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

TL;DR

This paper explicitly constructs logarithmic modules for the vertex operator algebra W_{p,p'} with nilpotent rank three, combining previous techniques and local systems theory, and introduces a new extension algebra.

Contribution

It provides an explicit realization of logarithmic W_{p,p'}-modules with nilpotent rank three and constructs a new extension algebra, confirming physics conjectures.

Findings

Explicit realization of logarithmic modules with nilpotent rank three

Construction of a new extension algebra of W_{p,p'}

Confirmation of physics literature claims on projective covers

Abstract

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. This approach can be applied to other models defined via a pair screenings.