilde{V}^{\mu\lambda}\right)\,,\quad\tilde{V}^{[\lambda\mu]}=\frac{1}{2}\left(\tilde{V}^{\lambda\mu}-\tilde{V}^{\mu\lambda}\right)\,.\end{split}$
(4.10)

The generators $\tilde{X}^{[\lambda\mu]}$ in (4.9) amount to
$\frac{k(k-1)}{2}$ generators, and the same holds for
$\tilde{X}^{\prime[\lambda\mu]}$, $\tilde{U}^{[\lambda\mu]}$,
$\tilde{Y}^{[\lambda\mu]}$, $\tilde{Y}^{\prime[\lambda\mu]}$, and
$\tilde{V}^{[\lambda\mu]}$, while the generators $\tilde{U}^{(\lambda\mu)}$
are $\frac{k(k+1)}{2}$ generators, and the same holds for the generators
$\tilde{V}^{(\lambda\mu)}$.

Furthermore, from the commutation relations involving the generators $T^{ij}$
and $S^{ij}$, we get the following non-vanishing ones (recall that we have
$\tilde{U}^{\prime\lambda\mu}=-\tilde{U}^{\mu\lambda}$ and
$\tilde{V}^{\prime\lambda\mu}=-\tilde{V}^{\mu\lambda}$):

$\begin{split}&\left[\tilde{T}^{\lambda\mu},\tilde{T}^{\nu\rho}\right]=\delta^{\mu\nu}\tilde{T}^{\lambda\rho}-\delta^{\lambda\nu}\tilde{T}^{\mu\rho}-\delta^{\mu\rho}\tilde{T}^{\lambda\nu}+\delta^{\lambda\rho}\tilde{T}^{\mu\nu}\,,\\\
&\left[\tilde{T}^{\lambda\mu},\tilde{U}^{\nu\rho}\right]=\delta^{\mu\rho}\tilde{U}^{\nu\lambda}-\delta^{\lambda\rho}\tilde{U}^{\nu\mu}\,,\\\
&\left[\tilde{T}^{\prime\lambda\mu},\tilde{T}^{\prime\nu\rho}\right]=\delta^{\mu\nu}\tilde{T}^{\prime\lambda\rho}-\delta^{\lambda\nu}\tilde{T}^{\prime\mu\rho}-\delta^{\mu\rho}\tilde{T}^{\prime\lambda\nu}+\delta^{\lambda\rho}\tilde{T}^{\prime\mu\nu}\,,\\\
&\left[\tilde{T}^{\prime\lambda\mu},\tilde{U}^{\nu\rho}\right]=\delta^{\mu\nu}\tilde{U}^{\lambda\rho}-\delta^{\lambda\nu}\tilde{U}^{\mu\rho}\,,\\\
&\left[\tilde{U}^{\lambda\mu},\tilde{U}^{\nu\rho}\right]=-\delta^{\lambda\nu}\tilde{T}^{\mu\rho}-\delta^{\mu\rho}\tilde{T}^{\prime\lambda\nu}\,,\\\
&\left[\tilde{T}^{\lambda\mu},\tilde{S}^{\nu\rho}\right]=\delta^{\mu\nu}\tilde{S}^{\lambda\rho}-\delta^{\lambda\nu}\tilde{S}^{\mu\rho}-\delta^{\mu\rho}\tilde{S}^{\lambda\nu}+\delta^{\lambda\rho}\tilde{S}^{\mu\nu}\,,\\\
&\left[\tilde{T}^{\lambda\mu},\tilde{V}^{\nu\rho}\right]=\delta^{\mu\rho}\tilde{V}^{\nu\lambda}-\delta^{\lambda\rho}\tilde{V}^{\nu\mu}\,,\\\
&\left[\tilde{T}^{\prime\lambda\mu},\tilde{S}^{\prime\nu\rho}\right]=\delta^{\mu\nu}\tilde{S}^{\prime\lambda\rho}-\delta^{\lambda\nu}\tilde{S}^{\prime\mu\rho}-\delta^{\mu\rho}\tilde{S}^{\prime\lambda\nu}+\delta^{\lambda\rho}\tilde{S}^{\prime\mu\nu}\,,\\\
&\left[\tilde{T}^{\prime\lambda\mu},\tilde{V}^{\nu\rho}\right]=\delta^{\mu\nu}\tilde{V}^{\lambda\rho}-\delta^{\lambda\nu}\tilde{V}^{\mu\rho}\,,\\\
&\left[\tilde{U}^{\lambda\mu},\tilde{S}^{\nu\rho}\right]=-\delta^{\mu\rho}\tilde{V}^{\lambda\nu}+\delta^{\mu\nu}\tilde{V}^{\lambda\rho}\,,\\\
&\left[\tilde{U}^{\lambda\mu},\tilde{S}^{\prime\nu\rho}\right]=-\delta^{\lambda\rho}\tilde{V}^{\nu\mu}+\delta^{\lambda\nu}\tilde{V}^{\rho\mu}\,,\\\
&\left[\tilde{U}^{\lambda\mu},\tilde{V}^{\nu\rho}\right]=-\delta^{\lambda\nu}\tilde{S}^{\mu\rho}-\delta^{\mu\rho}\tilde{S}^{\prime\lambda\nu}\,,\\\
&\left[\tilde{S}^{\lambda\mu},\tilde{S}^{\nu\rho}\right]=-\frac{1}{\ell}\left(\delta^{\mu\nu}\tilde{S}^{\lambda\rho}-\delta^{\lambda\nu}\tilde{S}^{\mu\rho}-\delta^{\mu\rho}\tilde{S}^{\lambda\nu}+\delta^{\lambda\rho}\tilde{S}^{\mu\nu}\right)\,,\\\
&\left[\tilde{S}^{\lambda\mu},\tilde{V}^{\nu\rho}\right]=-\frac{1}{\ell}\left(\delta^{\mu\rho}\tilde{V}^{\nu\lambda}-\delta^{\lambda\rho}\tilde{V}^{\nu\mu}\right)\,,\\\
&\left[\tilde{S}^{\prime\lambda\mu},\tilde{S}^{\prime\nu\rho}\right]=-\frac{1}{\ell}\left(\delta^{\mu\nu}\tilde{S}^{\prime\lambda\rho}-\delta^{\lambda\nu}\tilde{S}^{\prime\mu\rho}-\delta^{\mu\rho}\tilde{S}^{\prime\lambda\nu}+\delta^{\lambda\rho}\tilde{S}^{\prime\mu\nu}\right)\,,\\\
&\left[\tilde{S}^{\prime\lambda\mu},\tilde{V}^{\nu\rho}\right]=-\frac{1}{\ell}\left(\delta^{\mu\nu}\tilde{V}^{\lambda\rho}-\delta^{\lambda\nu}\tilde{V}^{\mu\rho}\right)\,,\\\
&\left[\tilde{V}^{\lambda\mu},\tilde{V}^{\nu\rho}\right]=\frac{1}{\ell}\left(\delta^{\lambda\nu}\tilde{S}^{\mu\rho}+\delta^{\mu\rho}\tilde{S}^{\prime\lambda\nu}\right)\,,\end{split}$
(4.11)

and

$\begin{split}&\left[\tilde{T}^{\lambda\mu},\tilde{Q}^{\pm\,\nu}_{\alpha}\right]=\frac{1}{2}\left[\delta^{\mu\nu}\left(\tilde{Q}^{+\,\lambda}_{\alpha}+\tilde{Q}^{-\,\lambda}_{\alpha}\right)-\delta^{\lambda\nu}\left(\tilde{Q}^{+\,\mu}_{\alpha}+\tilde{Q}^{-\,\mu}_{\alpha}\right)\right]\,,\\\
&\left[\tilde{T}^{\prime\lambda\mu},\tilde{Q}^{\pm\,\nu}_{\alpha}\right]=\pm\frac{1}{2}\left[\delta^{\mu\nu}\left(\tilde{Q}^{+\,\lambda}_{\alpha}-\tilde{Q}^{-\,\lambda}_{\alpha}\right)-\delta^{\lambda\nu}\left(\tilde{Q}^{+\,\mu}_{\alpha}-\tilde{Q}^{-\,\mu}_{\alpha}\right)\right]\,,\\\
&\left[\tilde{U}^{\lambda\mu},\tilde{Q}^{\pm\,\nu}_{\alpha}\right]=\mp\frac{1}{2}\left(\Gamma_{0}\right)_{\alpha\beta}\left[\delta^{\lambda\nu}\left(\tilde{Q}^{+\,\mu}_{\beta}+\tilde{Q}^{-\,\
\mu}_{\beta}\right)\pm\delta^{\mu\nu}\left(\tilde{Q}^{+\,\lambda}_{\beta}-\tilde{Q}^{-\,\lambda}_{\beta}\right)\right]\,.\end{split}$
(4.12)

Now, let us rescale the generators with a parameter $\sigma$ as

$\tilde{H}\rightarrow\sigma H\,,\quad\tilde{K}_{a}\rightarrow\sigma
K_{a}\,,\quad\tilde{S}^{\lambda\mu}\rightarrow\sigma
S^{\lambda\mu}\,,\quad\tilde{S}^{\prime\lambda\mu}\rightarrow\sigma
S^{\prime\lambda\mu}\,,\quad\tilde{V}^{\lambda\mu}\rightarrow\sigma
V^{\lambda\mu}\,,\quad\tilde{Q}^{\pm\,\lambda}_{\alpha}\rightarrow\sqrt{\sigma}Q^{\pm\,\lambda}_{\alpha}\,,$
(4.13)

where we have also removed the tilde symbol on the generators. Taking the
limit $\sigma\rightarrow\infty$ (and removing the tilde symbol also on the
generators that we have not rescaled), we end up with the
$\mathcal{N}=(\mathcal{N},0)$ AdS Carroll superalgebra (with $\mathcal{N}$
even), whose non-trivial (anti)commutation relations read as follows (recall
the definitions (4.8), (4.10), since we have expressed the anticommutation
relations in terms of the combinations given in that expressions, together
with the fact that $\tilde{U}^{\prime\lambda\mu}=-\tilde{U}^{\mu\lambda}$ and
$\tilde{V}^{\prime\lambda\mu}=-\tilde{V}^{\mu\lambda}$, and the
(anti)commutation relations (4.9), (4.11), and (4.12)):

$\begin{split}&\left[K_{a},J_{bc}\right]=\delta_{ab}K_{c}-\delta_{ac}K_{b}\,,\\\
&\left[J_{ab},P_{c}\right]=\delta_{bc}P_{a}-\delta_{ac}P_{b}\,,\\\
&\left[K_{a},P_{b}\right]=-\delta_{ab}H\,,\\\
&\left[P_{a},P_{b}\right]=\frac{1}{\ell^{2}}J_{ab}\,,\quad\quad\left[P_{a},H\right]=\frac{1}{\ell^{2}}K_{a}\,,\\\
&\left[{T}^{\lambda\mu},{T}^{\nu\rho}\right]=\delta^{\mu\nu}{T}^{\lambda\rho}-\delta^{\lambda\nu}{T}^{\mu\rho}-\delta^{\mu\rho}{T}^{\lambda\nu}+\delta^{\lambda\rho}{T}^{\mu\nu}\,,\\\
&\left[{T}^{\lambda\mu},{U}^{\nu\rho}\right]=\delta^{\mu\rho}{U}^{\nu\lambda}-\delta^{\lambda\rho}{U}^{\nu\mu}\,,\\\
&\left[{T}^{\prime\lambda\mu},{T}^{\prime\nu\rho}\right]=\delta^{\mu\nu}{T}^{\prime\lambda\rho}-\delta^{\lambda\nu}{T}^{\prime\mu\rho}-\delta^{\mu\rho}{T}^{\prime\lambda\nu}+\delta^{\lambda\rho}{T}^{\prime\mu\nu}\,,\\\
&\left[{T}^{\prime\lambda\mu},{U}^{\nu\rho}\right]=\delta^{\mu\nu}{U}^{\lambda\rho}-\delta^{\lambda\nu}{U}^{\mu\rho}\,,\\\
&\left[{U}^{\lambda\mu},{U}^{\nu\rho}\right]=-\delta^{\lambda\nu}{T}^{\mu\rho}-\delta^{\mu\rho}{T}^{\prime\lambda\nu}\,,\\\
&\left[{T}^{\lambda\mu},{S}^{\nu\rho}\right]=\delta^{\mu\nu}{S}^{\lambda\rho}-\delta^{\lambda\nu}{S}^{\mu\rho}-\delta^{\mu\rho}{S}^{\lambda\nu}+\delta^{\lambda\rho}{S}^{\mu\nu}\,,\\\
&\left[{T}^{\lambda\mu},{V}^{\nu\rho}\right]=\delta^{\mu\rho}{V}^{\nu\lambda}-\delta^{\lambda\rho}{V}^{\nu\mu}\,,\\\
&\left[{T}^{\prime\lambda\mu},{S}^{\prime\nu\rho}\right]=\delta^{\mu\nu}{S}^{\prime\lambda\rho}-\delta^{\lambda\nu}{S}^{\prime\mu\rho}-\delta^{\mu\rho}{S}^{\prime\lambda\nu}+\delta^{\lambda\rho}{S}^{\prime\mu\nu}\,,\\\
&\left[{T}^{\prime\lambda\mu},{V}^{\nu\rho}\right]=\delta^{\mu\nu}{V}^{\lambda\rho}-\delta^{\lambda\nu}{V}^{\mu\rho}\,,\\\
&\left[{U}^{\lambda\mu},{S}^{\nu\rho}\right]=-\delta^{\mu\rho}{V}^{\lambda\nu}+\delta^{\mu\nu}{V}^{\lambda\rho}\,,\\\
&\left[{U}^{\lambda\mu},{S}^{\prime\nu\rho}\right]=-\delta^{\lambda\rho}{V}^{\nu\mu}+\delta^{\lambda\nu}{V}^{\rho\mu}\,,\\\
&\left[{U}^{\lambda\mu},{V}^{\nu\rho}\right]=-\delta^{\lambda\nu}{S}^{\mu\rho}-\delta^{\mu\rho}{S}^{\prime\lambda\nu}\,,\\\
&\left[J_{ab},Q^{\pm\,\lambda}_{\alpha}\right]=-\frac{1}{2}\left(\Gamma_{ab}Q^{\pm\,\lambda}\right)_{\alpha}\,,\quad\quad\left[P_{a},Q^{\pm\,\lambda}_{\alpha}\right]=-\frac{1}{2\ell}\left(\Gamma_{a}Q^{\mp\,\lambda}\right)_{\alpha}\,,\\\
&\left[{T}^{\lambda\mu},{Q}^{\pm\,\nu}_{\alpha}\right]=\frac{1}{2}\left[\delta^{\mu\nu}\left({Q}^{+\,\lambda}_{\alpha}+{Q}^{-\,\lambda}_{\alpha}\right)-\delta^{\lambda\nu}\left({Q}^{+\,\mu}_{\alpha}+{Q}^{-\,\mu}_{\alpha}\right)\right]\,,\\\
&\left[{T}^{\prime\lambda\mu},{Q}^{\pm\,\nu}_{\alpha}\right]=\pm\frac{1}{2}\left[\delta^{\mu\nu}\left({Q}^{+\,\lambda}_{\alpha}-{Q}^{-\,\lambda}_{\alpha}\right)-\delta^{\lambda\nu}\left({Q}^{+\,\mu}_{\alpha}-{Q}^{-\,\mu}_{\alpha}\right)\right]\,,\\\
&\left[{U}^{\lambda\mu},{Q}^{\pm\,\nu}_{\alpha}\right]=\mp\frac{1}{2}\left(\Gamma_{0}\right)_{\alpha\beta}\left[\delta^{\lambda\nu}\left({Q}^{+\,\mu}_{\beta}+{Q}^{-\,\
\mu}_{\beta}\right)\pm\delta^{\mu\nu}\left({Q}^{+\,\lambda}_{\beta}-{Q}^{-\,\lambda}_{\beta}\right)\right]\,,\end{split}$
(4.14)
$\begin{split}&\\{{Q}^{+\,\lambda}_{\alpha},{Q}^{+\,\mu}_{\beta}\\}=\left(\Gamma^{0}C\right)_{\alpha\beta}\left(\delta^{\lambda\mu}{H}-{V}^{(\lambda\mu)}\right)+C_{\alpha\beta}{Y}^{[\lambda\mu]}\,,\\\
&\\{{Q}^{+\,\lambda}_{\alpha},{Q}^{-\,\mu}_{\beta}\\}=-\frac{1}{\ell}\delta^{\lambda\mu}\left(\Gamma^{a0}C\right)_{\alpha\beta}{K}_{a}+C_{\alpha\beta}{Y}^{\prime[\lambda\mu]}-\left(\Gamma^{0}C\right)_{\alpha\beta}{V}^{[\lambda\mu]}\,,\\\
&\\{{Q}^{-\,\lambda}_{\alpha},{Q}^{-\,\mu}_{\beta}\\}=\left(\Gamma^{0}C\right)_{\alpha\beta}\left(\delta^{\lambda\mu}{H}+{V}^{(\lambda\mu)}\right)+C_{\alpha\beta}{Y}^{[\lambda\mu]}\,.\end{split}$

