Duality via convolution of W-algebras

Feigin-Frenkel duality is the isomorphism between the principal W-algebras of a simple Lie algebra g and its Langlands dual Lie algebra Lg. A generalization of this duality to a larger family of W-algebras called hook-type was recently conjectured by Gaiotto and Rap.cak and proved by the first two authors. It says that the affine cosets of two different hook-type W-(super)algebras are isomorphic. A natural question is whether the duality between the affine cosets can be enhanced to reconstruct one W-algebra from the other. There is a convolution operation that maps a hook-type W-algebra W to a certain relative semi-infinite cohomology of W tensored with a suitable kernel VOA. The first two authors conjectured previously that this cohomology is isomorphic to the Feigin-Frenkel dual hook-type W-algebra. Our main result is a proof of this conjecture.