Holographic description of Narain CFTs and their code-based ensembles

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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.

Original languageEnglish
Article number343
JournalJournal of High Energy Physics
Issue number5
StatePublished - May 2024

Bibliographical note

Publisher Copyright:
© The Author(s) 2024.


  • AdS-CFT Correspondence
  • Chern-Simons Theories
  • Conformal Field Models in String Theory

ASJC Scopus subject areas

  • Nuclear and High Energy Physics


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