The original { \it membrane at the end of the universe} corresponds to a probe $M2$-brane of signature $(2,1)$ occupying the $S^{ 2} \times S^{ 1}$ boundary of the $(10,1)$ spacetime $AdS_{ 4} \times S^{ 7}$, and is described by an $OSp(4/8)$ SCFT. However, it was subsequently generalized to other worldvolume signatures $(s,t)$ and other spacetime signatures $(S,T)$. An interesting special case is provided by the $(3,0)$ brane at the end of the de Sitter universe $dS_{ 4}$ which has recently featured in the $dS/CFT$ correspondence. The resulting CFT contains the one recently proposed as the holographic dual of a four-dimensional de Sitter cosmology. Supersymmetry restricts $S,T,s,t$ by requiring that the corresponding bosonic symmetry $O(s+1,t+1) \times O(S-s,T-t)$ be a subgroup of a superconformal group. The case of $dS_{ 4} \times AdS_{ 7}$ is `doubly holographic' and may be regarded as the near horizon geometry of $N_{ 2}$ $M2$-branes or equivalently, under interchange of conformal and R symmetry, of $N_{ 5}$ $M5$-branes, provided $N_{ 2}=2N_{ 5}{ }^{ 2}$. The same correspondence holds in the pp-wave limit of conventional $M$-theory.
Categories: hep-th