On Kazama-Suzuki Duality between $\mathcal{W}_k(\mathfrak{sl}_4, f_{\rm sub})$ and $N=2$ Superconformal Vertex Algebra
Dražen Adamović, Ana Kontrec
TL;DR
This work establishes a complete Kazama--Suzuki duality classification between the $N=2$ superconformal vertex algebra and subregular $\mathcal{W}$-algebras for $\mathfrak{sl}_4$, showing duality occurs only at the pairs $(k,c)=(-1,-15)$ and $(-\tfrac{7}{3},1)$. It constructs explicit KS embeddings in these cases, proving $L^{N=2}_{c=-15}$ and $\mathcal{W}_{-1}(\mathfrak{sl}_4, f_{sub})$ are dual, and expresses $L^{N=2}_{c=-15}$ as a commuting subalgebra of $\mathcal{W}_{-1}(\mathfrak{sl}_4, f_{sub})\otimes \mathcal{F}_{-1}$ with a complementary realization of $\mathcal{W}_{-1}$ inside $L^{N=2}_{c=-15}\otimes \mathcal{F}_{1}$. From this duality, it derives a complete classification of irreducible $\mathcal{W}_{-1}(\mathfrak{sl}_4, f_{sub})$-modules, parameterized by explicit weight curves $S_1$ and $S_2$, and distinguishes 1- versus 2-dimensional top spaces. The methods combine coset/commutant analysis, universal $\mathcal{W}(c,\lambda)$ technology, and free-field realizations with lattice algebras to relate parafermionic and KS structures, offering a concrete realization and a template for KS dualities in other subregular-$\mathcal{W}$ settings.
Abstract
We classify all possible occurrences of Kazama-Suzuki duality between the ${N=2}$ superconformal algebra $L^{N=2}_c$ and the subregular $\mathcal{W}$-algebra $\mathcal{W}_{k}(\mathfrak{sl}_4, f_{\rm sub})$. We establish a new Kazama-Suzuki duality between the subregular $\mathcal{W}$-algebra $\mathcal{W}_k(\mathfrak{sl}_4, f_{\rm sub})$ and the $N = 2$ superconformal algebra $L^{N=2}_{c}$ for $c=-15$. As a consequence of the duality, we classify the irreducible $\mathcal{W}_{k=-1}(\mathfrak{sl}_4, f_{\rm sub})$-modules.