The W_3 algebra: modules, semi-infinite cohomology and BV-algebras

TL;DR

This paper explores the BRST quantization of the noncritical $D=4$ $W_3$ string, analyzing its physical spectrum and BV-algebra structure, and models the algebra using polyvector fields on a complex affine space.

Contribution

It provides a detailed analysis of the BRST cohomology and BV-algebra structure of the $W_3$ string, extending previous work to a higher-dimensional setting.

Findings

Calculated the physical spectrum via BRST cohomology.

Identified the BV-algebra structure related to polyvector fields.

Presented a comprehensive summary of the progress in $W_3$ algebra studies.

Abstract

The noncritical $D=4$ $W_3$ string is a model of $W_3$ gravity coupled to two free scalar fields. In this paper we discuss its BRST quantization in direct analogy with that of the $D=2$ (Virasoro) string. In particular, we calculate the physical spectrum as a problem in BRST cohomology. The corresponding operator cohomology forms a BV-algebra. We model this BV-algebra on that of the polyderivations of a commutative ring on six variables with a quadratic constraint, or, equivalently, on the BV-algebra of (polynomial) polyvector fields on the base affine space of $SL(3,C)$. In this paper we attempt to present a complete summary of the progress made in these studies. [...]