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

F. Lesage, P. Mathieu, J. Rasmussen, H. Saleur

TL;DR

The paper analyzes the bosonic βγ ghost system at central charge $c=-1$ through its kinship with the fractional-level $ su(2)_{-1/2}$ WZW model. By combining free-field realizations with Knizhnik–Zamolodchikov equations, it shows that the spectrum contains infinitely many operators with arbitrarily negative dimensions, arising from twisted sectors and their spectrally flowed images. Fusion rules and modular invariants are shown to require inclusion of these flowed and deeper-twist sectors, implying that the theories are not rational CFTs in the standard sense. The results provide a controlled, physical understanding of non-unitary CFTs and illuminate their role as building blocks in disordered systems and related constructions. The analysis also clarifies how modular invariants should be interpreted in the presence of spectral flow and an unbounded operator content.

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.

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

TL;DR

The paper analyzes the bosonic βγ ghost system at central charge through its kinship with the fractional-level WZW model. By combining free-field realizations with Knizhnik–Zamolodchikov equations, it shows that the spectrum contains infinitely many operators with arbitrarily negative dimensions, arising from twisted sectors and their spectrally flowed images. Fusion rules and modular invariants are shown to require inclusion of these flowed and deeper-twist sectors, implying that the theories are not rational CFTs in the standard sense. The results provide a controlled, physical understanding of non-unitary CFTs and illuminate their role as building blocks in disordered systems and related constructions. The analysis also clarifies how modular invariants should be interpreted in the presence of spectral flow and an unbounded operator content.

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.

Paper Structure

This paper contains 21 sections, 170 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The $\beta\gamma$ system
  3. Generalities
  4. Twist fields and monodromy considerations
  5. Free-field representation: $\beta\gamma$ twists vs $\eta\xi$ twists
  6. Twist correlators
  7. The $\widehat{su}(2)_{-1/2}$ WZW model
  8. Admissible representations of the $\widehat{su}(2)_{-1/2}$ model
  9. Generating the $\widehat{su}(2)_{-1/2}$ spectrum from the vacuum singular vector
  10. Fusion rules
  11. The $\widehat{su}(2)_{-1/2}$ model vs the $\beta\gamma$ system
  12. The $\beta\gamma$ representation of the current algebra
  13. Twist fields and representations of $\widehat{su}(2)_{-1/2}$
  14. The KZ equation and correlators
  15. Generating function for twist correlators
  16. ...and 6 more sections

Figures (1)

  • Figure 1: Diagram of the lowest-weight representation $\pi_{-1}(D_0)$, and its twist $\pi_{-2}(D_0)$. In the latter case, $L_{0}$ eigenvalues are not bounded from below.