Trialities of $\mathcal{W}$-algebras

Abstract

We prove the conjecture of Gaiotto and Rapčák that the $Y$-algebras $Y_{L,M,N}[\psi ]$ with one of the parameters $L,M,N$ zero, are simple one-parameter quotients of the universal two-parameter ${\mathcal{W}}_{1+\infty }$-algebra, and satisfy a symmetry known as triality. These $Y$-algebras are defined as the cosets of certain non-principal $\mathcal{W}$-algebras and $\mathcal{W}$-superalgebras by their affine vertex subalgebras, and triality is an isomorphism between three such algebras. Special cases of our result provide new and unified proofs of many theorems and open conjectures in the literature on $\mathcal{W}$-algebras of type $A$. This includes (1) Feigin-Frenkel duality, (2) the coset realization of principal $\mathcal{W}$-algebras due to Arakawa and us, (3) Feigin and Semikhatov's conjectured triality between subregular $\mathcal{W}$-algebras, principal $\mathcal{W}$-superalgebras, and affine vertex superalgebras, (4) the rationality of subregular $\mathcal{W}$-algebras due to Arakawa and van Ekeren, (5) the identification of Heisenberg cosets of subregular $\mathcal{W}$-algebras with principal rational $\mathcal{W}$-algebras that was conjectured in the physics literature over 25 years ago. Finally, we prove the conjectures of Procházka and Rapčák on the explicit truncation curves realizing the simple $Y$-algebras as ${\mathcal{W}}_{1+\infty }$-quotients, and on their minimal strong generating types.

Final version, to appear in Cambridge Journal of Mathematics

References in corpus (4)

• Duality of subregular W-algebras and principal W-superalgebras
• The vertex algebras ${\mathcal{R}}^{(p)}$ and ${\mathcal{V}}^{(p)}$
• On DS functor for affine Lie superalgebras
• Urod algebras and Translation of W-algebras

Cited by in corpus (20)

• Duality of subregular W-algebras and principal W-superalgebras
• Trialities of orthosymplectic $\mathcal{W}$-algebras
• Ribbon tensor structure on the full representation categories of the singlet vertex algebras
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• Whittaker vectors for $\mathcal{W}$-algebras from topological recursion
• Affine Laumon spaces and iterated W-algebras
• R-matrices and Miura operators in 5d Chern-Simons theory
• Cosets of free field algebras via arc spaces
• Quantum deformation of Feigin-Semikhatov's W-algebras and 5d AGT correspondence with a simple surface operator
• On rationality for $C_2$-cofinite vertex operator algebras
• Ordinary modules for vertex algebras of ${\mathfrak{osp}}_{1|2n}$
• On the structure of W-algebras in type A
• Center of affine ${\mathfrak{sl}}_{2|1}$ at the critical level
• Orbifolds of Gaiotto-Rapčák $Y$-algebras
• Orthosymplectic Feigin-Semikhatov duality
• Cosets from equivariant W-algebras
• Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras
• Duality via convolution of W-algebras
• Feigin-Semikhatov conjecture and related topics
• Vertex Algebras and Commutative Algebras