nLab
modular tensor category

Idea

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 this is also a ribbon category and such that “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 representative of simple objects $U_i$, $U_j$, the matrix

$$(s^{ij})=(U_i-\mathrm{circle}\mathrm{threading}\mathrm{through}\mathrm{the}{U}_j-\mathrm{circle})$$(s^{i j}) = (U_i-circle threading through the U_j-circle)

is non-degenerate.

Here on the right what is means is the diagram in the modular tensor category made from the identityie morphisms, the duality morphisms and the braiding morphism on the objects $U_i$ and $U_j$ that looks lik a fuigure-8 with one circle threading through the other, and this diagram is interpreted as an element in the endormorphism 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, $ℂ$-linear (i.e. ${\mathrm{Vect}}_{ℂ}$ enriched category), semisimple 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

  $$K(C){\otimes }_{\mathbb{Z}}\stackrel{\simeq }{\to }\mathrm{End}({\mathrm{Id}}_C)$$K(C) \otimes_{\mathbb{Z}} \stackrel{\simeq}{\to} End(Id_C)

  where

  $$[U]\mapsto {\alpha }_U$$[U] \mapsto \alpha_U

  where the transformation ${\alpha }_U$ is given on the simple object $V$ by

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

  (on the right we use string diagram notation)

Examples

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

References

General

section 2.1 of

• J. Fuchs, I. Runkel, C. Schweigert, TFT construction of RCFT correlators I: partition functions (arXiv)

Relation to modular tensor categories

A brief survey of results is in

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

See vertex operator algebra for details.

Examples

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