Correlator correspondences for subregular $\mathcal{W}$-algebras and principal $\mathcal{W}$-superalgebras

TL;DR

This paper explores dualities between certain $ ext{W}$-algebras and superalgebras, deriving correlator correspondences via path integrals, and extends these ideas to include fermionic dualities involving principal $ ext{W}$-superalgebras.

Contribution

It derives new correlator correspondences for generalized dualities involving subregular $ ext{W}$-algebras and principal $ ext{W}$-superalgebras, expanding understanding of their dualities.

Findings

Established correlator correspondences for a new series of dualities.

Extended fermionic FZZ-duality to include principal $ ext{W}$-superalgebras.

Connected dualities to the framework of corner vertex operator algebras.

Abstract

We examine a strong/weak duality between a Heisenberg coset of a theory with $\mathfrak{sl}_n$ subregular $\mathcal{W}$-algebra symmetry and a theory with a $\mathfrak{sl}_{n|1}$-structure. In a previous work, two of the current authors provided a path integral derivation of correlator correspondences for a series of generalized Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality. In this paper, we derive correlator correspondences in a similar way but for a different series of generalized duality. This work is a part of the project to realize the duality of corner vertex operator algebras proposed by Gaiotto and Rap\v{c}\'ak and partly proven by Linshaw and one of us in terms of two dimensional conformal field theory. We also examine another type of duality involving an additional pair of fermions, which is a natural generalization of the fermionic FZZ-duality. The generalization should be important since a principal $\mathcal{W}$-superalgebra appears as its symmetry and the properties of the superalgebra are less understood than bosonic counterparts.