Construct corrected normal forms for the reduced PBW character-counting monomials

Construct systematic corrected normal forms for the reduced monomial sets built from the five strong generators J, G, \widetilde G, W, and \widetilde W of the vertex operator algebra \(\mathcal{W}_{\mathbb{Z}_3}\), so that they account for relations and missing module directions at all levels while preserving the character-counting dimensions.

Background

The paper proposes a reduced character-counting set for the WZ3\mathcal{W}_{\mathbb{Z}_3}-module M0M_0, obtained by eliminating the generators TT, GWG_W, and G~W~\widetilde G_{\widetilde W} in favor of five generators. The resulting formal monomials reproduce the closed-form flavored character, but they cease to be linearly independent and do not always span the module in canonical order at higher levels.

At relative level $5/2$, the authors find one relation among eight formal monomials and a distinct missing module direction, with the two defects canceling in the graded character. They verify this cancellation only through relative level $9/2$, leaving the construction of a genuine all-level normal-form system unresolved.

References

The restricted monomials eq:five-letter-occupancies--eq:five-letter-W-restriction are thus best regarded as a character-counting set rather than as a literal PBW basis at all levels; we leave the systematic construction of the corrected normal forms to future work.

— Rank-one 4d $\mathcal N=3$ SCFTs: Schur index, VOA modules, and modularity  (2609.10685 - Guo et al., 9 Sep 2026) in Section 3, subsection “A reduced PBW basis”; Discussion

We have checked this cancellation sector by sector in the free field realization up to relative level \frac92, and conjecture that it persists for all \lambda, so that the formal span always carries the correct dimension, \begin{equation} \dim C_\lambda = \dim M_{0, \lambda} \ , \end{equation} even though the image of \Phi_\lambda may be a proper subspace of M_{0,\lambda} .

— Rank-one 4d $\mathcal N=3$ SCFTs: Schur index, VOA modules, and modularity  (2609.10685 - Guo et al., 9 Sep 2026) in Section 3, subsection “A reduced PBW basis,” paragraph “Level \(N=5/2\): failure of the canonical ordering”