# Two-gauge theory code[1]

## Description

A code whose codewords realize lattice two-gauge theory [2–7] for a finite two-group (a.k.a. a crossed module) in arbitrary spatial dimension. There exist several lattice-model formulations in arbitrary spatial dimension [1,8] as well as explicitly in 3D [9–12] and 4D [12], with the 3D case realizing the Yetter model [13–16].

A two-gauge theory generalizes ordinary gauge theory by replacing the gauge group with a two-group. Lattice formulations place gauge fields not only on edges of a lattice (as they do in ordinary gauge theory), but also on higher-dimensional structures such as faces.

Ground-state degeneracy is a topological invariant in the 3D case [8]. Excitations of the 3D models are studied in Refs. [17–19]. Generalizations of Ocneanu's tube algebras [20,21] can be used to characterize excitations [17,22].

## Parents

- Category-based quantum code
- Commuting-projector Hamiltonian code — Two-gauge theory codewords form ground-state subspaces of frustration-free commuting projector Hamiltonians.
- Frustration-free Hamiltonian code — Two-gauge theory codewords form ground-state subspaces of frustration-free commuting projector Hamiltonians.
- Topological code — Two-gauge theory codes realize lattice two-gauge theory for a finite two-group.

## Children

- Dijkgraaf-Witten gauge theory code — Replacing the two-group data in a two-gauge theory with a group and a cocycle reproduces the phase of the Dijkgraaf-Witten gauge theory, with the two theories equivalent in 2D [1]. Generalizations of Ocneanu's tube algebras [20,21] can be used to characterize excitations in both theories [17; Sec. 4.2]. A Dijkgraaf-Witten Lagrangian can also be re-expressed as a two-group gauge theory Lagrangian by relating the electric and magnetic gauge fields via the equations of motion [1,23,24].
- Cubic theory code — Cubic theory codes realize higher-form \(\mathbb{Z}_2^3\) gauge theories with non-Abelian excitations in arbitrary dimensions.
- Chen-Hsin invertible-order code — Chen-Hsin invertible-order codes realize beyond-group-cohomology invertible topological phases of order two and four in arbitrary dimensions. These phases are described by invertible two-gauge theories [25; pg. 11].

## Cousins

- \(G\)-enriched Walker-Wang model code — \(G\)-enriched Walker-Wang models realize 3D two-gauge theories [26].
- Walker-Wang model code — Two-gauge theory codes for particular two-groups are dual to certain Walker-Wang models based on Abelian groups [9; Sec. V][11; Sec. 7].
- Quantum-double code — Restricting 2-gauge theory constructions to a 2D manifold and replacing the 2-group with a group reproduces the phase of the Kitaev quantum double model [9].
- 3D subsystem color code — The 3D subsystem color code can be ungauged [27–29,29] to obtain six copies of \(\mathbb{Z}_2\) gauge theory with one-form symmetries [30].

