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sl^(2)_{-1/2}: A Case Study

David Ridout

TL;DR

This work analyzes the fractional-level affine algebra $\widehat{\mathfrak{sl}}(2)_{-1/2}$, showing it yields a non-logarithmic CFT equivalent to the $\beta\gamma$ ghost system when viewed through a simple-current extension. It derives the vacuum and admissible representations algebraically, identifies the spectrum as two infinite spectral-flow families, and verifies modular properties via a Grothendieck-ring formulation of the characters, with Verlinde-type structure constants matching this ring. The extended algebra construction clarifies associativity and monodromy, highlighting the role of adjoint choices (su(2) versus $\mathfrak{sl}(2,\mathbb{R})$) and the necessity of an auxiliary operator $\mathcal{S}$ in certain realizations. The results illuminate how fractional-level theories can be organized by modular-invariant Grothendieck data and offer a blueprint for extending these methods to other admissible levels, including the appearance of logarithmic behavior in related models.

Abstract

The construction of the non-logarithmic conformal field theory based on sl^(2)_{-1/2} is revisited. Without resorting to free-field methods, the determination of the spectrum and fusion rules is streamlined and the beta gamma ghost system is carefully derived as the extended algebra generated by the unique finite-order simple current. A brief discussion of modular invariance is given and the Verlinde formula is explicitly verified.

sl^(2)_{-1/2}: A Case Study

TL;DR

This work analyzes the fractional-level affine algebra , showing it yields a non-logarithmic CFT equivalent to the ghost system when viewed through a simple-current extension. It derives the vacuum and admissible representations algebraically, identifies the spectrum as two infinite spectral-flow families, and verifies modular properties via a Grothendieck-ring formulation of the characters, with Verlinde-type structure constants matching this ring. The extended algebra construction clarifies associativity and monodromy, highlighting the role of adjoint choices (su(2) versus ) and the necessity of an auxiliary operator in certain realizations. The results illuminate how fractional-level theories can be organized by modular-invariant Grothendieck data and offer a blueprint for extending these methods to other admissible levels, including the appearance of logarithmic behavior in related models.

Abstract

The construction of the non-logarithmic conformal field theory based on sl^(2)_{-1/2} is revisited. Without resorting to free-field methods, the determination of the spectrum and fusion rules is streamlined and the beta gamma ghost system is carefully derived as the extended algebra generated by the unique finite-order simple current. A brief discussion of modular invariance is given and the Verlinde formula is explicitly verified.

Paper Structure

This paper contains 14 sections, 143 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Algebraic Preliminaries
  3. Vacuum Module Structure
  4. Admissible Representations
  5. Fusion and the Spectrum
  6. Characters and Modular Invariants
  7. Extended Algebras and Ghosts
  8. A Simplification
  9. Extended Algebra Representation Theory
  10. Modular Invariance
  11. Spectral Flow
  12. Affine Weyl Group Translations
  13. Outer Automorphisms
  14. Jacobi Theta Functions

Figures (3)

  • Figure 1: The singular vector structure of the vacuum Verma module at level $\tfrac{-1}{2}$. Each singular vector is labelled by its $\mathfrak{sl} \left( 2 \right)$-weight and conformal dimension (respectively).
  • Figure 2: The multiplicities of the weights of the admissible representations of $\widehat{\mathfrak{sl}} \left( 2 \right)_{-1/2}$. In these pictures, the $\mathfrak{sl} \left( 2 \right)$-weight increases from right to left (in multiples of $2$) and the conformal dimension increases from top to bottom (in multiples of $1$).
  • Figure 3: Depictions of the modules appearing in the spectrum and the action of the spectral flow automorphism $\gamma$. Each "corner state" is labelled by its $\mathfrak{sl} \left( 2 \right)$-weight and conformal dimension (in that order).