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

We construct highest weight vectors of \(\widehat{\mathfrak {sl}_2}_{,k+1} \oplus \textsf{Vir}\) 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 \(\widehat{\mathfrak {sl}_2}_{k+1} \oplus \textsf{Vir}\). 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-dimensional CFT models.