Origin of the pure spinor and Green-Schwarz formalisms

Abstract

The pure spinor formalism for the superstring was recently obtained by gauge-fixing a purely bosonic classical action involving a twistor-like constraint ∂x m (γ m λ) α = 0 where λ α is a d=10 pure spinor. This twistor-like constraint replaces the usual Virasoro constraint ∂x m ∂x m = 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 λ α 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 $$ \frac{\mathrm{SO}(10)}{\mathrm{U}(5)} $$ , this action can be interpreted as a standard ĉ = 5 topological action where one integrates over the $$ \frac{\mathrm{SO}(10)}{\mathrm{U}(5)} $$ 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.