Cambridge Journal of Mathematics

Volume 10 (2022)

Number 1

Trialities of $\mathcal{W}$-algebras

Pages: 69 – 194



Thomas Creutzig (University of Alberta, Canada)

Andrew R. Linshaw (University of Denver, Colorado, U.S.A.)


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, and (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.


vertex algebra, $\mathcal{W}$-algebra, Poisson vertex algebra, Coset construction

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T. C. is supported by NSERC Discovery Grant #RES0019997.

A. L. is supported by Simons Foundation Grant #635650 and NSF Grant DMS-2001484.

Received 12 April 2021

Published 21 April 2022