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?
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2 Answers
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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 $Z$.
answered Jan 27, 2021 at 19:28
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2$\begingroup$ Where can I learn more about the example in the last paragraph? $\endgroup$ Commented Feb 12, 2021 at 16:44
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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
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2$\begingroup$ as a physicist I should mention that the expectation that "most QFTs are non-lagrangian" 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
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2$\begingroup$ @ArunDebray: I'm also interested in your example in the last paragraph, however I suspect that the non-uniqueness exhibited there is just an instance of integrating out (most of) the pair of CS gauge fields, and as such is a garden variety homotopy-equivalence/duality of apparently inequivalent field theory descriptions. (The interesting feature in this case, is that the resulting theory is again of Chern-Simons type, albeit apparently for a discrete gauge group.) $\endgroup$ Commented Mar 3, 2021 at 17:13
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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
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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.
