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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?

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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:

  1. 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.

  2. 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.

  3. 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.

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    $\begingroup$ Thank you for the detailed answer. My undergrad thesis involves reproducing the Becker sisters' calculation (which uses the Matrix model of M theory) to compute the scattering of D0 branes at two loops and comparing it the result to 11D supergravity. $\endgroup$
    – saad
    Jul 19, 2020 at 7:38
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    $\begingroup$ It's always a real pleasure. It sounds like a truly wonderful undergraduate thesis, a really nice training on important stuff. Stay curious, and the best of all lucks for your project. $\endgroup$ Jul 19, 2020 at 19:26
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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.

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    $\begingroup$ Does this basically mean that 11 D SUGRA is non renormalizable and has an ultraviolet cutoff and any M theory should provide a complete description? $\endgroup$
    – saad
    Jul 18, 2020 at 22:47
  • $\begingroup$ That's an interesting question saad, nothing guarantees the existence of an energy cut-off, even given the "effective" character of the eleven dimensional supergravity. It's part of the mystery. It could be that Newton's constant may be induced and the Planck scale (and spacetime) emerge from something else. In such a case the theory would exactly break at the Planck-scale. Recall that string theory has a natural energy scale above from what supergravity does not make sense, namely the string scale. $\endgroup$ Jul 19, 2020 at 4:35
  • $\begingroup$ Could it be the case for the eleven dimensional supergravity; it could be, but contrary to string theory, there is no natural parameter with units of lenght. The close analogue must be the square of the Planck lenght if you take seriously the idea that that serves as a "quantum" of area for the supermembrane; again, that has the incorrect units of lenght and you return to the suggestion that the theory breaks down exactly at the Planck´s lenght. $\endgroup$ Jul 19, 2020 at 4:37
  • $\begingroup$ @saad That's one way of looking at it. M-Theory should also tell us about comsology on the large scale too not just thinking about it in terms of particles. $\endgroup$
    – zooby
    Jul 19, 2020 at 12:59

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