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?
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 $Z$.

1$\begingroup$ Where can I learn more about the example in the last paragraph? $\endgroup$ – Konrad Waldorf Feb 12 at 16:44

$\begingroup$ @KonradWaldorf unfortunately I don't know a good reference. I learned this fact from a combination of physics papers which don't prove the equivalence. Sorry about that. $\endgroup$ – Arun Debray Feb 12 at 18:30