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

TL;DR

This paper investigates the $rak{su}(2)_{-1}$ WZW model, classifying its representations, developing a free field realization, and exploring modular invariants, with implications for string theory on AdS backgrounds.

Contribution

It provides a detailed analysis of the $rak{su}(2)_{-1}$ WZW model, including classification, free field realization, and modular invariants, advancing understanding of superalgebra models in string theory.

Findings

Classified representations of $rak{su}(2)_{-1}$

Developed a free field realization of the model

Identified continuous and discrete modular invariants

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.