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

TL;DR

This paper reveals a geometric interpretation of supergravity equations via a nilpotent vector field on a supermanifold, linking supergravity fields to the normal form of this vector field.

Contribution

It introduces a novel geometric framework connecting supergravity equations to the normal form of an odd nilpotent vector field in the pure spinor formalism.

Findings

Supergravity fields are encoded in the normal form of a nilpotent vector field.

The nilpotence condition relates to supergravity equations of motion.

A geometric interpretation of supergravity constraints is established.

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. All supergravity fields are encoded in this vector field, as coefficients in its normal form. The nilpotence implies, modulo some zero modes, that they satisfy the SUGRA equations of motion.