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

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 ⊗ 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 N = 4 super Liouville theory and a coset of the form Y (k 1 , k 2 )/SL(2) k1+k2 , where Y (k 1 , k 2 ) consists of two SL(2|1) ki 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 FZZtriality acts as a seed of the correspondence, which in particular implies a hidden SL(2. 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 ⊗ U (1)) for arbitrary n > 2.

Duality via convolution of W-algebras

“…In the spirit of the quantum geometric Langlands correspondence one also would like to get an equivalence of conformal blocks. A strong indication that such an equivalence might be true is that there is a conformal field theory approach using the path integral that indeed gives identifications of correlation functions [CH1,CH2,CHS,CH3]. We aim to pursue this direction further and are currently studying the behavior of characters under our convolution operation.…”