functorial quantum field theory
Reshetikhin?Turaev model? / Chern-Simons theory
FQFT and cohomology
The Reshetikhin-Turaev construction is the FQFT construction of a 3d TQFT from the data of a modular tensor category $\mathcal{C}$. It is something like the “square root” of the Turaev-Viro model on $\mathcal{C}$.
In the case that $C$ is a category of positive energy representations of a loop group $\Omega G$ of a Lie group $G$, then this algebraically defined QFT is thought to be the result of quantization of Chern-Simons theory over the group $G$.
The Reshetikhin-Turaev model is a boundary field theory of the 4d TQFT Crane-Yetter model (Barrett&Garci-Islas&Martins 04, theorem 2) Related discussion is in Freed “4-3-2 8-7-6”.
The RT-construction for group $G$ is expected to be the FQFT of $G$-Chern-Simons theory, though a fully explicit proof of this via quantization is currently not in the literature.
See at quantization of Chern-Simons theory for more on this.
The Fuchs-Runkel-Schweigert-construction builds from the RT-construction explicitly the rational 2-dimensional 2d CFT boundary theory (see at holographic principle).
Original articles include
A standard textbook account is
(See the dedicated page Help me! I'm trying to understand Bakalov and Kirillov for help with understanding the computations in this book.)
See also
Discussion that relates the geometric quantization of $G$-Chern-Simons theory to the Reshetikhin-Turaev construction of a 3d-TQFT from the modular tensor category induced by $G$ is in
and references cited there.
The relation to the Crane-Yetter model was discussed in