Abstract
Trialities of W-algebras are isomorphisms between the affine cosets of three different W-(super)algebras, and were first conjectured in the physics literature by Gaiotto and Rapčák. In this paper we prove trialities among eight families of 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 W∞-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 W-algebras of type D due to Arakawa and us. It also provides a new coset realization of principal W-algebras of types B and C. As an application, we prove the rationality of the affine vertex superalgebra Lk(osp1|2n), the minimal W-algebra Wk−1/2(sp2n+2,fmin), and the coset Com(Lk(sp2m),Lk(sp2n)), for all integers k,n,m≥1 with m<n. We also prove the rationality of some families of principal W-superalgebras of osp1|2n and osp2|2n, and subregular W-algebras of so2n+1.
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