Analytic treatment of irregular elements in compact Lie-group flat gauging

Develop a complete analytic treatment of irregular elements in a compact Lie group, particularly irregular elements of $SU(2)$ whose centralizers are larger than a Cartan torus, and thereby extend the formal flat-gauging analysis of the $text{Rep}(SU(2))$ symmetry beyond the regular stratum.

Background

Section 6 extends the finite-group Fourier-transform framework to the infinite representation category Rep(SU(2))\text{Rep}(SU(2)). Because commuting pairs of group elements form a singular moduli space, the analysis is straightforward only on the regular stratum, where the centralizer of an element is a maximal torus. For irregular elements, the centralizer changes discontinuously; for example, the centralizer of the identity is the entire group SU(2)SU(2).

The paper therefore treats the distributional manipulations and the resulting integrals formally on the regular stratum and does not provide a complete analysis of the singular or irregular strata. A remaining problem is to supply the analytic framework needed to define and control the Fourier and inverse-Fourier constructions there.

References

What does break down is the evaluation of that character at non-regular holonomy $k$: $\chi_\pi$ is a locally integrable function, but it diverges on the non-regular elements of $G$. For example, as $k_t \to e$, the character formally limits to the dimension $d_\pi$, which diverges for non-compact $G$. So the definition of the ribbon operators on states with non-regular holonomy, where the character blows up, has to be handled separately. However, because non-regular elements are measure zero within $G$, this may not be a problem, as this kind of divergence would not affect the matrix element of any physical operator. We leave a systematic treatment of states with non-regular holonomy to future work.

— Topological entanglement entropy in 3D gravity  (2609.19247 - Balasubramanian et al., 16 Sep 2026) in Appendix A, subsection “What labels the ribbons?”

Nevertheless, the treatment of irregular element $g\in G$, whose centralizer $C_G(g)$ is larger than Cartan torus $T$, still remains illusive and requires more analytic input. Accordingly, the distributional manipulations below are understood formally on the regular stratum and are used only to determine the form of the resulting integral up to an overall normalization. We do not attempt a complete analytic treatment of the irregular strata.

— On general background of quantum non-invertible symmetry in 2D  (2609.09550 - Chen et al., 9 Sep 2026) in Section 6, paragraph beginning “We emphasize that the construction in this section is a direct analogue of the finite-group gauging construction.”