Generalize one-loop lifting to the higher-rank non-Coulomb gaugino tower

Determine whether the one-loop quantum-supercharge pairing established for the non-Coulomb class \(\mathcal S^4\) extends to the entire non-Coulomb gaugino tower \(\mathcal S^m\) in higher-rank \(SO(N)\) theories by constructing the derivative completion \(F_{m,\partial}\) of the proposed \(\psi\phi\) source.

Background

The paper constructs an explicit fortuitous source F4F_4 whose one-loop quantum-supercharge image is proportional to the non-Coulomb monotone S4\mathcal S^4 in the SO(7)SO(7) theory. For higher powers Sm\mathcal S^m with r<m<hr<m<h, the proposed source has a ψϕ\psi\phi component with the desired quantum image, but it must be supplemented by derivative terms to become classically Q0Q_0-closed through a finite-rank trace relation.

The unresolved task is therefore to construct the derivative completion at higher rank. Establishing it for the first non-Coulomb power at a fixed rank would, according to the paper, propagate the construction to the remainder of the tower by multiplication with S\mathcal S.

References

The most immediate question is whether the pairing of $\mathcal S4$ extends to the entire non-Coulomb gaugino tower at higher ranks. As described in Section~\ref{ssec:higher-rank-predictions}, the remaining step involves constructing the derivative completion $F_{m,\partial}$ of the proposed $\psi\phi$ source.

— Non-Coulomb $Q$-Cohomology and Quantum Lifting  (2610.22332 - Kibe et al., 15 Oct 2026) in Section 5.4, “Higher-rank quantum pairings” (Section \ref{ssec:higher-rank-predictions})

It remains to determine whether this persists at higher rank. The one-dimensionality of the pure-gaugino target cohomology does not determine the dimension of this quotient.

— Non-Coulomb $Q$-Cohomology and Quantum Lifting  (2610.22332 - Kibe et al., 15 Oct 2026) in Section 5.4, “Higher-rank quantum pairings” (Section \ref{ssec:higher-rank-predictions})