nLab modular tensor category

Idea

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

$$(s^{i j}) = (U_i\text{-circle threading through the} \, U_j\text{ -circle})$$

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.

Definition

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

• fusion category

• braided monoidal category

• ribbon category, in particular objects have duals

• modularity a non-degeneracy condition on the braiding given by an isomorphism of algebras

  $$K(C) \otimes_{\mathbb{Z}} \stackrel{\simeq}{\to} End(Id_C)$$

  where

  $$[U] \mapsto \alpha_U$$

  where the transformation $\alpha_U$ is given on the simple object $V$ by

  $$\alpha_U(V) = \text{straight } V\text{-line encircled by } U\text{-loop}$$

  (on the right we use string diagram notation)

Examples

Rep categories of VOAs

General

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).

Relation to conformal blocks

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.

Examples

A database of examples is given by (Gannon & Höhn).

Related concepts

• tensor category

References

General

Original article:

• Vladimir Turaev, Modular categories and 3-manifold invariants, International Journal of Modern Physics B Vol. 06, No. 11n12, pp. 1807-1824 (1992) (doi:10.1142/S0217979292000876)

Review in the context of the Reshetikhin-Turaev construction of modular functors:

• Bojko Bakalov, Alexander Kirillov, Modular Tensor Categories, Ch. 3 in: Lectures on tensor categories and modular functors, University Lecture Series 21, Amer. Math. Soc. (2001) [pdf,web, ams:ulect/21]

Review in the context of 2d CFT/VOA

• Fuchs-Runkel-Schweigert 02, Sec. 2.1

• Ingo Runkel, Algebra in Braided Tensor Categories and Conformal Field Theory (pdf, pdf)

and with focus on relation to braid representations:

• Colleen Delaney, Lecture notes on modular tensor categories and braid group representations, 2019 (pdf, pdf)

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:

• Jürgen Fuchs, Christoph Schweigert, Simon Wood, Yang Yang, pp. 5 of: Algebraic structures in two-dimensional conformal field theory, Encyclopedia of Mathematical Physics [arXiv:2305.02773]

and generalization to logarithmic VOAs:

• Thomas Creutzig, Simon Lentner, Matthew Rupert, Characterizing braided tensor categories associated to logarithmic vertex operator algebras $[$arXiv:2104.13262$]$

A list of examples (with an emphasis on representation categories of rational vertex operator algebras) is in

• Terry Gannon, Gerald Höhn, Hiroshi Yamauchi, The online database of Vertex Operator Algebras and Modular Categories

Classification results

• Eric Rowell, Richard Stong, Zhenghan Wang, On classification of modular tensor categories, Comm. Math. Phys. 292 2 (2009) 343–389 [arXiv:0712.1377, doi:10.1007/s00220-009-0908-z]

On number theoretic aspects of modular tensor categories:

• Orit Davidovich, Tobias Hagge, Zhenghan Wang, On Arithmetic Modular Categories, arXiv preprit, 2013 [arXiv:1305.2229]

Relation to 3dCS/2dWZW quantum field theory

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

• Eric Rowell, From quantum groups to Unitary modular tensor categories, Contemporary Mathematics 2005 (arXiv:math/0503226)

See also

• Victor Ostrik, Tensor categories in conformal field theory, talk notes 2011 (pdf slides)

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

• James Lepowsky, From the representation theory of vertex operator algebras to modular tensor categories in conformal field theory (pdf)

Discussion from the point of view of the cobordism hypothesis (see also the discussion at fusion category) is in

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

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) [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]

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

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$]$