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.

42 pages, 1 figure. Version 2: some material from the version 1 removed to make the paper more uniform. Version 3: Introduction section supplement with a theorem summarizing the most important results; several commends in the main text and references added. Version accepted for publication in Communications in Mathematical Physics