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I don´t really understand what M-theory is supposed to be. Going beyond the dualities relating different string theories (for example the common $11-D$ limit of IIA and $E_{8}\times E_{8}$) I don't understand what is really M-theory and why is supposed to exist such a theory. I mean, what is incomplete the theory?

The motivation of this question arises in the context of string compactification and string landscape. The current paradigm includes such an enormous range of combinations (choice of: compactification manifold, version of the string theory, bundle/brane configuration, flux, critical point of $V_{eff}$...) which fixes the vacua in an approximate way that there would be interesting to know what does it lack in the final formulation of string/M-theory (in this case M-theory).

In this context I do not understand the interest of Matrix theory (I would be thankful if someone could explain to me its physical motivation). Although some of the calculations using that formalism are coherent with perturbative M-theory ones; in which sense it is expected to be a generalization of M-theory perturbative limit/11-D SUGRA? (Having in mind it is only formulated in Minkowski spacetime and toroidal compactifications).

The other path for "complete" M-theory is even more obscure for me: AdS/CFT dualities does not seem to suggest (for me) a physical idea/motivation for a different theory beyond the dualities between certain theories.

EDIT 1: Regarding the branes allowed in M-theory: how it can be a generalization of the five string theories if lots of the branes allowed in them are not present in M-theory (it only allows M2-branes and M5-branes (respectively) electrically and magnetically coupled to the four-field strength of the theory). What happens for example with D7-branes and D3-branes which are a fundamental ingredient to allow realistic string compactifications in IIB-theory? If M-theory is true, they just don't have physical existance?

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  • $\begingroup$ While you wait: encrypted.google.com/search?q=M-theory+non-pertubative - does that provide no answers? $\endgroup$ – Rob Feb 8 '18 at 0:55
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    $\begingroup$ @Chequez I edited my answer, you might want to have a look. Cheers!!! $\endgroup$ – Konstantinos Feb 17 '18 at 16:40
  • $\begingroup$ @Konstantinos Yes, I understood the way that M-branes include D-branes by reading the link mentioned in the previous comment (arxiv.org/abs/hep-th/9906108) and I have confirmed those explanations with your last edit. Thanks! $\endgroup$ – Chequez Feb 18 '18 at 19:26
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    $\begingroup$ Ignatios is a great physicist and a great person. You might also want to look the following link arxiv.org/abs/hep-th/9512062. Cheers!!! $\endgroup$ – Konstantinos Feb 18 '18 at 19:28
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I don't think that the questions have definite answers yet, but I think they are great for general discussion.

I am addressing the second question related to the AdS/CFT, and I am quoting -copying and pasting- some known facts from a recent paper. In case one wants to study all the details of the paper, find it below.

https://arxiv.org/abs/1712.08570

and the references therein of course; in particular the eight first ones.

People are trying to calculate beyond the SUGRA corrections to M-theory, in order to get some first lessons on this theory

M-theory arises in the strong coupling limit of type IIA string theory.

It is known that M-theory reduces to $11$d SUGRA at low energies, and its fundamental degrees of freedom are $2$d and $5$d objects known as M$2$-branes and M$5$-branes, respectively.

The worldvolume theory for M$2$-branes (on an orbifold) was found to be a $3$d superconformal Chern-Simons matter theory with classical $\mathcal{N} = 6$ supersymmetry.

The M$5$-brane worldvolume theory is expected to be a $6$d superconformal field theory with $OSp(8_{*}|4)$ symmetry; this is usually called the $6$d $(2, 0)$ theory.

If one considers a single M$5$-brane, one can formulate the theory in terms of an Abelian $(2, 0)$ tensor multiplet, consisting of a self-dual $2$-form gauge field, $5$ scalars, and $8$ fermions, but it is not known how to generalize this construction to describe multiple M$5$-branes.

This is where the AdS/CFT enters the game to give a hint. The statement is that the worldvolume theory for a stack of N M$5$-branes is dual to M-theory on $AdS_7 × S^4$ with N units of flux through the $4$-sphere, which reduces to $11$d SUGRA on this background in the limit large N limit; $N \rightarrow \infty$.

