Origin of the Pure Spinor and Green-Schwarz Formalisms

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The pure spinor formalism for the superstring was recently obtained by gauge-fixing a purely bosonic classical action involving a twistor-like constraint ∂ xᵐ (γₘλ)α=0 where λα is a d=10 pure spinor. This twistor-like constraint replaces the usual Virasoro constraint ∂ xᵐ ∂ xₘ =0, and the Green-Schwarz fermionic spacetime spinor variables θα arise as Faddeev-Popov ghosts for this constraint. In this paper, the purely bosonic classical action is simplified by replacing the classical d=10 pure spinor λα with a d=10 projective pure spinor. The pure spinor and Green-Schwarz formalisms for the superparticle and superstring are then obtained as different gauge-fixings of this purely bosonic classical action, and the Green-Schwarz kappa symmetry is directly related to the pure spinor BRST symmetry. Since a d=10 projective pure spinor parameterizes SO(10)U(5), this action can be interpreted as a standard c=5 topological action where one integrates over the SO(10)U(5) choice of complex structure. Finally, a purely bosonic action for the d=11 supermembrane is proposed which reduces upon double-dimensional reduction to the purely bosonic action for the d=10 Type IIA superstring.