Logarithmic lift of the [formula omitted] model

Abstract

This paper carries on the investigation of the non-unitary su(2) −1/2 WZW model. An essential tool in our first work on this topic was a free-field representation, based on a c=−2 ηξ ghost system, and a Lorentzian boson. It turns out that there are several ‘versions’ of the ηξ system, allowing different su(2) −1/2 theories. This is explored here in details. In more technical terms, we consider extensions (in the c=−2 language) from the small to the large algebra representation and, in a further step, to the full symplectic fermion theory. In each case, the results are expressed in terms of su(2) −1/2 representations. At the first new layer (large algebra), continuous representations appear which are interpreted in terms of relaxed modules. At the second step (symplectic formulation), we recover a logarithmic theory with its characteristic signature, the occurrence of indecomposable representations. To determine whether any of these three versions of the su(2) −1/2 WZW is well defined, one conventionally requires the construction of a modular invariant. This issue, however, is plagued with various difficulties, as we discuss.