Notice that if we restrict ourselves to the special case $\mathcal{N}=(2,0)$,
that is $x=1$, after some algebraic calculations, exploiting the definitions
(4.6), (4.8), (4.10), and the symmetry properties (4.7), we exactly reproduce
the $\mathcal{N}=(2,0)$ AdS Carroll superalgebra obtained in Section 2, given
by (2.10).888In particular, when restricting ourselves to $\mathcal{N}=(2,0)$,
namely $x=1$, we have $\lambda,\mu,\ldots=1$ and $T^{\lambda\mu}=T^{11}=0$,
$T^{\prime\lambda\mu}=T^{\prime 11}=0$, $S^{\lambda\mu}=S^{11}=0$,
$S^{\prime\lambda\mu}=S^{\prime 11}=0$,
$U^{\lambda\mu}=U^{11}=-T^{12}=-\epsilon^{12}T=-T$,
$V^{\lambda\mu}=V^{11}=-S^{12}=-\epsilon^{12}S=-S$, and (4.5) restricts itself
to (2.8) (when performing the Carroll limit, we also remove the tilde symbol
on the generators); then, one can show that the superalgebra (4.14) reduces to
the one given by (2.10).

In the sequel, we will construct a CS action in $2+1$ dimensions invariant
under (4.14).

###  4.2 $(\mathcal{N},0)$ AdS Carroll supergravity

We can now move to the formulation of a three-dimensional CS supergravity
action invariant under the superalgebra (4.14). We call this action
$(\mathcal{N},0)$ AdS Carroll CS supergravity action (where, in our analysis,
$\mathcal{N}$ is even).

To this aim, let us start by introducing the connection $1$-form $A$
associated with the superalgebra (4.14), that is

$\begin{split}A&=\frac{1}{2}\omega^{ab}J_{ab}+k^{a}K_{a}+V^{a}P_{a}+hH\\\
&+\frac{1}{2}t^{\lambda\mu}T_{\lambda\mu}+\frac{1}{2}t^{\prime\lambda\mu}T^{\prime}_{\lambda\mu}+u^{\lambda\mu}U_{\lambda\mu}+\frac{1}{2}s^{\lambda\mu}S_{\lambda\mu}+\frac{1}{2}s^{\prime\lambda\mu}S^{\prime}_{\lambda\mu}+v^{\lambda\mu}V_{\lambda\mu}\\\
&+\psi^{+}_{\lambda}Q^{+\,\lambda}+\psi^{-}_{\lambda}Q^{-\,\lambda}\,,\end{split}$
(4.15)

where $\omega^{ab}$, $k^{a}$, $V^{a}$, $h$, $t^{\lambda\mu}$,
$t^{\prime\lambda\mu}$, $u^{\lambda\mu}$, $s^{\lambda\mu}$,
$s^{\prime\lambda\mu}$, $v^{\lambda\mu}$, $\psi^{+}_{\lambda}$, and
$\psi^{-}_{\lambda}$ are the $1$-form fields dual to the generators $J_{ab}$,
$K_{a}$, $P_{a}$, $H$, $T_{\lambda\mu}$, $T^{\prime}_{\lambda\mu}$,
$U_{\lambda\mu}$, $S_{\lambda\mu}$, $S^{\prime}_{\lambda\mu}$,
$V_{\lambda\mu}$, $Q^{+\,\lambda}$, and $Q^{-\,\lambda}$, respectively.

The corresponding curvature $2$-form $F$ is

$\begin{split}F&=\frac{1}{2}\mathcal{R}^{ab}J_{ab}+\mathcal{K}^{a}K_{a}+R^{a}P_{a}+\mathcal{H}H\\\
&+\frac{1}{2}\mathcal{T}^{\lambda\mu}T_{\lambda\mu}+\frac{1}{2}\mathcal{T}^{\prime\lambda\mu}T^{\prime}_{\lambda\mu}+\mathcal{U}^{\lambda\mu}U_{\lambda\mu}+\frac{1}{2}\mathcal{S}^{\lambda\mu}S_{\lambda\mu}+\frac{1}{2}\mathcal{S}^{\prime\lambda\mu}S^{\prime}_{\lambda\mu}+\mathcal{V}^{\lambda\mu}V_{\lambda\mu}\\\
&+\nabla\psi^{+}_{\lambda}Q^{+\,\lambda}+\nabla\psi^{-}_{\lambda}Q^{-\,\lambda}\,,\end{split}$
(4.16)

with

$\begin{split}\mathcal{R}^{ab}&=d\omega^{ab}+\frac{1}{\ell^{2}}V^{a}V^{b}=R^{ab}+\frac{1}{\ell^{2}}V^{a}V^{b}\,,\\\
\mathcal{K}^{a}&=dk^{a}+\omega^{a}_{\phantom{a}b}k^{b}+\frac{1}{\ell^{2}}V^{a}h+\frac{1}{\ell}\bar{\psi}^{+\,\lambda}\Gamma^{a0}\psi^{-\,\lambda}=\mathfrak{K}^{a}+\frac{1}{\ell^{2}}V^{a}h+\frac{1}{\ell}\bar{\psi}^{+\,\lambda}\Gamma^{a0}\psi^{-\,\lambda}\,,\\\
R^{a}&=dV^{a}+\omega^{a}_{\phantom{a}b}V^{b}\,,\\\
\mathcal{H}&=dh+V^{a}k_{a}-\frac{1}{2}\bar{\psi}^{+\,\lambda}\Gamma^{0}\psi^{+\,\lambda}-\frac{1}{2}\bar{\psi}^{-\,\lambda}\Gamma^{0}\psi^{-\,\lambda}=\mathfrak{H}-\frac{1}{2}\bar{\psi}^{+\,\lambda}\Gamma^{0}\psi^{+\,\lambda}-\frac{1}{2}\bar{\psi}^{-\,\lambda}\Gamma^{0}\psi^{-\,\lambda}\,,\\\
\mathcal{T}^{\lambda\mu}&=dt^{\lambda\mu}+{t^{\lambda}}_{\nu}t^{\nu\mu}+{u^{\prime[\lambda}}_{\nu}u^{\nu|\mu]}\,,\\\
\mathcal{T}^{\prime\lambda\mu}&=dt^{\prime\lambda\mu}+{t^{\prime\lambda}}_{\nu}t^{\prime\nu\mu}+{u^{[\lambda}}_{\nu}u^{\prime\nu|\mu]}\,,\\\
\mathcal{U}^{\lambda\mu}&=du^{\lambda\mu}+{u^{\lambda}}_{\nu}t^{\nu\mu}+{t^{\prime\lambda}}_{\nu}u^{\nu\mu}\,,\\\
\mathcal{S}^{\lambda\mu}&=ds^{\lambda\mu}+2{t^{\lambda}}_{\nu}s^{\nu\mu}+2{u^{\prime[\lambda}}_{\nu}v^{\nu|\mu]}-\frac{1}{2}\bar{\psi}^{+\,\lambda}\psi^{+\,\mu}-\frac{1}{2}\bar{\psi}^{-\,\lambda}\psi^{-\,\mu}-\bar{\psi}^{+\,[\lambda}\psi^{-\,\mu]}\,,\\\
\mathcal{S}^{\prime\lambda\mu}&=ds^{\prime\lambda\mu}+2{t^{\prime\lambda}}_{\nu}s^{\prime\nu\mu}+2{u^{[\lambda}}_{\nu}v^{\prime\nu|\mu]}-\frac{1}{2}\bar{\psi}^{+\,\lambda}\psi^{+\,\mu}-\frac{1}{2}\bar{\psi}^{-\,\lambda}\psi^{-\,\mu}+\bar{\psi}^{+\,[\lambda}\psi^{-\,\mu]}\,,\\\
\mathcal{V}^{\lambda\mu}&=dv^{\lambda\mu}+{v^{\lambda}}_{\nu}t^{\nu\mu}+{t^{\prime\lambda}}_{\nu}v^{\nu\mu}+{u^{\lambda}}_{\nu}s^{\nu\mu}+{s^{\prime\lambda}}_{\nu}u^{\nu\mu}+\frac{1}{2}\bar{\psi}^{+\,\lambda}\Gamma^{0}\psi^{+\,\mu}-\frac{1}{2}\bar{\psi}^{-\,\lambda}\Gamma^{0}\psi^{-\,\mu}\\\
&+\bar{\psi}^{+\,[\lambda}\Gamma^{0}\psi^{-\,\mu]}\,,\\\
\nabla\psi^{+\,\lambda}&=d\psi^{+\,\lambda}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\psi^{+\,\lambda}+\frac{1}{2\ell}V^{a}\Gamma_{a}\psi^{-\,\lambda}+\frac{1}{2}t^{\lambda\mu}\psi^{+}_{\mu}+\frac{1}{2}t^{\lambda\mu}\psi^{-}_{\mu}+\frac{1}{2}t^{\prime\lambda\mu}\psi^{+}_{\mu}-\frac{1}{2}t^{\prime\lambda\mu}\psi^{-}_{\mu}\\\
&+u^{(\lambda\mu)}\Gamma_{0}\psi^{+}_{\mu}+u^{[\lambda\mu]}\Gamma_{0}\psi^{-}_{\mu}\,,\\\
\nabla\psi^{-\,\lambda}&=d\psi^{-\,\lambda}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\psi^{-\,\lambda}+\frac{1}{2\ell}V^{a}\Gamma_{a}\psi^{+\,\lambda}+\frac{1}{2}t^{\lambda\mu}\psi^{+}_{\mu}+\frac{1}{2}t^{\lambda\mu}\psi^{-}_{\mu}-\frac{1}{2}t^{\prime\lambda\mu}\psi^{+}_{\mu}+\frac{1}{2}t^{\prime\lambda\mu}\psi^{-}_{\mu}\\\
&-u^{(\lambda\mu)}\Gamma_{0}\psi^{-}_{\mu}-u^{[\lambda\mu]}\Gamma_{0}\psi^{+}_{\mu}\,,\end{split}$
(4.17)

where we have used

$\begin{split}&u^{\lambda\mu}=t^{\lambda+x\ \mu}=-t^{\mu\
\lambda+x}=-u^{\prime\mu\lambda}\,,\\\ &v^{\lambda\mu}=s^{\lambda+x\
\mu}=-s^{\mu\ \lambda+x}=-v^{\prime\mu\lambda}\,,\end{split}$ (4.18)

and where we have

$\begin{split}&u^{(\lambda\mu)}=\frac{1}{2}\left(u^{\lambda\mu}+u^{\mu\lambda}\right)\,,\\\
&u^{[\lambda\mu]}=\frac{1}{2}\left(u^{\lambda\mu}-u^{\mu\lambda}\right)\,.\end{split}$
(4.19)

In order to develop a CS action invariant under (4.14), we have to consider
the non-vanishing components of the invariant tensor in (4.4), decompose the
indices as in (2.5), exploit (4.5), (4.6), (4.7), and rescale not only the
generators in compliance with (4.13) but also the coefficients appearing in
(4.4) as in (2.16). Consequently, in the ultra-relativistic limit
$\sigma\rightarrow\infty$ we get the following non-vanishing components of an
invariant tensor for (4.14):

$\begin{split}&\langle
J_{ab}J_{cd}\rangle=\alpha_{0}\left(\delta_{ad}\delta_{bc}-\delta_{ac}\delta_{bd}\right)\,,\\\
&\langle J_{ab}H\rangle=\alpha_{1}\epsilon_{ab}\,,\\\ &\langle
K_{a}P_{b}\rangle=-\alpha_{1}\epsilon_{ab}\,,\\\ &\langle
P_{a}P_{b}\rangle=\frac{\alpha_{0}}{\ell^{2}}\delta_{ab}\,,\\\
&\langle{T}^{\lambda\mu}{T}^{\nu\rho}\rangle=\langle{T}^{\prime\lambda\mu}{T}^{\prime\nu\rho}\rangle=2\alpha_{0}\left(\delta^{\lambda\rho}\delta^{\nu\mu}-\delta^{\lambda\nu}\delta^{\rho\mu}\right)\,,\\\
&\langle{U}^{\lambda\mu}{U}^{\nu\rho}\rangle=-2\alpha_{0}\delta^{\lambda\nu}\delta^{\rho\mu}\,,\\\
&\langle{T}^{\lambda\mu}{S}^{\nu\rho}\rangle=\langle{T}^{\prime\lambda\mu}{S}^{\prime\nu\rho}\rangle=-2\alpha_{1}\left(\delta^{\lambda\rho}\delta^{\nu\mu}-\delta^{\lambda\nu}\delta^{\rho\mu}\right)\,,\\\
&\langle{U}^{\lambda\mu}{V}^{\nu\rho}\rangle=2\alpha_{1}\delta^{\lambda\nu}\delta^{\rho\mu}\,,\\\
&\langle Q^{+\,\lambda}_{\alpha}Q^{+\,\mu}_{\beta}\rangle=\langle
Q^{-\,\lambda}_{\alpha}Q^{-\,\mu}_{\beta}\rangle=2\alpha_{1}C_{\alpha\beta}\delta^{\lambda\mu}\,.\end{split}$
(4.20)

The invariant tensor for (4.14) above is non-degenerate if $\alpha_{1}\neq 0$.

