# Is string theory the boundary theory of M-Theory?

Looking at various AdS/CFT correspondences, we find that some (n-1) dimensional field theories on the boundary of $$AdS_n$$ with $$N=\frac{8}{n-3}$$ supersymmetries are equivalent to M-Theory in $$AdS_n \times S_{11-n}$$.

(e.g. for $$n=7$$ we get 6D $$N=(2,0)$$ superconformal CFT.)

Setting $$n=11$$, seems to suggest that there is a 10 dimensional N=1 supersymmetric theory on the boundary of $$AdS_{11}$$ that is equivalent to M-Theory in $$AdS_{11}$$.

The natural theory in 10 dimensions with $$N=1$$ supersymmetry is superstring theory (or one of them).

Therefor could 11D M-Theory and 10D superstring theory be related by the AdS/CFT correspondence? If not what 10D theory living on the boudary of $$AdS_{11}$$ would be equivalent to M-Theory on $$AdS_{11}$$?

Therefor is superstring theory not a theory of the bulk, but actually a theory of what happens on the surface of a black hole? i.e. string theory is just the theory of the hologram?

No, AdS/CFT relates a theory of quantum gravity in d dimensions to a QFT without gravity in d-1 dimensions. M theory and string theory are both quantum gravity theories.

The holographic dual to M theory is a type of CFT called ABJM theory.

However, there is a case unrelated to holography where your suggestion is valid in a different sense. In heterotic M theory, aka Horava-Witten theory, we imagine M theory with two 1+9D "end of the world" branes and the 11th dimension as a distance between the branes. The M2 branes are then open tubes ending on the branes. The endpoints of these M2 branes are now 1D closed strings, and the theory restricted to one or the other 1+9D brane is a heterotic string theory. In this sense, a string theory lives on the boundary of an M theory, but this is not a holographic boundary.

• ABJM is a 3 dimensional theory similar to Chern-Simons theory. Not a 10 dimensional theory. ABJM is only dual to M-Theory if 7 dimensions are on $S_7$ thereby it is the case where $n=4$ in my question. I'm not sure one can generalise this result to the uncompactified case. – zooby Sep 29 '20 at 22:06
• It is interesting about the M2 branes . – zooby Sep 29 '20 at 22:11
• Fair, but it is at least a reasonable ansatz that in the same way ABJM is the CFT living on an M2 worldvolume, the CFT dual of an AdS11 gravity theory would be the CFT living on a M9 volume, rather than a bulk string theory. But the high dimensional branes are notoriously difficult. The essential point is the d-1 dual has to be non-gravitational, i.e. no gravitons, which you necessarily get in a string theory – Felix Wotter Sep 29 '20 at 23:28
• That sounds fair by considering degrees of freedom I guess. It makes me wonder however, if there is any deep connection between this 10D CFT and 10D superstrings. And if the degrees of freedom are the same.... (or if the physics is the same.) Perhaps both represent the same thing but that the transformation from one to the other garbles up the information! e.g. is the theory of strings physically equivalent to the theory of 9 branes somehow. – zooby Sep 30 '20 at 10:50

Heres a simple way to see that the answer is no. The theory on the boundary has to be superconformal, but the classification of superconformal symmetry says that SCFTs terminate in dimension 6. For M-theory there can only be 3d and 6d SCFTs duals because these theories are the conformal fixed points of the worldvolume theories of M2 and M5 branes which are the only branes in the theory.

• Interesting. I didn't know about conformal fixed points. – zooby Oct 6 '20 at 18:30

The answer is no. Some arguments against the hypothesis of "string theory as the boundary of $$M$$-theory"

1) The boundary of an Anti de-Sitter space is a sphere. No consistent quantum theory of strings already exist over spheres.

2) The AdS/CFT duality is gravity/gauge duality. Both string and $$M$$-theory contain gravity. No single example of gravity/gravity duality has been constructed, so far.

3) The strong coupling limit of the type $$IIA$$ string is $$M$$-theory. This affirmation is very precise, and the direction that emerges (the one parametrizing the value of the $$IIA$$ dilaton) is a circle. I cannot see how does anyone can construct an AdS space as a circle fibration over its boundary.

4) The strong coupling limit of the $$E_{8}\times E_{8}$$ string is heterotic $$M$$-theory. The problem here is, again, that it's pretty difficult (if not impossible) to generalize the Horava-Witten construction (two parallel $$M9$$-planes separated by a distance proportional to the VeV of the heterotic string dilaton) in a hyperbolic space.

5) All the known examples of the AdS/CFT type are constructed as the near horizon geometry of a stack of $$D$$-branes. You are suggesting an $$AdS_{11}$$/String theory correspondence; well, the problem is that Nham's theorem rule out the possibility of an holographic dual for $$AdS_{11}$$ because no interacting SCFTs theories exist above six dimensions.

6) Recall that the AdS/CFT correspondence is indeed an AdS $$\times X$$ /CFT correspondence, where $$X$$ is some manifold (typically a sphere). Then you are suggesting the existence of a higher dimensional supersymmetric theory. And it is a well known fact that no supersymmetric supergravities exist above twelve dimensions.

7) There are no solutions for the eleven dimensional supergravity in $$AdS_{11}$$.

8) An overall $$RR$$ flux is needed for the correspondence to work. What is the analogue in this case? there is none. All the possible central extensions of the eleven dimensional super-Poincarè algebra have been studied, and no one produce a negative value for the cosmological constant.

9) Large $$N$$-limits are required for the correspondence. What is the propossed limit in this case?

Many more arguments can be elaborated. But I find those enough to ensure the implausibility of the proposal.

• That's a lot of interesting reasons! Thanks for your answer. I have to separate them out into which are saying impossible and which are saying unlikely. I guess number 8 is the most compelling. – zooby Oct 6 '20 at 18:26
• It was a pleasure :) Glad that it was useful for You. – Ramiro Hum-Sah Oct 6 '20 at 22:27
• What is Nham's theorem: do you have a reference? Google gives nothing. – Harry Wilson Jul 1 at 16:44
• @HarryWilson In short Nahm's theorems says "D = 6: it is the highest spacetime dimension in which we can combine supersymmetry and conformal symmetry" Here is the original Nham's paper cds.cern.ch/record/132743/files/197709213.pdf and I recommend the discussion in section 2 of arxiv.org/abs/1805.06467 to learn about. – Ramiro Hum-Sah Jul 2 at 0:29
• @ RamiroHum-Sah Thank you, that's very useful. – Harry Wilson Jul 2 at 21:19