# nLab modular tensor category

Contents

### Context

#### Monoidal categories

monoidal categories

# Contents

## 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).

## References

### General

Original article:

Review in the context of 2d CFT/VOA

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:

and generalization to logarithmic VOAs:

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

Partial classification:

### 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

More specific discussion in the context of 2d CFT is in

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

### Anyonic topological order in terms of braided fusion categories

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 may be:

• 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):

Further discussion (all without attribution):

• 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$]$

• Willie Aboumrad, Quantum computing with anyons: an F-matrix and braid calculator $[$arXiv:2212.00831$]$

Relation to ZX-calculus:

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:

Exposition and review:

• Sachin Valera, A Quick Introduction to the Algebraic Theory of Anyons, talk at CQTS Initial Researcher Meeting (Sep 2022) $[$pdf$]$

• Liang Kong, Topological Wick Rotation and Holographic Dualities, talk at CQTS (Oct 2022) $[$pdf$]$