On Kazama-Suzuki Duality between ${\mathcal{W}}_k({\mathfrak{sl}}_4,f_{\mathrm{sub}})$ and $N=2$ Superconformal Vertex Algebra

Abstract

We classify all possible occurrences of Kazama-Suzuki duality between the $N=2$ superconformal algebra $L_c^{N=2}$ and the subregular $\mathcal{W}$-algebra ${\mathcal{W}}_k({\mathfrak{sl}}_4,f_{\mathrm{sub}})$. We establish a new Kazama-Suzuki duality between the subregular $\mathcal{W}$-algebra ${\mathcal{W}}_k({\mathfrak{sl}}_4,f_{\mathrm{sub}})$ and the $N=2$ superconformal algebra $L_c^{N=2}$ for $c=-15$. As a consequence of the duality, we classify the irreducible ${\mathcal{W}}_{k=-1}({\mathfrak{sl}}_4,f_{\mathrm{sub}})$-modules.