The Complete Cohomology of the W3 String

TL;DR

This paper introduces a straightforward method to determine the complete cohomology of the BRST operator in W3 string theories, using invertible physical operators to classify all physical states.

Contribution

It provides a novel, simplified procedure for constructing the full cohomology of the BRST operator in W3 string models, enabling exhaustive classification of physical operators.

Findings

Constructed two level-15 physical operators that are invertible.

Mapped all physical operators into a fundamental unit cell.

Solved the entire cohomology problem for the W3 string.

Abstract

We present a simple procedure for constructing the complete cohomology of the BRST operator of the two-scalar and multi-scalar $W_3$ strings. The method consists of obtaining two level--15 physical operators in the two-scalar $W_3$ string that are invertible, and that can normal order with all other physical operators. They can be used to map all physical operators into non-trivial physical operators whose momenta lie in a fundamental unit cell. By carrying out an exhaustive analysis of physical operators in this cell, the entire cohomology problem is solved.