Completeness and realizability of 2D anyon data

A cluster of invariants is used to label topologically ordered phases. The question is whether they form a complete invariant: do two systems with identical data necessarily lie in the same phase, and does every consistent set of data arise from a physical Hamiltonian (realizability)?

What is known: For 2D bosonic topological order the working conjecture is that a unitary modular tensor category (UMTC) together with the chiral central charge $c_-$ completely classifies the phase; much structure (fusion, braiding, modular data) is understood, and the relation to conformal field theory boundaries is well developed. Rank-finiteness (Bruillard–Ng–Rowell–Wang) shows only finitely many UMTCs exist at each rank. Symmetry-enriched theories need additional symmetry action, fractionalization, defect and obstruction data; fractionalization alone is not a general classification.

Completeness and realizability are not proved in this generality. Modular $S,T$ matrices alone do not determine a UMTC (Mignard–Schauenburg); equality of those matrices is not equality of the full categorical input. The full chiral central charge matters, not only its value modulo eight, because invertible bosonic layers can change it without adding anyons. Symmetry enrichment, fermionic systems and 3D loop excitations are separate branches of B8 rather than hidden assumptions of this target.

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.