Explaining the Pure Spinor Formalism for the Superstring

TL;DR

This paper elucidates the pure spinor formalism for superstrings by reformulating the BRST operator through gauge fixing, similarity transformations, and connections to RNS formalism, clarifying its structure and vertex operator construction.

Contribution

It provides a detailed explanation of the pure spinor formalism, including the derivation of the BRST operator and the construction of GSO(-) vertex operators, linking it to Green-Schwarz and RNS formalisms.

Findings

BRST operator expressed as a conventional form with Virasoro and fermionic constraints

GSO(-) vertex operators constructed using spin fields and picture-changing

Connection established between pure spinor, Green-Schwarz, and RNS formalisms

Abstract

After adding a pair of non-minimal fields and performing a similarity transformation, the BRST operator in the pure spinor formalism is expressed as a conventional-looking BRST operator involving the Virasoro constraint and (b,c) ghosts, together with 12 fermionic constraints. This BRST operator can be obtained by gauge-fixing the Green-Schwarz superstring where the 8 first-class and 8 second-class Green-Schwarz constraints are combined into 12 first-class constraints. Alternatively, the pure spinor BRST operator can be obtained from the RNS formalism by twisting the ten spin-half RNS fermions into five spin-one and five spin-zero fermions, and using the SO(10)/U(5) pure spinor variables to parameterize the different ways of twisting. GSO(-) vertex operators in the pure spinor formalism are constructed using spin fields and picture-changing operators in a manner analogous to Ramond vertex operators in the RNS formalism.