FQFT and cohomology
The local data for a CFT in dimension allows to assign to each -dimensional cobordism a vector space of “possible correlator”s: those functions on the space of conformal structures on that have the correct behaviour to qualify as the (chiral) correlator of a CFT. This is called a space of conformal blocks . 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 an element such that these choices glue under composition of cobordism: such that they solve the sewing constraints.
A generalization of the modular functors is in