Determine whether pairing conditions guide the exclusion of massless RG flows

Determine whether the pairing condition for unbroken mathbb{Z}_{N} symmetries and defects can serve as a guiding principle for prohibiting massless renormalization-group flows between SU(N)_{K} and SU(N)_{K'} WessZuminoWitten models.

Background

The paper distinguishes the pairing condition, derived from half-integer-spin simple currents in folded topological quantum field theories, from the weaker anomaly-matching condition in conformal field theories. Although the pairing condition yields no-go results under additional assumptions, the authors do not establish that it universally excludes massless renormalization-group flows lacking the corresponding pairing structure.

References

The pairing condition canonically relates symmetries and defects, as in category theory, whereas it is unclear whether it can be a guiding principle in prohibiting the massless renormalization group flows.

— Difference between pair creation and anomaly matching: Nontrivial aspects of invertible symmetries and defects in $SU(N)$ models  (2610.21478 - Fukusumi et al., 15 Oct 2026) in Abstract

However, the realization of the pairing and anomaly matching conditions in the lower dimensional perturbative QFT and the corresponding critical lattice models is less clear, and we leave this as an open problem.

— Difference between pair creation and anomaly matching: Nontrivial aspects of invertible symmetries and defects in $SU(N)$ models  (2610.21478 - Fukusumi et al., 15 Oct 2026) in Section 1, Introduction, subsection “Summary of results”

We keep the physical interpretation of the mismatch for a general parameter $N$ as an open problem, because the field theoretic discussions in are also very convincing comparable to .

— Difference between pair creation and anomaly matching: Nontrivial aspects of invertible symmetries and defects in $SU(N)$ models  (2610.21478 - Fukusumi et al., 15 Oct 2026) in Section 5, Conclusion

The precise relationship between the three fundamental structures, the anomaly matching conditions ($\mathbb{Q}=\mathbb{Q'}$), the correspondence between defects and symmetries ($\rho_{\mathcal{Q}=\rho_{\mathcal{D}$), and the integer spin nonsimple current condition ($h_{\alpha_{\text{ub}\pm h_{\alpha'_{\text{ub}\in \mathbb{Z}/2$) has not been studied sufficiently. Their correspondence has been assumed, more or less implicitly or heuristically in the literature, but we would like to propose this correspondence as a fundamental open problem classifying CFTs and TQFTs.

— Difference between pair creation and anomaly matching: Nontrivial aspects of invertible symmetries and defects in $SU(N)$ models  (2610.21478 - Fukusumi et al., 15 Oct 2026) in Section 5, Conclusion