The $\mathfrak{u}(2\mathrm{\mid }2{)}_1$ WZW model

WZW models based on super Lie algebras play an important role for the description of string theory on AdS spaces. In particular, for the case of ${\mathrm{A}\mathrm{d}\mathrm{S}}_3\times {\mathrm{S}}^3$ with pure NS-NS flux the super Lie algebra of $\mathfrak{p}\mathfrak{s}\mathfrak{u}(1,1\mathrm{\mid }2{)}_k$ appears in the hybrid formalism, and higher dimensional AdS spaces can be described in terms of related supergroup cosets. In this paper we study the WZW models based on $\mathfrak{u}(2\mathrm{\mid }2{)}_1$ and $\mathfrak{p}\mathfrak{s}\mathfrak{u}(2\mathrm{\mid }2{)}_1$ that may play a role for the worldsheet theory that is dual to free super Yang-Mills in 4D.