Super M-brane actions in ${\mathrm{AdS}}_4\times {\mathrm{S}}^7$ and ${\mathrm{AdS}}_7\times {\mathrm{S}}^4$

The world-volume action of the M2-brane and the M5-brane in an ${\mathrm{AdS}}_4\times {\mathrm{S}}^7$ and an ${\mathrm{AdS}}_7\times {\mathrm{S}}^4$ background is derived to all orders in anticommuting superspace coordinates $\theta .$ Contrary to recent constructions of super p-brane actions relying on supercoset methods, we only use 11-dimensional supergravity torsion and curvature constraints. Complete agreement between the two methods is found. The possible simplification of the action by choosing a suitable $\kappa$ gauge is discussed.