On the BRST structure of W_3 gravity coupled to c=2 matter

TL;DR

This paper investigates the structure of singular vectors in $c=2$ Verma modules of the $ ext{W}_3$ algebra, constructing resolutions and computing BRST cohomology for $ ext{W}_3$ gravity coupled to $c=2$ matter, including the ground ring states.

Contribution

It provides explicit constructions of singular vectors, resolutions of irreducible modules, and calculations of BRST cohomology in the context of $ ext{W}_3$ gravity with $c=2$ matter, advancing understanding of the algebra's structure.

Findings

Explicit singular vector structures in $c=2$ Verma modules

Resolutions of $c=2$ irreducible modules

Identification of states in the ground ring

Abstract

We present some explicit results on the structure of singular vectors in $c=2$ Verma modules of the $\cW_3$ algebra. Using the embedding patterns of those vectors we construct resolutions for the $c=2$ irreducible modules, and thus are able to compute some of the BRST cohomology of $\cW_3$ gravity coupled to $c=2$ matter. In particular, we determine the states in the ground ring of the theory. (To appear in the proceedings of the AMS Special Session on "Geometry and Physics", USC, Los Angeles, November 5-6, 1992)