A note on W_{2,s} strings

TL;DR

This paper analyzes BRST operators for two-dimensional theories with higher-spin currents, identifying their algebraic structures and cohomological properties, and extends the construction to the case s=7.

Contribution

It provides a detailed analysis of known higher-spin BRST operators, links them to specific W-algebras, and introduces the BRST operator for s=7.

Findings

s=4 BRST operators linked to W B_2 algebra

s=6 BRST operators linked to W G_2 algebra

Constructed BRST operator for s=7

Abstract

BRST operators for two-dimensional theories with spin-2 and spin-$s$ currents, generalising the $W_3$ BRST operator of Thierry-Mieg, have previously been obtained. The construction was based on demanding nilpotence of the BRST operators, making no reference to whether or not an underlying $W$ algebra exists. In this paper, we analyse the known cases ($s=3$, 4, 5 and 6), showing that the two $s=4$ BRST operators are associated with the $W\!B_2$ algebra, and that two of the four $s=6$ BRST operators are associated with the $W\!G_2$ algebra. We discuss the cohomology of all the known higher-spin BRST operators, the Weyl symmetry of their physical states, and their relation with certain minimal models. We also obtain the BRST operator for the case $s=7$.