, \\
& \delta \psi \; : \quad \quad \alpha_1 \Psi = 0 \,  ,
\end{split}$$ respectively.[^8] In particular, when $\alpha_1 \neq 0$, the equations of motion in reduce to the vanishing of the super-AdS Carroll curvature $2$-forms, namely $$\label{eomvac}
\mathcal{R}^{ab} = 0 \,  , \quad \mathcal{K}^a = 0 \, , \quad R^a = 0 \, , \quad \mathcal{H}=0 \, , \quad \Psi =0 \, .$$ Let us observe that $\alpha_1 \neq 0$ is a sufficient condition to recover , meaning that one could consistently set $\alpha_0=0$, which corresponds to the vanishing of the exotic term in the CS action .

Notice also that if we restrict ourselves to the purely bosonic part of the action (that is if we neglect supersymmetry), we get $$\label{CSACbos}
\begin{split}
I^{\text{AdS Carroll}}_{CS} & = \frac{k}{4 \pi} \int_\mathcal{M} \Bigg \lbrace \frac{\alpha_0}{2} \left( \omega^a_{\phantom{a} b} R^b_{\phantom{b} a}  + \frac{2}{\ell^2} V^a R_a  \right) + \alpha_1 \left( \epsilon_{ab} R^{ab} h - 2 \epsilon_{ab} \mathfrak{K}^a V^b + \frac{1}{\ell^2} \epsilon_{ab} V^a V^b h \right) \\
& - d \left( \frac{\alpha_1}{2} \epsilon_{ab} \omega^{ab} h - \alpha_1 \epsilon_{ab} k^a V^b \right) \Bigg \rbrace \, ,
\end{split}$$ which is the three-dimensional CS gravity action invariant under the $D=3$ AdS Carroll algebra [@Matulich:2019cdo] (see 