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.

This post imported from StackExchange Physics at 2015-08-04 14:38 (UTC), posted by SE-user leastaction