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 Drinfeľd-Sokolov reductions of $BCD$ series of simple Lie algebras, orthogonal and symplectic cosets as well as orthosymplectic $Y$-algebras of Gaiotto and Rapčák. Based on explicit calculations we conjecture a complete list of co-dimension $1$ truncations of the algebra.