Holographic description of Narain CFTs and their code-based ensembles

TL;DR

This paper establishes a holographic duality between ensembles of Narain CFTs and sums over Chern-Simons theories on handlebodies, connecting code theory, topological quantum field theory, and holography.

Contribution

It introduces a precise holographic duality linking Narain CFT ensembles to Chern-Simons theories and reformulates the sum over topologies using Hecke operators.

Findings

Ensemble averaging corresponds to a sum over handlebody topologies in the bulk.

Large k limit recovers the ensemble of all Narain theories and a 'U(1)-gravity' dual.

Reformulation with Hecke operators enables potential generalizations.

Abstract

We provide a precise relation between an ensemble of Narain conformal field theories (CFTs) with central charge $c=n$, and a sum of $(U(1) \times U(1))^n$ 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 ${\mathbb Z}_k \times {\mathbb Z}_k$ for prime $k$. We show that averaging over this ensemble is holographically dual to a level-$k$ $(U(1) \times U(1))^n$ 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.