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By extending the methods used in our earlier work, in this paper, we presentan explicit realization of logarithmic $\mathcal{W}_{p,p'$}-modules that haveL(0) nilpotent rank three. This was achieved by combining the techniquesdeveloped in \cite{AdM-2009} with the theory of local systems of vertexoperators \cite{LL}. In addition, we also construct a new type of extension of$\mathcal{W}_{p,p'}$, denoted by $\mathcal{V}$. Our results confirm severalclaims in the physics literature regarding the structure of projective coversof certain irreducible representations in the principal block. This approachcan be applied to other models defined via a pair screenings.

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