# Rectangular W-(super)algebras and their representations

## TL;DR

This paper explores the structure and representations of rectangular W-algebras derived from quantum Hamiltonian reduction of sl(Mn), analyzing their operator product expansions, degenerate representations, and supersymmetric extensions in the context of higher spin gravity.

## Contribution

It extends previous work by computing OPEs for all n, analyzing degenerate representations and their geometric implications, and introducing N=2 supersymmetry into the framework.

## Key findings

- Computed OPEs for low spin generators with n ≠ 2.
- Analyzed degenerate representations and their relation to coset spectra.
- Extended the algebra to include N=2 supersymmetry.

## Abstract

We study W-algebras obtained by quantum Hamiltonian reduction of $sl(Mn)$ associated to the $sl(2)$ embedding of rectangular type. The algebra can be realized as the asymptotic symmetry of higher spin gravity with $M \times M$ matrix valued fields. In our previous work, we examined the basic properties of the W-algebra and claimed that the algebra can be realized as the symmetry of Grassmannian-like coset even with finite central charge based on a proposal of holography. In this paper, we extend the analysis in the following ways. First, we compute the operator product expansions among low spin generators removing the restriction of $n = 2$. Second, we investigate the degenerate representations in several ways, and see the relations to the coset spectrum and the conical defect geometry of the higher spin gravity. For these analyses, we mainly set $M=n=2$. Finally, we extend the previous analysis by introducing $\mathcal{N}=2$ supersymmetry.