## References

- [1]
- A. Kapustin and R. Thorngren, “Higher symmetry and gapped phases of gauge theories”, (2015) arXiv:1309.4721
- [2]
- J. C. Baez and A. D. Lauda, “Higher-Dimensional Algebra V: 2-Groups”, (2004) arXiv:math/0307200
- [3]
- J. Baez and U. Schreiber, “Higher Gauge Theory: 2-Connections on 2-Bundles”, (2004) arXiv:hep-th/0412325
- [4]
- J. C. Baez and U. Schreiber, “Higher Gauge Theory”, (2006) arXiv:math/0511710
- [5]
- J. C. Baez and J. Huerta, “An invitation to higher gauge theory”, General Relativity and Gravitation 43, 2335 (2010) arXiv:1003.4485 DOI
- [6]
- S. Gukov and A. Kapustin, “Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories”, (2013) arXiv:1307.4793
- [7]
- A. Kapustin and R. Thorngren, “Topological Field Theory on a Lattice, Discrete Theta-Angles and Confinement”, (2013) arXiv:1308.2926
- [8]
- A. Bullivant et al., “Higher lattices, discrete two-dimensional holonomy and topological phases in (3 + 1)D with higher gauge symmetry”, Reviews in Mathematical Physics 32, 2050011 (2019) arXiv:1702.00868 DOI
- [9]
- A. Bullivant et al., “Topological phases from higher gauge symmetry in3+1dimensions”, Physical Review B 95, (2017) arXiv:1606.06639 DOI
- [10]
- C. Delcamp and A. Tiwari, “From gauge to higher gauge models of topological phases”, Journal of High Energy Physics 2018, (2018) arXiv:1802.10104 DOI
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- C. Delcamp and A. Tiwari, “On 2-form gauge models of topological phases”, Journal of High Energy Physics 2019, (2019) arXiv:1901.02249 DOI
- [12]
- Z. Wan, J. Wang, and Y. Zheng, “Quantum 4d Yang-Mills theory and time-reversal symmetric 5d higher-gauge topological field theory”, Physical Review D 100, (2019) arXiv:1904.00994 DOI
- [13]
- D. N. YETTER, “TQFT’S FROM HOMOTOPY 2-TYPES”, Journal of Knot Theory and Its Ramifications 02, 113 (1993) DOI
- [14]
- T. Porter, “Topological Quantum Field Theories from Homotopy n -Types”, Journal of the London Mathematical Society 58, 723 (1998) DOI
- [15]
- T. PORTER, “INTERPRETATIONS OF YETTER’S NOTION OF G-COLORING: SIMPLICIAL FIBRE BUNDLES AND NON-ABELIAN COHOMOLOGY”, Journal of Knot Theory and Its Ramifications 05, 687 (1996) DOI
- [16]
- M. Mackaay, “Finite groups, spherical 2-categories, and 4-manifold invariants”, (1999) arXiv:math/9903003
- [17]
- A. Bullivant and C. Delcamp, “Excitations in strict 2-group higher gauge models of topological phases”, Journal of High Energy Physics 2020, (2020) arXiv:1909.07937 DOI
- [18]
- J. Huxford and S. H. Simon, “Excitations in the higher-lattice gauge theory model for topological phases. III. The (3+1)-dimensional case”, Physical Review B 109, (2024) arXiv:2206.09941 DOI
- [19]
- J. Huxford and S. H. Simon, “Excitations in the higher-lattice gauge theory model for topological phases. I. Overview”, Physical Review B 108, (2023) arXiv:2202.08294 DOI
- [20]
- Ocneanu, Adrian. "Chirality for operator algebras." Subfactors (Kyuzeso, 1993) 39 (1994).
- [21]
- A. Ocneanu, “Operator Algebras, Topology and Subgroups of Quantum Symmetry – Construction of Subgroups of Quantum Groups –”, Advanced Studies in Pure Mathematics DOI
- [22]
- T. Bartsch, M. Bullimore, and A. Grigoletto, “Representation theory for categorical symmetries”, (2023) arXiv:2305.17165
- [23]
- L. Lootens et al., “Mapping between Morita-equivalent string-net states with a constant depth quantum circuit”, Physical Review B 105, (2022) arXiv:2112.12757 DOI
- [24]
- P. S. Hsin, private communication, 2024.
- [25]
- Y.-A. Chen and P.-S. Hsin, “Exactly solvable lattice Hamiltonians and gravitational anomalies”, SciPost Physics 14, (2023) arXiv:2110.14644 DOI
- [26]
- D. J. Williamson and Z. Wang, “Hamiltonian models for topological phases of matter in three spatial dimensions”, Annals of Physics 377, 311 (2017) arXiv:1606.07144 DOI
- [27]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [28]
- L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter”, Journal of High Energy Physics 2017, (2017) arXiv:1605.01640 DOI
- [29]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
- [30]
- A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836

## Page edit log

- Victor V. Albert (2024-06-11) — most recent

## Cite as:

“Two-gauge theory code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/yetter_gauge_theory