Tensor categories of weight modules of ${\widehat{\mathfrak{s}\mathfrak{l}}}_2$ at admissible level

The category of weight modules $L_k({\mathfrak{s}\mathfrak{l}}_2)\text{-wtmod}$ of the simple affine vertex algebra of ${\mathfrak{s}\mathfrak{l}}_2$ at an admissible level $k$ is neither finite nor semisimple and modules are usually not lower-bounded and have infinite dimensional conformal weight subspaces. However this vertex algebra enjoys a duality with $W_{\mathrm{\ell }}({\mathfrak{s}\mathfrak{l}}_{2\mathrm{\mid }1})$, the simple prinicipal $W$-algebra of ${\mathfrak{s}\mathfrak{l}}_{2\mathrm{\mid }1}$ at level $\mathrm{\ell }$ (the $N=2$ super conformal algebra) where the levels are related via $(k+2)(\mathrm{\ell }+1)=1$. Every weight module of $W_{\mathrm{\ell }}({\mathfrak{s}\mathfrak{l}}_{2\mathrm{\mid }1})$ is lower-bounded and has finite-dimensional conformal weight spaces. The main technical result is that every weight module of $W_{\mathrm{\ell }}({\mathfrak{s}\mathfrak{l}}_{2\mathrm{\mid }1})$ is $C_1$-cofinite. The existence of a vertex tensor category follows and the theory of vertex superalgebra extensions implies the existence of vertex tensor category structure on $L_k({\mathfrak{s}\mathfrak{l}}_2)\text{-wtmod}$ for any admissible level $k$. As applications, the fusion rules of ordinary modules with any weight module are computed and it is shown that $L_k({\mathfrak{s}\mathfrak{l}}_2)\text{-wtmod}$ is a ribbon category if and only if $L_{k+1}({\mathfrak{s}\mathfrak{l}}_2)\text{-wtmod}$ is, in particular it follows that for admissible levels $k=-2+\frac{u}{v}$ and $v\in \{2,3\}$ and $u=-1\mathrm{m}\mathrm{o}\mathrm{d}v$ the category $L_k({\mathfrak{s}\mathfrak{l}}_2)\text{-wtmod}$ is a ribbon category.