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We show that Berry data measured on a torus are enough to determine the full all-genus Abelian topological theory, including its defects. This puts a long-standing piece of folklore in Abelian topological order on rigorous footing.

We show that toral Chern–Simons theory and its Reshetikhin–Turaev counterpart are equivalent as TQFTs. Earlier work established relations only at the level of partition functions or mapping-class-group representations; to our knowledge, this is the first explicit proof of the expected equivalence at the level of the full, nonperturbative TQFT.

We construct toral Chern–Simons theory through geometric quantization and show that it leads to a well-defined nonperturbative TQFT. This provides a rigorous mathematical foundation for describing Abelian topological order on surfaces of arbitrary genus.