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The membrane at the end of the (de Sitter) universe

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.