A Kazhdan-Lusztig correspondence for $L_{-\frac{3}{2}}(\mathfrak{sl}_3)$

The abelian and monoidal structure of the category of smooth weight modules over a non-integrable affine vertex algebra of rank greater than one is an interesting, difficult and essentially wide open problem. Even conjectures are lacking. This work details and tests such a conjecture for $L_{-\frac{3}{2}}(\mathfrak{sl}_3)$ via a logarithmic Kazhdan-Lusztig correspondence. We first investigate the representation theory of $\overline{U}^H_i(\mathfrak{sl}_3)$, the unrolled restricted quantum group of $\mathfrak{sl}_3$ at fourth root of unity. In particular, we analyse its finite-dimensional weight category, determining Loewy diagrams for all projective indecomposables and decomposing all tensor products of irreducibles. Our motivation is that this category is conjecturally braided tensor equivalent to a category of $W_{A_2}^0(2)$-modules. Here, $W_{A_2}^0(2)$ is an orbifold of the octuplet vertex algebra $W_{A_2}(2)$ of Semikhatov, the latter being the natural $\mathfrak{sl}_3$-analogue of the well known triplet algebra. Moreover, $W_{A_2}^0(2)$ is the parafermionic coset of the affine vertex algebra $L_{-\frac{3}{2}}(\mathfrak{sl}_3)$. We formulate an explicit conjecture relating the representation theory of $W_{A_2}^0(2)$ and $\overline{U}^H_i(\mathfrak{sl}_3)$ and work out the resulting structures of the corresponding $L_{-\frac{3}{2}}(\mathfrak{sl}_3)$-modules. In particular, we obtain conjectural Loewy diagrams for the latter's projective indecomposables and decompositions for the fusion products of its irreducibles. These products coincide with those recently computed via Verlinde's formula. Finally, we give analogous results for $W_{A_2}(2)$.