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 $-3$.
Categories: math.QA hep-th math-ph math.MP math.RT