Normal form of nilpotent vector field near the tip of the pure spinor cone

Abstract

Pure spinor formalism implies that supergravity equations in space-time are equivalent to the requirement that the worldsheet sigma-model satisfies certain properties. Here we point out that one of these properties has a particularly transparent geometrical interpretation. Namely, there exists an odd nilpotent vector field on some singular supermanifold, naturally associated to space-time. Is it true that all supergravity fields are encoded in this vector field, as coefficients in its normal form, and the nilpotence is equivalent to the target space equations of motion? We show that this is approximately correct. The normal form is parametrized by some tensor fields, which satisfy hyperbolic equations. These equations are slightly weaker than the full supergravity equations.