Alternative names: Cohomological gauge theory code.
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. There exist lattice-model formulations in arbitrary spatial dimension [3]. Boundaries and excitations have been studied for arbitrary dimension [4].Member of code lists
Primary Hierarchy
Parents
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 [5]. Generalizations of Ocneanu's tube algebras [6,7] can be used to characterize excitations in both theories [8; 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 [5,9,10].
Dijkgraaf-Witten gauge theory code
Children
Restricting Dijkgraaf-Witten gauge theory to a 2D manifold reproduces the phase of the TQD model [11]. 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 \(D^{\omega}(G)\) [4; pg. 41]. TQD codewords are gauge-invariant boundary states of a 3D Dijkgraaf-Witten theory [12; Sec. IX].
Restricting Dijkgraaf-Witten gauge theory to a 3D manifold reproduces the phase of the TQT model.
An untwisted Dijkgraaf-Witten theory in 4D for the group \(G=\mathbb{Z}_2\) is a \((1,3)\) 4D toric code.
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]
- 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
- [5]
- A. Kapustin and R. Thorngren, “Higher symmetry and gapped phases of gauge theories”, (2015) arXiv:1309.4721
- [6]
- Ocneanu, Adrian. "Chirality for operator algebras." Subfactors (Kyuzeso, 1993) 39 (1994).
- [7]
- A. Ocneanu, “Operator Algebras, Topology and Subgroups of Quantum Symmetry – Construction of Subgroups of Quantum Groups –”, Advanced Studies in Pure Mathematics 235 DOI
- [8]
- 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
- [9]
- L. Lootens, B. Vancraeynest-De Cuiper, N. Schuch, and F. Verstraete, “Mapping between Morita-equivalent string-net states with a constant depth quantum circuit”, Physical Review B 105, (2022) arXiv:2112.12757 DOI
- [10]
- P. S. Hsin, private communication, 2024.
- [11]
- 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
- [12]
- 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
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