LOGARITHMIC INTERTWINING OPERATORS AND W(2,2p − 1)-ALGEBRAS

For every p ≥ 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 cp,1 = 1 − 6(p−1) 2 p , p ≥ 2. 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. 0. INTRODUCTION The Virasoro algebra is the most fundamental structure in two-dimensional conformal field theory. The most important family of Virasoro algebra modules are certainly the minimal models , because these models give rise to rational conformal field theories. Interestingly, many non-rational models have recently appeared in studies of W-algebras, which are certain extensions of Virasoro vertex algebras. Since there are several different types of W-algebras (see for instance [FKRW] for W-algebras of positive integer central charge), in this paper we limit ourselves to W-algebras closely related to representations of Virasoro algebra with central charge c p,1 = 1 − 6(p − 1) 2 p , p ∈ N ≥2. These central charges, belonging to the boundary of Kac's table, are relevant in logarithmic con-formal field theory [Gu],[F2], [F3], [Ga]. If we denote by L(c p,1 , 0) the simple lowest weight Virasoro algebra module of central charge c p,1 , then we have the following embedding of W-where W(2, 2p − 1) is also known as the singlet W-algebra, and W(2, (2p − 1) 3) is the triplet W-algebra. The theory of W-algebras could be understood much better from vertex algebra point of view. In this setup, the singlet vertex algebra W(2, 2p − 1) is generated by the Virasoro element ω and another element H of conformal weight 2p − 1 (cf. [H], [A2], [EFHHNV], [KW]). The singlet vertex algebra admits infinitely many nonisomorphic irreducible modules, so it fails to be rational. On the other hand, the triplet algebra W(2, (2p − 1) 3) (cf. [Ka1], [Ka2]) …