General non-degeneracy of the extended current algebra

Prove that, for theories to which the parafermion-embedding theorem applies, the extended VOA constructed in its proof contains no weight-one currents beyond the three currents generating the affine \(\mathfrak{su}(2)_N\) algebra and the currents already present in the fixed-point VOA.

Background

For the Leech lattice examples with twisted-sector ground states of conformal weight $1-1/N$, the authors find that the extended VOA used to prove the parafermion embedding has exactly the expected three additional su(2)N\mathfrak{su}(2)_N currents, together with the currents of the fixed-point algebra, and no further weight-one fields.

They suggest that this absence of extra currents may be a general phenomenon for constructions satisfying the hypotheses of the parafermion-embedding theorem, because additional currents could obstruct the consistency of the intended current-algebra structure. The paper does not establish this claim in general.

References

It is tempting to conjecture that this non-degeneracy is a general feature of theories where theorem $\ref{th:paraf}$ applies, because it seems difficult to obtain a consistent current algebra in $\tilde V$ otherwise.

— Monstrous parafermionic defects and other non-invertible symmetries in chiral CFTs  (2609.03043 - Volpato, 2 Sep 2026) in Section 3.3, “Parafermions in the Leech lattice VOA and in Schellekens theories”