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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, parametrizing the fusion rules of condensable TQFT anyons. The boundary theories dual to TQFTs with a maximal symmetry subgroup gauged, i.e., with the corresponding anyons condensed, 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. 

A<sc>bstract</sc> We provide a precise relation between an ensemble of Narain conformal field theories (CFTs) with central chargec=n, and a sum of (U(1) × U(1))ⁿChern-Simons theories on different handlebody topologies. We begin by reviewing the general relation of additive codes to Narain CFTs. Then we describe a holographic duality between any given Narain theory and a pure Chern-Simons theory on a handlebody manifold. We proceed to consider an ensemble of Narain theories, defined in terms of an ensemble of codes of lengthnoverℤ_k × ℤ_kfor primek. We show that averaging over this ensemble is holographically dual to a level-k(U(1) × U(1))ⁿChern-Simons theory, summed over a finite number of inequivalent classes of handlebody topologies. In the limit of largekthe ensemble approaches the ensemble of all Narain theories, and its bulk dual becomes equivalent to “U(1)-gravity” — the sum of the pertubative part of the Chern-Simons wavefunction over all possible handlebodies — providing a bulk microscopic definition for this theory. Finally, we reformulate the sum over handlebodies in terms of Hecke operators, paving the way for generalizations.