Then, substituting the connection $1$-form in (4.15) and the non-zero
components of the invariant tensor (4.20) into (2.11), we end up with the
three-dimensional $(\mathcal{N},0)$ AdS Carroll CS supergravity action, which
reads

$\begin{split}I^{(\mathcal{N},0)}_{CS}&=\frac{k}{4\pi}\int_{\mathcal{M}}\Bigg{\\{}\frac{\alpha_{0}}{2}\bigg{(}\omega^{a}_{\phantom{a}b}R^{b}_{\phantom{b}a}+\frac{2}{\ell^{2}}V^{a}R_{a}+2{t^{\lambda}}_{\mu}{dt^{\mu}}_{\lambda}+\frac{4}{3}{t^{\lambda}}_{\mu}{t^{\mu}}_{\nu}{t^{\nu}}_{\lambda}+2{t^{\prime\lambda}}_{\mu}{dt^{\prime\mu}}_{\lambda}+\frac{4}{3}{t^{\prime\lambda}}_{\mu}{t^{\prime\mu}}_{\nu}{t^{\prime\nu}}_{\lambda}\\\
&+4{u^{\lambda}}_{\mu}{du^{\prime\mu}}_{\lambda}-4t_{\lambda\mu}{u^{\prime\lambda}}_{\nu}u^{\nu\mu}-4t^{\prime}_{\lambda\mu}{u^{\lambda}}_{\nu}u^{\prime\nu\mu}\bigg{)}+\alpha_{1}\bigg{[}\epsilon_{ab}R^{ab}h-2\epsilon_{ab}\mathfrak{K}^{a}V^{b}+\frac{1}{\ell^{2}}\epsilon_{ab}V^{a}V^{b}\\\
&-2{t^{\lambda}}_{\mu}\left({ds^{\mu}}_{\lambda}+{t^{\mu}}_{\nu}{s^{\nu}}_{\lambda}\right)-2{t^{\prime\lambda}}_{\mu}\left({ds^{\prime\mu}}_{\lambda}+{t^{\prime\mu}}_{\nu}{s^{\prime\nu}}_{\lambda}\right)-4{u^{\lambda}}_{\mu}{dv^{\prime\mu}}_{\lambda}-2{u^{\prime\lambda}}_{\mu}{u^{\mu}}_{\nu}{s^{\nu}}_{\lambda}-2{u^{\lambda}}_{\mu}{u^{\prime\mu}}_{\nu}{s^{\prime\nu}}_{\lambda}\\\
&-4{u^{\prime\lambda}}_{\mu}{v^{\mu}}_{\nu}{t^{\nu}}_{\lambda}-4{u^{\lambda}}_{\mu}{v^{\prime\mu}}_{\nu}{t^{\prime\nu}}_{\lambda}+2\bar{\psi}^{+\,\lambda}\nabla\psi^{+\,\lambda}+2\bar{\psi}^{-\,\lambda}\nabla\psi^{-\,\lambda}\bigg{]}\\\
&-d\left(\frac{\alpha_{1}}{2}\epsilon_{ab}\omega^{ab}h-\alpha_{1}\epsilon_{ab}k^{a}V^{b}+\alpha_{1}{t^{\lambda}}_{\mu}{s^{\mu}}_{\lambda}+\alpha_{1}{t^{\prime\lambda}}_{\mu}{s^{\prime\mu}}_{\lambda}+2\alpha_{1}{u^{\lambda}}_{\mu}{v^{\prime\mu}}_{\lambda}\right)\Bigg{\\}}\,,\end{split}$
(4.21)

where we have also exploited (4.18). The action (4.21) has been written in
terms of the curvatures appearing in (4.17) and it involves two coupling
constants, that are $\alpha_{0}$ and $\alpha_{1}$. Up to boundary terms,
(4.21) can be rewritten as

$\begin{split}I^{(\mathcal{N},0)}_{CS}&=\frac{k}{4\pi}\int_{\mathcal{M}}\Bigg{\\{}\frac{\alpha_{0}}{2}\bigg{(}\omega^{a}_{\phantom{a}b}R^{b}_{\phantom{b}a}+\frac{2}{\ell^{2}}V^{a}R_{a}+2{t^{\lambda}}_{\mu}{dt^{\mu}}_{\lambda}+\frac{4}{3}{t^{\lambda}}_{\mu}{t^{\mu}}_{\nu}{t^{\nu}}_{\lambda}+2{t^{\prime\lambda}}_{\mu}{dt^{\prime\mu}}_{\lambda}+\frac{4}{3}{t^{\prime\lambda}}_{\mu}{t^{\prime\mu}}_{\nu}{t^{\prime\nu}}_{\lambda}\\\
&+4{u^{\lambda}}_{\mu}{du^{\prime\mu}}_{\lambda}-4t_{\lambda\mu}{u^{\prime\lambda}}_{\nu}u^{\nu\mu}-4t^{\prime}_{\lambda\mu}{u^{\lambda}}_{\nu}u^{\prime\nu\mu}\bigg{)}+\alpha_{1}\bigg{[}\epsilon_{ab}R^{ab}h-2\epsilon_{ab}\mathfrak{K}^{a}V^{b}+\frac{1}{\ell^{2}}\epsilon_{ab}V^{a}V^{b}\\\
&-2{t^{\lambda}}_{\mu}\left({ds^{\mu}}_{\lambda}+{t^{\mu}}_{\nu}{s^{\nu}}_{\lambda}\right)-2{t^{\prime\lambda}}_{\mu}\left({ds^{\prime\mu}}_{\lambda}+{t^{\prime\mu}}_{\nu}{s^{\prime\nu}}_{\lambda}\right)-4{u^{\lambda}}_{\mu}{dv^{\prime\mu}}_{\lambda}-2{u^{\prime\lambda}}_{\mu}{u^{\mu}}_{\nu}{s^{\nu}}_{\lambda}-2{u^{\lambda}}_{\mu}{u^{\prime\mu}}_{\nu}{s^{\prime\nu}}_{\lambda}\\\
&-4{u^{\prime\lambda}}_{\mu}{v^{\mu}}_{\nu}{t^{\nu}}_{\lambda}-4{u^{\lambda}}_{\mu}{v^{\prime\mu}}_{\nu}{t^{\prime\nu}}_{\lambda}+2\bar{\psi}^{+\,\lambda}\nabla\psi^{+\,\lambda}+2\bar{\psi}^{-\,\lambda}\nabla\psi^{-\,\lambda}\bigg{]}\Bigg{\\}}\,.\end{split}$
(4.22)

The contribution proportional to $\alpha_{0}$ corresponds to the exotic
Lagrangian, and, in the present case, it involves, besides the Lorentz and
torsional terms, also pieces including the $1$-form fields $t^{\lambda\mu}$,
$t^{\prime\lambda\mu}$, and $u^{\lambda\mu}$. On the other hand, the
contribution proportional to $\alpha_{1}$ also includes terms involving the
$1$-form fields $s^{\lambda\mu}$, $s^{\prime\lambda\mu}$, and
$v^{\lambda\mu}$, plus the spinor $1$-form fields $\psi^{+\,\lambda}$ and
$\psi^{-\,\lambda}$.

The CS action (4.21) is invariant by construction under (4.14), and the local
gauge transformations $\delta_{\lambda}A=d\lambda+\left[A,\lambda\right]$ with
gauge parameter

$\begin{split}\lambda&=\frac{1}{2}\lambda^{ab}J_{ab}+\kappa^{a}K_{a}+\lambda^{a}P_{a}+\tau
H+\frac{1}{2}\varrho^{\lambda\mu}t_{\lambda\mu}+\frac{1}{2}\varrho^{\prime\lambda\mu}t^{\prime}_{\lambda\mu}+\varphi^{\lambda\mu}u_{\lambda\mu}\\\
&+\frac{1}{2}\vartheta^{\lambda\mu}s_{\lambda\mu}+\frac{1}{2}\vartheta^{\prime\lambda\mu}s_{\lambda\mu}+\varsigma^{\lambda\mu}v_{\lambda\mu}+\varepsilon^{+\,\lambda}Q^{+}_{\lambda}+\varepsilon^{-\,\lambda}Q^{-}_{\lambda}\end{split}$
(4.23)

are given by

$\begin{split}\delta\omega^{ab}&=d\lambda^{ab}+\frac{2}{\ell^{2}}V^{[a}\lambda^{b]}\,,\\\
\delta
k^{a}&=d\kappa^{a}-\lambda^{a}_{\phantom{a}b}k^{b}+\omega^{a}_{\phantom{a}b}\kappa^{b}-\frac{1}{\ell^{2}}\lambda^{a}h+\frac{1}{\ell^{2}}V^{a}\tau-\frac{1}{\ell}\bar{\varepsilon}^{+\,\lambda}\Gamma^{a0}\psi^{-\,\lambda}-\frac{1}{\ell}\bar{\varepsilon}^{-\,\lambda}\Gamma^{a0}\psi^{+\,\lambda}\,,\\\
\delta
V^{a}&=d\lambda^{a}-\lambda^{a}_{\phantom{a}b}V^{b}+\omega^{a}_{\phantom{a}b}\lambda^{b}\,,\\\
\delta
h&=d\tau-\lambda^{a}k_{a}+V^{a}\kappa_{a}+\bar{\varepsilon}^{+\,\lambda}\Gamma^{0}\psi^{+\,\lambda}+\bar{\varepsilon}^{-\,\lambda}\Gamma^{0}\psi^{-\,\lambda}\,,\\\
\delta
t^{\lambda\mu}&=d\varrho^{\lambda\mu}-2{\varrho^{[\lambda}}_{\nu}t^{\nu|\mu]}-2{\varphi^{\prime[\lambda}}_{\nu}u^{\nu|\mu]}\,,\\\
\delta
t^{\prime\lambda\mu}&=d\varrho^{\prime\lambda\mu}-2{\varrho^{\prime[\lambda}}_{\nu}t^{\prime\nu|\mu]}-2{\varphi^{[\lambda}}_{\nu}u^{\prime\nu|\mu]}\,,\\\
\delta
u^{\lambda\mu}&=d\varphi^{\lambda\mu}-{\varphi^{\lambda}}_{\nu}t^{\nu\mu}+{u^{\lambda}}_{\nu}\varrho^{\nu\mu}-{\varrho^{\prime\lambda}}_{\nu}u^{\nu\mu}+{t^{\prime\lambda}}_{\nu}\varphi^{\nu\mu}\,,\\\
\delta
s^{\lambda\mu}&=d\vartheta^{\lambda\mu}-2{\varrho^{[\lambda}}_{\nu}s^{\nu|\mu]}+2{t^{[\lambda}}_{\nu}\vartheta^{\nu|\mu]}-2{\varphi^{\prime[\lambda}}_{\nu}v^{\nu|\mu]}+2{u^{\prime[\lambda}}_{\nu}\varsigma^{\nu|\mu]}+\bar{\varepsilon}^{+\,[\lambda}\psi^{+\,\mu]}+\bar{\varepsilon}^{-\,[\lambda}\psi^{-\,\mu]}\\\
&+\bar{\varepsilon}^{+\,[\lambda}\psi^{-\,\mu]}+\bar{\varepsilon}^{-\,[\lambda}\psi^{+\,\mu]}\,,\\\
\delta
s^{\prime\lambda\mu}&=d\vartheta^{\prime\lambda\mu}-2{\varrho^{\prime[\lambda}}_{\nu}s^{\prime\nu|\mu]}+2{t^{\prime[\lambda}}_{\nu}\vartheta^{\prime\nu|\mu]}-2{\varphi^{[\lambda}}_{\nu}v^{\prime\nu|\mu]}+2{u^{[\lambda}}_{\nu}\varsigma^{\prime\nu|\mu]}+\bar{\varepsilon}^{+\,[\lambda}\psi^{+\,\mu]}+\bar{\varepsilon}^{-\,[\lambda}\psi^{-\,\mu]}\\\
&-\bar{\varepsilon}^{+\,[\lambda}\psi^{-\,\mu]}-\bar{\varepsilon}^{-\,[\lambda}\psi^{+\,\mu]}\,,\\\
\delta
v^{\lambda\mu}&=d\varsigma^{\lambda\mu}-{\varsigma^{\lambda}}_{\nu}t^{\nu\mu}+{v^{\lambda}}_{\nu}\varrho^{\nu\mu}-{\varrho^{\prime\lambda}}_{\nu}v^{\nu\mu}+{t^{\prime\lambda}}_{\nu}\varsigma^{\nu\mu}-{\varphi^{\lambda}}_{\nu}s^{\nu\mu}+{u^{\lambda}}_{\nu}\vartheta^{\nu\mu}-{\vartheta^{\prime\lambda}}_{\nu}u^{\nu\mu}+{s^{\prime\lambda}}_{\nu}\varphi^{\nu\mu}\\\
&-\bar{\varepsilon}^{+\,\lambda}\Gamma^{0}\psi^{+\,\mu}+\bar{\varepsilon}^{-\,\lambda}\Gamma^{0}\psi^{-\,\mu}-\bar{\varepsilon}^{+\,[\lambda}\Gamma^{0}\psi^{-\,\mu]}+\bar{\varepsilon}^{-\,[\lambda}\Gamma^{0}\psi^{+\,\mu]}\,,\\\
\delta\psi^{+\,\lambda}&=d\varepsilon^{+\,\lambda}-\frac{1}{4}\lambda^{ab}\Gamma_{ab}\psi^{+\,\lambda}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{+\,\lambda}-\frac{1}{2\ell}\lambda^{a}\Gamma_{a}\psi^{-\,\lambda}+\frac{1}{2\ell}V^{a}\Gamma_{a}\varepsilon^{-\,\lambda}\\\
&-\frac{1}{2}\varrho^{\lambda\mu}\psi^{+}_{\mu}+\frac{1}{2}t^{\lambda\mu}\varepsilon^{+}_{\mu}-\frac{1}{2}\varrho^{\lambda\mu}\psi^{-}_{\mu}+\frac{1}{2}t^{\lambda\mu}\varepsilon^{-}_{\mu}-\frac{1}{2}\varrho^{\prime\lambda\mu}\psi^{+}_{\mu}+\frac{1}{2}t^{\prime\lambda\mu}\varepsilon^{+}_{\mu}+\frac{1}{2}\varrho^{\prime\lambda\mu}\psi^{-}_{\mu}-\frac{1}{2}t^{\prime\lambda\mu}\varepsilon^{-}_{\mu}\\\
&-\varphi^{(\lambda\mu)}\Gamma_{0}\psi^{+}_{\mu}+u^{(\lambda\mu)}\Gamma_{0}\varepsilon^{+}_{\mu}-\varphi^{[\lambda\mu]}\Gamma_{0}\psi^{-}_{\mu}+u^{[\lambda\mu]}\Gamma_{0}\varepsilon^{-}_{\mu}\,,\\\
\delta\psi^{-\,\lambda}&=d\varepsilon^{-\,\lambda}-\frac{1}{4}\lambda^{ab}\Gamma_{ab}\psi^{-\,\lambda}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{-\,\lambda}-\frac{1}{2\ell}\lambda^{a}\Gamma_{a}\psi^{+\,\lambda}+\frac{1}{2\ell}V^{a}\Gamma_{a}\varepsilon^{+\,\lambda}\\\
&-\frac{1}{2}\varrho^{\lambda\mu}\psi^{+}_{\mu}+\frac{1}{2}t^{\lambda\mu}\varepsilon^{+}_{\mu}-\frac{1}{2}\varrho^{\lambda\mu}\psi^{-}_{\mu}+\frac{1}{2}t^{\lambda\mu}\varepsilon^{-}_{\mu}+\frac{1}{2}\varrho^{\prime\lambda\mu}\psi^{+}_{\mu}-\frac{1}{2}t^{\prime\lambda\mu}\varepsilon^{+}_{\mu}-\frac{1}{2}\varrho^{\prime\lambda\mu}\psi^{-}_{\mu}+\frac{1}{2}t^{\prime\lambda\mu}\varepsilon^{-}_{\mu}\\\
&+\varphi^{(\lambda\mu)}\Gamma_{0}\psi^{-}_{\mu}-u^{(\lambda\mu)}\Gamma_{0}\varepsilon^{-}_{\mu}+\varphi^{[\lambda\mu]}\Gamma_{0}\psi^{+}_{\mu}-u^{[\lambda\mu]}\Gamma_{0}\varepsilon^{+}_{\mu}\,,\end{split}$
(4.24)

where we have also used the properties and definitions

$\begin{split}&\varphi^{\lambda\mu}=-\varphi^{\prime\mu\lambda}\,,\\\
&\varphi^{(\lambda\mu)}\equiv\frac{1}{2}\left(\varphi^{\lambda\mu}+\varphi^{\mu\lambda}\right)\,,\\\
&\varphi^{[\lambda\mu]}=\frac{1}{2}\left(\varphi^{\lambda\mu}-\varphi^{\mu\lambda}\right)\,.\end{split}$
(4.25)

