The $\mathfrak{su}(2)_{-1}$ WZW model

Abstract

Some WZW models on affine Lie superalgebras at critical level describe string theory on AdS backgrounds at critical values of the string tension. This is the case of $\mathfrak{psu}(1,1|2)_1$ for ${\rm AdS}_3 \times {\rm S}^3$ and potentially of $\mathfrak{u}(2|2)_1$ (or related algebras) for ${\rm AdS}_5 \times {\rm S}^5$. Many interesting features of these superalgebra models are already captured by their affine subalgebra $\mathfrak{su}(2)_{-1}$. In this paper we study the WZW model on $\mathfrak{su}(2)_{-1}$: we classify the representations, introduce a free field realisation, and decompose the free field modules in terms of $\mathfrak{su}(2)_{-1}$. We find continuous and discrete modular invariants and see that the latter naturally leads to considering superalgebra extensions of $\mathfrak{su}(2)_{-1}$. Lastly, we find an invariant for the free field theory of four symplectic bosons.