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arXiv · 2312.14695

Decomposition of ${\widehat{\mathfrak{sl}_2}}_{,k} \ \oplus \ {\widehat{\mathfrak{sl}_2}}_{,1}$ highest weight representations for generic level $k$ and equivalence between two dimensional CFT models

Leszek Hadasz·Błażej Ruba
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Abstract

We construct highest weight vectors of ${\widehat{\mathfrak{sl}_2}}_{,k+1} \oplus \mathsf{Vir}$ in tensor products of highest weight modules of ${\widehat{\mathfrak{sl}_2}}_{,k}$ and ${\widehat{\mathfrak{sl}_2}}_{,1}$, and thus for generic weights we find the decomposition of the tensor product into irreducibles of ${\widehat{\mathfrak{sl}_2}}_{,k+1} \oplus \mathsf{Vir}$. The construction uses Wakimoto representations of ${\widehat{\mathfrak{sl}_2}}_{,k}$, but the obtained vectors can be mapped back to Verma modules. Singularities of this mapping are cancelled by a renormalization. A detailed study of ``degenerations'' of Wakimoto modules allowed to find the renormalization factor explicitly. The obtained result is a significant step forward in a proof of equivalence of certain two-dimesnional CFT models.

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Leszek Hadasz, Błażej Ruba. 2023-12-22. Decomposition of ${\widehat{\mathfrak{sl}_2}}_{,k} \ \oplus \ {\widehat{\mathfrak{sl}_2}}_{,1}$ highest weight representations for generic level $k$ and equivalence between two dimensional CFT models. https://arxiv.org/abs/2312.14695

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