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

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.