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.

Simple examples:

Example

(graded vector spaces)
For

• $G$ a finite group,

• $\mathbb{K}$ a field,

the category of finite-dimensional $G$-graded $\mathbb{K}$-vector spaces

$$Vect_{G}^{fdim}$$

is a fusion category, with tensor product given by the tensor product of vector spaces and the binary operation of the group:

$$V_g \otimes W_h \;\; \coloneqq \;\; (V \otimes W)_{g \cdot h} \,.$$

More generally, for

• $\omega \,\in\, H^3_{Grp}(G, \mathbb{K}^\times)$ a 3-cocycle in the group cohomology of $G$ with coefficients in the group of units $\mathbb{K}^\times$,

the above construction but with associator multiplied by the 3-cocycle applied to the $G$-degrees of the 3 factors is again a fusion category

$$Vect_{G, \omega}^{fdim} \,.$$

(Etingof, Nikshych & Ostrik 2005, item 1. on p. 584)

Example

For $\mathbb{K}$ a field and $G$ a finite group (or finite super-group), whose order is relatively prime to the characteristic of $\mathbb{K}$, then the category of representations $Rep(G, \mathbb{K})$ is a fusion category.

(Etingof, Nikshych & Ostrik 2005, item 2. on p. 584)

Properties

Relation to weak Hopf algebras

Under Tannaka duality, every fusion category $C$ arises as the representation category of a weak Hopf algebra (Ostrik). However, this does not mean that every fusion category admits a fiber functor to the category of vector spaces $\text{Vect}= k-Mod$.

Given any multi-fusion category $C$, one can always construct a fiber functor $F:C\to RMod$ for $R$ the algebra spanned by a basis of orthogonal idempotents $\{v_i\}_{i\in I}$ for $I$ the equivalence classes of simple objects of $C$. This functor is referred to in some sources as a generalized fiber functor. The endomorphisms of this functor then give a weak Hopf algebra that represents $C$. In Hayashi 1999 (see there for the relevant definitions), this is computed as a coend, where one has that $C\cong Rep(A)$ for $A= \text{coend}(F^*\otimes F: C^{op} \times C \to Bmd(E))$, where $E=\dot R\otimes R$ is equipped with a coalgebra structure

$$\Delta(\dot\lambda \mu) = \sum_{\nu\in I} \dot\lambda \nu\otimes \dot\nu \mu$$

$$\epsilon (\dot\lambda \mu) = \delta_{\lambda,\mu}$$

It is important to note that, generally speaking, $C$ may admit other fiber functor to different module categories $RMod$, as is the case for fusion categories of the form $Rep(H)$ for $H$ a Hopf algebra, which admits both the fiber functor described above, as well as a fiber functor to $\text{Vect}$.

Relation to pivotal and spherical categories

Fusion categories were first systematically studied by Etingof, Nikshych & Ostrik 2005. 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.

Tube (Ocneanu) algebra

Every fusion category $\mathcal{C}$ has a tube algebra, which encodes an action by conjugation of $\mathcal{C}$ on itself. See there for more.

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).

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

Motivation as generalization of (the representation theory of) finite groups:

• Greg Kuperberg (2009) [MO:a/6183]

Original articles:

• Pavel Etingof, Dmitri Nikshych, Victor Ostrik, On fusion categories, Annals of Mathematics Second Series 162 2 (2005) 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 1 (2010) 1–119 [arXiv:0906.0620, 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 Schommer-Pries, The Structure of Fusion Categories via 3D TQFTs (2001) [DSSFusionSlides.pdf]

• Chris Douglas, Chris Schommer-Pries, Noah Snyder, Dualizable tensor categories, Memoirs of the AMS 268 1308 (2021) [arXiv:1312.7188, ams:memo-268-1308]

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? [MO:q/6180]

Further work on their classification using finite groups:

• Agustina Czenky: Diagramatics for cyclic pointed fusion categories [arXiv:2404.08084]

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]

On fusion categories invertible with respect to the Deligne tensor product:

• Sean Sanford, Noah Snyder: Invertible Fusion Categories [arXiv:2407.02597]

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) [doi:10.1103/RevModPhys.80.1083, arXiv:0707.1888]

• 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]

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]

• K. Beer et al.: From categories to anyons: a travelogue [arXiv:1811.06670]

• Colleen Delaney, A categorical perspective on symmetry, topological order, and quantum information (2019) [uc:5z384290>, pdf]

• 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) [doi:10.1103/PhysRevResearch.3.033110, arXiv:2102.05677]

• Zhuan Li, Roger S.K. Mong, Detecting topological order from modular transformations of ground states on the torus [arXiv:2203.04329]

• Liang Kong, Zhi-Hao Zhang: An invitation to topological orders and category theory [arXiv:2205.05565]

• 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]

• Steven H. Simon, Topological Quantum, Oxford University Press (2023) [ISBN:9780198886723, pdf, webpage]

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]

• Sunjin Choi, Dongmin Gang, Hee-Cheol Kim: Infrared phases of 3D Class R theories, J. High Energ. Phys. 2022 151 (2022) [doi:10.1007/JHEP11(2022)151, arXiv:2206.11982]

• Federico Bonetti, Sakura Schäfer-Nameki, Jingxiang Wu, $MTC[M_3,G]$: 3d Topological Order Labeled by Seifert Manifolds [arXiv:2403.03973]

• Hisham Sati, Urs Schreiber: Anyons on M5-Probes of Seifert 3-Orbifolds [arXiv:2411.16852]

• Hisham Sati, Urs Schreiber: Engineering of Anyons on M5-Probes via Flux-Quantization [arXiv:2501.17927]

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]