Origin of the Pure Spinor and Green-Schwarz Formalisms

The pure spinor formalism for the superstring was recently obtained by gauge-fixing a purely bosonic classical action involving a twistor-like constraint $\partial x^m (\gamma_m\lambda)_\alpha =0$ where $\lambda^\alpha$ is a d=10 pure spinor. This twistor-like constraint replaces the usual Virasoro constraint $\partial x^m \partial x_m =0$, and the Green-Schwarz fermionic spacetime spinor variables $\theta^\alpha$ 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 $\lambda^\alpha$ 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 ${{SO(10)}\over{U(5)}}$, this action can be interpreted as a standard $\hat c=5$ topological action where one integrates over the ${{SO(10)}\over{U(5)}}$ 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.