# Rectangular W-algebras of types $so(M)$ and $sp(2M)$ and dual coset CFTs

## TL;DR

This paper studies rectangular W-algebras with $so(M)$ or $sp(2M)$ symmetry, analyzing their structure, central charges, and dual coset CFT realizations, including supersymmetric extensions, to support higher spin holography.

## Contribution

It provides explicit calculations of W-algebra structures, central charges, and dual CFT correspondences, including supersymmetric cases, advancing understanding of higher spin holography.

## Key findings

- Computed central charges and levels for the algebras.
- Derived operator product expansions for generators up to spin two.
- Supported dual coset CFT descriptions of the W-algebras.

## Abstract

We examine rectangular W-algebras with $so(M)$ or $sp(2M)$ symmetry, which can be realized as the asymptotic symmetry of higher spin gravities with restricted matrix extensions. We compute the central charges of the algebras and the levels of $so(M)$ or $sp(2M)$ affine subalgebras by applying the Hamiltonian reductions of $so$ or $sp$ type Lie algebras. For simple cases with generators of spin up to two, we obtain their operator product expansions by requiring the associativity. We further claim that the W-algebras can be realized as the symmetry algebras of dual coset CFTs and provide several strong supports. The analysis can be regarded as a check of extended higher spin holographies including full quantum corrections. We also extend the analysis by introducing $\mathcal{N}=1$ supersymmetry.