modular tensor category



Monoidal categories

monoidal categories

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Internal monoids



In higher category theory



A modular tensor category is roughly a 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,jIi,j \in I indices for representatives of simple objects U iU_i, U jU_j, the matrix

(s ij)=(U i-circle threading through theU j -circle) (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 iU_i and U jU_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 Vect_{\mathbb{C}} enriched category), semisimple category tensor category

  • the tensor unit is a simple object, II 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) End(Id C) K(C) \otimes_{\mathbb{Z}} \stackrel{\simeq}{\to} End(Id_C)


    [U]α U [U] \mapsto \alpha_U

    where the transformation α U\alpha_U is given on the simple object VV by

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

    (on the right we use string diagram notation)


Modular tensor categories arise as representation categories of vertex operator algebras (see there for more details). A database of examples is given by (GannonHöhn).



A review is for instance in section 2.1 of (Fuchs-Runkel-Schweigert 02).

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

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

See also

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


Last revised on March 20, 2016 at 05:13:19. See the history of this page for a list of all contributions to it.