Restricting ourselves to supersymmetry, we get the following supersymmetry
transformation laws:

$\begin{split}\delta\omega^{ab}&=0\,,\\\ \delta
k^{a}&=-\frac{1}{\ell}\bar{\varepsilon}^{+\,\lambda}\Gamma^{a0}\psi^{-\,\lambda}-\frac{1}{\ell}\bar{\varepsilon}^{-\,\lambda}\Gamma^{a0}\psi^{+\,\lambda}\,,\\\
\delta V^{a}&=0\,,\\\ \delta
h&=\bar{\varepsilon}^{+\,\lambda}\Gamma^{0}\psi^{+\,\lambda}+\bar{\varepsilon}^{-\,\lambda}\Gamma^{0}\psi^{-\,\lambda}\,,\\\
\delta t^{\lambda\mu}&=0\,,\quad\quad\delta
t^{\prime\lambda\mu}=0\,,\quad\quad\delta u^{\lambda\mu}=0\,,\\\ \delta
s^{\lambda\mu}&=\bar{\varepsilon}^{+\,[\lambda}\psi^{+\,\mu]}+\bar{\varepsilon}^{-\,[\lambda}\psi^{-\,\mu]}+\bar{\varepsilon}^{+\,[\lambda}\psi^{-\,\mu]}+\bar{\varepsilon}^{-\,[\lambda}\psi^{+\,\mu]}\,,\\\
\delta
s^{\prime\lambda\mu}&=\bar{\varepsilon}^{+\,[\lambda}\psi^{+\,\mu]}+\bar{\varepsilon}^{-\,[\lambda}\psi^{-\,\mu]}-\bar{\varepsilon}^{+\,[\lambda}\psi^{-\,\mu]}-\bar{\varepsilon}^{-\,[\lambda}\psi^{+\,\mu]}\,,\\\
\delta
v^{\lambda\mu}&=-\bar{\varepsilon}^{+\,\lambda}\Gamma^{0}\psi^{+\,\mu}+\bar{\varepsilon}^{-\,\lambda}\Gamma^{0}\psi^{-\,\mu}-\bar{\varepsilon}^{+\,[\lambda}\Gamma^{0}\psi^{-\,\mu]}+\bar{\varepsilon}^{-\,[\lambda}\Gamma^{0}\psi^{+\,\mu]}\,,\\\
\delta\psi^{+\,\lambda}&=d\varepsilon^{+\,\lambda}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{+\,\lambda}+\frac{1}{2\ell}V^{a}\Gamma_{a}\varepsilon^{-\,\lambda}+\frac{1}{2}t^{\lambda\mu}\varepsilon^{+}_{\mu}+\frac{1}{2}t^{\lambda\mu}\varepsilon^{-}_{\mu}+\frac{1}{2}t^{\prime\lambda\mu}\varepsilon^{+}_{\mu}-\frac{1}{2}t^{\prime\lambda\mu}\varepsilon^{-}_{\mu}\\\
&+u^{(\lambda\mu)}\Gamma_{0}\varepsilon^{+}_{\mu}+u^{[\lambda\mu]}\Gamma_{0}\varepsilon^{-}_{\mu}\,,\\\
\delta\psi^{-\,\lambda}&=d\varepsilon^{-\,\lambda}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{-\,\lambda}+\frac{1}{2\ell}V^{a}\Gamma_{a}\varepsilon^{+\,\lambda}+\frac{1}{2}t^{\lambda\mu}\varepsilon^{+}_{\mu}+\frac{1}{2}t^{\lambda\mu}\varepsilon^{-}_{\mu}-\frac{1}{2}t^{\prime\lambda\mu}\varepsilon^{+}_{\mu}+\frac{1}{2}t^{\prime\lambda\mu}\varepsilon^{-}_{\mu}\\\
&-u^{(\lambda\mu)}\Gamma_{0}\varepsilon^{-}_{\mu}-u^{[\lambda\mu]}\Gamma_{0}\varepsilon^{+}_{\mu}\,.\end{split}$
(4.26)

Finally, one can prove that from the variation of the action (4.21) with
respect to the $1$-form fields $\omega^{ab}$, $k^{a}$, $V^{a}$, $h$,
$t^{\lambda\mu}$, $t^{\prime\lambda\mu}$, $u^{\lambda\mu}$, $s^{\lambda\mu}$,
$s^{\prime\lambda\mu}$, $v^{\lambda\mu}$, $\psi^{+\,\lambda}$, and
$\psi^{-\,\lambda}$, we get, respectively, the equations of motion

$\begin{split}\delta\omega^{ab}&:\quad\alpha_{0}\mathcal{R}^{ab}+\alpha_{1}\epsilon^{ab}\mathcal{H}=0\,,\\\
\delta k^{a}&:\quad\alpha_{1}R^{a}=0\,,\\\ \delta
V^{a}&:\quad\frac{\alpha_{0}}{\ell^{2}}R^{a}+2\alpha_{1}\epsilon_{ab}\mathcal{K}^{b}=0\,,\\\
\delta h&:\quad\alpha_{1}\mathcal{R}^{ab}=0\,,\\\ \delta
t^{\lambda\mu}&:\quad\alpha_{0}\mathcal{T}^{\lambda\mu}+\alpha_{1}\mathcal{S}^{\lambda\mu}=0\,,\\\
\delta
t^{\prime\lambda\mu}&:\quad\alpha_{0}\mathcal{T}^{\prime\lambda\mu}+\alpha_{1}\mathcal{S}^{\prime\lambda\mu}=0\,,\\\
\delta
u^{\lambda\mu}&:\quad-\alpha_{0}\mathcal{U}^{\lambda\mu}+\alpha_{1}\mathcal{V}^{\lambda\mu}=0\,,\\\
\delta s^{\lambda\mu}&:\quad\alpha_{1}\mathcal{T}^{\lambda\mu}=0\,,\\\ \delta
s^{\prime\lambda\mu}&:\quad\alpha_{1}\mathcal{T}^{\prime\lambda\mu}=0\,,\\\
\delta v^{\lambda\mu}&:\quad\alpha_{1}\mathcal{U}^{\lambda\mu}=0\,,\\\
\delta\psi^{+\,\lambda}&:\quad\alpha_{1}\nabla\psi^{+\,\lambda}=0\,,\\\
\delta\psi^{-\,\lambda}&:\quad\alpha_{1}\nabla\psi^{-\,\lambda}=0\,,\end{split}$
(4.27)

written up to boundary contributions. We observe that, for $\alpha_{1}\neq 0$,
the equations (4.27) reduce precisely to the vanishing of the
$(\mathcal{N},0)$ super-AdS Carroll curvature $2$-forms given in (4.16), that
is to say

$\begin{split}&\mathcal{R}^{ab}=0\,,\quad\mathcal{K}^{a}=0\,,\quad
R^{a}=0\,,\quad\mathcal{H}=0\,,\\\
&\mathcal{T}^{\lambda\mu}=0\,,\quad\mathcal{T}^{\prime\lambda\mu}=0\,,\quad\mathcal{U}^{\lambda\mu}=0\,,\\\
&\mathcal{S}^{\lambda\mu}=0\,,\quad\mathcal{S}^{\prime\lambda\mu}=0\,,\quad\mathcal{V}^{\lambda\mu}=0\,,\\\
&\nabla\psi^{+\,\lambda}=0\,,\quad\nabla\psi^{-\,\lambda}=0\,.\end{split}$
(4.28)

Also in this case, we notice that $\alpha_{1}\neq 0$ is a sufficient condition
to recover (4.28), which means that we can consistently impose $\alpha_{0}=0$
in the CS action (4.21), making the exotic term disappear.

Let us also mention that, restricting ourselves to the special case
$\mathcal{N}=(2,0)$, that is $x=1$, after some algebraic calculations, we
exactly reproduce the results of Section 2, that is to say, in fact, the
$(2,0)$ AdS Carroll supergravity theory.

##  5 $(p,q)$ AdS Carroll supergravity theories in $2+1$ dimensions

We finally extend our analysis to the $(p,q)$ case, where both $p$ and $q$ are
integers $>0$. We first derive the $\mathcal{N}=(p,q)$ AdS Carroll
superalgebra as the Carrollian contraction of the direct sum of an
$\mathfrak{so}(p)\oplus\mathfrak{so}(q)$ algebra and
$\mathfrak{osp}(2|p)\otimes\mathfrak{osp}(2,q)$. This allows us to end up with
a non-degenerate invariant tensor in the ultra-relativistic limit, and to
consequently construct the three-dimensional CS supergravity theory invariant
under the aforesaid $\mathcal{N}=(p,q)$ AdS Carroll superalgebra.

###  5.1 $\mathcal{N}=(p,q)$ AdS Carroll superalgebra

Let us begin by considering the direct sum of the
$\mathfrak{osp}(2|p)\otimes\mathfrak{osp}(2,q)$ superalgebra and an
$\mathfrak{so}(p)\oplus\mathfrak{so}(q)$ algebra. The non-zero
(anti)commutation relations are the following ones (see [40]):

$\begin{split}&\left[\tilde{J}_{AB},\tilde{J}_{CD}\right]=\eta_{BC}\tilde{J}_{AD}-\eta_{AC}\tilde{J}_{BD}-\eta_{BD}\tilde{J}_{AC}+\eta_{AD}\tilde{J}_{BC}\,,\\\
&\left[\tilde{J}_{AB},\tilde{P}_{C}\right]=\eta_{BC}\tilde{P}_{A}-\eta_{AC}\tilde{P}_{B}\,,\quad\quad\left[\tilde{P}_{A},\tilde{P}_{B}\right]=\frac{1}{\ell^{2}}\tilde{J}_{AB}\,,\\\
&\left[\tilde{Z}^{ij},\tilde{Z}^{kl}\right]=\delta^{jk}\tilde{Z}^{il}-\delta^{ik}\tilde{Z}^{jl}-\delta^{jl}\tilde{Z}^{ik}+\delta^{il}\tilde{Z}^{jk}\,,\\\
&\left[\tilde{Z}^{IJ},\tilde{Z}^{KL}\right]=\delta^{JK}\tilde{Z}^{IL}-\delta^{IK}\tilde{Z}^{JL}-\delta^{JL}\tilde{Z}^{IK}+\delta^{IL}\tilde{Z}^{JK}\,,\\\
&\left[\tilde{S}^{ij},\tilde{S}^{kl}\right]=-\frac{1}{\ell}\left(\delta^{jk}\tilde{S}^{il}-\delta^{ik}\tilde{S}^{jl}-\delta^{jl}\tilde{S}^{ik}+\delta^{il}\tilde{S}^{jk}\right)\,,\\\
&\left[\tilde{S}^{IJ},\tilde{S}^{KL}\right]=-\frac{1}{\ell}\left(\delta^{JK}\tilde{S}^{IL}-\delta^{IK}\tilde{S}^{JL}-\delta^{JL}\tilde{S}^{IK}+\delta^{IL}\tilde{S}^{JK}\right)\,,\\\
&\left[\tilde{J}_{AB},\tilde{Q}^{i}_{\alpha}\right]=-\frac{1}{2}\left(\Gamma_{AB}\tilde{Q}^{i}\right)_{\alpha}\,,\quad\quad\left[\tilde{J}_{AB},\tilde{Q}^{I}_{\alpha}\right]=-\frac{1}{2}\left(\Gamma_{AB}\tilde{Q}^{I}\right)_{\alpha}\,,\\\
&\left[\tilde{P}_{A},\tilde{Q}^{i}_{\alpha}\right]=-\frac{1}{2\ell}\left(\Gamma_{A}\tilde{Q}^{i}\right)_{\alpha}\,,\quad\quad\left[\tilde{P}_{A},\tilde{Q}^{I}_{\alpha}\right]=\frac{1}{2\ell}\left(\Gamma_{A}\tilde{Q}^{I}\right)_{\alpha}\,,\end{split}$
(5.1)
$\begin{split}&\left[\tilde{Z}^{ij},\tilde{Q}^{k}_{\alpha}\right]=\delta^{jk}\tilde{Q}^{i}_{\alpha}-\delta^{ik}\tilde{Q}^{j}_{\alpha}\,,\quad\quad\left[\tilde{Z}^{IJ},\tilde{Q}^{K}_{\alpha}\right]=\delta^{JK}\tilde{Q}^{I}_{\alpha}-\delta^{IK}\tilde{Q}^{J}_{\alpha}\,,\\\
&\left\\{\tilde{Q}^{i}_{\alpha},\tilde{Q}^{j}_{\beta}\right\\}=\delta^{ij}\left[-\frac{1}{2\ell}\left(\Gamma^{AB}C\right)_{\alpha\beta}\tilde{J}_{AB}+\left(\Gamma^{A}C\right)_{\alpha\beta}\tilde{P}_{A}\right]+\frac{1}{\ell}C_{\alpha\beta}\tilde{Z}^{ij}\,,\\\
&\left\\{\tilde{Q}^{I}_{\alpha},\tilde{Q}^{J}_{\beta}\right\\}=\delta^{IJ}\left[\frac{1}{2\ell}\left(\Gamma^{AB}C\right)_{\alpha\beta}\tilde{J}_{AB}+\left(\Gamma^{A}C\right)_{\alpha\beta}\tilde{P}_{A}\right]-\frac{1}{\ell}C_{\alpha\beta}\tilde{Z}^{IJ}\,,\end{split}$

with $A,B,\ldots=0,1,2$, $i,j,\ldots=1,\ldots,p$, $I,J,\ldots=1,\ldots,q$, and
where $\tilde{Z}^{ij}=-\tilde{Z}^{ji}$, $\tilde{S}^{ij}=-\tilde{S}^{ij}$, and
$\tilde{Z}^{IJ}=-\tilde{Z}^{JI}$, $\tilde{S}^{IJ}=-\tilde{S}^{JI}$.

