On Projections to the Pure Spinor Space

TL;DR

This paper introduces covariant non-linear projections from SO(10) Weyl spinors to pure spinors, analyzing their properties and applications to string theory ghost actions, potentially simplifying gauge fixing and measure calculations.

Contribution

It presents a family of covariant projections onto pure spinor space, identifying a Hermitian case derived from a scalar potential, and applies this to improve understanding of pure spinor string theory.

Findings

Projection simplifies ghost constraints in pure spinor string theory

Hermitian projection derived from a scalar potential

Potential for new gauge choices to clarify ghost origin

Abstract

A family of covariant non-linear projections from the space of SO(10) Weyl spinors onto the space of pure SO(10) Weyl spinors is presented. The Jacobian matrices of these projections are related to a linear projector which was previously discussed in pure spinor string literature and which maps the antighost to its gauge invariant part. Only one representative of the family leads to a Hermitian Jacobian matrix and can itself be derived from a scalar potential. Comments on the SO(1,9) case are given as well as on the non-covariant version of the projection map. The insight is applied to the ghost action of pure spinor string theory, where the constraints on the fields can be removed using the projection, while introducing new gauge symmetries. This opens the possibility of choosing different gauges which might help to clarify the origin of the pure spinor ghosts. Also the measure of the pure spinor space is discussed from the projection point of view. The appendix contains the discussion of a toy model which served as a guideline for the pure spinor case.