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I have read that the worldvolume theory of the M5 brane is a $(2, 0)$ superconformal field theory (SCFT). But I have also learnt from talks that the $(2, 0)$ theory lacks a Lagrangian description.

There are candidate actions for the M5 brane worldvolume theory, but if one would take any such candidate action, would it contradict the above statement (or is there some caveat about the action being well defined without the Lagrangian being well defined)?

Also, would pushing to get a worldvolume action for the M5 brane OR advances in understanding $(2, 0)$ SCFTs imply some deeper understanding of M theory?

Finally, compactifying a six-dimensional $(2, 0)$ SCFT on a suitable Riemann surface can yield more "familiar" theories, which do have Lagrangian descriptions. How does one explain the fact that a compactification of apparently a non-Lagrangian theory can yield a Lagrangian theory in lower dimensions?

I'd also appreciate if someone can share any pedagogical or at least introductory references about these topics, particularly the advances in the $(2, 0)$ theories.

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    $\begingroup$ This is a very interesting question with lots of dimensions and subtleties and implicit errors in the question etc. First of all, (2,0) is just the low energy, deep infrared limit of the dynamics of coincident M5-branes, not "everything" about M5-branes in a generic situation. Second, the non-existence of a Lagrangian description is a conjecture, or a claim about the invalidity of the usual simplest Lagrangians we could guess. The problem is with the self-duality of the 3-form. $\endgroup$ – Luboš Motl Jul 28 '15 at 12:50
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    $\begingroup$ Also, there is no contradiction between the non-existence of a Lagrangian of a full theory, and the existence of a Lagrangian for a dimensionally reduced (compactified, taken to the limit) theory. And yes, the behavior of M5-branes must have some more contrived, bootstrap-like, cubic and higher-order etc. structures that are seeds of those in all of M-theory. There are some hints in the literature, but no complete answer. Each paper about (2,0), and there are lots of them, addresses these questions to one extent or another but there's no known complete answer. $\endgroup$ – Luboš Motl Jul 28 '15 at 12:51
  • $\begingroup$ Thanks for the response @LubošMotl. I don't mean to imply that there is a contradiction between the non-existence of a Lagrangian for a full theory and the existence of one for a compactified theory. But, as a student I only studied the Kaluza-Klein reduction, and in all the standard examples, there is an algorithm connecting the dimensionally reduced action to the uncompactified action. How is the compactification here different? Again, maybe this is a wrong question. $\endgroup$ – leastaction Jul 30 '15 at 23:08
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    $\begingroup$ I am no expert, but I found Witten's paper 'Conformal Field Theory in Four and Six Dimensions' to be particularly helpful in understanding the 6d (2,0) SCFT. Here he discusses the counter-intuitive dimensional reduction of the 6d theory to 4d N=4 Super Yang-Mills, in particular how integration over the sixth dimension is a quantum operation which gives a factor of 1/R instead of R, and how this leads to S-duality in the 4d theory. $\endgroup$ – Meer Ashwinkumar Jan 12 '16 at 4:10
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    $\begingroup$ It also explains why the usual action one would expect for such a theory does not work, due to the self-duality of the three-form field strength of the two-form field in the theory, as Luboš explained. Witten's other papers which are related to this topic might also be helpful, such as 'Some Comments on String Dynamics', 'Five-brane Effective Action in M-theory', and 'Geometric Langlands from Six Dimensions'. $\endgroup$ – Meer Ashwinkumar Jan 12 '16 at 5:20

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