We now perform, on the same lines of [40], the redefinition

$\tilde{T}^{ij}\equiv\tilde{Z}^{ij}-\ell\tilde{S}^{ij}\,,\quad\tilde{T}^{IJ}\equiv\tilde{Z}^{IJ}-\ell\tilde{S}^{IJ}\,,$
(5.2)

which is analogous to the ones we have done in Sections 2 and 4. Consequently,
we can rewrite the (anti)commutation relations (5.1) as

$\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]=\frac{1}{\ell^{2}}\tilde{J}_{AB}\,,\\\
&\left[\tilde{T}^{ij},\tilde{T}^{kl}\right]=\delta^{jk}\tilde{T}^{il}-\delta^{ik}\tilde{T}^{jl}-\delta^{jl}\tilde{T}^{ik}+\delta^{il}\tilde{T}^{jk}\,,\\\
&\left[\tilde{T}^{IJ},\tilde{T}^{KL}\right]=\delta^{JK}\tilde{T}^{IL}-\delta^{IK}\tilde{T}^{JL}-\delta^{JL}\tilde{T}^{IK}+\delta^{IL}\tilde{T}^{JK}\,,\\\
&\left[\tilde{T}^{ij},\tilde{S}^{kl}\right]=\delta^{jk}\tilde{S}^{il}-\delta^{ik}\tilde{S}^{jl}-\delta^{jl}\tilde{S}^{ik}+\delta^{il}\tilde{S}^{jk}\,,\\\
&\left[\tilde{T}^{IJ},\tilde{S}^{KL}\right]=\delta^{JK}\tilde{S}^{IL}-\delta^{IK}\tilde{S}^{JL}-\delta^{JL}\tilde{S}^{IK}+\delta^{IL}\tilde{S}^{JK}\,,\\\
&\left[\tilde{S}^{ij},\tilde{S}^{kl}\right]=-\frac{1}{\ell}\left(\delta^{jk}\tilde{S}^{il}-\delta^{ik}\tilde{S}^{jl}-\delta^{jl}\tilde{S}^{ik}+\delta^{il}\tilde{S}^{jk}\right)\,,\\\
&\left[\tilde{S}^{IJ},\tilde{S}^{KL}\right]=-\frac{1}{\ell}\left(\delta^{JK}\tilde{S}^{IL}-\delta^{IK}\tilde{S}^{JL}-\delta^{JL}\tilde{S}^{IK}+\delta^{IL}\tilde{S}^{JK}\right)\,,\\\
&\left[\tilde{J}_{AB},\tilde{Q}^{i}_{\alpha}\right]=-\frac{1}{2}\left(\Gamma_{AB}\tilde{Q}^{i}\right)_{\alpha}\,,\\\
&\left[\tilde{J}_{AB},\tilde{Q}^{I}_{\alpha}\right]=-\frac{1}{2}\left(\Gamma_{AB}\tilde{Q}^{I}\right)_{\alpha}\,,\\\
&\left[\tilde{P}_{A},\tilde{Q}^{i}_{\alpha}\right]=-\frac{1}{2\ell}\left(\Gamma_{A}\tilde{Q}^{i}\right)_{\alpha}\,,\\\
&\left[\tilde{P}_{A},\tilde{Q}^{I}_{\alpha}\right]=\frac{1}{2\ell}\left(\Gamma_{A}\tilde{Q}^{I}\right)_{\alpha}\,,\\\
&\left[\tilde{T}^{ij},\tilde{Q}^{k}_{\alpha}\right]=\delta^{jk}\tilde{Q}^{i}_{\alpha}-\delta^{ik}\tilde{Q}^{j}_{\alpha}\,,\\\
&\left[\tilde{T}^{IJ},\tilde{Q}^{K}_{\alpha}\right]=\delta^{JK}\tilde{Q}^{I}_{\alpha}-\delta^{IK}\tilde{Q}^{J}_{\alpha}\,,\\\
&\left\\{\tilde{Q}^{i}_{\alpha},\tilde{Q}^{j}_{\beta}\right\\}=\delta^{ij}\left[-\frac{1}{2\ell}\left(\Gamma^{AB}C\right)_{\alpha\beta}\tilde{J}_{AB}+\left(\Gamma^{A}C\right)_{\alpha\beta}\tilde{P}_{A}\right]+C_{\alpha\beta}\left(\frac{1}{\ell}\tilde{T}^{ij}+\tilde{S}^{ij}\right)\,,\\\
&\left\\{\tilde{Q}^{I}_{\alpha},\tilde{Q}^{J}_{\beta}\right\\}=\delta^{IJ}\left[\frac{1}{2\ell}\left(\Gamma^{AB}C\right)_{\alpha\beta}\tilde{J}_{AB}+\left(\Gamma^{A}C\right)_{\alpha\beta}\tilde{P}_{A}\right]-C_{\alpha\beta}\left(\frac{1}{\ell}\tilde{T}^{IJ}+\tilde{S}^{IJ}\right)\,.\end{split}$
(5.3)

Notice that taking the flat limit $\ell\rightarrow\infty$ of (5.3), one
recovers the $\mathcal{N}=(p,q)$ Poincaré superalgebra of [40].

The non-vanishing components of an invariant tensor for the superalgebra
(5.3), that will be useful in the following study, are given by

$\begin{split}&\langle\tilde{J}_{AB}\tilde{J}_{CD}\rangle=\alpha_{0}\left(\eta_{AD}\eta_{BC}-\eta_{AC}\eta_{BD}\right)\,,\\\
&\langle\tilde{J}_{AB}\tilde{P}_{C}\rangle=\alpha_{1}\epsilon_{ABC}\,,\\\
&\langle\tilde{P}_{A}\tilde{P}_{B}\rangle=\frac{\alpha_{0}}{\ell^{2}}\eta_{AB}\,,\\\
&\langle\tilde{T}^{ij}\tilde{T}^{kl}\rangle=2\alpha_{0}\left(\delta^{il}\delta^{kj}-\delta^{ik}\delta^{lj}\right)\,,\\\
&\langle\tilde{T}^{IJ}\tilde{T}^{KL}\rangle=2\alpha_{0}\left(\delta^{IL}\delta^{KJ}-\delta^{IK}\delta^{LJ}\right)\,,\\\
&\langle\tilde{T}^{ij}\tilde{S}^{kl}\rangle=-2\alpha_{1}\left(\delta^{il}\delta^{kj}-\delta^{ik}\delta^{lj}\right)\,,\\\
&\langle\tilde{T}^{IJ}\tilde{S}^{KL}\rangle=2\alpha_{1}\left(\delta^{IL}\delta^{KJ}-\delta^{IK}\delta^{LJ}\right)\,,\\\
&\langle\tilde{S}^{ij}\tilde{S}^{kl}\rangle=\frac{2\alpha_{1}}{\ell}\left(\delta^{il}\delta^{kj}-\delta^{ik}\delta^{lj}\right)\,,\\\
&\langle\tilde{S}^{IJ}\tilde{S}^{KL}\rangle=-\frac{2\alpha_{1}}{\ell}\left(\delta^{IL}\delta^{KJ}-\delta^{IK}\delta^{LJ}\right)\,,\\\
&\langle\tilde{Q}^{i}_{\alpha}\tilde{Q}^{j}_{\beta}\rangle=2\left(\alpha_{1}-\frac{\alpha_{0}}{\ell}\right)C_{\alpha\beta}\delta^{ij}\,,\\\
&\langle\tilde{Q}^{I}_{\alpha}\tilde{Q}^{J}_{\beta}\rangle=2\left(\alpha_{1}+\frac{\alpha_{0}}{\ell}\right)C_{\alpha\beta}\delta^{IJ}\,,\end{split}$
(5.4)

where $\alpha_{0}$ and $\alpha_{1}$ are arbitrary constants.

In order to take the ultra-relativistic contraction of the superalgebra (5.3),
we decompose, as usual, the indices $A,B,\ldots=0,1,2$ as in (2.5), which
induces the decomposition (2.6), together with (2.7). After that, we rescale
the generators with a parameter $\sigma$ as

$\tilde{H}\rightarrow\sigma H\,,\quad\tilde{K}_{a}\rightarrow\sigma
K_{a}\,,\quad\tilde{S}^{ij}\rightarrow\sigma
S^{ij}\,,\quad\tilde{S}^{IJ}\rightarrow\sigma
S^{IJ}\,,\quad\tilde{Q}^{i}_{\alpha}\rightarrow\sqrt{\sigma}Q^{i}_{\alpha}\,,\quad\tilde{Q}^{I}_{\alpha}\rightarrow\sqrt{\sigma}Q^{I}_{\alpha}\,.$
(5.5)

Then, taking the limit $\sigma\rightarrow\infty$ (and removing the tilde
symbol also on the generators that we have not rescaled), we end up with the
$\mathcal{N}=(p,q)$ AdS Carroll superalgebra whose non-trivial
(anti)commutation relations read

$\begin{split}&\left[K_{a},J_{bc}\right]=\delta_{ab}K_{c}-\delta_{ac}K_{b}\,,\quad\quad\left[J_{ab},P_{c}\right]=\delta_{bc}P_{a}-\delta_{ac}P_{b}\,,\\\
&\left[K_{a},P_{b}\right]=-\delta_{ab}H\,,\quad\quad\left[P_{a},P_{b}\right]=\frac{1}{\ell^{2}}J_{ab}\,,\quad\quad\left[P_{a},H\right]=\frac{1}{\ell^{2}}K_{a}\,,\\\
&\left[{T}^{ij},{T}^{kl}\right]=\delta^{jk}{T}^{il}-\delta^{ik}{T}^{jl}-\delta^{jl}{T}^{ik}+\delta^{il}{T}^{jk}\,,\\\
&\left[{T}^{IJ},{T}^{KL}\right]=\delta^{JK}{T}^{IL}-\delta^{IK}{T}^{JL}-\delta^{JL}{T}^{IK}+\delta^{IL}{T}^{JK}\,,\\\
&\left[{T}^{ij},{S}^{kl}\right]=\delta^{jk}{S}^{il}-\delta^{ik}{S}^{jl}-\delta^{jl}{S}^{ik}+\delta^{il}{S}^{jk}\,,\\\
&\left[{T}^{IJ},{S}^{KL}\right]=\delta^{JK}{S}^{IL}-\delta^{IK}{S}^{JL}-\delta^{JL}{S}^{IK}+\delta^{IL}{S}^{JK}\,,\\\
&\left[J_{ab},Q^{i}_{\alpha}\right]=-\frac{1}{2}\left(\Gamma_{ab}Q^{i}\right)_{\alpha}\,,\quad\quad\left[J_{ab},Q^{I}_{\alpha}\right]=-\frac{1}{2}\left(\Gamma_{ab}Q^{I}\right)_{\alpha}\,,\\\
&\left[P_{a},Q^{i}_{\alpha}\right]=-\frac{1}{2\ell}\left(\Gamma_{a}Q^{i}\right)_{\alpha}\,,\quad\quad\left[P_{a},Q^{I}_{\alpha}\right]=\frac{1}{2\ell}\left(\Gamma_{a}Q^{I}\right)_{\alpha}\,,\end{split}$
(5.6)
$\begin{split}&\left[{T}^{ij},{Q}^{k}_{\alpha}\right]=\delta^{jk}{Q}^{i}_{\alpha}-\delta^{ik}{Q}^{j}_{\alpha}\,,\quad\quad\left[{T}^{IJ},{Q}^{K}_{\alpha}\right]=\delta^{JK}{Q}^{I}_{\alpha}-\delta^{IK}{Q}^{J}_{\alpha}\,,\\\
&\left\\{Q^{i}_{\alpha},Q^{j}_{\beta}\right\\}=\delta^{ij}\left[-\frac{1}{\ell}\left(\Gamma^{a0}C\right)_{\alpha\beta}K_{a}+\left(\Gamma^{0}C\right)_{\alpha\beta}H\right]+C_{\alpha\beta}S^{ij}\,,\\\
&\left\\{Q^{I}_{\alpha},Q^{J}_{\beta}\right\\}=\delta^{ij}\left[\frac{1}{\ell}\left(\Gamma^{a0}C\right)_{\alpha\beta}K_{a}+\left(\Gamma^{0}C\right)_{\alpha\beta}H\right]-C_{\alpha\beta}S^{IJ}\,.\end{split}$

Notice that if we restrict ourselves to the special case $\mathcal{N}=(1,1)$,
that is $p=q=1$, we exactly reproduce the $\mathcal{N}=(1,1)$ AdS Carroll
superalgebra obtained in Section 3, namely (3.4).

In the following, we will construct a three-dimensional CS action invariant
under (5.6).

###  5.2 $(p,q)$ AdS Carroll supergravity

We will now construct a three-dimensional CS supergravity action invariant
under the superalgebra (5.6) just introduced. We call this action $(p,q)$ AdS
Carroll CS supergravity action.

To this aim, let us first introduce the connection $1$-form $A$ associated
with (5.6), namely

$A=\frac{1}{2}\omega^{ab}J_{ab}+k^{a}K_{a}+V^{a}P_{a}+hH+\frac{1}{2}t^{ij}T_{ij}+\frac{1}{2}t^{IJ}T_{IJ}+\frac{1}{2}s^{ij}S_{ij}+\frac{1}{2}s^{IJ}S_{IJ}+\psi_{i}Q^{i}+\psi_{I}Q^{I}\,,$
(5.7)

being $\omega^{ab}$, $k^{a}$, $V^{a}$, $h$, $t^{ij}$, $t^{IJ}$, $s^{ij}$,
$s^{IJ}$, $\psi_{i}$, and $\psi_{I}$ the $1$-form fields respectively dual to
the generators $J_{ab}$, $K_{a}$, $P_{a}$, $H$, $T_{ij}$, $T_{IJ}$, $S_{ij}$,
$S_{IJ}$, $Q^{i}$, and $Q^{I}$ (obeying the (anti)commutation relations given
in (5.6)), and the related curvature $2$-form $F$, that is

$F=\frac{1}{2}\mathcal{R}^{ab}J_{ab}+\mathcal{K}^{a}K_{a}+R^{a}P_{a}+\mathcal{H}H+\frac{1}{2}\mathcal{T}^{ij}T_{ij}+\frac{1}{2}\mathcal{T}^{IJ}T_{IJ}+\frac{1}{2}\mathcal{S}^{ij}S_{ij}+\frac{1}{2}\mathcal{S}^{IJ}S_{IJ}+\nabla\psi_{i}Q^{i}+\nabla\psi_{I}Q^{I}\,,$
(5.8)

with

$\begin{split}\mathcal{R}^{ab}&=d\omega^{ab}+\frac{1}{\ell^{2}}V^{a}V^{b}=R^{ab}+\frac{1}{\ell^{2}}V^{a}V^{b}\,,\\\
\mathcal{K}^{a}&=dk^{a}+\omega^{a}_{\phantom{a}b}k^{b}+\frac{1}{\ell^{2}}V^{a}h+\frac{1}{2\ell}\bar{\psi}^{i}\Gamma^{a0}\psi^{i}-\frac{1}{2\ell}\bar{\psi}^{I}\Gamma^{a0}\psi^{I}=\mathfrak{K}^{a}+\frac{1}{2\ell}\bar{\psi}^{i}\Gamma^{a0}\psi^{i}-\frac{1}{2\ell}\bar{\psi}^{I}\Gamma^{a0}\psi^{I}\,,\\\
R^{a}&=dV^{a}+\omega^{a}_{\phantom{a}b}V^{b}\,,\\\
\mathcal{H}&=dh+V^{a}k_{a}-\frac{1}{2}\bar{\psi}^{i}\Gamma^{0}\psi^{i}-\frac{1}{2}\bar{\psi}^{I}\Gamma^{0}\psi^{I}=\mathfrak{H}-\frac{1}{2}\bar{\psi}^{i}\Gamma^{0}\psi^{i}-\frac{1}{2}\bar{\psi}^{I}\Gamma^{0}\psi^{I}\,,\\\
\mathcal{T}^{ij}&=dt^{ij}+{t^{i}}_{k}t^{kj}\,,\\\
\mathcal{T}^{IJ}&=dt^{IJ}+{t^{I}}_{K}t^{KJ}\,,\\\
\mathcal{S}^{ij}&=ds^{ij}+2{t^{i}}_{k}s^{kj}-\bar{\psi}^{i}\psi^{j}\,,\\\
\mathcal{S}^{IJ}&=ds^{IJ}+2{t^{I}}_{K}s^{KJ}+\bar{\psi}^{I}\psi^{J}\,,\\\
\nabla\psi^{i}&=d\psi^{i}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\psi^{i}+\frac{1}{2\ell}V^{a}\Gamma_{a}\psi^{i}+t^{ij}\psi_{j}\,,\\\
\nabla\psi^{I}&=d\psi^{I}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\psi^{I}-\frac{1}{2\ell}V^{a}\Gamma_{a}\psi^{I}+t^{IJ}\psi_{J}\,.\end{split}$
(5.9)

We can now move to the explicit construction of a CS action invariant under
(5.6). To this aim, consider the non-vanishing components of the invariant
tensor given in (5.4), decompose the indices as in (2.5), and rescale not only
the generators in compliance with (3.3) but also the coefficients appearing in
(5.4) as in (2.16). Consequently, the Carroll limit $\sigma\rightarrow\infty$
leads to the following non-vanishing components of an invariant tensor for the
superalgebra (5.6):

$\begin{split}&\langle
J_{ab}J_{cd}\rangle=\alpha_{0}\left(\delta_{ad}\delta_{bc}-\delta_{ac}\delta_{bd}\right)\,,\\\
&\langle J_{ab}H\rangle=\alpha_{1}\epsilon_{ab}\,,\\\ &\langle
K_{a}P_{b}\rangle=-\alpha_{1}\epsilon_{ab}\,,\\\ &\langle
P_{a}P_{b}\rangle=\frac{\alpha_{0}}{\ell^{2}}\delta_{ab}\,,\\\
&\langle{T}^{ij}{T}^{kl}\rangle=2\alpha_{0}\left(\delta^{il}\delta^{kj}-\delta^{ik}\delta^{lj}\right)\,,\\\
&\langle{T}^{IJ}{T}^{KL}\rangle=2\alpha_{0}\left(\delta^{IL}\delta^{KJ}-\delta^{IK}\delta^{LJ}\right)\,,\\\
&\langle{T}^{ij}{S}^{kl}\rangle=-2\alpha_{1}\left(\delta^{il}\delta^{kj}-\delta^{ik}\delta^{lj}\right)\,,\\\
&\langle{T}^{IJ}{S}^{KL}\rangle=2\alpha_{1}\left(\delta^{IL}\delta^{KJ}-\delta^{IK}\delta^{LJ}\right)\,,\\\
&\langle
Q^{i}_{\alpha}Q^{j}_{\beta}\rangle=2\alpha_{1}C_{\alpha\beta}\delta^{ij}\,,\\\
&\langle
Q^{I}_{\alpha}Q^{J}_{\beta}\rangle=2\alpha_{1}C_{\alpha\beta}\delta^{IJ}\,.\end{split}$
(5.10)

The invariant tensor whose components are given in (5.10) is non-degenerate
when $\alpha_{1}\neq 0$.

