theory.

AdS Carroll Chern-Simons supergravity in 2+1 dimensions {#S2}
=======================================================

We can now move to the construction of the three-dimensional CS supergravity theory invariant under the so-called AdS Carroll superalgebra (in $D=3$) introduced in [@Bergshoeff:2015wma].

In [@Bergshoeff:2015wma] the $\mathcal{N}=1$ AdS Carroll superalgebra was obtained as a contraction of the $\mathcal{N}=1$ AdS superalgebra. In particular, as shown in [@Bergshoeff:2015wma], the AdS Carroll superalgebra is obtained by performing the indices decomposition in the AdS superalgebra, which in $D=3$ yields , and subsequently, to make the Carroll contraction, by rescaling the generators with a parameter, let us call it $\sigma$, as follows: $$\label{resc}
\tilde{H} \rightarrow \sigma H \, , \quad \tilde{K}_a \rightarrow \sigma K_a \,  , \quad \tilde{Q}_\alpha \rightarrow \sqrt{\sigma} Q_\alpha  \, .$$ Then, taking the limit $\sigma \rightarrow \infty$,[^6] in $D=3$ we get the following $\mathcal{N}=1$ AdS Carroll superalgebra generated by the set of generators $\lbrace J_{ab}, K_a, P_a, H, Q_\alpha \rbrace$ (spatial rotations, Carrollian boosts, space translations, time translations, and a two-components Majorana spinor charge, respectively)[^7] fulfilling the following non-trivial (anti)commutation relations: $$\label{adscarrollsuper}
\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} \, , \quad \quad
\left[ K_{a}, 