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

Elia Mazzucchelli

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arXiv · 2410.13017

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

Elia Mazzucchelli

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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.

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Elia Mazzucchelli. 2024-10-16. The $\mathfrak{su}(2)_{-1}$ WZW model. https://arxiv.org/abs/2410.13017

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