We provide a precise relation between an ensemble of Narain conformal field
theories (CFTs) with central charge c=n, and a sum of (U(1)ΓU(1))n
Chern-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 length n over ZkβΓZkβ for prime k. We show that averaging over this
ensemble is holographically dual to a level-k(U(1)ΓU(1))n
Chern-Simons theory, summed over a finite number of inequivalent classes of
handlebody topologies. In the limit of large k the 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.Comment: 53 page