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?


I don't think there's a way to extract a Lagrangian from the Reshetikhin-Turaev construction. There's certainly not a unique way to do so.

Physicists believe that most QFTs are "non-Lagrangian," 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 Reshetikhin-Turaev construction is a very general construction of 3d anomalous TFTs: Bartlett-Douglas-Schommer-Pries-Vicary show that it sees all once-extended 3d TFTs for a particular target 2-category. So I'd expect that unless there's some good reason to believe otherwise, there are 3d TFTs produced by the Reshetikhin-Turaev construction that are not Chern-Simons 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 Chern-Simons theories.

One thing which we do have examples of is 3d TFTs admitting multiple inequivalent descriptions as Chern-Simons 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 Chern-Simons 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; Chern-Simons 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

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