Universal two-parameter 𝒲∞-algebra and vertex algebras of type 𝒲(2, 3, …, N). (10th February 2021) - Linshaw, Andrew R.
Abstract : We prove the longstanding physics conjecture that there exists a unique two-parameter W_∞ -algebra which is freely generated of type W(2, 3, … ), and generated by the weights 2 and 3 fields. Subject to some mild constraints, all vertex algebras of type W(2, 3, …, N) for some N can be obtained as quotients of this universal algebra. As an application, we show that for n≥ 3, the structure constants for the principal W -algebras W^k(sl_n, f_prin) are rational functions of k and n, and we classify all coincidences among the simple quotients W_k(sl_n, f_prin) for n≥ 2 . We also obtain many new coincidences between W_k(sl_n, f_prin) and other vertex algebras of type W(2, 3, …, N) which arise as cosets of affine vertex algebras or nonprincipal W -algebras.