Ten-Dimensional Supergravity Constraints from the Pure Spinor Formalism for the Superstring

It has recently been shown that the ten-dimensional superstring can be quantized using the BRST operator $Q=\oint\lambda^\alpha d_\alpha$ where $\lambda^\alpha$ is a pure spinor satisfying $\lambda \gamma^m \lambda=0$ and $d_\alpha$ is the fermionic supersymmetric derivative. In this paper, the pure spinor version of superstring theory is formulated in a curved supergravity background and it is shown that nilpotency and holomorphicity of the pure spinor BRST operator imply the on-shell superspace constraints of the supergravity background. This is shown to lowest order in $\alpha'$ for the heterotic and Type II superstrings, thus providing a compact pure spinor version of the ten-dimensional superspace constraints for N=1, Type IIA and Type IIB supergravities. Since quantization is straightforward using the pure spinor version of the superstring, it is expected that these methods can also be used to compute higher-order $\alpha'$ corrections to the ten-dimensional superspace constraints.