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.

纯旋量形式主义意味着时空中的超重力方程等价于世界表 sigma 模型满足某些属性的要求。在这里，我们指出这些属性之一具有特别透明的几何解释。也就是说，在某个奇异的超流形上存在一个奇幂零向量场，自然地与时空相关联。是不是所有的超重力场都在这个向量场中编码，作为其正规形式的系数，并且幂零等价于目标空间运动方程？我们证明这大致正确。正规形式由一些​​满足双曲方程的张量场参数化。这些方程比完整的超重力方程略弱。