Correspondences of Categories for Subregular $${{\mathcal {W}}}$$-Algebras and Principal $${\mathcal {W}}$$-Superalgebras

Abstract

Based on the Kazama-Suzuki type coset construction and its inverse coset between the subregular $\mathcal{W}$-algebras for $\mathfrak{sl}_n$ and the principal $\mathcal{W}$-superalgebras for $\mathfrak{sl}_{1|n}$, we prove weight-wise linear equivalences of their representation categories. Our main results are then improvements of these correspondences incorporating the monoidal structures. Firstly, in the rational case, we obtain the classification of simple modules and their fusion rules via simple current extensions from their Heisenberg cosets. Secondly, beyond the rational case, we use certain kernel VOAs together with relative semi-infinite cohomology functors to get functors from categories of modules for the subregular $\mathcal{W}$-algebras for $\mathfrak{sl}_n$ to categories of modules for the principal $\mathcal{W}$-superalgebras for $\mathfrak{sl}_{1|n}$ and vice versa. We study these functors and in particular prove isomorphisms between the superspaces of logarithmic intertwining operators. As a corollary, we obtain correspondences of representation categories in the monoidal sense beyond the rational case as well.