Global Symmetries, Code Ensembles, and Sums over Geometries

Area of Science:

• Condensed matter physics
• Quantum field theory
• Quantum information theory

Background:

• Topological quantum field theories (TQFTs) in 3D exhibit invertible global symmetries.
• Additive codes are a significant class of quantum error-correcting codes.

Purpose of the Study:

• To explore the connection between gauging symmetries in 3D TQFTs and the emergence of additive codes.
• To establish a holographic bridge between 1-form symmetries of TQFTs and 2D CFTs.
• To propose a holographic description for ensembles of boundary CFTs derived from general 3D TQFTs.

Main Methods:

• Analysis of Abelian 3D TQFTs and their invertible global symmetries.
• Identification of nonanomalous subgroups of 1-form symmetry groups parametrizing anyon fusion rules.
• Investigation of boundary theories dual to TQFTs with maximal symmetry subgroup gaugings.
• Consideration of ensembles of maximal gaugings in general 3D TQFTs.

Main Results:

• Gauging invertible global symmetries in 3D TQFTs naturally yields additive codes.
• These additive codes correspond to nonanomalous subgroups of the 1-form symmetry group.
• Boundary theories of TQFTs with condensed anyons are identified as "code" conformal field theories (CFTs).
• A holographic relationship is established between 3D TQFT symmetries and 2D CFTs.

Conclusions:

• The study establishes a novel connection between TQFTs, additive codes, and CFTs through the lens of symmetry gauging.
• The findings suggest that condensed anyons in TQFTs directly relate to the structure of additive codes.
• The proposed holographic description of boundary CFT ensembles as summed gravitational theories opens new avenues for research.