Two-gauge theory code[1] 

Also known as Higher gauge theory code.

Description

A code whose codewords realize lattice two-gauge theory [27] 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 [912] and 4D [12], with the 3D case realizing the Yetter model [1316].

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. [1719]. Generalizations of Ocneanu's tube algebras [20,21] can be used to characterize excitations [17,22].

Parents

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

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
[11]
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
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Zoo Code ID: yetter_gauge_theory

Cite as:
“Two-gauge theory code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/yetter_gauge_theory
BibTeX:
@incollection{eczoo_yetter_gauge_theory, title={Two-gauge theory code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/yetter_gauge_theory} }
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Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/categories/gauge/yetter_gauge_theory.yml.