Global Symmetries, Code Ensembles, and Sums Over Geometries

We consider Abelian topological quantum field theories (TQFTs) in 3d and show that gaugings of invertible global symmetries naturally give rise to additive codes. These codes emerge as nonanomalous subgroups of the 1-form symmetry group, parameterizing the fusion rules of condensable TQFT anyons. The boundary theories dual to TQFTs with the anyons condensed, i.e. with the corresponding symmetry subgroup gauged, are “code” conformal field theories (CFTs). This observation bridges together, in the holographic sense, results on 1-form symmetries of 3d TQFTs with developments connecting codes to 2d CFTs. Building on this relationship, we proceed to consider the ensemble of maximal gaugings (topological boundary conditions) in a general, not necessarily Abelian 3d TQFT, and propose that the resulting ensemble of boundary CFTs has a holographic description as a gravitational theory – the bulk TQFT summed over topologies.