Feigin-Semikhatov conjecture and related topics

TL;DR

This paper reviews the proof of the Feigin-Semikhatov conjecture, establishing algebraic isomorphisms between subregular W-algebras and principal W-superalgebras of type A, and discusses related developments in algebraic structures and fusion rules.

Contribution

It provides a comprehensive review of the recent proof of the Feigin-Semikhatov conjecture and explores its implications for algebraic isomorphisms and module structures.

Findings

Confirmed algebraic isomorphisms between subregular W-algebras and principal W-superalgebras.

Detailed analysis of modules and intertwining operators related to the conjecture.

Insights into fusion rules and their role in the algebraic correspondence.

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.