# nLab unitary fusion category

Contents

### Context

#### Monoidal categories

monoidal categories

# Contents

## Idea

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

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## 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

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 may be:

• 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):

• 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$]$

Relation to ZX-calculus:

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:

Exposition and review:

• Sachin Valera, A Quick Introduction to the Algebraic Theory of Anyons, talk at CQTS Initial Researcher Meeting (Sep 2022) $[$pdf$]$

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