Completeness of ribbon operators for transformable groups

Prove that the closed ribbon operators constructed from a conjugacy class and a representation of its centralizer exhaust the complete set of ribbon operators for a general transformable group.

Background

The appendix generalizes the finite-group quantum-double construction of ribbon operators to arbitrary transformable Lie groups. The proposed operators are labeled by a regular conjugacy class and a character of the associated Cartan subgroup, with a Weyl-group identification.

For finite groups, ribbon operators are known to provide the complete anyon data. The paper constructs a corresponding class for general transformable groups but does not establish that no additional ribbon operators exist.

References

It would be interesting to demonstrate explicitly that the ribbons we construct exhaust the complete set of ribbon operators when $G$ is a general transformable group, as they do when $G$ is a finite group, but we will leave this analysis for future work.

— Topological entanglement entropy in 3D gravity  (2609.19247 - Balasubramanian et al., 16 Sep 2026) in Appendix A, subsection “Definition of the closed ribbon operator”