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 $(U(1)\times U(1){)}^n$ 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.

53 pages

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Cited by in corpus (9)

• TQFT gravity and ensemble holography
• Global Symmetries, Code Ensembles, and Sums Over Geometries
• Fluctuating Ensemble Averages and the BTZ Threshold
• Non-Chiral Vertex Operator Algebra Associated To Lorentzian Lattices And Narain CFTs
• Fermionic CFTs from topological boundaries in abelian Chern-Simons theories
• Algebraic constructions of code lattices in Narain conformal field theories
• Algebraic Realisation of the Zamolodchikov Metric in Narain Theories
• Code CFTs and Topological Matter
• A universal sum over topologies in 3d gravity