Duality via convolution of W-algebras

Abstract

Feigin-Frenkel duality is the isomorphism between the principal $\mathcal{W}$-algebras of a simple Lie algebra $\mathfrak{g}$ and its Langlands dual Lie algebra ${}^L\mathfrak{g}$. A generalization of this duality to a larger family of $\mathcal{W}$-algebras called hook-type was recently conjectured by Gaiotto and Rapčák and proved by the first two authors. It says that the affine cosets of two different hook-type $\mathcal{W}$-algebras are isomorphic. A natural question is whether the duality between affine cosets can be enhanced to a duality between the full $\mathcal{W}$-algebras. There is a convolution operation that maps a hook-type $\mathcal{W}$-algebra $\mathcal{W}$ to a certain relative semi-infinite cohomology of $\mathcal{W}$ tensored with a suitable kernel VOA. The first two authors conjectured previously that this cohomology is isomorphic to the Feigin-Frenkel dual hook-type $\mathcal{W}$-algebra. Our main result is a proof of this conjecture.

Revised, 24 pages

References in corpus (21)

• Quantum Reduction for Affine Superalgebras
• Finite vs. affine W-algebras
• Chiral de Rham complex
• W-algebras as coset vertex algebras
• $W_n^{(2)}$ algebras
• Trialities of $\mathcal{W}$-algebras
• Duality of subregular W-algebras and principal W-superalgebras
• Screening operators for W-algebras
• Gluing vertex algebras
• Universal two-parameter ${\mathcal{W}}_{\infty }$-algebra and vertex algebras of type $\mathcal{W}(2,3,\dots ,N)$
• Universal two-parameter even spin ${\mathcal{W}}_{\infty }$-algebra
• S-duality for the large $N=4$ superconformal algebra
• Trialities of orthosymplectic $\mathcal{W}$-algebras
• Correspondences of categories for subregular W-algebras and principal W-superalgebras
• $N=4$ superconformal algebras and diagonal cosets
• Orthosymplectic Satake equivalence
• Mirabolic Satake equivalence and supergroups
• Quantum coordinate ring in WZW model and affine vertex algebra extensions
• Higher rank FZZ-dualities
• Ordinary modules for vertex algebras of ${\mathfrak{osp}}_{1|2n}$
• FZZ-triality and large $\mathcal{N}=4$ super Liouville theory