Logarithmic lift of the su(2)_{-1/2} model

TL;DR

This paper explores different formulations of the non-unitary su(2)_{-1/2} WZW model using free-field representations, revealing continuous and indecomposable representations and discussing challenges in constructing modular invariants.

Contribution

It provides a detailed analysis of multiple versions of the su(2)_{-1/2} model through free-field representations and explores their representation theory and modular properties.

Findings

Identification of continuous and relaxed modules in the large algebra extension.

Recovery of a logarithmic theory with indecomposable representations.

Discussion of difficulties in constructing modular invariants for these models.

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 \eta\xi ghost system, and a Lorentzian boson. It turns out that there are several ``versions'' of the \eta\xi 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.