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
algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
A modular tensor category is, roughly, a braided monoidal category that encodes the topological structure underlying a rational 2-dimensional conformal field theory. In other words, it is a basis-independent formulation of Moore-Seiberg data.
It is in particular a fusion category that is also a ribbon category such that the “modularity operation” is non-degenerate (this is what the name “modular tensor category” comes from):
this means that for $i,j \in I$ indices for representatives of simple objects $U_i$, $U_j$, the matrix
is non-degenerate.
Here on the right what is meant is the diagram in the modular tensor category made from the identity morphisms, the duality morphisms and the braiding morphism on the objects $U_i$ and $U_j$ that looks like a figure-eight with one circle threading through the other, and this diagram is interpreted as an element in the endomorphism space of the tensor unit object, which in turn is canonically identified with the ground field.
In the description of 2-dimensional conformal field theory in the FFRS-formalism it is manifestly this kind of modular diagram that encodes the torus partition function of the CFT. This explains the relevance of modular tensor categories in the description of conformal field theory.
Since 2-dimensional conformal field theory is related by a holographic principle to 3-dimensional TQFT, modular tensor categories also play a role there, which was in fact understood before the full application in conformal field theory was: in the Reshetikhin-Turaev model.
A modular tensor category is a category with the following long list of extra structure.
needs to be put in more coherent form, just a stub
it is an abelian category, $\mathbb{C}$-linear (i.e. $Vect_{\mathbb{C}}$ enriched category), semisimple category monoidal category (tensor category)
the tensor unit is a simple object, $I$ a finite set of representatives of isomorphism classes of simple objects
ribbon category, in particular objects have duals
modularity a non-degeneracy condition on the braiding given by an isomorphism of algebras
where
where the transformation $\alpha_U$ is given on the simple object $V$ by
(on the right we use string diagram notation)
Many modular tensor categories arise as representation categories of vertex operator algebras (Huang 2005, Sec. 1; Huang 2008; Huang, Lepowski & Zhang 2014 see EGNO 15, Sec. 8.27.6), hence of chiral fields of 2d conformal field theories.
(For logarithmic CFTs one still gets braided tensor categories, see Creutzig, Lentner & Rupert 2021).
In this case the monoidal- and the braided structure (hence the modular tensor structure) on the underlying representation category is entirely fixed by the space of conformal blocks of the 2d CFT on the Riemann sphere (the “genus zero conformal blocks”).
This may be found highlighted in EGNO 15, p. 266, Runkel, Sec. 4.3. The essentially equivalent fact that the genus=0 conformal blocks already determine the modular functor of the CFT is proven in Andersen & Ueno 2012.
A database of examples is given by (Gannon & Höhn).
Original article:
Review in the context of the Reshetikhin-Turaev construction of modular functors:
Review in the context of 2d CFT/VOA
Ingo Runkel, Algebra in Braided Tensor Categories and Conformal Field Theory (pdf, pdf)
and with focus on relation to braid representations:
Construction of modular tensor categories from vertex operator algebras:
Yi-Zhi Huang, Vertex operator algebras, the Verlinde conjecture and modular tensor categories, Proc. Nat. Acad. Sci. 102 (2005) 5352-5356 $[$arXiv:math/0412261, doi:10.1073/pnas.0409901102$]$
Yi-Zhi Huang, Rigidity and modularity of vertex tensor categories, Communications in Contemporary Mathematics 10 supp01 (2008) 871-911 $[$arXiv:math/0502533, doi:10.1142/S0219199708003083$]$
Yi-Zhi Huang, James Lepowsky, Lin Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules, In: Bai, Fuchs, Huang, Kong, Runkel, Schweigert (eds.), Conformal Field Theories and Tensor Categories Mathematical Lectures from Peking University. Springer (2014) $[$arXiv:1012.4193, doi:10.1007/978-3-642-39383-9_5$]$
Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, Section 8.27.6 of: Tensor Categories, AMS Mathematical Surveys and Monographs 205 (2015) [ISBN:978-1-4704-3441-0]
brief survey:
and generalization to logarithmic VOAs:
A list of examples (with an emphasis on representation categories of rational vertex operator algebras) is in
Classification results
On number theoretic aspects of modular tensor categories:
Discussion of modular tensor categories in quantum field theory (3d TQFT and 2d CFT, as well as their relation via the CS/WZW correspondence) includes the following.
A general survey of the literature is in
See also
More specific discussion in the context of 2d CFT is in
Jürgen Fuchs, Ingo Runkel, Christoph Schweigert, TFT construction of RCFT correlators I: partition functions (arXiv:hep-th/0204148)
(for more along these lines see at FRS formalism)
Review of construction of MTCs from vertex operator algebras is in
Discussion from the point of view of the cobordism hypothesis (see also the discussion at fusion category) is in
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 May 7, 2023 at 04:57:01. See the history of this page for a list of all contributions to it.