FZZ-triality and large $\mathcal{N}=4$ super Liouville theory

TL;DR

This paper explores dualities among 2D conformal field theories, establishing a new triality relation called FZZ-triality, and applies it to connect large N=4 super Liouville theory with coset models, revealing hidden symmetries and constructing boundary actions.

Contribution

It introduces the FZZ-triality, a novel duality connecting three models, and extends the duality framework to higher-rank supergroups, advancing understanding of 2D conformal field theories.

Findings

Derived duality between coset and sine-Liouville models.

Established FZZ-triality linking three models.

Generalized triality to higher n|1 supergroups.

Abstract

We examine dualities of two dimensional conformal field theories by applying the methods developed in previous works. We first derive the duality between $SL(2|1)_k/(SL(2)_k \otimes U(1))$ coset and Witten's cigar model or sine-Liouville theory. The latter two models are Fateev-Zamolodchikov-Zamolodchikov (FZZ-)dual to each other, hence the relation of the three models is named FZZ-triality. These results are used to study correlator correspondences between large $\mathcal{N}=4$ super Liouville theory and a coset of the form $Y(k_1,k_2)/SL(2)_{k_1 +k_2}$, where $Y(k_1 , k_2)$ consists of two $SL(2|1)_{k_i}$ and free bosons or equivalently two $U(1)$ cosets of $D(2,1;k_i -1)$ at level one. These correspondences are a main result of this paper. The FZZ-triality acts as a seed of the correspondence, which in particular implies a hidden $SL(2)_{k'}$ in $SL(2|1)_k$ or $D(2,1 ; k-1)_1$. The relation of levels is $k' -1 = 1/(k-1)$. We also construct boundary actions in sine-Liouville theory as another use of the FZZ-triality. Furthermore, we generalize the FZZ-triality to the case with $SL(n|1)_k/(SL(n)_k \otimes U(1))$ for arbitrary $n>2$.