Singular vectors for the WN algebras and the BRST cohomology for relaxed highest-weight Lk(sl(2)) modules

Abstract

This thesis presents the computation of singular vectors of the W_n algebras and the BRST cohomology of modules of the simple vertex operator algebra L_k(sl2) associated to the affine Lie algebra of sl2 in the relaxed category

We will first recall some general theory on vertex operator algebras. We will then introduce the module categories that are relevant for conformal field theory. They are the category O of highest-weight modules and the relaxed category which contains O as well as the relaxed highest-weight modules with spectral flow and non-split extensions. We will then introduce the W_n algebras and the simple vertex operator algebra L_k(sl2). Properties of the Heisenberg algebra, the bosonic and the fermionic ghosts will be discussed as they are required in the free field realisations of W_n and L_k(sl2) as well as the construction of the BRST complex.

We will then compute explicitly the singular vectors of W_n algebras in their Fock representations. In particular, singular vectors can be realised as the image of screening operators of the W_n algebras. One can then realise screening operators in terms of Jack functions when acting on a highest-weight state, thereby obtaining explicit formulae of the singular vectors in terms of symmetric functions.

We will then discuss the BRST construction and the BRST cohomology for modules in category O. Lastly we compute the BRST cohomology for L_k(sl2) modules in the relaxed category. In particular, we compute the BRST cohomology for the highest-weight modules with positive spectral flow for all degrees and the BRST cohomology for the highest-weight modules with negative spectral flow for one degree.