An explicit realization of logarithmic modules for the vertex operator algebra W_{p,p'}

Understand

By extending the methods used in our earlier work, in this paper, we present an explicit realization of logarithmic $\mathcal{W}_{p,p'$}-modules that have L(0) nilpotent rank three.

• This was achieved by combining the techniques developed in \cite{AdM-2009} with the theory of local systems of vertex operators \cite{LL}.
• In addition, we also construct a new type of extension of $\mathcal{W}_{p,p'}$, denoted by $\mathcal{V}$.
• Our results confirm several claims in the physics literature regarding the structure of projective covers of certain irreducible representations in the principal block.