FQFT and cohomology
In this is also called a homological conformal field theory. The passage to homology forgets the conformal structure. The refinement of this concept from homology groups to chain complexes is called TCFT.
Z : H_\bullet(Bord) \to Ab \,.
For 2-dimensional cobordisms with closed boundary HCFT has been considered in
A detailed treatment in involving arbitrary sets of branes is in section 2 of