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In a recent paper (JHEP {\bf 1508} (2015) 021), we have investigatedhypersymmetry bounds in the context of simple anti-de Sitter hypergravity in$2+1$ dimensions. We showed that these bounds involved non linearly thespin-$2$ and spin-$4$ charges, and were saturated by a class of extremal blackholes, which are $\frac14$-hypersymmetric. We continue the analysis here byconsidering $(M,N)$-extended anti-de Sitter hypergravity models, based on thesuperalgebra $osp(M \vert 4) \oplus osp(N \vert 4)$. The asymptotic symmetrysuperalgebra is then the direct sum of two-copies of a $W$-superalgebra thatcontains $so(M)$ (or $so(N)$) Kac-Moody currents of conformal weight $1$,fermionic generators of conformal weight $5/2$ and bosonic generators ofconformal weight $4$ in addition to the Virasoro generators. The nonlinearhypersymmetry bounds on the conserved charges are derived and shown to besaturated by a class of extreme hypersymmetric black holes which we explicitlyconstruct.

Extended anti-de Sitter Hypergravity in $2+1$ Dimensions and Hypersymmetry Bounds