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An explicit realization of logarithmic modules for the vertex operator algebra W(p, p)'

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 [AM3] with the theory of local systems of vertex operators [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. This approach can be applied to other models defined via a pair screenings.

vertex algebras; logarithmic conformal field theory; triplet vertex algebras; extended vertex algebras; projective covers; intertwining operators; screening operators; c=0 triplet model

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