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

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