Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 030, 26 pages      arXiv:2509.04795      https://doi.org/10.3842/SIGMA.2026.030
Contribution to the Special Issue on Recent Advances in Vertex Operator Algebras in honor of James Lepowsky

The Principal W-Algebra of $\mathfrak{psl}_{2|2}$

Zachary Fehily a, Christopher Raymond b and David Ridout a
a) School of Mathematics and Statistics, University of Melbourne, Australia
b) Department of Mathematics, University of Hamburg, Germany

Received September 08, 2025, in final form February 18, 2026; Published online March 25, 2026

Abstract
We study the structure and representation theory of the principal W-algebra $\mathsf{W}_{\rm pr}^{\mathsf{k}}$ 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 $\mathsf{k} = \pm \frac{1}{2}$, $\mathsf{W}_{\rm pr}^{\mathsf{k}}$ 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 $-3$.

Key words: vertex-operator algebras; conformal field theory; representation theory.

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