Tensionless supersymmetric M2 branes in $Ad{S}_4\times S^7$ and Giant Diabolo

We find various supersymmetric configurations of toroidal M2 brane solutions in ${\text{AdS}}_4\times S^7$ or, more generally, in ${\text{AdS}}_4\times S^7\mathrm{/}{Z}_k$. In this class we identify solutions preserving 1/4 and 1/8 supersymmetries of the background. The supersymmetric M2 branes have angular momenta and winding on $S^7$, and null world-volumes. In certain cases they collapse to string-like configurations. These configurations can be viewed as a higher-dimensional (membrane) analog of BMN states. We compute the energy and angular momenta, showing that all supersymmetric configurations obey the BPS relation $E=J\mathrm{/}R$,
$J={\sum }_{i=1}^4\mathrm{\mid }{J}_i\mathrm{\mid }$ with $E,J\to \mathrm{\infty }$. Finally, we also study another class of supersymmetric M2-branes, including uncompact rotating membranes of “diabolo” shape.