The semi-infinite cohomology of affine Lie algebras

We study the semi-infinite or BRST cohomology of affine Lie algebras in detail. This cohomology is relevant in the BRST approach to gauged WZNW models. Our main result is to prove necessary and sufficient conditions on ghost numbers and weights for non-trivial elements in the cohomology. In particular 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’fandGel’fand, for finite dimensional Lie algebras, on the structure and multiplicities of Verma modules. The present work gives the theoretical basis of the explicit construction of the elements in cohomology presented previously. Our analysis proves and makes use of the close relationship between highest weight null-vectors and elements of the cohomology. [email protected]