One of the results from the ReshetikhinTuraev 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 ChernSimons theory. Now, the latter has a wellknown 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?
2 Answers
I don't think there's a way to extract a Lagrangian from the ReshetikhinTuraev construction. There's certainly not a unique way to do so.
Physicists believe that most QFTs are "nonLagrangian," meaning that these QFTs cannot be produced by quantizing classical field theories. To make this statement precise, one has to say what exactly is meant by QFT, and this is out of reach for now. But what this means is that if there were some fully general classification of QFTs in a given dimension, people expect that QFTs studied with a Lagrangian and path integral don't give you all of them.
The ReshetikhinTuraev construction is a very general construction of 3d anomalous TFTs: BartlettDouglasSchommerPriesVicary show that it sees all onceextended 3d TFTs for a particular target 2category. So I'd expect that unless there's some good reason to believe otherwise, there are 3d TFTs produced by the ReshetikhinTuraev construction that are not ChernSimons theories. However, I do not know of an example asserting this, so it's in principle possible that there could be a proof that all RT theories arise from ChernSimons theories.
One thing which we do have examples of is 3d TFTs admitting multiple inequivalent descriptions as ChernSimons theories. For example, if $G = \mathbb Z/2$, the possible choices of level are given by $H^4(B\mathbb Z/2;\mathbb Z)\cong\mathbb Z/2$; let $Z$ be the ChernSimons theory for the nonzero level. We could also consider $G = \mathrm U(1)\times\mathrm U(1)$, whose levels are given by $2\times 2$ matrices; ChernSimons theory for the level
$$\begin{pmatrix}2 & 0\\0 & 2\end{pmatrix}$$
is isomorphic to $Z$.

2$\begingroup$ Where can I learn more about the example in the last paragraph? $\endgroup$ Commented Feb 12, 2021 at 16:44

2$\begingroup$ I hope you don't mind a cheeky remark, I can't resist re: your second paragraph ... to make this statement precise, one has to say what exactly is meant by a physicist, and this is out of reach for now ... $\endgroup$ Commented Mar 3, 2021 at 16:36

2$\begingroup$ as a physicist I should mention that the expectation that "most QFTs are nonlagrangian" is only based on physicist failures at writing down lagrangians for certain examples. In fact, BLG and ABJM theories were widely conjectured to be nonlagrangian... and then BLG was written down. The loophole in those cases is that the coupling constant takes values in integers, because the coupling is a CS level. $\endgroup$ Commented Mar 3, 2021 at 17:09

2$\begingroup$ @ArunDebray: I'm also interested in your example in the last paragraph, however I suspect that the nonuniqueness exhibited there is just an instance of integrating out (most of) the pair of CS gauge fields, and as such is a garden variety homotopyequivalence/duality of apparently inequivalent field theory descriptions. (The interesting feature in this case, is that the resulting theory is again of ChernSimons type, albeit apparently for a discrete gauge group.) $\endgroup$ Commented Mar 3, 2021 at 17:13

1$\begingroup$ @ArunDebray ... in other words, I expect the example in the last paragraph is an instance of the construction here: arxiv.org/pdf/0903.0995.pdf $\endgroup$ Commented Mar 3, 2021 at 17:16
It's an open conjecture by Moore and Seiberg (originally in the context of conformal field theory) that every MTC can be obtained from ChernSimons 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 ChernSimons 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 ReshetikhinTuraev TQFT, although not unique.
There have been some math/physics works on potential "exotic" MTCs, basically counterexamples of the MooreSeiberg 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.