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On Singleton Composites in Non-compact WZW Models

J. Engquist, P. Sundell, L. Tamassia

TL;DR

The paper analyzes the subcritical $\ig\widehat{\mathfrak{so}}(2,D-1)\,$ WZW model at $k=-(D-3)/2$ and shows that the decoupling of a singular vacuum vector enforces an affine hyperlightlikeness condition that encodes the scalar singleton's equation of motion in AdS. By employing spectral flow, twisted primary fields, and free-field realizations, the authors construct a spectrum that includes the singleton and its tensor products (multipletons) and, in $D=4$, an extended model with spinor singletons realized via $\,sp(4)$. They further develop a symplectic-boson realization for $D=4$, derive fusion rules, and demonstrate how singleton composites reproduce massless higher-spin spectra, offering a bridge between affine algebras, phase-space quantization, and unfolded higher-spin theories. The work proposes that gauged versions of these models could describe topological, tensionless backgrounds and phase-space formulations of partonic branes and higher-spin gauge theories, with significant implications for AdS/CFT and SFT-like constructions in AdS spaces.

Abstract

We examine the so(2,D-1) WZW model at the subcritical level -(D-3)/2. It has a singular vacuum vector at Virasoro level 2. Its decoupling constitutes an affine extension of the equation of motion of the (D+1)-dimensional conformal particle, i.e. the scalar singleton. The admissible (spectrally flowed) representations contain the singleton and its direct products, consisting of massless and massive particles in AdS_D. In D=4 there exists an extended model containing both scalar and spinor singletons of sp(4). Its realization in terms of 4 symplectic-real bosons contains the spinor-oscillator constructions of the 4D singletons and their composites. We also comment on the prospects of relating gauged versions of the models to the phase-space quantization of partonic branes and higher-spin gauge theory.

On Singleton Composites in Non-compact WZW Models

TL;DR

The paper analyzes the subcritical WZW model at and shows that the decoupling of a singular vacuum vector enforces an affine hyperlightlikeness condition that encodes the scalar singleton's equation of motion in AdS. By employing spectral flow, twisted primary fields, and free-field realizations, the authors construct a spectrum that includes the singleton and its tensor products (multipletons) and, in , an extended model with spinor singletons realized via . They further develop a symplectic-boson realization for , derive fusion rules, and demonstrate how singleton composites reproduce massless higher-spin spectra, offering a bridge between affine algebras, phase-space quantization, and unfolded higher-spin theories. The work proposes that gauged versions of these models could describe topological, tensionless backgrounds and phase-space formulations of partonic branes and higher-spin gauge theories, with significant implications for AdS/CFT and SFT-like constructions in AdS spaces.

Abstract

We examine the so(2,D-1) WZW model at the subcritical level -(D-3)/2. It has a singular vacuum vector at Virasoro level 2. Its decoupling constitutes an affine extension of the equation of motion of the (D+1)-dimensional conformal particle, i.e. the scalar singleton. The admissible (spectrally flowed) representations contain the singleton and its direct products, consisting of massless and massive particles in AdS_D. In D=4 there exists an extended model containing both scalar and spinor singletons of sp(4). Its realization in terms of 4 symplectic-real bosons contains the spinor-oscillator constructions of the 4D singletons and their composites. We also comment on the prospects of relating gauged versions of the models to the phase-space quantization of partonic branes and higher-spin gauge theory.

Paper Structure

This paper contains 21 sections, 194 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Singletons and Multipletons
  3. Elements of $\mathfrak{so}(2,D-1)$ Representation Theory
  4. Masslessness and Hyperlightlikeness
  5. Compositeness and Higher-Spin Algebra
  6. On Subcritical $\widehat{\mathfrak{so}}(2,D-1)$ WZW Models
  7. Affine Hyperlightlikeness
  8. Twisted Primary Scalars and Multipletons
  9. Spectrum-Generating Flow and Fusion
  10. Twisted Primary Spinors in $D=4$
  11. The Extended $\widehat{\mathfrak{sp}}(4)$ Model
  12. Realization Using Symplectic Bosons
  13. Generating the Spectrum by Spectral Flow
  14. Oscillator realization of the Multipleton Subsectors
  15. Free-Field Realization of the Extended $\widehat{\mathfrak{sp}}(4)$ Model
  16. ...and 6 more sections

Figures (1)

  • Figure 1: The conformal dimensions $h_{[P]}$ as a function of the spin $m_1=m$ of the ground states of the $\mathfrak{so}(2,3)\simeq\mathfrak{sp}(4)$ subspaces ${\mathfrak D}_{[P];h}(e_0,\mathbf m_0)$ defined in (\ref{['DP']}). Note that there are two scalar ground states having $h=-1$ in each of the $P=\pm2$ sectors, one of which corresponds to the twisted primary state $|[\pm2];-1;\pm1,0\rangle$ and the other to the pseudoscalar $|[\pm2];-1;\pm2,0\rangle$.