nLab fusion category

Definition

Definition

A fusion category is a rigid semisimple linear (Vect-enriched) monoidal category (“tensor category”), with only finitely many isomorphism classes of simple objects, such that the endomorphisms of the unit object form just the ground field $k$.

Often one also assumes a braiding and speaks of a braided fusion category.

Examples

The name “fusion category” comes from the central examples of structures whose canonical tensor product is called a “fusion product”, notably representations of loop groups and of Hopf algebras and of vertex operator algebras.

Some of the easiest examples are:

• Representations of a finite group or (finite super-group)
• For a given finite group $G$ and a 3-cocycle on $G$ with values in (the multiplicative group of units of) a field $k$ (an element of $H^3(G,k^\times)$), take $G$-graded vector spaces with the cocycle as associator.

Properties

Relation to weak Hopf algebras

Under Tannaka duality, every fusion category $C$ arises as the representation category of a weak Hopf algebra. (Ostrik)

Relation to pivotal and spherical categories

Fusion categories were first systematically studied by Etingof, Nikshych and Ostrik in On fusion categories. This paper listed many examples and proved many properties of fusion categories. One of the important conjectures made in that paper was the following:

Conjecture (Etingof, Nikshych, and Ostrik)

Every fusion category admits a pivotal structure.

Providing a certain condition is satisfied, a pivotal structure on a fusion category can be shown to correspond to a ‘twisted’ monoidal natural endotransformation of the identity functor on the category, where the twisting is given by the pivotal symbols.

Relation to extended 3d TQFT

Given the data of a fusion category one can build a 3d extended TQFT by various means. This is explained by the fact, see below, that fusion categories are (probably precisely) the fully dualizable objects in the 3-category $MonCat$ of monoidal categories. By the homotopy hypothesis this explains how they induce 3d TQFTs.

Definition

Write $MonCat_{bim}$ for the (infinity,3)-category which has as

• objects monoidal categories,

• morphism bimodules of these,

• and so on.

Proposition

With its natural tensor product, $MonCat$ is a symmetric monoidal (infinity,3)-category.

Proposition

A monoidal category which is fusion is fully dualizable in the (infinity,3)-category $MonCat_{bim}$, def. .

This is due to (Douglas & Schommer-Pries & Snyder 13).

Remark

Via the cobordism theorem prop. implies that fusion categories encode extended TQFTs on 3-dimensional framed cobordisms, while their $O(3)$-homotopy fixed points encode extended 3d TQFTs on general (not framed) cobordisms.

These 3d TQFTs hence arise from similar algebraic data as the Turaev-Viro model and the Reshetikhin-Turaev construction, however there are various slight differences. See (Douglas & Schommer-Pries & Snyder 13, p. 5).

Suggestions

Here are three things such that it’d be awesome if they were sorted out on this page:

1. Kuperberg’s theorem saying that abelian semisimple implies linear over some field. Finite, connected, semisimple, rigid tensor categories are linear

2. Some correct version of the claim that abelian semisimple is the same as idempotent complete and nondegenerate. Math Overflow question

3. Good notation distinguishing simple versus absolutely simple? (is $End(V) = k$ or just $V$ has no nontrivial proper subobjects).

Together 1 and 2 let you go between the two different obvious notions of semisimple which seem a bit muddled here when I clicked through the links.

Related concepts

• fusion category

  • unitary fusion category

  • fusion 2-category

  • fusion ring, Frobenius-Perron dimension

• pivotal category

• spherical category

• modular tensor category

• Deligne's theorem on tensor categories

• graded fusion category

• Tambara-Yamagami category

References

General

Original articles:

• Pavel Etingof, Dmitri Nikshych, Victor Ostrik, On fusion categories, Annals of Mathematics Second Series, Vol. 162, No. 2 (Sep., 2005), pp. 581-642 (arXiv:math/0203060, jstor:20159926)

• Pavel Etingof and Damien Calaque, Lectures on tensor categories.

• Michael Müger, Tensor categories: A selective guided tour.

• Pavel Etingof, D. Nikshych and V. Ostrik, Fusion categories and homotopy theory , Quantum Topology, 1(2010), 209-273. (Earlier version available as ArXiv:0909.3140

• Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, On braided fusion categories I, Selecta Mathematica. New Series 16 (2010), no. 1, 1–119 (doi:10.1007/s00029-010-0017-z)

Further review:

• Bruce Bartlett, chapter 6 of

  On unitary 2-representations of finite groups and topological quantum field theory.

On the Tannaka duality to weak Hopf algebras:

• Takahiro Hayashi, A canonical Tannaka duality for finite seimisimple tensor categories (arXiv:math/9904073)

• Victor Ostrik, Module categories, weak Hopf algebras and modular invariants (arXiv:math/0111139)

The relation to 3d TQFT clarified via the cobordism hypothesis:

• Chris Douglas, Chris Schommer-Pries, Noah Snyder, The Structure of Fusion Categories via 3D TQFTs (talk pdf)

• Chris Douglas, Chris Schommer-Pries, Noah Snyder, Dualizable tensor categories (arXiv:1312.7188)

and for the case of modular tensor categories:

• Bruce Bartlett, Christopher Douglas, Chris Schommer-Pries, Jamie Vicary, Modular categories as representations of the 3-dimensional bordism 2-category (arXiv:1509.06811)

Discussion in terms of skein relations:

• Anup Poudel, Sachin J. Valera, Skein-Theoretic Methods for Unitary Fusion Categories [arXiv:2008.07129]

See also:

• Math Overflow, Why are fusion categories interesting? .

On a notion of fusion 2-categories:

• Christopher L. Douglas, David J. Reutter, Fusion 2-categories and a state-sum invariant for 4-manifolds [arXiv:1812.11933]

• Thibault Decoppet, Matthew Yu. Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories. (2023) [arXiv:2306.08117]

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) $[$

  arXiv:2004.06282, doi:10.1016/j.aop.2021.168471$]$

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. (20223) $[$arXiv:2206.13563, doi:10.1142/S0129055X23500010$]$

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$]$