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

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.

14 pages, to appear in conference proceedings for AMS Special Session on Vertex Algebras and Geometry

References in corpus (2)

• Braided tensor categories and extensions of vertex operator algebras
• Logarithmic intertwining operators and W(2,2p-1)-algebras

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• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• Cosets of free field algebras via arc spaces
• On the structure of W-algebras in type A
• On the representation theory of the vertex algebra $L_{-5/2}(sl(4))$
• Invariant subalgebras of the small $\mathcal{N}=4$ superconformal algebra
• Correlator correspondences for subregular $\mathcal{W}$-algebras and principal $\mathcal{W}$-superalgebras