Logarithmic Intertwining Operators and Vertex Operators

This is the first in a series of papers where we study logarithmic intertwining operators for various vertex subalgebras of Heisenberg vertex operator algebras. In this paper we examine logarithmic intertwining operators associated with rank one Heisenberg vertex operator algebra M(1)a, of central charge 1 − 12a . We classify these operators in terms of depth and provide explicit constructions in all cases. Furthermore, for a = 0 we focus on the vertex operator subalgebra L(1, 0) of M(1)0 and obtain logarithmic intertwining operators among indecomposable Virasoro algebra modules. In particular, we construct explicitly a family of hidden logarithmic intertwining operators, i.e., those that operate among two ordinary and one genuine logarithmic L(1, 0)-module. In the sequel we shall consider logarithmic intertwiners for various vertex subalgebras of M(1)a. 0. Introduction The theory of vertex algebras continues to be very effective in proving rigorous results in two-dimensional Conformal Field Theory (CFT) (for some recent breakthrough see [H1], [H2], [Le]). In 1993, Gurarie [Gu] constructed a CFT-like structure with two peculiar features absent in the ordinary CFT: logarithmic behavior of matrix coefficients and appearance of indecomposable representations of the Virasoro algebra underlying the theory. There are many examples of ”logarithmic” models with similar properties that have been discovered since then (see for instance [GK1], [F1], [G1], and especially [FFHST], [F2], [G2] and references therein). By now, a structure that involves a family of modules closed under the fusion, with logarithmic terms in the operator product expansion is usually called a logarithmic conformal field theory (LCFT). In spite of the progress made, it is still unclear to us how to define a rational LCFT and how to formulate an analogue of the Verlinde formula. Nevertheless, important examples of interest have been studied in great details from different points of view (e.g., the triplet model [GK1], [GK2], [GK3], [BF], etc.). In [M1] we proposed a purely algebraic approach to LCFT based on the notion of logarithmic modules and logarithmic intertwining operators. The key idea is to introduce a deformation parameter log(x) and to define logarithmic intertwining operators as expressions involving intertwining-like operators multiplied with appropriate powers of log(x), such that the translation invariance is preserved. In our setup we do not require an extension of the space of ”states” for the underlying vertex algebra. There are other proposals in the literature [FFHST] where log(x) is also viewed as a deformation parameter, but with an important difference that log(x) is also part of an extended chiral algebra (or an OPE algebra). Even though the construction in [FFHST] has been shown successful in explaining various logarithmic behaviors of CFTs, it is unclear to us if the approach in [FFHST] can be used to address the problem of fusion. On 1 2 ANTUN MILAS the other hand, the logarithmic intertwining operators have been used by Huang Lepowsky and Zhang in [HLZ] as a convenient tool to develop a generalization of Huang-Lepowsky’s tensor product theory [HL] to non-semisimple tensor categories. In addition, Miyamoto [My] found a generalization of the modular invariance theorem for logarithmic modules of VOAs, which satisfy the C2-condition. From everything being said it appears that several important aspects of LCFTs can be studied in the framework of vertex (operator) algebras. This paper continues naturally on [M1] and [M2]. In the present work we focus on a simple, yet interesting class of vertex operator algebras-those associated with Heisenberg Lie algebras and the Virasoro algebra. The aim is to construct a family of logarithmic intertwining operators associated with certain weak M(1)a-modules, which can be used for building logarithmic intertwining operators among indecomposable representations of the Virasoro algebra (and possibly various W-algebras). The most interesting part of our work is an explicit construction of the socalled hidden logarithmic intertwining operators, i.e., those which intertwine a pair of ordinary and one logarithmic module. Let us elaborate this on an example. Consider the Feigin-Fuchs moduleM(1, λ)a of central charge c = 1−12a 2 and lowest conformal weight λ 2 2 − aλ. Tensor the module M(1, λ)a with a two-dimensional space Ω, where h(0) acts on Ω (in some basis) as