Pure Spinors for General Backgrounds

TL;DR

This paper demonstrates the geometric equivalence of various pure spinor constraints in supergravity, computes their solutions, and confirms the degrees of freedom match those used in superstring theory.

Contribution

It provides a geometric understanding of pure spinor constraints, computes their solutions in different covariant decompositions, and establishes their equivalence across various formulations.

Findings

Pure spinor constraints are geometrically equivalent.

Solutions have 22 degrees of freedom, matching superstring constraints.

FDA type IIA/B constraints are equivalent to Poincare pure spinor constraints.

Abstract

We show the equivalence of the different types of pure spinor constraints geometrically derived from the Free Differential Algebras of N=2 d=10 supergravities. Firstly, we compute the general solutions of these constraints, using both a G_2 and an SO(8) covariant decomposition of the 10d chiral spinors. Secondly, we verify that the number of independent degrees of freedom is equal to that implied by the Poincare' pure spinor constraints so-far used for superstrings, namely twenty two. Thirdly, we show the equivalence between the FDA type IIA/B constraints among each other and with the Poincare' ones.