Cosets of the $\mathcal{W}^k(\mathfrak{sl}_4, f_{\text{subreg}})$-algebra (1711.11109v1)

Abstract: Let $\mathcal {W}^k(\mathfrak{sl}_4, f_{\text {subreg}})$ be the universal $\mathcal{W}$-algebra associated to $\mathfrak{sl}4$ with its subregular nilpotent element, and let $\mathcal {W}_k(\mathfrak{sl}_4, f{\text {subreg}})$ be its simple quotient. There is a Heisenberg subalgebra $\mathcal{H}$, and we denote by $\mathcal{C}^k$ the coset $\text{Com}(\mathcal{H}, \mathcal {W}^k(\mathfrak{sl}_4, f_{\text {subreg}}))$, and by $\mathcal{C}k$ its simple quotient. We show that for $k=-4+(m+4)/3$ where $m$ is an integer greater than $2$ and $m+1$ is coprime to $3$, $\mathcal{C}_k$ is isomorphic to a rational, regular $\mathcal W$-algebra $\mathcal{W}(\mathfrak{sl}_m, f{\text{reg}})$. In particular, $\mathcal{W}k(\mathfrak{sl}_4, f{\text {subreg}})$ is a simple current extension of the tensor product of $\mathcal{W}(\mathfrak{sl}m, f{\text{reg}})$ with a rank one lattice vertex operator algebra, and hence is rational.