Then, substituting the connection $1$-form in (5.7) and the non-zero
components of the invariant tensor (5.10) into the general expression (2.11),
we end up with the $(p,q)$ AdS Carroll CS supergravity action in three
dimensions, that is

$\begin{split}I^{(p,q)}_{CS}&=\frac{k}{4\pi}\int_{\mathcal{M}}\Bigg{\\{}\frac{\alpha_{0}}{2}\left(\omega^{a}_{\phantom{a}b}R^{b}_{\phantom{b}a}+\frac{2}{\ell^{2}}V^{a}R_{a}+2{t^{i}}_{j}{dt^{j}}_{i}+\frac{4}{3}{t^{i}}_{j}{t^{j}}_{k}{t^{k}}_{i}+2{t^{I}}_{J}{dt^{J}}_{I}+\frac{4}{3}{t^{I}}_{J}{t^{J}}_{K}{t^{K}}_{I}\right)\\\
&+\alpha_{1}\bigg{[}\epsilon_{ab}R^{ab}h-2\epsilon_{ab}\mathfrak{K}^{a}V^{b}+\frac{1}{\ell^{2}}\epsilon_{ab}V^{a}V^{b}h-2{t^{i}}_{j}\left({ds^{j}}_{i}+{t^{j}}_{k}{s^{k}}_{i}\right)+2{t^{I}}_{J}\left({ds^{J}}_{I}+{t^{J}}_{K}{s^{K}}_{I}\right)\\\
&+2\bar{\psi}^{i}\nabla\psi^{i}+2\bar{\psi}^{I}\nabla\psi^{I}\bigg{]}-d\left(\frac{\alpha_{1}}{2}\epsilon_{ab}\omega^{ab}h-\alpha_{1}\epsilon_{ab}k^{a}V^{b}+\alpha_{1}{t^{i}}_{j}{s^{j}}_{i}-\alpha_{1}{t^{I}}_{J}{s^{J}}_{I}\right)\Bigg{\\}}\,,\end{split}$
(5.11)

which is written in terms of the curvatures appearing in (5.9) and it involves
two coupling constants, that are $\alpha_{0}$ and $\alpha_{1}$. Up to boundary
terms, the action (5.11) can be reworked as follows:

$\begin{split}I^{(p,q)}_{CS}&=\frac{k}{4\pi}\int_{\mathcal{M}}\Bigg{\\{}\frac{\alpha_{0}}{2}\left(\omega^{a}_{\phantom{a}b}R^{b}_{\phantom{b}a}+\frac{2}{\ell^{2}}V^{a}R_{a}+2{t^{i}}_{j}{dt^{j}}_{i}+\frac{4}{3}{t^{i}}_{j}{t^{j}}_{k}{t^{k}}_{i}+2{t^{I}}_{J}{dt^{J}}_{I}+\frac{4}{3}{t^{I}}_{J}{t^{J}}_{K}{t^{K}}_{I}\right)\\\
&+\alpha_{1}\bigg{[}\epsilon_{ab}R^{ab}h-2\epsilon_{ab}\mathfrak{K}^{a}V^{b}+\frac{1}{\ell^{2}}\epsilon_{ab}V^{a}V^{b}h-2{t^{i}}_{j}\left({ds^{j}}_{i}+{t^{j}}_{k}{s^{k}}_{i}\right)+2{t^{I}}_{J}\left({ds^{J}}_{I}+{t^{J}}_{K}{s^{K}}_{I}\right)\\\
&+2\bar{\psi}^{i}\nabla\psi^{i}+2\bar{\psi}^{I}\nabla\psi^{I}\bigg{]}\Bigg{\\}}\,.\end{split}$
(5.12)

As usual, the contribution proportional to $\alpha_{0}$ corresponds to the
exotic Lagrangian, and we can see that it involves, besides the Lorentz and
torsional terms, also pieces including the $1$-form fields $t^{ij}$ and
$t^{IJ}$. However, it does not contain terms involving $\psi^{i}$ and
$\psi^{I}$. On the other hand, the contribution proportional to $\alpha_{1}$
also includes pieces involving the $1$-form fields $s^{ij}$, $s^{IJ}$,
$\psi^{i}$, and $\psi^{I}$.

The action (5.11) is invariant by construction under (5.6), and the local
gauge transformations $\delta_{\lambda}A=d\lambda+\left[A,\lambda\right]$ with
gauge parameter

$\lambda=\frac{1}{2}\lambda^{ab}J_{ab}+\kappa^{a}K_{a}+\lambda^{a}P_{a}+\tau
H+\frac{1}{2}\varphi^{ij}t_{ij}+\frac{1}{2}\varphi^{IJ}t_{IJ}+\frac{1}{2}\varsigma^{ij}s_{ij}+\frac{1}{2}\varsigma^{IJ}s_{IJ}+\varepsilon^{i}Q_{i}+\varepsilon^{I}Q_{I}$
(5.13)

are given by

$\begin{split}\delta\omega^{ab}&=d\lambda^{ab}+\frac{2}{\ell^{2}}V^{[a}\lambda^{b]}\,,\\\
\delta
k^{a}&=d\kappa^{a}-\lambda^{a}_{\phantom{a}b}k^{b}+\omega^{a}_{\phantom{a}b}\kappa^{b}-\frac{1}{\ell^{2}}\lambda^{a}h+\frac{1}{\ell^{2}}V^{a}\tau-\frac{1}{\ell}\bar{\varepsilon}^{i}\Gamma^{a0}\psi^{i}+\frac{1}{\ell}\bar{\varepsilon}^{I}\Gamma^{a0}\psi^{I}\,,\\\
\delta
V^{a}&=d\lambda^{a}-\lambda^{a}_{\phantom{a}b}V^{b}+\omega^{a}_{\phantom{a}b}\lambda^{b}\,,\\\
\delta
h&=d\tau-\lambda^{a}k_{a}+V^{a}\kappa_{a}+\bar{\varepsilon}^{i}\Gamma^{0}\psi^{i}+\bar{\varepsilon}^{I}\Gamma^{0}\psi^{I}\,,\\\
\delta t^{ij}&=d\varphi^{ij}-2{\varphi^{[i}}_{k}t^{k|j]}\,,\\\ \delta
t^{IJ}&=d\varphi^{IJ}-2{\varphi^{[I}}_{K}t^{K|J]}\,,\\\ \delta
s^{ij}&=d\varsigma^{ij}-2{\varphi^{[i}}_{k}s^{k|j]}+2{t^{[i}}_{k}\varsigma^{k|j]}+2\bar{\varepsilon}^{[i}\psi^{j]}\,,\\\
\delta
s^{IJ}&=d\varsigma^{IJ}+2{\varphi^{[I}}_{K}s^{K|J]}+2{t^{[I}}_{K}\varsigma^{K|J]}-2\bar{\varepsilon}^{[I}\psi^{J]}\,,\\\
\delta\psi^{i}&=d\varepsilon^{+}-\frac{1}{4}\lambda^{ab}\Gamma_{ab}\psi^{i}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{i}-\frac{1}{2\ell}\lambda^{a}\Gamma_{a}\psi^{i}+\frac{1}{2\ell}V^{a}\Gamma_{a}\varepsilon^{i}-\varphi^{ij}\psi_{j}+t^{ij}\varepsilon_{j}\,,\\\
\delta\psi^{I}&=d\varepsilon^{-}-\frac{1}{4}\lambda^{ab}\Gamma_{ab}\psi^{I}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{I}+\frac{1}{2\ell}\lambda^{a}\Gamma_{a}\psi^{I}-\frac{1}{2\ell}V^{a}\Gamma_{a}\varepsilon^{I}-\varphi^{IJ}\psi_{J}+t^{IJ}\varepsilon_{J}\,.\end{split}$
(5.14)

Thus, the restriction to supersymmetry transformations gives us

$\begin{split}\delta\omega^{ab}&=0\,,\\\ \delta
k^{a}&=-\frac{1}{\ell}\bar{\varepsilon}^{i}\Gamma^{a0}\psi^{i}+\frac{1}{\ell}\bar{\varepsilon}^{I}\Gamma^{a0}\psi^{I}\,,\\\
\delta V^{a}&=0\,,\\\ \delta
h&=\bar{\varepsilon}^{i}\Gamma^{0}\psi^{i}+\bar{\varepsilon}^{I}\Gamma^{0}\psi^{I}\,,\\\
\delta t^{ij}&=0\,,\\\ \delta t^{IJ}&=0\,,\\\ \delta
s^{ij}&=2\bar{\varepsilon}^{[i}\psi^{j]}\,,\\\ \delta
s^{IJ}&=-2\bar{\varepsilon}^{[I}\psi^{J]}\,,\\\
\delta\psi^{i}&=d\varepsilon^{+}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{i}+\frac{1}{2\ell}V^{a}\Gamma_{a}\varepsilon^{i}+t^{ij}\varepsilon_{j}\,,\\\
\delta\psi^{I}&=d\varepsilon^{-}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{I}-\frac{1}{2\ell}V^{a}\Gamma_{a}\varepsilon^{I}+t^{IJ}\varepsilon_{J}\,.\end{split}$
(5.15)

Finally, the equations of motion obtained from the variation of (5.11) with
respect to the $1$-form fields $\omega^{ab}$, $k^{a}$, $V^{a}$, $h$, $t^{ij}$,
$t^{IJ}$, $s^{ij}$, $s^{IJ}$, $\psi^{i}$, and $\psi^{I}$ are, respectively (up
to boundary contributions),

$\begin{split}\delta\omega^{ab}&:\quad\alpha_{0}\mathcal{R}^{ab}+\alpha_{1}\epsilon^{ab}\mathcal{H}=0\,,\\\
\delta k^{a}&:\quad\alpha_{1}R^{a}=0\,,\\\ \delta
V^{a}&:\quad\frac{\alpha_{0}}{\ell^{2}}R^{a}+2\alpha_{1}\epsilon_{ab}\mathcal{K}^{b}=0\,,\\\
\delta h&:\quad\alpha_{1}\mathcal{R}^{ab}=0\,,\\\ \delta
t^{ij}&:\quad-\alpha_{0}\mathcal{T}^{ij}+\alpha_{1}\mathcal{S}^{ij}=0\,,\\\
\delta
t^{IJ}&:\quad-\alpha_{0}\mathcal{T}^{IJ}-\alpha_{1}\mathcal{S}^{IJ}=0\,,\\\
\delta s^{ij}&:\quad\alpha_{1}\mathcal{T}^{ij}=0\,,\\\ \delta
s^{IJ}&:\quad\alpha_{1}\mathcal{T}^{IJ}=0\,,\\\
\delta\psi^{i}&:\quad\alpha_{1}\nabla\psi^{i}=0\,,\\\
\delta\psi^{I}&:\quad\alpha_{1}\nabla\psi^{I}=0\,,\end{split}$ (5.16)

and, for $\alpha_{1}\neq 0$, they reduce precisely to the vanishing of the
$(p,q)$ super-AdS Carroll curvature $2$-forms given in (5.8), that is to say

$\begin{split}&\mathcal{R}^{ab}=0\,,\quad\mathcal{K}^{a}=0\,,\quad
R^{a}=0\,,\quad\mathcal{H}=0\,,\quad\mathcal{T}^{ij}=0\,,\quad\mathcal{T}^{IJ}=0\,,\quad\mathcal{S}^{ij}=0\,,\quad\mathcal{S}^{IJ}=0\,,\\\
&\nabla\psi^{i}=0\,,\quad\nabla\psi^{I}=0\,.\end{split}$ (5.17)

We observe that, as usual, $\alpha_{1}\neq 0$ is a sufficient condition to
recover (5.17), meaning that $\alpha_{0}$ can be consistently set to zero,
making the exotic term in the CS action (5.11) disappear.

Let us finally mention that, if we restrict ourselves to the case $p=q=1$, we
exactly reproduce the results of Section 3, that is to say, as properly
expected, the $(1,1)$ AdS Carroll supergravity theory.

##  6 Study of the flat limit $\ell\rightarrow\infty$

In the sequel, we study the flat limit $\ell\rightarrow\infty$, which can be
directly applied to the $\mathcal{N}$-extended AdS Carroll superalgebras
(2.10), (3.4), (4.14), and (5.6), to the corresponding curvature $2$-forms,
respectively given by (2.15), (3.7), (4.17), and (5.9), to the related CS
actions (2.18), (3.9), (4.21), and (5.11), to the transformation laws (2.21),
(3.12), (4.24), (5.14) (and, in particular to the supersymmetry transformation
laws (2.22), (3.13), (4.26), (5.15)), and to the field equations of the
respective theories, namely (2.23), (3.14), (4.27), and (5.16).

###  6.1 $(2,0)$ Carroll supergravity from the $\ell\rightarrow\infty$ limit

In the limit $\ell\rightarrow\infty$, the (anti)commutation relations of the
$\mathcal{N}=(2,0)$ AdS Carroll superalgebra (2.10) reduce to the following
non-vanishing ones:

$\begin{split}&\left[K_{a},J_{bc}\right]=\delta_{ab}K_{c}-\delta_{ac}K_{b}\,,\\\
&\left[J_{ab},P_{c}\right]=\delta_{bc}P_{a}-\delta_{ac}P_{b}\,,\\\
&\left[K_{a},P_{b}\right]=-\delta_{ab}H\,,\\\
&\left[J_{ab},Q^{\pm}_{\alpha}\right]=-\frac{1}{2}\left(\Gamma_{ab}Q^{\pm}\right)_{\alpha}\,,\\\
&\left[T,Q^{+}_{\alpha}\right]=\left(\Gamma_{0}\right)_{\alpha\beta}Q^{+}_{\beta}\,,\\\
&\left[T,Q^{-}_{\alpha}\right]=-\left(\Gamma_{0}\right)_{\alpha\beta}Q^{-}_{\beta}\,,\\\
&\left\\{Q^{+}_{\alpha},Q^{+}_{\beta}\right\\}=\left(\Gamma^{0}C\right)_{\alpha\beta}\left(H+S\right)\,,\\\
&\left\\{Q^{-}_{\alpha},Q^{-}_{\beta}\right\\}=\left(\Gamma^{0}C\right)_{\alpha\beta}\left(H-S\right)\,.\end{split}$
(6.1)

These are the (anti)commutation relations of a new $\mathcal{N}=(2,0)$, $D=3$
super-Carroll algebra, involving an extra generator $S$, which could also have
been derived by considering the ultra-relativistic contraction of the
$\mathcal{N}=(2,0)$, $D=3$ Poincaré superalgebra supplemented with an $so(2)$
extension consisting in the extra generator $\tilde{S}$ introduced at the
relativistic level.

