Torus Berry Data Determine All-Genus Abelian Topological Orders

Abstract

We show that for Abelian Chern-Simons topological orders, torus Berry matrices determine the all-genus extended TQFT. We identify the topological part of the projective Berry holonomy under metric deformations with the mapping-class-group representation of the Abelian Chern-Simons TQFT and prove that the normalized torus data reconstruct its finite quadratic module $(G,q)$. Recent work showed that $(G,q)$ classifies the extended theory up to symmetric monoidal natural isomorphism, then genus-one data determine the all-genus theory without choosing a $K$-matrix presentation. We also prove that, for normalized character row errors $δ<21.96$%, nearest-row decoding recovers the Abelian fusion algebra independently of the number of anyons. The result applies to Abelian fractional quantum Hall and spin-liquid phases described by even-lattice Chern-Simons theories.