Higher rank FZZ-dualities

TL;DR

This paper generalizes the FZZ-duality in 2D conformal field theories to higher ranks, establishing new strong/weak dualities between coset models and superalgebra structures, with explicit derivations for specific cases.

Contribution

It introduces higher rank dualities extending the FZZ-duality, connecting $ ext{sl}(N+1)/( ext{sl}(N) imes u(1))$ models to $ ext{sl}(N+1|N)$ theories, and derives these dualities explicitly for N=2,3.

Findings

Derived dualities for N=2,3 models.

Connected coset models to superalgebra theories.

Provided a conformal field theoretic proof of Gaiotto-Rapčák dualities.

Abstract

We propose new strong/weak dualities in two dimensional conformal field theories by generalizing the Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality between Witten's cigar model described by the $\mathfrak{sl}(2)/\mathfrak{u}(1)$ coset and sine-Liouville theory. In a previous work, a proof of the FZZ-duality was provided by applying the reduction method from $\mathfrak{sl}(2)$ Wess-Zumino-Novikov-Witten model to Liouville field theory and the self-duality of Liouville field theory. In this paper, we work with the coset model of the type $\mathfrak{sl}(N+1)/(\mathfrak{sl}(N) \times \mathfrak{u}(1))$ and propose that the model is dual to a theory with an $\mathfrak{sl}(N+1|N)$ structure. We derive the duality explicitly for $N=2,3$ by applying recent works on the reduction method extended for $\mathfrak{sl}(N)$ and the self-duality of Toda field theory. Our results can be regarded as a conformal field theoretic derivation of the duality of the Gaiotto-Rap\v{c}\'ak corner vertex operator algebras $Y_{0, N, N+1}[\psi]$ and $Y_{N, 0, N+1}[\psi^{-1}]$.