On even spin $W_\infty$

Abstract

We study the even spin $\mathcal{W}_\infty$ which is a universal $\mathcal{W}$-algebra for orthosymplectic series of $\mathcal{W}$-algebras. We use the results of Fateev and Lukyanov to embed the algebra into $\mathcal{W}_{1+\infty}$. Choosing the generators to be quadratic in those of $\mathcal{W}_{1+\infty}$, we find that the algebra has quadratic operator product expansions. Truncations of the universal algebra include principal Drinfe\v{l}d-Sokolov reductions of $BCD$ series of simple Lie algebras, orthogonal and symplectic cosets as well as orthosymplectic $Y$-algebras of Gaiotto and Rap\v{c}\'{a}k. Based on explicit calculations we conjecture a complete list of co-dimension $1$ truncations of the algebra.