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

Abstract

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

20 pages, 4 figures, comments welcome! v2 is the authors' version and includes a new Theorem 2.2 on the simplicity of the algebra. v3 is the published version from Symmetry, Integrability and Geometry: Methods and Applications