Null-state structure of the universal \(\mathcal{W}_{\infty}^{s,s}\) algebra at finite rank

Establish that the universal \(\mathcal{W}_{\infty}^{s,s}\) algebra at central charge \(c=3(1-N^2)\), for every integer \(N\geq2\), contains null states and, in particular, that all generators in the multiplets \(\mathbb{W}_p\) with \(p>N\) are null.

Background

The proposed VOA associated with N=4\mathcal{N}=4 SU(N)SU(N) super Yang–Mills theory is obtained by taking a simple quotient of the universal nonlinear W∞s,s\mathcal{W}_{\infty}^{s,s} algebra. The quotient is defined using an ideal generated by the purported null generators Wp>NW_{p>N}.

The truncation and the resulting finite-rank VOA therefore depend on proving the conjectured null-state structure at the special central charges c=3(1−N2)c=3(1-N^2). Establishing this would give an algebraic foundation for the proposed VOA construction.

References

It is conjectured that when the central charge is set to be c=3(1-N{2}), N=2,3,\cdots, the \mathcal{W}{\infty}{s,s} algebra contains null states. In particular, all generators in the multiplet \mathbb{W}{p} for p>N are null.

— Non-invertible symmetry and vertex operator algebra outer-automorphism  (2608.18926 - Maruyoshi et al., 19 Aug 2026) in Section 2.2, subsection “VOA automorphism”