Ito's conjecture and the coset construction for ${\mathcal{W}}^k(\mathfrak{sl}(3|2))$

Abstract

Many $\mathcal{W}$-(super)algebras which are defined by the generalized Drinfeld-Sokolov reduction are also known or expected to have coset realizations. For example, it was conjectured by Ito that the principal $\mathcal{W}$-superalgebra ${\mathcal{W}}^k(\mathfrak{sl}(n+1|n))$ is isomorphic to the coset of $V^{l+1}({\mathfrak{gl}}_n)$ inside $V^l({\mathfrak{sl}}_{n+1})\otimes \mathcal{E}(n)$ for generic values of $l$. Here $\mathcal{E}(n)$ denotes the rank $n$ $bc$-system, which carries an action of $V^1({\mathfrak{gl}}_n)$, and $k$ and $l$ are related by $(k+1)(l+n+1)=1$. This conjecture is known in the case $n=1$, which is somewhat degenerate, and we shall prove it in the first nontrivial case $n=2$. As a consequence, we show that the simple quotient ${\mathcal{W}}_k(\mathfrak{sl}(3|2))$ is lisse and rational for all positive integers $l>1$. These are new examples of rational $\mathcal{W}$-superalgebras.

Lemmas 3.1 and 3.2 are added, final version, to appear in RIMS Kokyuroku Bessatsu