functorial quantum field theory
Reshetikhin?Turaev model? / Chern-Simons theory
FQFT and cohomology
A homological TQFT is a representation of the cobordism category that depends only on the homology of the hom-spaces.
In $d = 2$ 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.
Write $Bord$ for any given cobordism category, regarded as a Top-enriched category. A homological TQFT is a symmetric monoidal Ab-enriched functor
For 2-dimensional cobordisms with closed boundary HCFT has been considered in
A detailed treatment in $d = 2$ involving arbitrary sets of branes is in section 2 of