# Tensor Categories and Modular Functors

## Main Objects and Result

• Tensor categories: Abelian categories with associative tensor product, unit objects, with some additional properties.
• Moreover, semisimple and the commutativity condition is weaken such that the commutativity isomorphism $$\sigma_{VW}:V\otimes W \rightarrow W\otimes V$$ exists but $$\sigma^2 = id$$ is not required.
• In particular, Modular tensor categories(MTC)
• 3D TQFT (defined as functors from the category of 3-manifold cobordisms to that of vector spaces)
• 2D Modular functor:
• topological: functors from 2-manifold with boundary to a finite-dimensional vector space, with additional data assigned to the boundaries, and assignment of an isomorphism between the corresponding vector spaces to every homotopy class of homeomorphisms between such manifolds. Also, vector spaces behave nicely under gluing.
• complex-analytic: a collection of vector bundles with flat connection on the moduli spapce of complex curbes with marked points + gluing axiom describing the behavior of these flat connections near the boundary of the moduli space in Deligne-Mumford compactification. Appear naturally in CFTs, every rational CFT gives rise to a complex-analytic modular functor. Example: WZW model.

Result: The notions of a modular tensor category, 3d TQFT and 2d MF are equivalent.