category with duals (list of them)
dualizable object (what they have)
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.
this means that for indices for representatives of simple objects , , the matrix
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 and 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
the tensor unit is a simple object, 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 the transformation is given on the simple object by
(on the right we use string diagram notation)
A review is for instance in section 2.1 of (Fuchs-Runkel-Schweigert 02).
A general survey of the literature is in
More specific discussion in the context of 2d CFT is in
(for more along these lines see at FRS formalism)
Review of construction of MTCs from vertex operator algebras is in