The su(2)_{-1/2} WZW model and the beta-gamma system

TL;DR

This paper provides a comprehensive analysis of the su(2)_{-1/2} WZW model and its associated beta-gamma system, revealing an infinite spectrum of negative-dimension fields and clarifying their algebraic and modular properties.

Contribution

It offers the first complete physical analysis of the su(2)_{-1/2} WZW model and elucidates the role of spectral flow and twisted sectors in the spectrum.

Findings

Spectrum includes infinitely many fields with negative dimensions.

Twisted sectors are related to spectrally flowed representations.

Modular invariance is reinterpreted in light of the spectrum.

Abstract

The bosonic beta-gamma ghost system has long been used in formal constructions of conformal field theory. It has become important in its own right in the last few years, as a building block of field theory approaches to disordered systems, and as a simple representative -- due in part to its underlying su(2)_{-1/2} structure -- of non-unitary conformal field theories. We provide in this paper the first complete, physical, analysis of this beta-gamma system, and uncover a number of striking features. We show in particular that the spectrum involves an infinite number of fields with arbitrarily large negative dimensions. These fields have their origin in a twisted sector of the theory, and have a direct relationship with spectrally flowed representations in the underlying su(2)_{-1/2} theory. We discuss the spectral flow in the context of the operator algebra and fusion rules, and provide a re-interpretation of the modular invariant consistent with the spectrum.