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

For every p2, 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 cp,1 =1- [6 (p-1)2 p], p2. 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.