nLab
modular functor

Idea

The local data for a CFT in dimension $d$ allows to assign to each $d$-dimensional cobordism $\Sigma$ a vector space of “possible correlator”s: those functions on the space of conformal structures on $\Sigma$ that have the correct behaviour to qualify as the (chiral) correlator of a CFT. This is called a space of conformal blocks $\mathrm{Bl}(\Sigma )$. This assignment is functorial under diffeomorphism. The corresponding functor is called a modular functor.

To get an actual collection of correlators one has to choose from each space of conformal blocks $\mathrm{Bl}(\Sigma )$ an element such that these choices glue under composition of cobordism: such that they solve the sewing constraints.

References

• Bojko Bakalov, Alexander Kirillov, Lectures on tensor categories and modular functor (web)