Three questions:
What can be some applications of AdS/CFT correspondence on M-theory? For example, would it be possible to represent 11-dimensional world using 10-dimensional surface?
It seems that using AdS/CFT correspondence, we can project our world as being 100-dimensional or 1000 or more. What is wrong with this?
Is AdS/CFT correspondence just a conjecture, or a proven theorem?
Thanks.
The traditional application of AdS/CFT to M-theory is to reproduce the full 11 dimensional theory from a zero dimensional effective quantum mechanics on point black holes which emerge in a type IIA compactification (D0 branes), and this is called Matrix String Theory, or BFSS theory (Banks Fischler Schenker Susskind). Matrix theory predated AdS/CFT, and the matheamtics of Matrix theory is in many ways the ultimate AdS/CFT, since all the space is emergent from the matrices.
The number of dimensions cannot grow to large in fully consistent quantum AdS/CFT because too much degrees of freedom make the description inconsistent. If you look at a string world sheet, the number of oscillations it can support is at most 26, in the bosonic case, and usually 10, in the supersymmetric case. The matrix theory business uses SUSY to deduce the effective theory, and it ends up describing 11 dimensions, which is the biggest SUSY dimension, but it is pretty clear that you couldn't get 1000 dimensions, just because of the too-many-degrees-of-freedom of oscillations problem.
Every dimension is an extra wriggle that the black hole can have, and every extra wriggle adds to the entropy. There is a precise consistency relation in string theory (at least for line black holes, for actual strings) that requires the central charge (the degree of freedom count) to exactly cancel the reparametrization freedom negative-central charge on the sheet. This constraint picks out 26 scalar degrees of freedom. It should be the same for other AdS/CFT models, and so far, all of these have been in the same universe that string theory describes. This might just be an accident of the fact that they are all supersymmetric, but I don't think so.
The AdS/CFT, in a mathematical sense, is not quite a conjecture, because it doesn't relate two completely precise things. The gauge theory side is precise, but the string theory side is not 100% precise, because the correspondence only becomes fully classical in large t'Hooft coupling limits, where the strings are strongly interacting. But there are sub-relations of AdS/CFT where you can compute both sides, and in those cases, the idea works.
It is a physical principle, motivated by holography, and verified by many different precise regimes where it works (most astonishingly, by matrix theory). So it cannot be doubted anymore.
It might be possible to reconstruct 11 dimensions using a non-gravitational 10 dimensional thing, but this is a little wacky, because space-filling branes are not intuitive. The current reconstructions of M-theory are from Matrix theory (which is not really M-theory, by strong coupling IIA theory), from stacking M2 branes (2+1 dimensions), which gives the recent Chern-Simons theory studied by Schwarz and others, which does reproduce M-theory as far as people believe.
You might also reproduce M-theory from stacking the magnetically dual M5 branes, which is usually studied in the NS5 brane context (5+1 dimensions), and gives little strings. I am not sure if the emergence of M-theory from little-strings is understood.
1) There is a version of AdS/CFT which relates M-theory on $AdS_7 \times S^4$ background and 6D ${\cal N}=(2,0)$ SCFT which is an effective theory describing the worldvolume physics of M5 branes. See, for example, this classical review. There is analogous duality between certain ${\cal N}=6$ Chern-Simons-matter theory, which governs the worldvolume physics of M2 branes, and M-theory on $AdS_4 \times S^7$ background (ABJM theory).
2) We cannot change the dimensionality of spacetime on which string theory lives in AdS/CFT correspondence arbitrarily. It is fixed by internal consistency of the theory - 10D for string theory, 11D for M-theory. There are also some more exotic cases. For example, Chern-Simons theory on $S^3$ is holographically dual to topological strings on certain 6D conifold (see the original paper by Ooguri and Vafa). Moreover, it is possible to formulate such dualities more phenomenologically between certain gravity theories and low-dimensional systems (see this discussion on SE on SYK model).
3) AdS/CFT correspondence was proven in the simplest case of topological theories on both sides (see the aforementioned Ooguri and Vafa paper). Non-topological cases are less trivial, and were not proven, but were checked for various theories. Such checks were performed even experimentally for the field theories describing certain condensed matter systems and strong interactions between elementary particles. See this book for a review of this subject.
Something obvious to me is that this relation, so-called ADS/CFT, is not a fundamental concept, and at smaller scales (Planck scale) this correspondence doesn't apply anymore. I can firmly say all or almost all fundamental and important questions of quantum gravity remained unsolved within this framework. On the other hand this is a disappointing fact, that despite almost 15 years of such huge efforts (during this period more than 6500 papers just have published on this narrow subject), we don't have any complete and ultimate answers to our basic questions (questions of quantum gravity and quantum cosmology).
