Weak modules and logarithmic intertwining operators for vertex operator algebras

We consider a class of weak modules for vertex operator algebras that we call logarithmic modules. We also construct nontrivial examples of intertwining operators between certain logarithmic modules for the Virasoro vertex operator algebra. At the end we speculate about some possible logarithmic intertwiners at the level c = 0. Introduction This work is an attempt to explain an algebraic reformulation of “logarithmic conformal field theory” from the vertex operator algebra point of view. The theory of vertex operator algebras, introduced in works of Borcherds (cf. [Bo]), Frenkel, Huang, Lepowsky and Meurman ([FHL], [FLM]) , is the mathematical counterpart of conformal field theory, introduced in [BPZ]. One usually studies rational vertex operator algebras, which satisfy a certain semisimplicity condition for modules. If we want to go beyond rational vertex operator algebras we encounter several difficulties. First, we have to study indecomposable, reducible modules, for which there is no classification theory. Another problem is that the notion of intertwining operator, as defined in [FHL], is not the most natural. We can illustrate this with the following example. Let L(c, 0) (cf. [FZ]) be a non–rational vertex operator algebra associated to a lowest weight representation of the Virasoro algebra and Y1,Y2 (1) a pair of intertwining operators between certain triples of L(c, 0)–modules. It is well known that one can study matrix coefficients 〈w 3,Y1(w1, x)w2〉 (2) and 〈v 4,Y1(v1, x1)Y2(v2, x2)v3〉, (3) ∗The author is partially supported by NSF grants.