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Finite $W$-superalgebras and quadratic spacetime supersymmetries

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