The above limit is the SUGRA approximation.

Recently, people have started thinking ways of going beyond the SUGRA approximation, by using the conformal bootstrap programme. This is how people hope to gain new insight into M-theory beyond the SUGRA approximation.

It is elusive yet how one can formulate the $6$d $(2, 0)$ theory, and this is why people are using the superconformal and crossing symmetry to deduce the structure of the $4$-point correlators in this theory.

The implications of the aforementioned symmetries for the $4$-point correlators of stress tensor multiplets in the $6$d $(2, 0)$, adapting the holographic arguments, have showed that these solutions scale in the large-N limit. The spin-$0$ solution scales like $N^{−5}$, all higher-spin solutions are suppressed by at least $N^{−19/3}$. As a result, the spin-$0$ solution corresponds to the leading correction to the supergravity prediction for the $4$-point correlator (which scales like $N^{−3}$) and all higher-spin solutions are subleading compared to the $1$-loop supergravity prediction (which scales like $N^{−6}$ but has not been computed yet).

The lessons are about the low energy effective action of M-theory on $AdS^7 \times S_4$. In particular, noting that the anomalous dimensions of the spin-$0$ solution scale like $n^6$ compared to those of the SUGRA prediction (where n is the twist), this suggests that the corresponding terms in the effective action contain six more derivatives than the SUGRA Lagrangian, and are subsequently of the form $\mathcal{R}^4$, where $\mathcal{R}$ is the Riemann tensor.

Edit: M-branes and D-branes.

Punchline: The M-branes realize all D-branes, and this is why D-branes are consistent objects that you don't need to throw away.

Eleven-dimensional supergravity has only an anti-symmetric tensor field $A_{(3)}$ -the rank of the tensor is 3- and the realisation of branes is restricted.

The consistent supergravity solutions are known as M$2$ and M$5$. The are further solutions to classical supergravity known as F$1$ -the fundamental string- and its magnetic dual the NS$5$-brane.

You can classify the theories based on the type of supergravity theory you get in the classical limit.

The way that M-theory sees D-branes is via the net of dualities.

All of the D-branes and the NS$5$ brane are solutions to type II theories, both A and B.

When you reduce the M-theory on a circle, you get back to TypeIIA. TypeIIA is the same as TypeIIB when you perform a T-duality. The M$2$ and M$5$ branes reduce to the various D-branes and S-dualizing the D$5$ brane you get the NS$5$. I don't remember the actual dualities you need to apply, but you can go to the seminal papers and look there for more detailed answers.

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    $\begingroup$ Minor comment to the post (v1): In the future please link to abstract pages rather than pdf files. $\endgroup$ – Qmechanic Feb 9 '18 at 10:05
  • $\begingroup$ Sorry about that. I thought it would be ok, since it is an arXiv file. I will keep that in mind. Thanks $\endgroup$ – Konstantinos Feb 9 '18 at 10:20
  • $\begingroup$ Thanks for the information, it is very useful. Some questions, though: regarding the branes allowed in M-theory: how it can be a generalization of the five string theories if lots of the branes allowed in them are not present in M-theory (it only allows M2-branes and M5-branes (respectively) electrically and magnetically coupled to the four-field strength of the theory). What happens for example with D7-branes and D3-branes which are a fundamental ingredient to allow realistic string compactifications in IIB-theory? If M-theory is true, they just don't have physical existance ? $\endgroup$ – Chequez Feb 9 '18 at 12:35
  • $\begingroup$ No worries. Glad you liked it. I will try to answer to your question on M and D branes when I get back home, as I am swamped right now. Sorry for that. By the way, I think -I am not quite sure- that it is one of the rules that each question will have a different topic, so you might want to ask a new one about branes. Cheers!!! $\endgroup$ – Konstantinos Feb 9 '18 at 13:02
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    $\begingroup$ Sorry, I have not replied yet, but I am working on a project that consumes most of my time and energy; I saw the edit thoug. I will get back to this question as soon as possible. Thanks for the understanding. Cheers!!! $\endgroup$ – Konstantinos Feb 10 '18 at 21:35

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