also [@Bergshoeff:2015wma]) $$\label{adscarroll}
\begin{split}
& \left[ K_a , J_{bc} \right] = \delta_{ab} K_c - \delta_{ac} K_b \, , \quad \quad
\left[ J_{ab}, P_{c}\right] =\delta_{bc}P_{a}-\delta _{ac}P_{b} \, , \quad \quad
\left[ K_{a}, P_{b}\right] = - \delta_{ab} H \, , \\
& \left[ P_a , P_b \right] = \frac{1}{\ell^2} J_{ab} \, , \quad \quad
\left[ P_a , H \right] = \frac{1}{\ell^2} K_{a} \, , 
\end{split}$$ bosonic subalgebra of the AdS-Carroll superalgebra .

Study of the flat limit {#S3}
=======================

In the sequel, we finally study the flat limit ($\ell \rightarrow \infty$), which can be directly applied to the AdS Carroll superalgebra , to the curvature $2$-forms , to the CS action , to the transformation laws , , and , and to the field equations .

In the limit $\ell \rightarrow \infty$, the (anti)commutation relations of the superalgebra reduce to the following non-trivial ones: $$\label{adscarrollsuperF}
\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 , H \right]  = \frac{1}{\ell^2} K_{a} \, , \\
& \left[ J_{ab},Q_{\alpha }\right] =-\frac{1}{2}\left( \Gamma _{ab} Q \right)_{\alpha 