Orthosymplectic Feigin-Semikhatov duality

TL;DR

This paper establishes a duality between certain W-algebras related to orthosymplectic Lie superalgebras, generalizing known dualities and classifying simple modules with explicit character formulas.

Contribution

It introduces an orthosymplectic analogue of Feigin-Semikhatov duality, extending the correspondence of modules and free field representations to new algebraic structures.

Findings

Block-wise equivalence of weight modules established

Classification of simple modules at exceptional levels

Derivation of explicit character formulas

Abstract

We study the representation theory of the subregular W-algebra $\mathcal{W}^k(\mathfrak{so}_{2n+1},f_{sub})$ of type B and the principal W-superalgebra $\mathcal{W}^\ell(\mathfrak{osp}_{2|2n})$, which are related by an orthosymplectic analogue of Feigin-Semikhatov duality in type A. We establish a block-wise equivalence of weight modules over the W-superalgebras by using the relative semi-infinite cohomology functor and spectral flow twists, which generalizes the result of Feigin-Semikhatov-Tipunin for the N=2 superconformal algebra. In particular, the correspondence of Wakimoto type free field representations is obtained. When the level of the subregular W-algebra is exceptional, we classify the simple modules over the simple quotients $\mathcal{W}_k(\mathfrak{so}_{2n+1},f_{sub})$ and $\mathcal{W}_\ell(\mathfrak{osp}_{2|2n})$ and derive the character formulae.