The semi-infinite cohomology of affine Lie algebras

TL;DR

This paper investigates the semi-infinite cohomology of affine Lie algebras, revealing an infinite sequence of cohomology elements and implications for the state space in BRST quantization of WZNW models.

Contribution

It provides a detailed analysis of semi-infinite cohomology for affine Lie algebras and generalizes Verma module structure results from finite to affine cases.

Findings

Existence of an infinite sequence of cohomology elements for non-zero ghost numbers

BRST approach admits more states than traditional coset models

Generalization of Verma module structure results to affine Lie algebras

Abstract

We study in detail the semi-infinite or BRST cohomology of general affine Lie algebras. This cohomology is relevant in the BRST approach to gauged WZNW models. We prove the existence of an infinite sequence of elements in the cohomology for non-zero ghost numbers. This will imply that the BRST approach to topological WZNW model admits many more states than a conventional coset construction. This conclusion also applies to some non-topological models. Our work will also contain results on the structure of Verma modules over affine Lie algebras. In particular, we generalize the results of Verma and Bernstein-Gel'fand-Gel'fand,for finite dimensional Lie algebras, on the structure and multiplicities of Verma modules.