Holographic duality from Howe duality: Chern-Simons gravity as an ensemble of code CFTs

Abstract: We discuss the holographic correspondence between 3d "Chern-Simons gravity" and an ensemble of 2d Narain code CFTs. Starting from 3d abelian Chern-Simons theory, we construct an ensemble of boundary CFTs defined by gauging all possible maximal subgroups of the bulk one-form symmetry. Each maximal non-anomalous subgroup is isomorphic to a classical even self-dual error-correcting code over $\mathbb Z_p\times \mathbb Z_p$, providing a way to define a boundary "code CFT." The average over the ensemble of such theories is holographically dual to Chern-Simons gravity, a bulk theory summed over 3d topologies sharing the same boundary. In the case of prime $p$, the sum reduces to that over handlebodies, i.e. becomes the Poincar\'e series akin to that in semiclassical gravity. As the main result of the paper, we show that the mathematical identity underlying this holographic duality can be understood and rigorously proven using the framework of Howe duality over finite fields. This framework is concerned with the representation theory of two commuting groups forming a dual pair: the symplectic group of modular transformations of the boundary, and an orthogonal group mapping codes to each other. Finally, we reformulate the holographic duality as an identity between different averages over quantum stabilizer states, providing an interpretation in terms of quantum information theory.

Holographic Duality from Howe Duality: Chern--Simons Gravity as an Ensemble of Code CFTs

The paper under discussion delves into the intricate interplay between quantum field theories and gravity, specifically investigating a novel approach that connects 3-dimensional "Chern--Simons gravity" to an ensemble of 2-dimensional Narain code conformal field theories (CFTs). The framework employed to bridge these realms is Howe duality over finite fields, providing a rigorous mathematical underpinning for the proposed duality.

The essence of the study is to establish a holographic correspondence between the bulk Chern--Simons theory and the boundary ensemble of code CFTs. This is achieved by considering the 3-dimensional abelian Chern--Simons theory on a manifold with boundary and constructing an ensemble of boundary CFTs through the gauging of all possible maximal subgroups of the bulk's one-form symmetry. Each subgroup corresponds to a classical even self-dual error-correcting code, which serves as a foundation for defining a boundary "code CFT."

The paper's main result highlights that the mathematical identity underlying this holographic duality can be rigorously proven via Howe duality, which deals with the representation theory of two commuting groups forming a dual pair: the symplectic group of modular transformations and an orthogonal group mapping codes. Furthermore, the holographic duality is reformulated in terms of quantum information theory as an identity between different averages over quantum stabilizer states.

Numerically, the results demonstrate that the ensemble average of code CFTs on arbitrary Riemann surfaces matches the Poincaré series of the bulk theory's vacuum character. This is a significant finding, as the bulk sum over topologies reduces to a sum over handlebodies, particularly when the field parameter ( p ) is a prime. The Poincaré series is thus expressed as a sum over all modular transformations of the vacuum character, leading to an equality that has deep implications for our understanding of quantum gravity and holography.

Practically, this research suggests new ways to construct gravitational theories from TQFTs by leveraging ensemble averages of CFTs. Theoretically, it opens avenues for exploring quantum gravity through the lens of quantum information theory, offering insights into the structure of space-time itself.

The implications of this work are vast, suggesting that similar techniques could apply to more complex gauge groups or other finite fields, potentially unraveling new classes of holographic dualities. Additionally, exploring larger or continuous gauge groups might bridge this framework with conventional string theoretic constructions, paving paths to new theories of holography that encompass a broader class of models. Overall, the paper provides a comprehensive and mathematically robust framework for exploring the connections between quantum field theories and gravity, with substantial potential for future exploration in theoretical physics.