The $\mathcal{N}=1$ algebra ${\mathcal{W}}_{\infty }[\mu ]$ and its truncations

Abstract

The main objective of this work is to construct and classify the most general classical and quantum $\mathcal{N}=1$ ${\mathcal{W}}_{\infty }$-algebras generated by the same spins as the singlet algebra of $N$ fermions and $N$ bosons in the vector representation of $O(N)$ in the $N\to \infty$ limit. This type of algebras appears in a recent $\mathcal{N}=1$ version of the minimal model holography. Our analysis strongly suggests that there is a one parameter family ${\mathcal{W}}_{\infty }[\mu ]$ of such algebras at every given central charge. Relying on this assumption, we identify various truncations of ${\mathcal{W}}_{\infty }[\mu ]$ with, on the one hand, (orbifolds of) the Drinfel'd-Sokolov reductions of the Lie superalgebras $B(n,n)$, $B(n-1,n)$, $D(n,n)$ and $D(n+1,n)$, and, on the other hand, (orbifolds of) three $\mathcal{N}=1$ cosets. After a closer inspection we show that these cosets can be realized as a Drinfel'd-Sokolov reduction of $B(n,n)$, $D(n,n)$ and $D(n+1,n)$. We then discuss the implications of our findings for the quantum version of the $\mathcal{N}=1$ minimal model holography.

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References in corpus (5)

• Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields
• Nonlinear W(infinity) Algebra as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity
• Symmetries of Holographic Minimal Models
• Even spin minimal model holography
• Higher Spin Gravity and Exact Holography

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