Quasi-lisse extension of affine ${\mathfrak{s}\mathfrak{l}}_2$ à la Feigin--Tipunin

We study the affine analogue ${\mathrm{F}\mathrm{T}}_p({\mathfrak{s}\mathfrak{l}}_2)$ of the triplet algebra. We show that ${\mathrm{F}\mathrm{T}}_p({\mathfrak{s}\mathfrak{l}}_2)$ is quasi-lisse and the associated variety is the nilpotent cone of ${\mathfrak{s}\mathfrak{l}}_2$. We realize ${\mathrm{F}\mathrm{T}}_p({\mathfrak{s}\mathfrak{l}}_2)$ as the global sections of a sheaf of vertex algebras in the spirit of Feigin–Tipunin and thereby construct infinitely many simple modules and, in particular solve a conjecture by Semikhatov and Tipunin. We introduce the Kazama–Suzuki dual superalgebra $s{\mathcal{W}}_p({\mathfrak{s}\mathfrak{l}}_{2\mathrm{\mid }1})$ of ${\mathrm{F}\mathrm{T}}_p({\mathfrak{s}\mathfrak{l}}_2)$ and their singlet type subalgebras $s{\mathcal{M}}_p({\mathfrak{s}\mathfrak{l}}_{2\mathrm{\mid }1})$ and ${\mathcal{M}}_p({\mathfrak{s}\mathfrak{l}}_2)$ and show their correspondence of categories. For $p=1$, we show the logarithmic Kazhdan–Lusztig correspondence for these (super)algebras and, in particular, show that the quantum group corresponding to $s{\mathcal{M}}_1({\mathfrak{s}\mathfrak{l}}_{2\mathrm{\mid }1})$ is the unrolled restricted quantum supergroup $u_{-1}^H({\mathfrak{s}\mathfrak{l}}_{2\mathrm{\mid }1})$ as suggested by Semikhatov and Tipunin.