Origin of the pure spinor and Green-Schwarz formalisms

Abstract

The pure spinor formalism for the superstring was recently obtained by gaugefixing 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 θ α 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 GreenSchwarz 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 standardĉ = 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 doubledimensional reduction to the purely bosonic action for the d=10 Type IIA superstring.

On the field-antifield (a)symmetry of the pure spinor superstring

“…A deeper analysis shows that this is the most general construction for the measure in SO (8). The particular case with A = −1, B = − 9 2 and C = 5 is displayed in equation (3.39).…”

“…A full "ungauged" description has been presented in [7,8], involving a twistor-like symmetry. A master action was proposed and shown to give rise to the usual pure spinor BRST symmetry or to the Green-Schwarz κ-symmetry [9], depending on the chosen gauge fixing.…”

Notes on the ambitwistor pure spinor string

“…If one requires the physical states to be annihilated by H B , that would automatically project out any possible massive BRST-closed state. As it turns out, H B is BRST-exact [12]. To show that, consider first the following:…”

“…On the other hand, the action S has further symmetries. One of them, in particular, is generated by the particle-like Hamiltonian 12) which can be interpreted as the mass operator and commutes with the BRST charge. If one requires the physical states to be annihilated by H B , that would automatically project out any possible massive BRST-closed state.…”

The non-minimal heterotic pure spinor string in a curved background

“…The answer is yes. A way to get b L is to first define a RNS-like vector 12) where N np = 1 2 (λ L γ np ω L ), that transforms under Q L as 13) and…”