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
TL;DR
The paper advances a twofold goal: first, to decompose the tensor product of a generic $\widehat{\mathfrak{sl}_2}$ Verma module at level $k$ with a level-1 module into irreducibles of $\widehat{\mathfrak{sl}_2}_{,k+1}$ and the Virasoro algebra, and second, to provide explicit highest-weight vectors realizing this decomposition.Using Wakimoto free-field realizations and noncommutative localization to handle singular vectors, it constructs vectors $v^n_{\kappa,j,\epsilon}$ that are highest-weight for $\mathsf{Vir} \oplus \widehat{\mathfrak{sl}_2}_{,k+1}$ with conformal weights $\Delta^n_{k,j,\epsilon}$ and $\widehat{\mathfrak{sl}_2}_{,k+1}$ charge $j+\epsilon+n$, proving that the tensor product generically decomposes into a direct sum of Virasoro$\oplus$affine modules. The work leverages two Wakimoto realizations, studies determinant structures (Kac–Kazhdan) via a factorization into two blocks, and uses screening charges to generate singular vectors with explicit normalization factors. The explicit states and the corresponding character identity underpin a program to establish an exact equivalence between Liouville/H^+3 and $\widehat{\mathfrak{sl}_2}$ coset models, moving toward a complete comparison of correlation functions across the two CFT frameworks. Overall, the results provide a concrete, general construction of Virasoro$\oplus$affine decompositions at generic weights and set the stage for a full CFT equivalence proof in this non-rational setting.
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.