category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
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 $i,j \in I$ 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 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)
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
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