Identification of the finite-rank VOA quotient with \(\mathcal{N}=4\) SYM

Prove that the simple quotient of \(\mathcal{W}_{\infty}^{s,s}\) by the ideal generated by \(W_{p>N}\) is isomorphic to the vertex operator algebra associated with four-dimensional \(\mathcal{N}=4\) \(SU(N)\) super Yang–Mills theory.

Background

The paper denotes the ideal generated by the conjecturally null generators $W_{p>N}$ by $\mathcal{I}_N$, and denotes the corresponding simple quotient by $\mathcal{V}(A_{N-1})$. This quotient is then used throughout the paper as the VOA governing the Schur and Macdonald sectors of $\mathcal{N}=4$ $SU(N)$ SYM.

The proposed isomorphism is a structural assumption underlying the comparison between outer-automorphism-twisted VOA characters and non-invertible-symmetry-twisted four-dimensional indices.