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Logarithmic intertwining operators and W(2, 2p− 1) algebras

For every $p\ge 2$, we obtained an explicit construction of a family of W(2, 2p− 1) modules, which decompose as direct sum of simple Virasoro algebra modules. Furthermore, we classified all irreducible self-dual W(2, 2p− 1) modules, we described their internal structure, and computed their graded dimensions. In addition, we constructed certain hidden logarithmic intertwining operators among two ordinary and one logarithmic W(2, 2p− 1) modules. This work, in particular, gives a mathematically precise formulation and interpretation of what physicists have been referring to as “ logarithmic conformal field theory” of central charge c_{;p, 1};. Our explicit construction can be easily applied for computations of correlation functions. Techniques from this paper can be used to study the triplet vertex operator algebra W(2, (2p− 1)^3) and other logarithmic models.

W-algebras; vertex operator algebras; logarithmic intertwining operators

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