BRST cohomology of the critical $W_4$ string

TL;DR

This paper analyzes the cohomology of the critical $W_4$ string using a specialized BRST charge, revealing a stepwise construction of physical operators linked to minimal models and comparing with existing results.

Contribution

It introduces a novel basis for the $W_4$ BRST charge, enabling a three-step cohomology analysis and detailed structure description of the critical $W_4$ string.

Findings

Cohomology associated with a spin-four constraint contains $c=4/5$ $W_3$ minimal model operators.

Adding the spin-three constraint dresses operators to the $c=7/10$ Virasoro minimal model.

Complete cohomology of the critical $W_4$ string is explicitly described and compared with prior work.

Abstract

We study the cohomology of the critical $W_4$ string using the $W_4$ BRST charge in a special basis in which it contains three separately nilpotent BRST charges. This allows us to obtain the physical operators in three steps. In the first step we obtain the cohomology associated to a spin-four constraint only, and it contains operators of the $c={4\over5}$ $W_3$ minimal model. In the next step, where the spin-three constraint is added, these operators get dressed to operators of the $c={7\over10}$ Virasoro minimal model. Finally, the Virasoro constraint is added to obtain the cohomology of the critical $W_4$ string. We describe the structure of the complete cohomology and compare with other results.