nLab unitary fusion category

Idea

A unitary fusion category is a C*-fusion category.

(…)

Related concept

• unitary operator

• dagger-category, C*-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.

References

General

• 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 $\mathbb{Z}/2\mathbb{Z}$-quadratic unitary fusion categories $[$arXiv:2108.01564$]$

Anyonic topological order in terms of braided fusion categories

Claim and status

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:

• Alexei Kitaev, Section 8 and Appendix E of: Anyons in an exactly solved model and beyond, Annals of Physics 321 1 (2006) 2-111 $[$doi:10.1016/j.aop.2005.10.005$]$

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:

• Sachin J. Valera, Fusion Structure from Exchange Symmetry in (2+1)-Dimensions, Annals of Physics 429 (2021) [doi:10.1016/j.aop.2021.168471, arXiv:2004.06282]

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:

• Hisham Sati, Urs Schreiber, Anyonic topological order in TED K-theory, Rev. Math. Phys. 35 03 (2023) 2350001 [doi:10.1142/S0129055X23500010,arXiv:2206.13563]

In string/M-theory

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, $MTC[M_3,G]$: 3d Topological Order Labeled by Seifert Manifolds [arXiv:2403.03973]

Via 3-brane defects:

• Hisham Sati, Urs Schreiber: Anyonic Defect Branes and Conformal Blocks in Twisted Equivariant Differential K-Theory, Reviews in Mathematical Physics 35 06 (2023) 2350009 [arXiv:2203.11838, doi:10.1142/S0129055X23500095]

Further discussion

Relation to ZX-calculus:

• Fatimah Rita Ahmadi, Aleks Kissinger, Topological Quantum Computation Through the Lens of Categorical Quantum Mechanics $[$arXiv:2211.03855$]$

On detection of topological order by observing modular transformations on the ground state:

• Zhuan Li, Roger S. K. Mong, Detecting topological order from modular transformations of ground states on the torus, Phys. Rev. B 106 (2022) 235115 $[$doi:10.1103/PhysRevB.106.235115, arXiv:2203.04329$]$

See also:

• Liang Kong, Topological Wick Rotation and Holographic Dualities, talk at CQTS (Oct 2022) $[$pdf$]$