Correlator correspondences for Gaiotto-Rap\v{c}\'ak dualities and first order formulation of coset models

TL;DR

This paper develops a first order formulation of coset models in two-dimensional conformal field theories, enabling the derivation of correlation function correspondences and dualities, including conjectural equivalences and higher rank FZZ-duality.

Contribution

It introduces a new first order formulation of coset models using BRST formalism, facilitating the derivation of duality correspondences and extending previous results.

Findings

Derived correlator correspondences for dual conformal field theories

Established a first order formulation of coset models with BRST formalism

Extended higher rank FZZ-duality to more cases

Abstract

We derive correspondences of correlation functions among dual conformal field theories in two dimensions by developing a "first order formulation" of coset models. We examine several examples, and the most fundamental one may be a conjectural equivalence between a coset $(SL(n)_k \otimes SL(n)_{-1})/SL(n)_{k-1}$ and $\mathfrak{sl}(n)$ Toda field theory with generic level $k$. Among others, we also complete the derivation of higher rank FZZ-duality involving a coset $SL(n+1)_k /(SL(n)_{k} \otimes U(1))$, which could be done only for $n=2,3$ in our previous paper. One obstacle in the previous work was our poor understanding of a first order formulation of coset models. In this paper, we establish such a formulation using the BRST formalism. With our better understanding, we successfully derive correlator correspondences of dual models including the examples mentioned above. The dualities may be regarded as conformal field theory realizations of some of the Gaiotto-Rap\v{c}\'ak dualities of corner vertex operator algebras.