Classification of non-chiral topological orders by anyon data

Prove that every non-chiral two-dimensional topological order is fully characterized by its anyon content, including fusion and braiding statistics encoded by a modular tensor category (e.g., the Drinfeld centre Z(D)), thereby establishing that this data completely determines the phase.

Background

The lecture notes discuss intrinsic topological order and its characterization through anyons, fusion rules, and braiding statistics. They state the widely held belief that non-chiral topological orders are fully determined by such anyon data.

A general proof would confirm the modular tensor category framework as a complete invariant for non-chiral topological phases and connect microscopic lattice realizations to categorical classifications.