in in the general expression , we can finally write the three-dimensional AdS Carroll CS supergravity action, which reads $$\label{CSAC}
\begin{split}
& I^{\text{super-AdS Carroll}}_{CS} = \frac{k}{4 \pi} \int_\mathcal{M} \Bigg \lbrace \frac{\alpha_0}{2} \left( \omega^a_{\phantom{a} b} R^b_{\phantom{b} a}  + \frac{2}{\ell^2} V^a R_a  \right) \\
& + \alpha_1 \left( \epsilon_{ab} R^{ab} h - 2 \epsilon_{ab} \mathfrak{K}^a V^b + \frac{1}{\ell^2} \epsilon_{ab} V^a V^b h+ 2 \bar{\psi} \Psi \right) - d \left( \frac{\alpha_1}{2} \epsilon_{ab} \omega^{ab} h - \alpha_1 \epsilon_{ab} k^a V^b \right) \Bigg \rbrace \,  ,
\end{split}$$ written in terms of the curvatures appearing in . Notice that the term proportional to $\alpha_0$ correspond to the exotic Lagrangian involving the Lorentz contribution and a torsional piece, while it does not contain any contribution from the gravitino $1$-form field $\psi$.

The CS action is characterized by two coupling constants and it is invariant by construction under the super-AdS Carroll group. In particular, the local gauge transformations $\delta_\varrho A = d \varrho + \left[A, \varrho \right]$ with gauge parameter $$\label{gpar}
\varrho = \frac{1}{2} \varrho^{ab} J_{ab} + \kappa^a K_a + \varrho^a P_a + \tau H + \varepsilon^\alpha Q_\alpha$$ are given by $$\label{gaugetr}
\begin{split}
\delta \omega^{ab} & = d \varrho^{ab} + \frac{2}{\ell^2} V^a \varrho^b \, , \\
\delta k^a 