TITLE: Lagrangian of Reshetikhin-Turaev TFT's
QUESTION [13 upvotes]: One of the results from the Reshetikhin-Turaev package is that given a modular tensor category $\mathscr{C}$ one can construct a TFT $Z$. In the case where $\mathscr{C}$ is the category of positive energy representations of the loop group, it is accepted that $Z$ coincides with the Chern-Simons theory. Now, the latter has a well-known Lagrangian. Is there a way to recover this Lagrangian from the RT construction? More generally, can one always have a Lagrangian for any of these RT theories? How are these constructed?

REPLY [8 votes]: It's an open conjecture by Moore and Seiberg (originally in the context of conformal field theory) that every MTC can be obtained from Chern-Simons theory of simple Lie groups with known constructions. The constructions in the CFT context include orbifolding, coset and chiral algebra extension, which correspond to gauging symmetry and condensation for MTCs. Note that their original statement is that every MTC is the Chern-Simons theory of compact Lie groups, which I believe is equivalent to the above. If the conjecture is true, there will be a Lagrangian description for each Reshetikhin-Turaev TQFT, although not unique.
There have been some math/physics works on potential "exotic" MTCs, basically counter-examples of the Moore-Seiberg conjecture. Two examples were considered in https://arxiv.org/abs/0710.5761, but one quickly shot down, and the other, quantum double of the even sectors of the Haagerup subfactor, still remains.