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

TL;DR

This work shows that nilpotency and holomorphicity of the pure spinor BRST operator in a curved ten-dimensional supergravity background constrain the background superfields to the on-shell equations of motion, providing compact pure-spinor formulations for $N=1$ and $N=2$ (Type II) ten-dimensional supergravity. By analyzing both heterotic and Type II sigma models to leading order in $\alpha'$, it derives explicit superspace constraints that reproduce the standard supergravity/Yang–Mills equations and expresses background fields entirely in terms of spinor superfields. The paper also establishes the compatibility of the pure-spinor description of Type IIB with the Howe–West superspace via a Weyl- and Lorentz-reducing mapping, clarifying the role of the $SL(2,\mathbb{R})$ scalars. Finally, it outlines how higher-order $\alpha'$ corrections could be computed within this framework, including the potential roles of Fradkin–Tseytlin terms and Chern–Simons modifications, and anticipates corrections such as $R^4$ terms at higher loops.

Abstract

It has recently been shown that the ten-dimensional superstring can be quantized using the BRST operator $Q=\ointλ^αd_α$ where $λ^α$ is a pure spinor satisfying $λγ^m λ=0$ and $d_α$ 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 $α'$ 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 $α'$ corrections to the ten-dimensional superspace constraints.