With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
A unitary fusion category is a C*-fusion category.
(…)
In the condensed matter theory of topological phases of matter (especially in contexts related to topological quantum computation, see the references there) it is folklore that the anyon-content in topological orders is characterized by unitary fusion categories.
David Penneys, Unitary fusion categories, Section 3 in: Topological Phases of Matter (2021) pdf
MO discussion: Unitary structures on fusion categories MO:q/131546/
Cain Edie-Michell, Masaki Izumi, David Penneys, Classification of -quadratic unitary fusion categories arXiv:2108.01564
In condensed matter theory it is folklore that species of anyonic topological order correspond to braided unitary fusion categories/modular tensor categories.
The origin of the claim is:
Early accounts re-stating this claim (without attribution):
Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, Sankar Das Sarma, pp. 28 of: Non-Abelian Anyons and Topological Quantum Computation, Rev. Mod. Phys. 80 1083 (2008) arXiv:0707.1888, doi:10.1103/RevModPhys.80.1083
Zhenghan Wang, Section 6.3 of: Topological Quantum Computation, CBMS Regional Conference Series in Mathematics 112, AMS (2010) ISBN-13: 978-0-8218-4930-9, pdf
Further discussion (mostly review and mostly without attribution):
Simon Burton, A Short Guide to Anyons and Modular Functors arXiv:1610.05384
(this one stands out as still attributing the claim to Kitaev (2006), Appendix E)
Eric C. Rowell, Zhenghan Wang, Mathematics of Topological Quantum Computing, Bull. Amer. Math. Soc. 55 (2018), 183-238 (arXiv:1705.06206, doi:10.1090/bull/1605)
Tian Lan, A Classification of (2+1)D Topological Phases with Symmetries arXiv:1801.01210
From categories to anyons: a travelogue [arXiv:1811.06670]
Colleen Delaney, A categorical perspective on symmetry, topological order, and quantum information (2019) pdf, uc:5z384290
Colleen Delaney, Lecture notes on modular tensor categories and braid group representations (2019) pdf, pdf
Liang Wang, Zhenghan Wang, In and around Abelian anyon models, J. Phys. A: Math. Theor. 53 505203 (2020) doi:10.1088/1751-8121/abc6c0
Parsa Bonderson, Measuring Topological Order, Phys. Rev. Research 3, 033110 (2021) arXiv:2102.05677, doi:10.1103/PhysRevResearch.3.033110
Zhuan Li, Roger S.K. Mong, Detecting topological order from modular transformations of ground states on the torus arXiv:2203.04329
Eric C. Rowell, Braids, Motions and Topological Quantum Computing arXiv:2208.11762
Sachin Valera, A Quick Introduction to the Algebraic Theory of Anyons, talk at CQTS Initial Researcher Meeting (Sep 2022) pdf
Willie Aboumrad, Quantum computing with anyons: an F-matrix and braid calculator arXiv:2212.00831
Emphasis that the expected description of anyons by braided fusion categories had remained folklore, together with a list of minimal assumptions that would need to be shown:
An argument that the statement at least for SU(2)-anyons does follow from an enhancement of the K-theory classification of topological phases of matter to interacting topological order:
Arguments realizing such anyonic topological order in the worldvolume-field theory on M5-branes:
Via KK-compactification on closed 3-manifolds (Seifert manifolds) analogous to the 3d-3d correspondence (which instead uses hyperbolic 3-manifolds):
Gil Young Cho, Dongmin Gang, Hee-Cheol Kim: M-theoretic Genesis of Topological Phases, J. High Energ. Phys. 2020 115 (2020) [arXiv:2007.01532, doi:10.1007/JHEP11(2020)115]
Shawn X. Cui, Paul Gustafson, Yang Qiu, Qing Zhang, From Torus Bundles to Particle-Hole Equivariantization, Lett Math Phys 112 15 (2022) [doi:10.1007/s11005-022-01508-3, arXiv:2106.01959]
Shawn X. Cui, Yang Qiu, Zhenghan Wang, From Three Dimensional Manifolds to Modular Tensor Categories, Commun. Math. Phys. 397 (2023) 1191–1235 [doi:10.1007/s00220-022-04517-4, arXiv:2101.01674]
Federico Bonetti, Sakura Schäfer-Nameki, Jingxiang Wu, : 3d Topological Order Labeled by Seifert Manifolds [arXiv:2403.03973]
Relation to ZX-calculus:
On detection of topological order by observing modular transformations on the ground state:
See also:
Last revised on June 7, 2022 at 19:46:35. See the history of this page for a list of all contributions to it.