Feigin-Semikhatov conjecture and related topics

Abstract: Feigin-Semikhatov conjecture, now established, states algebraic isomorphisms between the cosets of the subregular $\mathcal{W}$-algebras and the principal $\mathcal{W}$-superalgebras of type A by their full Heisenberg subalgebras. It can be seen as a variant of Feigin-Frenkel duality between the $\mathcal{W}n$-algebras and also as a generalization of the connection between the $\mathcal{N}=2$ superconformal algebra and the affine algebra $\hat{\mathfrak{sl}}{2,k}$.We review the recent developments on the correspondence of the subregular W-algebras and the principal W-superalgebras of type A at the level of algebras, modules and intertwining operators, including fusion rules.