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Simplicity of the principal W-algebra U^pr versus level k

Establish whether the universal principal W-algebra U^pr, obtained by quantum Hamiltonian reduction of the universal affine vertex superalgebra V_k(psl(2|2)) at level k, is simple precisely when k is not a nonintegral rational number. Equivalently, determine the exact set of levels k for which U^pr fails to be simple and prove that U^pr is simple for all k in R \ (Q \ Z).

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Background

After computing strong generators and operator product expansions for the universal principal W-algebra Upr associated to psl(2|2), the authors observe numerically the existence of singular vectors at specific rational levels (e.g., k=±1/2, ±1/3, ±3/2, ±1/4, ±2/3, ±5/2). This motivates a conjectural characterization of the levels at which Upr is not simple.

Clarifying the simplicity regime of Upr is central to understanding its structure and representation theory and to identifying when its simple quotient collapses to known vertex algebras (e.g., symplectic fermions at k=±1/2).

References

Numerical investigations with the analogue of the Shapovalov form on $\upr$ show that there are singular vectors of conformal weight $2$, if $k=\pm\frac{1}{2}$, weight $4$ if $k=\pm\frac{1}{3}, \pm\frac{3}{2}$, and weight $6$, if $k=\pm\frac{1}{4}, \pm\frac{2}{3}, \pm\frac{5}{2}$. We therefore conjecture that $\upr$ is a simple \ unless $k \in Q \setminus Z$.

— The principal W-algebra of $\mathfrak{psl}_{2|2}$ (2509.04795 - Fehily et al., 5 Sep 2025) in Section 2.3 (Operator product expansions), end of itemized observations