The b Ghost of the Pure Spinor Formalism is Nilpotent

TL;DR

This paper demonstrates that the ghost variable in the pure spinor formalism is nilpotent by showing its operator product expansion with itself has no singularities, clarifying its algebraic properties.

Contribution

It proves the nilpotency of the ghost variable in the pure spinor formalism, which was previously not established.

Findings

The ghost variable commutes with the BRST operator to produce the stress tensor.

The operator product expansion of the ghost with itself is non-singular.

The nilpotency of the ghost simplifies the understanding of the formalism.

Abstract

The ghost for world-sheet reparametrization invariance is not a fundamental field in the pure spinor formalism. It is written as a combination of pure spinor variables which have conformal dimension two and such that it commutes with the BRST operator to give the world-sheet stress tensor. We show that the ghost variable defined in this way is nilpotent since the OPE of b with itself does not have singularities.