Holographic description of Narain CFTs and their code-based ensembles

Abstract

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) × U(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 ℤk × ℤk for prime k. We show that averaging over this ensemble is holographically dual to a level-k (U(1) × U(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.