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 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.
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:
(graded vector spaces)
For
$G$ a finite group,
$\mathbb{K}$ a field,
the category of finite-dimensional $G$-graded $\mathbb{K}$-vector spaces
is a fusion category, with tensor product given by the tensor product of vector spaces and the binary operation of the group:
More generally, for
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
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.
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
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 $\text{Vect}$.
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:
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.
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.
Write $MonCat_{bim}$ for the (infinity,3)-category which has as
objects monoidal categories,
morphism bimodules of these,
and so on.
With its natural tensor product, $MonCat$ is a symmetric monoidal (infinity,3)-category.
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).
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).
Here are three things such that it’d be awesome if they were sorted out on this page:
Kuperberg’s theorem saying that abelian semisimple implies linear over some field. Finite, connected, semisimple, rigid tensor categories are linear
Some correct version of the claim that abelian semisimple is the same as idempotent complete and nondegenerate. Math Overflow question
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.
fusion category
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.
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:
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:
Discussion in terms of skein relations:
See also:
Further work on their classification using finite groups:
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:
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, $MTC[M_3,G]$: 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 July 4, 2024 at 16:52:01. See the history of this page for a list of all contributions to it.