The $W_{3}$ algebra: modules, semi-infinite cohomology and BV algebras

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. [...]