## Description

A code whose codewords realize \(D\)-dimensional lattice Dijkgraaf-Witten gauge theory [1,2] for a finite group \(G\) and a \(D+1\)-cocycle \(\omega\) in the cohomology class \(H^{D+1}(G,U(1))\). When the cocycle is non-trivial, the gauge theory is called a twisted gauge theory. For trivial cocycles in 3D, the model can be called a quantum triple model, in allusion to being a 3D version of the quantum double model. There exist lattice-model formulations in arbitrary spatial dimension [3] as well as explicitly in 3D [4,5].

Boundaries and excitations have been studied in 3D [6–8] and arbitrary dimension [9]. Generalizations of Ocneanu's tube algebras [10,11] can be used to characterize excitations, which are described by the tube algebra of the category \(\text{Vec}^{\omega}(G)\) for 3D models [9,12]. Gapped boundaries of the 3D models are classified by a subgroup \(K \subseteq G\) and a particular three-cochain [6].

## Parents

- Group-based quantum code
- Two-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 [13]. Generalizations of Ocneanu's tube algebras [10,11] can be used to characterize excitations in both theories [14; 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 [13,15,16].

## Children

- Twisted quantum double (TQD) code — Restricting Dijkgraaf-Witten gauge theory to a 2D manifold reproduces the phase of the TQD model [17]. The Drinfield center of the category \(\text{Vec}^{\omega}(G)\) is used to describe bulk excitations of 3D Dijkgraaf-Witten models, and this center is equivalent to the twisted quantum double \(D^{\omega}(G)\) [9; pg. 41]. TQD codewords are gauge-invariant boundary states of a 3D Dijkgraaf-Witten theory [18; Sec. IX].
- Chiral semion Walker-Wang model code — When treated as ground states of the code Hamiltonian, the code states realize 3D double-semion topological order, a topological phase of matter that exists as the deconfined phase of the 3D twisted \(\mathbb{Z}_2\) gauge theory [19].

## References

- [1]
- R. Dijkgraaf and E. Witten, “Topological gauge theories and group cohomology”, Communications in Mathematical Physics 129, 393 (1990) DOI
- [2]
- D. S. Freed and F. Quinn, “Chern-Simons theory with finite gauge group”, Communications in Mathematical Physics 156, 435 (1993) arXiv:hep-th/9111004 DOI
- [3]
- A. Mesaros and Y. Ran, “Classification of symmetry enriched topological phases with exactly solvable models”, Physical Review B 87, (2013) arXiv:1212.0835 DOI
- [4]
- J. C. Wang and X.-G. Wen, “Non-Abelian string and particle braiding in topological order: ModularSL(3,Z)representation and(3+1)-dimensional twisted gauge theory”, Physical Review B 91, (2015) arXiv:1404.7854 DOI
- [5]
- Y. Wan, J. C. Wang, and H. He, “Twisted gauge theory model of topological phases in three dimensions”, Physical Review B 92, (2015) arXiv:1409.3216 DOI
- [6]
- H. Wang et al., “Gapped boundary theory of the twisted gauge theory model of three-dimensional topological orders”, Journal of High Energy Physics 2018, (2018) arXiv:1807.11083 DOI
- [7]
- A. Bullivant and C. Delcamp, “Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases”, Journal of High Energy Physics 2021, (2021) arXiv:2006.06536 DOI
- [8]
- J. Huxford, D. X. Nguyen, and Y. B. Kim, “Twisted Lattice Gauge Theory: Membrane Operators, Three-loop Braiding and Topological Charge”, (2024) arXiv:2401.13042
- [9]
- A. Bullivant and C. Delcamp, “Tube algebras, excitations statistics and compactification in gauge models of topological phases”, Journal of High Energy Physics 2019, (2019) arXiv:1905.08673 DOI
- [10]
- Ocneanu, Adrian. "Chirality for operator algebras." Subfactors (Kyuzeso, 1993) 39 (1994).
- [11]
- A. Ocneanu, “Operator Algebras, Topology and Subgroups of Quantum Symmetry – Construction of Subgroups of Quantum Groups –”, Advanced Studies in Pure Mathematics DOI
- [12]
- T. Bartsch, M. Bullimore, and A. Grigoletto, “Representation theory for categorical symmetries”, (2023) arXiv:2305.17165
- [13]
- A. Kapustin and R. Thorngren, “Higher symmetry and gapped phases of gauge theories”, (2015) arXiv:1309.4721
- [14]
- 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
- [15]
- 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
- [16]
- P. S. Hsin, private communication, 2024.
- [17]
- D. Naidu and D. Nikshych, “Lagrangian Subcategories and Braided Tensor Equivalences of Twisted Quantum Doubles of Finite Groups”, Communications in Mathematical Physics 279, 845 (2008) arXiv:0705.0665 DOI
- [18]
- Y. Hu, Y. Wan, and Y.-S. Wu, “Twisted quantum double model of topological phases in two dimensions”, Physical Review B 87, (2013) arXiv:1211.3695 DOI
- [19]
- R. Dijkgraaf and E. Witten, “Topological gauge theories and group cohomology”, Communications in Mathematical Physics 129, 393 (1990) DOI

## Page edit log

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

## Cite as:

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