Tensor category $K{L}_k({\mathfrak{sl}}_{2n})$ via minimal affine $W$-algebras at the non-admissible level $k=-\frac{2n+1}{2}$

Abstract

We prove that $K{L}_k({\mathfrak{sl}}_m)$ is a semi-simple, rigid braided tensor category for all even $m\ge 4$, and $k=-\frac{m+1}{2}$ which generalizes result from arXiv:2103.02985 obtained for $m=4$. Moreover, all modules in $K{L}_k({\mathfrak{sl}}_m)$ are simple-currents and they appear in the decomposition of conformal embeddings ${\mathfrak{gl}}_m\hookrightarrow {\mathfrak{sl}}_{m+1}$ at level $k=-\frac{m+1}{2}$ from arXiv:1509.06512. For this we inductively identify minimal affine $W$-algebra $W_{k-1}({\mathfrak{sl}}_{m+2},\theta )$ as simple current extension of $L_k({\mathfrak{sl}}_m)\otimes \mathcal{H}\otimes \mathcal{M}$, where $\mathcal{H}$ is the rank one Heisenberg vertex algebra, and $\mathcal{M}$ the singlet vertex algebra for $c=-2$. The proof uses previously obtained results for the tensor categories of singlet algebra from arXiv:2202.05496. We also classify all irreducible ordinary modules for $W_{k-1}({\mathfrak{sl}}_{m+2},\theta )$. The semi-simple part of the category of $W_{k-1}({\mathfrak{sl}}_{m+2},\theta )$-modules comes from $K{L}_{k-1}({\mathfrak{sl}}_{m+2})$, using quantum Hamiltonian reduction, but this $W$-algebra also contains indecomposable ordinary modules.

20 pages