Why is 11 Dimensional SUGRA not a consistent quantum theory? I am studying the $S$ duality between 11D supergravity and type IIA superstring theory from Adel Bilal’s paper. The author says:

Of course, eleven dimensional supergravity is not expected to yield a consistent quantum theory. It should only be the low-energy limit of some consistent theory, baptised M-theory.

I don’t understand what this really means. What is a “consistent quantum theory”? And how do we know that 11 dimensional supergravity is the low energy limit of M theory?
 A: Essentially the answer is that the eleven dimensional supergravity is non-renormalizable; to be precise, above two loops, the graviton-graviton scattering amplitude is divergent. A nice review on the specifics of maximal supergravity is Kaluza-Klein supergravity.
Some comments to gain intuition about the UV problems in eleven dimensional supergravity:

*

*They only known way to avoid the UV divergences in all known consistent theories of gravity is the UV/IR mixing mechanism of string theory amplitudes (see my answer to this question). Such a mechanism is absent in the maximal supergravity.


*Non-perturbative states in its spectrum exist, namely membranes, three-branes and fivebranes. It is known that the worldvolume theory of the M5-brane is always strongly coupled (the moduli space of its parameters is just a point). A consequence of this is that there is no possibilty to derive its physics from a lagrangian theory, in particular not from the lagrangian of the maximal supergravity. Something more is needed.


*As in any supergravity, it's not clear that a lagrangian description must be the correct description of the theory at the Planck scale. Again, an unknown UV-regulator (new physics) is needed.
A: By definition 11 dimensional supergravity is the low energy limit of M-Theory!
When calculating loop diagrams, those with higher loops are thought to diverge. (i.e. give infinite results).
One can correct these divergences by adding "counter-terms" to the theory which cancel out the divergences. But then the divergences happen at even higher loops. So we add more "counter terms".
If you add an infinite number of counter terms, you get something with a new infinite-dimensional symmetry. Which one can call M-Theory. Alternatively you can try to think of a theory of super-membranes and see if the low energy limit is 11D supergravity.
Unfortunately, nobody knows how to do either of these things! (So maybe M-Theory doesn't exist at all!)
The only thing people can do at the moment is know that the 10D superstring theories have a low energy limit one of the 10D supergravities.