As $\ell\rightarrow\infty$, the $2$-form curvatures (2.15) become

$\begin{split}\mathcal{R}^{ab}&=d\omega^{ab}=R^{ab}\,,\\\
\mathcal{K}^{a}&=dk^{a}+\omega^{a}_{\phantom{a}b}k^{b}=\mathfrak{K}^{a}\,,\\\
R^{a}&=dV^{a}+\omega^{a}_{\phantom{a}b}V^{b}\,,\\\
\mathcal{H}&=dh+V^{a}k_{a}-\frac{1}{2}\bar{\psi}^{+}\Gamma^{0}\psi^{+}-\frac{1}{2}\bar{\psi}^{-}\Gamma^{0}\psi^{-}=\mathfrak{H}-\frac{1}{2}\bar{\psi}^{+}\Gamma^{0}\psi^{+}-\frac{1}{2}\bar{\psi}^{-}\Gamma^{0}\psi^{-}\,,\\\
\mathcal{T}&=dt\,,\\\
\mathcal{S}&=ds-\frac{1}{2}\bar{\psi}^{+}\Gamma^{0}\psi^{+}+\frac{1}{2}\bar{\psi}^{-}\Gamma^{0}\psi^{-}\,,\\\
\nabla\psi^{+}&=d\psi^{+}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\psi^{+}-t\Gamma_{0}\psi^{+}\,,\\\
\nabla\psi^{-}&=d\psi^{-}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\psi^{-}+t\Gamma_{0}\psi^{-}\,.\end{split}$
(6.2)

On the other hand, by applying the $\ell\rightarrow\infty$ limit to the three-
dimensional CS action (2.18), we end up with

$\begin{split}I^{(2,0)}_{CS}|_{\ell\rightarrow\infty}&=\frac{k}{4\pi}\int_{\mathcal{M}}\Bigg{\\{}\frac{\alpha_{0}}{2}\left(\omega^{a}_{\phantom{a}b}R^{b}_{\phantom{b}a}-4tdt\right)+\alpha_{1}\bigg{(}\epsilon_{ab}R^{ab}h-2\epsilon_{ab}\mathfrak{K}^{a}V^{b}+4tds\\\
&+2\bar{\psi}^{+}\nabla\psi^{+}+2\bar{\psi}^{-}\nabla\psi^{-}\bigg{)}-d\left(\frac{\alpha_{1}}{2}\epsilon_{ab}\omega^{ab}h-\alpha_{1}\epsilon_{ab}k^{a}V^{b}-2\alpha_{1}ts\right)\Bigg{\\}}\,,\end{split}$
(6.3)

which is written in terms of the super-Carroll curvatures appearing in (6.2).
The latter must not be confused with the super-AdS Carroll ones given in
(2.15), since (6.2) correspond to the flat limit of (2.15). Here we signal
that we have done a little abuse of notation. The action (6.3) can also be
derived by using the following non-vanishing components of the invariant
tensor:

$\begin{split}&\langle
J_{ab}J_{cd}\rangle=\alpha_{0}\left(\delta_{ad}\delta_{bc}-\delta_{ac}\delta_{bd}\right)\,,\\\
&\langle J_{ab}H\rangle=\alpha_{1}\epsilon_{ab}\,,\\\ &\langle
K_{a}P_{b}\rangle=-\alpha_{1}\epsilon_{ab}\,,\\\ &\langle
TT\rangle=-2\alpha_{0}\,,\\\ &\langle TS\rangle=2\alpha_{1}\,,\\\ &\langle
Q^{+}_{\alpha}Q^{+}_{\beta}\rangle=\langle
Q^{-}_{\alpha}Q^{-}_{\beta}\rangle=2\alpha_{1}C_{\alpha\beta}\,,\end{split}$
(6.4)

which are obtained by taking the limit $\ell\rightarrow\infty$ of (2.17), and
the connection $1$-form for the $\mathcal{N}=(2,0)$ (flat) Carroll
superalgebra (6.1) in the general expression (2.11). Notice that the exotic
term, which is the one proportional to $\alpha_{0}$ in (6.3), now reduces
purely to the so-called Lorentz Lagrangian.

The CS action (6.3) is invariant by construction under the super-Carroll group
associated with (6.1). In particular, concerning the flat limit of the gauge
transformations (2.21), we get the local gauge transformations

$\begin{split}\delta\omega^{ab}&=d\lambda^{ab}\,,\\\ \delta
k^{a}&=d\kappa^{a}-\lambda^{a}_{\phantom{a}b}k^{b}+\omega^{a}_{\phantom{a}b}\kappa^{b}\,,\\\
\delta
V^{a}&=d\lambda^{a}-\lambda^{a}_{\phantom{a}b}V^{b}+\omega^{a}_{\phantom{a}b}\lambda^{b}\,,\\\
\delta
h&=d\tau-\lambda^{a}k_{a}+V^{a}\kappa_{a}+\bar{\varepsilon}^{+}\Gamma^{0}\psi^{+}+\bar{\varepsilon}^{-}\Gamma^{0}\psi^{-}\,,\\\
\delta t&=d\varphi\,,\\\ \delta
s&=d\varsigma+\bar{\varepsilon}^{+}\Gamma^{0}\psi^{+}-\bar{\varepsilon}^{-}\Gamma^{0}\psi^{-}\,,\\\
\delta\psi^{+}&=d\varepsilon^{+}-\frac{1}{4}\lambda^{ab}\Gamma_{ab}\psi^{+}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{+}+\varphi\Gamma_{0}\psi^{+}-t\Gamma_{0}\varepsilon^{+}\,,\\\
\delta\psi^{-}&=d\varepsilon^{-}-\frac{1}{4}\lambda^{ab}\Gamma_{ab}\psi^{-}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{-}-\varphi\Gamma_{0}\psi^{-}+t\Gamma_{0}\varepsilon^{-}\,.\end{split}$
(6.5)

The restriction to supersymmetry transformations reads

$\begin{split}\delta\omega^{ab}&=0\,,\\\ \delta k^{a}&=0\,,\\\ \delta
V^{a}&=0\,,\\\ \delta
h&=\bar{\varepsilon}^{+}\Gamma^{0}\psi^{+}+\bar{\varepsilon}^{-}\Gamma^{0}\psi^{-}\,,\\\
\delta t&=0\,,\\\ \delta
s&=\bar{\varepsilon}^{+}\Gamma^{0}\psi^{+}-\bar{\varepsilon}^{-}\Gamma^{0}\psi^{-}\,,\\\
\delta\psi^{+}&=d\varepsilon^{+}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{+}-t\Gamma_{0}\varepsilon^{+}\,,\\\
\delta\psi^{-}&=d\varepsilon^{-}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{-}+t\Gamma_{0}\varepsilon^{-}\,.\end{split}$
(6.6)

Finally, the equations of motion for the action (6.3) (flat limit of the
equations of motion given in (2.23)) are

$\begin{split}\delta\omega^{ab}&:\quad\alpha_{0}\mathcal{R}^{ab}+\alpha_{1}\epsilon^{ab}\mathcal{H}=0\,,\\\
\delta k^{a}&:\quad\alpha_{1}R^{a}=0\,,\\\ \delta
V^{a}&:\quad\alpha_{1}\epsilon_{ab}\mathcal{K}^{b}=0\,,\\\ \delta
h&:\quad\alpha_{1}\mathcal{R}^{ab}=0\,,\\\ \delta
t&:\quad-\alpha_{0}\mathcal{T}+\alpha_{1}\mathcal{S}=0\,,\\\ \delta
s&:\quad\alpha_{1}\mathcal{T}=0\,,\\\
\delta\psi^{+}&:\quad\alpha_{1}\nabla\psi^{+}=0\,,\\\
\delta\psi^{-}&:\quad\alpha_{1}\nabla\psi^{-}=0\,,\end{split}$ (6.7)

and we can see that, when $\alpha_{1}\neq 0$, they exactly reduce to the
vanishing of the curvature $2$-forms given in (6.2). We can also observe that,
in analogy with the AdS case of Section 2, also in the flat limit
$\alpha_{1}\neq 0$ results to be a sufficient condition to recover the
vanishing of the curvature $2$-forms (6.2) obtained in the flat limit, which
means that one could consistently set $\alpha_{0}=0$ and thus neglect the
exotic term (i.e., the Lorentz Lagrangian) in the CS action (6.3).999Let us
also observe that if we now restrict ourselves to the purely bosonic part of
the action (6.3), we get a three-dimensional CS Carroll gravity action that is
different from the one obtained in [35] (see also [13]) by considering the
purely bosonic contributions, due to the presence of the bosonic $1$-form
fields $t$ and $s$ dual to the generators $T$ and $S$, respectively.
Nevertheless, if we consider the purely bosonic level and set $t=s=0$ through
an IW contraction, we have that the aforementioned actions coincide.

###  6.2 $(1,1)$ Carroll supergravity from the $\ell\rightarrow\infty$ limit

The limit $\ell\rightarrow\infty$ performed on the (anti)commutation relations
of the $\mathcal{N}=(1,0)$ AdS Carroll superalgebra (3.4) leads to the
following non-vanishing ones:

$\begin{split}&\left[K_{a},J_{bc}\right]=\delta_{ab}K_{c}-\delta_{ac}K_{b}\,,\\\
&\left[J_{ab},P_{c}\right]=\delta_{bc}P_{a}-\delta_{ac}P_{b}\,,\\\
&\left[K_{a},P_{b}\right]=-\delta_{ab}H\,,\\\
&\left[J_{ab},Q^{\pm}_{\alpha}\right]=-\frac{1}{2}\left(\Gamma_{ab}Q^{\pm}\right)_{\alpha}\,,\\\
&\left\\{Q^{\pm}_{\alpha},Q^{\pm}_{\beta}\right\\}=\left(\Gamma^{0}C\right)_{\alpha\beta}H\,.\end{split}$
(6.8)

These are the (anti)commutation relations of the $\mathcal{N}=(1,1)$, $D=3$
super-Carroll algebra (see [14], where (6.8) corresponds to the superalgebra
obtained in the $R\rightarrow\infty$ limit of the $\mathcal{N}=(1,1)$ AdS-
Carroll superalgebra of Section C.4 of the same paper). It could be also
obtained by considering the ultra-relativistic contraction of the
$\mathcal{N}=(1,1)$, $D=3$ Poincaré superalgebra.

Taking $\ell\rightarrow\infty$, the $2$-form curvatures (3.7) reduce to

$\begin{split}\mathcal{R}^{ab}&=d\omega^{ab}=R^{ab}\,,\\\
\mathcal{K}^{a}&=dk^{a}+\omega^{a}_{\phantom{a}b}k^{b}=\mathfrak{K}^{a}\,,\\\
R^{a}&=dV^{a}+\omega^{a}_{\phantom{a}b}V^{b}\,,\\\
\mathcal{H}&=dh+V^{a}k_{a}-\frac{1}{2}\bar{\psi}^{+}\Gamma^{0}\psi^{+}-\frac{1}{2}\bar{\psi}^{-}\Gamma^{0}\psi^{-}=\mathfrak{H}-\frac{1}{2}\bar{\psi}^{+}\Gamma^{0}\psi^{+}-\frac{1}{2}\bar{\psi}^{-}\Gamma^{0}\psi^{-}\,,\\\
\nabla\psi^{+}&=d\psi^{+}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\psi^{+}\,,\\\
\nabla\psi^{-}&=d\psi^{-}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\psi^{-}\,,\end{split}$
(6.9)

and the $\ell\rightarrow\infty$ limit of the CS action (3.9) leads us to the
following three-dimensional one:

$\begin{split}I^{(1,1)}_{CS}|_{\ell\rightarrow\infty}&=\frac{k}{4\pi}\int_{\mathcal{M}}\Bigg{\\{}\frac{\alpha_{0}}{2}\left(\omega^{a}_{\phantom{a}b}R^{b}_{\phantom{b}a}\right)+\alpha_{1}\left(\epsilon_{ab}R^{ab}h-2\epsilon_{ab}\mathfrak{K}^{a}V^{b}+2\bar{\psi}^{+}\nabla\psi^{+}+2\bar{\psi}^{-}\nabla\psi^{-}\right)\\\
&-d\left(\frac{\alpha_{1}}{2}\epsilon_{ab}\omega^{ab}h-\alpha_{1}\epsilon_{ab}k^{a}V^{b}\right)\Bigg{\\}}\,,\end{split}$
(6.10)

written in terms of the curvatures appearing in (6.9) (again, we are doing a
little abuse of notation). The action (6.10) can also be derived by using the
connection $1$-form for the $\mathcal{N}=(1,1)$ (flat) Carroll superalgebra
(6.8) together with the non-vanishing components of the invariant tensor

$\begin{split}&\langle
J_{ab}J_{cd}\rangle=\alpha_{0}\left(\delta_{ad}\delta_{bc}-\delta_{ac}\delta_{bd}\right)\,,\\\
&\langle J_{ab}H\rangle=\alpha_{1}\epsilon_{ab}\,,\\\ &\langle
K_{a}P_{b}\rangle=-\alpha_{1}\epsilon_{ab}\,,\\\ &\langle
Q^{+}_{\alpha}Q^{+}_{\beta}\rangle=\langle
Q^{-}_{\alpha}Q^{-}_{\beta}\rangle=2\alpha_{1}C_{\alpha\beta}\,\end{split}$
(6.11)

in the general expression for a three-dimensional CS action (2.11).
Analogously to what happened in the $(2,0)$ flat theory, also in the current
case the exotic term, proportional to $\alpha_{0}$, now reduces purely to the
Lorentz Lagrangian.

By construction, the CS action (6.10) is invariant under the $(1,1)$ super-
Carroll group, that is associated with the superalgebra given in (6.8). In
particular, taking the flat limit of the gauge transformations (3.12), we get
the following local gauge transformations:

$\begin{split}\delta\omega^{ab}&=d\lambda^{ab}\,,\\\ \delta
k^{a}&=d\kappa^{a}-\lambda^{a}_{\phantom{a}b}k^{b}+\omega^{a}_{\phantom{a}b}\kappa^{b}\,,\\\
\delta
V^{a}&=d\lambda^{a}-\lambda^{a}_{\phantom{a}b}V^{b}+\omega^{a}_{\phantom{a}b}\lambda^{b}\,,\\\
\delta
h&=d\tau-\lambda^{a}k_{a}+V^{a}\kappa_{a}+\bar{\varepsilon}^{+}\Gamma^{0}\psi^{+}+\bar{\varepsilon}^{-}\Gamma^{0}\psi^{-}\,,\\\
\delta\psi^{+}&=d\varepsilon^{+}-\frac{1}{4}\lambda^{ab}\Gamma_{ab}\psi^{+}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{+}\,,\\\
\delta\psi^{-}&=d\varepsilon^{-}-\frac{1}{4}\lambda^{ab}\Gamma_{ab}\psi^{-}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{-}\,,\end{split}$
(6.12)

and restricting ourselves to supersymmetry transformations, we are left with

$\begin{split}\delta\omega^{ab}&=0\,,\\\ \delta k^{a}&=0\,,\\\ \delta
V^{a}&=0\,,\\\ \delta
h&=\bar{\varepsilon}^{+}\Gamma^{0}\psi^{+}+\bar{\varepsilon}^{-}\Gamma^{0}\psi^{-}\,,\\\
\delta\psi^{+}&=d\varepsilon^{+}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{+}\,,\\\
\delta\psi^{-}&=d\varepsilon^{-}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\varepsilon^{-}\,.\end{split}$
(6.13)

Concluding, the equations of motion for the action (6.10) (flat limit of the
equations of motion given in (3.14)) read as follows:

$\begin{split}\delta\omega^{ab}&:\quad\alpha_{0}\mathcal{R}^{ab}+\alpha_{1}\epsilon^{ab}\mathcal{H}=0\,,\\\
\delta k^{a}&:\quad\alpha_{1}R^{a}=0\,,\\\ \delta
V^{a}&:\quad\alpha_{1}\epsilon_{ab}\mathcal{K}^{b}=0\,,\\\ \delta
h&:\quad\alpha_{1}\mathcal{R}^{ab}=0\,,\\\
\delta\psi^{+}&:\quad\alpha_{1}\nabla\psi^{+}=0\,,\\\
\delta\psi^{-}&:\quad\alpha_{1}\nabla\psi^{-}=0\,.\end{split}$ (6.14)

When $\alpha_{1}\neq 0$, the eqs. (6.14) exactly reduce to the vanishing of
the curvature $2$-forms in (6.9) ($\alpha_{1}\neq 0$ is a sufficient condition
to recover the vanishing of the curvature $2$-forms (6.9), meaning that one
could consistently set $\alpha_{0}=0$, omitting the exotic term, that is the
Lorentz Lagrangian, in the CS action (6.10)).

