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

TL;DR

This paper formulates the pure spinor superstring in curved supergravity backgrounds, showing that BRST operator properties lead to on-shell supergravity constraints, facilitating quantization and potential higher-order correction calculations.

Contribution

It provides a pure spinor formalism in curved backgrounds that derives supergravity constraints directly from BRST nilpotency and holomorphicity.

Findings

Derivation of supergravity constraints from pure spinor BRST conditions

Formulation applicable to heterotic and Type II superstrings

Framework for computing higher-order $oldsymbol{eta'}$ corrections

Abstract

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.