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The principal W-algebra of $\mathfrak{psl}_{2|2}$

Zachary Fehily, Christopher Raymond, David Ridout

Abstract

We study the structure and representation theory of the principal W-algebra $\mathsf{W}^{\mathsf{k}}_{\mathrm{pr}}$ of $\mathsf{V}^{\mathsf{k}}(\mathfrak{psl}_{2|2})$. The defining operator product expansions are computed, as is the Zhu algebra, and these results are used to classify irreducible highest-weight modules. In particular, for $\mathsf{k} = \pm \frac{1}{2}$, $\mathsf{W}^{\mathsf{k}}_{\mathrm{pr}}$ is not simple and the corresponding simple quotient is the symplectic fermion vertex algebra. We use this fact, along with inverse hamiltonian reduction, to study relaxed highest-weight and logarithmic modules for the small $N=4$ superconformal algebra at central charges $-9$ and $-3$.

The principal W-algebra of $\mathfrak{psl}_{2|2}$

Abstract

We study the structure and representation theory of the principal W-algebra of . The defining operator product expansions are computed, as is the Zhu algebra, and these results are used to classify irreducible highest-weight modules. In particular, for , is not simple and the corresponding simple quotient is the symplectic fermion vertex algebra. We use this fact, along with inverse hamiltonian reduction, to study relaxed highest-weight and logarithmic modules for the small superconformal algebra at central charges and .

Paper Structure

This paper contains 15 sections, 9 theorems, 64 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Outline and results
  3. Computing the principal W-algebra
  4. The Lie superalgebra $\mathfrak{psl}_{2 \vert 2}$
  5. The principal reduction
  6. Operator product expansions
  7. Irreducible $\mathsf{W}^{\mathsf{k}}_{\textnormal{pr}}$-modules
  8. The principal Zhu algebra
  9. Modules of the Zhu algebra
  10. Irreducible lower-bounded $\mathsf{W}^{\mathsf{k}}_{\textnormal{pr}}$- and $\mathsf{W}_{\mathsf{k}}^{\textnormal{pr}}$-modules
  11. Inverse reduction and the $N=4$ superconformal algebra
  12. Inverse quantum hamiltonian reduction
  13. Constructing weight $\mathsf{W}_{\pm 1/2}^{\textnormal{min}}$-modules
  14. Degenerations and spectral flows
  15. Logarithmic $\mathsf{W}_{\pm 1/2}^{\textnormal{min}}$-modules

Key Result

Theorem 2.1

The principal quantum hamiltonian reduction $\mathsf{W}^{\mathsf{k}}_{\textnormal{pr}}$ of $\mathsf{V}^{\mathsf{k}}(\mathfrak{psl}_{2 \vert 2})$ is strongly and freely generated by two even elements, $L$ and $H$, and four odd elements, $\chi$, $\widebar{\chi}{}$, $\psi$ and $\widebar{\psi}{}$. Here,

Figures (4)

  • Figure 1: The roots of the simple Lie superalgebra $\mathfrak{psl}_{2 \vert 2}$, labelled by their root vectors (the odd roots have multiplicity $2$). The horizontal axis indicates the eigenvalue under the adjoint action of $H^1$ while the vertical axis records it for $H^2$.
  • Figure 2: An illustration of the structure of the $\mathsf{W}_{1/2}^{\textnormal{min}}$-module $\mathcal{M}^{\textnormal{NS}}_{\newline{\llbracket1\rrbracket}}$. The dots represent weight vectors, with the $J^0_0$-eigenvalue increasing from left to right and the $T_0$-eigenvalue $\Delta$ from top to bottom.
  • Figure 3: An illustration of the structure of the top spaces of the $\mathsf{W}_{\pm 1/2}^{\textnormal{min}}$-modules $\mathcal{V}^{\textnormal{R}}_{\newline{\llbracket1/2\rrbracket}}$ (left) and $\mathcal{V}^{\textnormal{R}}_{\newline{\llbracket-1/2\rrbracket}}$ (right). The dots represent weight vectors, with the $J^0_0$-eigenvalue increasing from left to right. The top row corresponds to vectors with label "$m$" and the bottom row has label "$b$". The shading suggests the composition factors.
  • Figure 4: Loewy diagrams and top-space structures of the modules $\mathcal{P}^{\textnormal{R}}_{\newline{\llbracket\pm1/2\rrbracket}}$ when $\mathsf{k} = -\frac{1}{2}$. The top row of the structures corresponds to vectors with label "$t$", the middle row to those with labels "$m$" and "$\bm$", and the bottom row to those with labels "$b$". The shading again suggests the composition factors.

Theorems & Definitions (12)

  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • Theorem 3.4
  • Theorem 4.1
  • ...and 2 more