Notice that the restriction to the purely bosonic part of the action (6.10)
yields exactly the three-dimensional CS gravity action invariant under the
$D=3$ Carroll algebra [1, 2]. The aforesaid CS action involving purely bosonic
terms is equivalent, as argued in [13], to the action found in [13] if we take
the $D=3$ case in the same paper.

###  6.3 $(\mathcal{N},0)$ Carroll supergravity theories from the
$\ell\rightarrow\infty$ limit

Taking the flat limit $\ell\rightarrow\infty$ of the (anti)commutation
relations (4.14), we get the following non-trivial ones:

$\begin{split}&\left[K_{a},J_{bc}\right]=\delta_{ab}K_{c}-\delta_{ac}K_{b}\,,\\\
&\left[J_{ab},P_{c}\right]=\delta_{bc}P_{a}-\delta_{ac}P_{b}\,,\\\
&\left[K_{a},P_{b}\right]=-\delta_{ab}H\,,\\\
&\left[{T}^{\lambda\mu},{T}^{\nu\rho}\right]=\delta^{\mu\nu}{T}^{\lambda\rho}-\delta^{\lambda\nu}{T}^{\mu\rho}-\delta^{\mu\rho}{T}^{\lambda\nu}+\delta^{\lambda\rho}{T}^{\mu\nu}\,,\\\
&\left[{T}^{\lambda\mu},{U}^{\nu\rho}\right]=\delta^{\mu\rho}{U}^{\nu\lambda}-\delta^{\lambda\rho}{U}^{\nu\mu}\,,\\\
&\left[{T}^{\prime\lambda\mu},{T}^{\prime\nu\rho}\right]=\delta^{\mu\nu}{T}^{\prime\lambda\rho}-\delta^{\lambda\nu}{T}^{\prime\mu\rho}-\delta^{\mu\rho}{T}^{\prime\lambda\nu}+\delta^{\lambda\rho}{T}^{\prime\mu\nu}\,,\\\
&\left[{T}^{\prime\lambda\mu},{U}^{\nu\rho}\right]=\delta^{\mu\nu}{U}^{\lambda\rho}-\delta^{\lambda\nu}{U}^{\mu\rho}\,,\\\
&\left[{U}^{\lambda\mu},{U}^{\nu\rho}\right]=-\delta^{\lambda\nu}{T}^{\mu\rho}-\delta^{\mu\rho}{T}^{\prime\lambda\nu}\,,\\\
&\left[{T}^{\lambda\mu},{S}^{\nu\rho}\right]=\delta^{\mu\nu}{S}^{\lambda\rho}-\delta^{\lambda\nu}{S}^{\mu\rho}-\delta^{\mu\rho}{S}^{\lambda\nu}+\delta^{\lambda\rho}{S}^{\mu\nu}\,,\\\
&\left[{T}^{\lambda\mu},{V}^{\nu\rho}\right]=\delta^{\mu\rho}{V}^{\nu\lambda}-\delta^{\lambda\rho}{V}^{\nu\mu}\,,\\\
&\left[{T}^{\prime\lambda\mu},{S}^{\prime\nu\rho}\right]=\delta^{\mu\nu}{S}^{\prime\lambda\rho}-\delta^{\lambda\nu}{S}^{\prime\mu\rho}-\delta^{\mu\rho}{S}^{\prime\lambda\nu}+\delta^{\lambda\rho}{S}^{\prime\mu\nu}\,,\\\
&\left[{T}^{\prime\lambda\mu},{V}^{\nu\rho}\right]=\delta^{\mu\nu}{V}^{\lambda\rho}-\delta^{\lambda\nu}{V}^{\mu\rho}\,,\\\
&\left[{U}^{\lambda\mu},{S}^{\nu\rho}\right]=-\delta^{\mu\rho}{V}^{\lambda\nu}+\delta^{\mu\nu}{V}^{\lambda\rho}\,,\\\
&\left[{U}^{\lambda\mu},{S}^{\prime\nu\rho}\right]=-\delta^{\lambda\rho}{V}^{\nu\mu}+\delta^{\lambda\nu}{V}^{\rho\mu}\,,\\\
&\left[{U}^{\lambda\mu},{V}^{\nu\rho}\right]=-\delta^{\lambda\nu}{S}^{\mu\rho}-\delta^{\mu\rho}{S}^{\prime\lambda\nu}\,,\\\
&\left[J_{ab},Q^{\pm\,\lambda}_{\alpha}\right]=-\frac{1}{2}\left(\Gamma_{ab}Q^{\pm\,\lambda}\right)_{\alpha}\,,\\\
&\left[{T}^{\lambda\mu},{Q}^{\pm\,\nu}_{\alpha}\right]=\frac{1}{2}\left[\delta^{\mu\nu}\left({Q}^{+\,\lambda}_{\alpha}+{Q}^{-\,\lambda}_{\alpha}\right)-\delta^{\lambda\nu}\left({Q}^{+\,\mu}_{\alpha}+{Q}^{-\,\mu}_{\alpha}\right)\right]\,,\\\
&\left[{T}^{\prime\lambda\mu},{Q}^{\pm\,\nu}_{\alpha}\right]=\pm\frac{1}{2}\left[\delta^{\mu\nu}\left({Q}^{+\,\lambda}_{\alpha}-{Q}^{-\,\lambda}_{\alpha}\right)-\delta^{\lambda\nu}\left({Q}^{+\,\mu}_{\alpha}-{Q}^{-\,\mu}_{\alpha}\right)\right]\,,\\\
&\left[{U}^{\lambda\mu},{Q}^{\pm\,\nu}_{\alpha}\right]=\mp\frac{1}{2}\left(\Gamma_{0}\right)_{\alpha\beta}\left[\delta^{\lambda\nu}\left({Q}^{+\,\mu}_{\beta}+{Q}^{-\,\
\mu}_{\beta}\right)\pm\delta^{\mu\nu}\left({Q}^{+\,\lambda}_{\beta}-{Q}^{-\,\lambda}_{\beta}\right)\right]\,,\\\
&\\{{Q}^{+\,\lambda}_{\alpha},{Q}^{+\,\mu}_{\alpha}\\}=\left(\Gamma^{0}C\right)_{\alpha\beta}\left(\delta^{\lambda\mu}{H}-{V}^{(\lambda\mu)}\right)+C_{\alpha\beta}{Y}^{[\lambda\mu]}\,,\\\
&\\{{Q}^{+\,\lambda}_{\alpha},{Q}^{-\,\mu}_{\beta}\\}=C_{\alpha\beta}{Y}^{\prime[\lambda\mu]}-\left(\Gamma^{0}C\right)_{\alpha\beta}{V}^{[\lambda\mu]}\,,\\\
&\\{{Q}^{-\,\lambda}_{\alpha},{Q}^{-\,\mu}_{\beta}\\}=\left(\Gamma^{0}C\right)_{\alpha\beta}\left(\delta^{\lambda\mu}{H}+{V}^{(\lambda\mu)}\right)+C_{\alpha\beta}{Y}^{[\lambda\mu]}\,.\end{split}$
(6.15)

The (anti)commutation relations (6.15) are those of the
$\mathcal{N}=(\mathcal{N},0)$, $D=3$ super-Carroll algebra (with $\mathcal{N}$
even), and one could also obtain it by taking the ultra-relativistic limit of
the $\mathfrak{so}(\mathcal{N})$ extension of the
$\mathcal{N}=(\mathcal{N},0)$, $D=3$ Poincaré superalgebra.

Moreover, as $\ell\rightarrow\infty$, the $2$-form curvatures (4.17) become

$\begin{split}\mathcal{R}^{ab}&=d\omega^{ab}=R^{ab}\,,\\\
\mathcal{K}^{a}&=dk^{a}+\omega^{a}_{\phantom{a}b}k^{b}=\mathfrak{K}^{a}\,,\\\
R^{a}&=dV^{a}+\omega^{a}_{\phantom{a}b}V^{b}\,,\\\
\mathcal{H}&=dh+V^{a}k_{a}-\frac{1}{2}\bar{\psi}^{+\,\lambda}\Gamma^{0}\psi^{+\,\lambda}-\frac{1}{2}\bar{\psi}^{-\,\lambda}\Gamma^{0}\psi^{-\,\lambda}=\mathfrak{H}-\frac{1}{2}\bar{\psi}^{+\,\lambda}\Gamma^{0}\psi^{+\,\lambda}-\frac{1}{2}\bar{\psi}^{-\,\lambda}\Gamma^{0}\psi^{-\,\lambda}\,,\\\
\mathcal{T}^{\lambda\mu}&=dt^{\lambda\mu}+{t^{\lambda}}_{\nu}t^{\nu\mu}+{u^{\prime[\lambda}}_{\nu}u^{\nu|\mu]}\,,\\\
\mathcal{T}^{\prime\lambda\mu}&=dt^{\prime\lambda\mu}+{t^{\prime\lambda}}_{\nu}t^{\prime\nu\mu}+{u^{[\lambda}}_{\nu}u^{\prime\nu|\mu]}\,,\\\
\mathcal{U}^{\lambda\mu}&=du^{\lambda\mu}+{u^{\lambda}}_{\nu}t^{\nu\mu}+{t^{\prime\lambda}}_{\nu}u^{\nu\mu}\,,\\\
\mathcal{S}^{\lambda\mu}&=ds^{\lambda\mu}+2{t^{\lambda}}_{\nu}s^{\nu\mu}+2{u^{\prime[\lambda}}_{\nu}v^{\nu|\mu]}-\frac{1}{2}\bar{\psi}^{+\,\lambda}\psi^{+\,\mu}-\frac{1}{2}\bar{\psi}^{-\,\lambda}\psi^{-\,\mu}-\bar{\psi}^{+\,[\lambda}\psi^{-\,\mu]}\,,\\\
\mathcal{S}^{\prime\lambda\mu}&=ds^{\prime\lambda\mu}+2{t^{\prime\lambda}}_{\nu}s^{\prime\nu\mu}+2{u^{[\lambda}}_{\nu}v^{\prime\nu|\mu]}-\frac{1}{2}\bar{\psi}^{+\,\lambda}\psi^{+\,\mu}-\frac{1}{2}\bar{\psi}^{-\,\lambda}\psi^{-\,\mu}+\bar{\psi}^{+\,[\lambda}\psi^{-\,\mu]}\,,\\\
\mathcal{V}^{\lambda\mu}&=dv^{\lambda\mu}+{v^{\lambda}}_{\nu}t^{\nu\mu}+{t^{\prime\lambda}}_{\nu}v^{\nu\mu}+{u^{\lambda}}_{\nu}s^{\nu\mu}+{s^{\prime\lambda}}_{\nu}u^{\nu\mu}+\frac{1}{2}\bar{\psi}^{+\,\lambda}\Gamma^{0}\psi^{+\,\mu}-\frac{1}{2}\bar{\psi}^{-\,\lambda}\Gamma^{0}\psi^{-\,\mu}\\\
&+\bar{\psi}^{+\,[\lambda}\Gamma^{0}\psi^{-\,\mu]}\,,\\\
\nabla\psi^{+\,\lambda}&=d\psi^{+\,\lambda}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\psi^{+\,\lambda}+\frac{1}{2}t^{\lambda\mu}\psi^{+}_{\mu}+\frac{1}{2}t^{\lambda\mu}\psi^{-}_{\mu}+\frac{1}{2}t^{\prime\lambda\mu}\psi^{+}_{\mu}-\frac{1}{2}t^{\prime\lambda\mu}\psi^{-}_{\mu}\\\
&+u^{(\lambda\mu)}\Gamma_{0}\psi^{+}_{\mu}+u^{[\lambda\mu]}\Gamma_{0}\psi^{-}_{\mu}\,,\\\
\nabla\psi^{-\,\lambda}&=d\psi^{-\,\lambda}+\frac{1}{4}\omega^{ab}\Gamma_{ab}\psi^{-\,\lambda}+\frac{1}{2}t^{\lambda\mu}\psi^{+}_{\mu}+\frac{1}{2}t^{\lambda\mu}\psi^{-}_{\mu}-\frac{1}{2}t^{\prime\lambda\mu}\psi^{+}_{\mu}+\frac{1}{2}t^{\prime\lambda\mu}\psi^{-}_{\mu}\\\
&-u^{(\lambda\mu)}\Gamma_{0}\psi^{-}_{\mu}-u^{[\lambda\mu]}\Gamma_{0}\psi^{+}_{\mu}\,.\end{split}$
(6.16)

Applying the $\ell\rightarrow\infty$ limit to the CS action (4.21), we obtain

$\begin{split}I^{(\mathcal{N},0)}_{CS}|_{\ell\rightarrow\infty}&=\frac{k}{4\pi}\int_{\mathcal{M}}\Bigg{\\{}\frac{\alpha_{0}}{2}\bigg{(}\omega^{a}_{\phantom{a}b}R^{b}_{\phantom{b}a}+2{t^{\lambda}}_{\mu}{dt^{\mu}}_{\lambda}+\frac{4}{3}{t^{\lambda}}_{\mu}{t^{\mu}}_{\nu}{t^{\nu}}_{\lambda}+2{t^{\prime\lambda}}_{\mu}{dt^{\prime\mu}}_{\lambda}+\frac{4}{3}{t^{\prime\lambda}}_{\mu}{t^{\prime\mu}}_{\nu}{t^{\prime\nu}}_{\lambda}\\\
&+4{u^{\lambda}}_{\mu}{du^{\prime\mu}}_{\lambda}-4t_{\lambda\mu}{u^{\prime\lambda}}_{\nu}u^{\nu\mu}-4t^{\prime}_{\lambda\mu}{u^{\lambda}}_{\nu}u^{\prime\nu\mu}\bigg{)}+\alpha_{1}\bigg{[}\epsilon_{ab}R^{ab}h-2\epsilon_{ab}\mathfrak{K}^{a}V^{b}\\\
&-2{t^{\lambda}}_{\mu}\left({ds^{\mu}}_{\lambda}+{t^{\mu}}_{\nu}{s^{\nu}}_{\lambda}\right)-2{t^{\prime\lambda}}_{\mu}\left({ds^{\prime\mu}}_{\lambda}+{t^{\prime\mu}}_{\nu}{s^{\prime\nu}}_{\lambda}\right)-4{u^{\lambda}}_{\mu}{dv^{\prime\mu}}_{\lambda}-2{u^{\prime\lambda}}_{\mu}{u^{\mu}}_{\nu}{s^{\nu}}_{\lambda}\\\