# M-Theory? What's wrong with you! Jeff?

1. Apr 15, 2004

### Jeebus

I have been recently reading a lot on Super Strings and M-Theory, and I discussed this with selfAdjoint earlier but he said I should advise to someone say, like jeff, Doc Al, Integral, anyone with a very experienced knowledge of this on this matter of perplexed dimensional frustration.

In M-Theory which is trying to explain the theory of everything theorists speak about 11 dimensions, adding one because of string theory. But, what I don't understand is:

1. How does M-theory work though?
2. Why does everything work out perfectly when there are 11 dimensions instead of say, . . .8. Why 11, why not another number?

Conclusion: Specifically saying how do theorists determine 11? What is the math behind 11?

I know now that nonsupersymmetric bosonic string theory for solving gives you that D = 26 and then that supersymmetry "eats" 16, giving you a duality of five and somewhere adding one dimension, but that's in abstract laymen's terms, translating to how do this happen and somebody explain-speak.

Note: The above is only because of help from sA. Otherwise I wouldn't have a clue!

I have listened to selfAdjoint and I believe he was on the right path because it made sense but he said his study group was limited on increasing his already vast knowledge, so he said I should make this into a thread, so I did.

Now, I have been referring to this site: http://www.theory.caltech.edu/people/jhs/strings/str154.html to try to clear some confusion, but it isn't working, so I need your help! It explains a lot of concepts, but not the concept of 11.

I appreciate it if you can resolve this for me!

Thanks,
Jeebus

Last edited: Apr 15, 2004
2. Apr 15, 2004

### Haelfix

I cant speak for M theory, but in general spaces that have maximal supersymmetry are much easier to deal with mathematically.

Integration and the like are simplified tremendously, due to some rather beautiful cancellations, and the measure itself becomes much more tractable.

What's hard for String theorists, is getting 3+1 large dimensions, since there is in principle many complicated and nonequivalent ways to go from 11 --> 4

3. Apr 16, 2004

Staff Emeritus
Actually your link does explain the concept of 11 dimensions, though you may not have noticed it. It speaks of dimensions blowing up into a circle in one kind of superstring theory and into a line segment in another kind.

The point is that those dimensions were already there, but compacted. Just as the extra dimensions of string physics are compacted on tiny manifolds in our spacetime so we can't see them, so the perturbative (weak interaction) theories couldn't see their own compacted dimension, which only shows up in the nonperturbative (strong interaction) area of the physics. So we have kind of a double compacting here.

4. Apr 17, 2004

### Javier

What are you looking for here?
When you quantize classical string theory in the covariant fashion, covariance with respect to the Lorentz group is manifest. However, unitarity of the theory is not. To ensure unitarity, there is a constraint that must be satisfied. For string theories without worldsheet supersymmetry, this constraint says that the number of spacetime dimensions must be 26. If there is worldsheet supersymmetry, D must be 10.

There is another method of quantization where covariance wrt Lorentz group is not manifest, but unitarity is. To ensure covariance, there is a constraint, with the same constraint on the spacetime dimensions as before.

Therefore, the dimension 10 comes from the fact that it is the only dimension in which one can formulate a Lorentz invariant, unitary string theory on a non-compact spacetime. Of course, one may then compactify some of the spatial dimensions to get string theories in lower dimensions, but the string interpretation demands that you keep all non-trivial dependence of the string modes on the compact dimensions...if you truncate to just the trivial modes, you must necessarily consider only the low energy limit of the string theory, which is a supergravity theory. Supergravity can be consistently formulated in 11 or fewer spacetime dimensions (the constraint on the dimension here has to do with the maximum particle spin in a consistent quantum field theory being 2).

Remarkably, perturbative string theory hides an extra dimension...the number 11 appears again. The full non-perturbative theory appears to be 11 dimensional. This is hinted by the fact that 11-d supergravity can be dimensionally reduced on a circle. You get a 10d supergravity theory with a coupling constant (coupling strength) that has an interpretation as parametrizing the size of the compact spatial dimension. But the 10d supergravity theory is a low energy limit of a 10d string theory. Therefore, it seems as though the strong coupling limit of this 10d string theory corresponds to enlarging the size of the compact dimension...keep going and you get an eleven dimensional theory (of which 11d supergravity was a low energy limit). However, note that we cannot speak of a consistent 11d string theory, so this strong coupling limit of 10d string theory must have other ingredients. Some of these ingredients are what are called 1-branes all the way up to 10-branes. The importance of these things is another story.