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

Abstract

We study the affine analogue ${\mathrm{FT}}_p({\mathfrak{sl}}_2)$ of the triplet algebra. We show that ${\mathrm{FT}}_p({\mathfrak{sl}}_2)$ is quasi-lisse and the associated variety is the nilpotent cone of ${\mathfrak{sl}}_2$. We realize ${\mathrm{FT}}_p({\mathfrak{sl}}_2)$ 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 $s{\mathcal{W}}_p({\mathfrak{sl}}_{2|1})$ of ${\mathrm{FT}}_p({\mathfrak{sl}}_2)$ and their singlet type subalgebras $s{\mathcal{M}}_p({\mathfrak{sl}}_{2|1})$ and ${\mathcal{M}}_p({\mathfrak{sl}}_2)$ 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 $s{\mathcal{M}}_1({\mathfrak{sl}}_{2|1})$ is the unrolled restricted quantum supergroup $u_{-1}^H({\mathfrak{sl}}_{2|1})$ as suggested by Semikhatov and Tipunin.

arXiv comment45 pages, comments are welcome!

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• On the structure of W-algebras in type A