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We consider Lie superalgebras under constraints of Hamiltonian reduction,yielding finite $W$-superalgebras which provide candidates for quadraticspacetime superalgebras. These have an undeformed bosonic symmetry algebra(even generators) graded by a fermionic sector (supersymmetry generators) withanticommutator brackets which are quadratic in the even generators. We analyzethe reduction of several Lie superalgebras of type $gl(M|N)$ or $osp(M|2N)$ atthe classical (Poisson bracket) level, and also establish their quantum (Liebracket) equivalents. Purely bosonic extensions are also considered. As aspecial case we recover a recently identified quadratic superconformal algebra,certain of whose unitary irreducible massless representations (in fourdimensions) are "zero-step" multiplets, with no attendant superpartners. Othercases studied include a six dimensional quadratic superconformal algebra withvectorial odd generators, and a variant quadratic superalgebra with undeformed$osp(1|2N)$ singleton supersymmetry, and a triplet of spinorial supercharges.

Finite $W$-superalgebras and quadratic spacetime supersymmetries