Trialities of orthosymplectic $\mathcal{W}$-algebras

Trialities of $\mathcal{W}$-algebras are isomorphisms between the affine cosets of three different $\mathcal{W}$-(super)algebras, and were first conjectured in the physics literature by Gaiotto and Rap\v{c}\'ak. In this paper we prove trialities among eight families of $\mathcal{W}$-(super)algebras of types $B$, $C$, and $D$. The key idea is to identify the affine cosets of these algebras with one-parameter quotients of the universal two-parameter even spin $\mathcal{W}_{\infty}$-algebra which was recently constructed by Kanade and the second author. Our result is a vast generalization of both Feigin-Frenkel duality in types $B$, $C$, and $D$, and the coset realization of principal $\mathcal{W}$-algebras of type $D$ due to Arakawa and us. It also provides a new coset realization of principal $\mathcal{W}$-algebras of types $B$ and $C$. As an application, we prove the rationality of the affine vertex superalgebra $L_k(\mathfrak{osp}_{1|2n})$, the minimal $\mathcal{W}$-algebra $\mathcal{W}_{k-1/2}(\mathfrak{sp}_{2n+2}, f_{\text{min}})$, and the coset $\text{Com}(L_k(\mathfrak{sp}_{2m}), L_k(\mathfrak{sp}_{2n}))$, for all integers $k,n,m \geq 1$ with $m<n$. We also prove the rationality of some families of principal $\mathcal{W}$-superalgebras of $\mathfrak{osp}_{1|2n}$ and $\mathfrak{osp}_{2|2n}$, and subregular $\mathcal{W}$-algebras of $\mathfrak{so}_{2n+1}$