Virasoro Singular Vectors via Quantum DS Reduction

TL;DR

This paper uses BRST quantisation and quantum Drinfeld-Sokolov reduction to systematically derive all Virasoro algebra singular vectors from affine algebra counterparts, revealing their equivalence.

Contribution

It introduces a quantum DS gauge transformation method to connect Virasoro singular vectors with affine algebra vectors, providing a new systematic derivation approach.

Findings

All Virasoro singular vectors can be obtained from affine algebra vectors.

Virasoro singular vectors are equivalent to affine ones modulo trivial terms.

Quantum DS gauge transformation is a key tool in this derivation.

Abstract

The BRST quantisation of the Drinfeld - Sokolov reduction is exploited to recover all singular vectors of the Virasoro algebra Verma modules from the corresponding $A^{(1)}_1\,$ ones. The two types of singular vectors are shown to be identical modulo terms trivial in the $Q_{BRST}$ cohomology. The main tool is a quantum version of the DS gauge transformation.