Decomposition of 𝔰𝔩_2_,k ⊕𝔰𝔩_2_,1 highest weight representations for generic level k and equivalence between two dimensional CFT models

We construct highest weight vectors of 𝔰𝔩_2_,k+1⊕𝖵𝗂𝗋 in tensor products of highest weight modules of 𝔰𝔩_2_,k and 𝔰𝔩_2_,1, and thus for generic weights we find the decomposition of the tensor product into irreducibles of 𝔰𝔩_2_,k+1⊕𝖵𝗂𝗋. The construction uses Wakimoto representations of 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.