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

TL;DR

This paper constructs and classifies a family of $ ext{N}=1$ $ ext{W}_ ext{infinity}$ algebras, exploring their truncations and connections to superalgebra reductions and holography, advancing understanding of supersymmetric higher-spin theories.

Contribution

It introduces a one-parameter family of $ ext{N}=1$ $ ext{W}_ ext{infinity}$ algebras and identifies their truncations with superalgebra reductions and coset models, linking to holography.

Findings

Identification of a one-parameter family $ ext{W}_ ext{infinity}[ ext{mu}]$ at each central charge.

Truncations correspond to Drinfel'd-Sokolov reductions of specific Lie superalgebras.

Connections established between these algebras, coset models, and holographic dualities.

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.