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### Kaluza-Klein compactification

In the original Kaluza-Klein theory from the 1920s, had five spacetime dimensions, but the extra space dimension was rolled up into a tiny circle, existing everywhere but too small to see with existing instruments. The vibrations of the gravitational field in the rolled up extra space dimension would look to observers like vibrations in an electromagnetic field and a scalar field in the remaining three space and one time dimensions.
So the underlying five dimensional spacetime only has one kind of force: a gravitational force. But the four dimensional spacetime that is seen at large distances appears to have three kinds of forces: a gravitational and an electromagnetic and a scalar force. This is one way of making a unified theory -- derive all the other forces from higher-dimensional modes of the graviton.
In string theory, Kaluza-Klein compactification of the extra dimensions has one important difference from the particle theory version. A closed string can get wound several times around a rolled up dimension. When a string does this, the string oscillations have a winding mode. The winding modes add a symmetry to the theory not present in particle physics. A theory with a rolled up dimension with size R turns out to be equivalent to a theory with a rolled up dimension of size Ls2/R with the winding modes and momentum modes in the extra dimension exchanged. (Ls is the string length scale.)
This is a very important symmetry because it equates theories with very small extra dimensions to theories with very large extra dimensions. This symmetry is called T duality. It shows that seemingly different different string theories can turn out to be the same theory looked at in a different way.
In superstring theory, Kaluza-Klein compactification has to be done on six space dimensions at once. One proposed model has been to take the heterotic string theory and make six space dimensions form a small compact space called a Calabi-Yau space. The geometry and topology of the Calabi-Yau space determine the symmetries and spectrum of the particle theory measured at low energy or large distance.
There have been many interesting models, but if a realistic model had been found, we wouldn't need to discuss the next option: braneworlds.

### Braneworlds

Pretend you lived on your computer screen and could only move on that two dimensional surface. The computer exists in three space dimensions but you can only move on a two dimensional subspace made by the screen, so the spacetime that you experience would look like three dimensions (two space plus time) rather than four.
That's sort of the idea in a braneworld higher dimensional theory. Our observed four dimensional spacetime is like the computer screen, a subspace of some bigger space that we can't see because all matter and forces are constrained to move (mainly) on our subspace, or brane (as in membrane).The total space is called the bulk and the subspace or brane on which we would live is called the brane.
Light is made of electromagnetic radiation, and in a braneworld model, the charges and the fields should propagate only on the brane. So there wouldn't be any way to probe the extra dimensions in the bulk by using light, even if the extra dimensions were large.
Gravity is the force that determines the shape of spacetime. Therefore, at least in principle, gravity should propagate in all dimensions equally. That means we should be able to detect large extra dimensions by looking for suspicious behavior in the gravitational force.
But in certain braneworld models, gravity can actually be confined or bound to our brane so that it doesn't propagate very far in the bulk. That makes the extra dimensions harder to detect using gravity.
Braneworld models are conceptually different from compactified Kaluza-Klein models because they don't attempt to derive nongravitational forces from the gravitational oscillations in the extra dimensions. On the contrary, if the extra dimensions are large, the gravitational oscillations have to die out quickly in those directions, so that we can't detect them. There are still Kaluza-Klein modes of oscillation in the extra dimensions, but because they couple through gravity, and gravity is mostly confined to the brane, they are effectively invisible to our world on the brane.

### How could they be observed?

One problem with theoretical models of gravity and particle physics is that before they can make unique testable predictions of new physics, they have to be worked on so that they don't contradict any existing theoretical or experimental knowledge. That can be a long process, and it's not really over for superstring theories or for braneworld models, especially not braneworld models derived from superstring theories.
In superstring theory with Kaluza-Klein compactification, there are several different energy scales that come into play in going from a string theory to a low energy effective particle theory that is consistent with observed particle physics and cosmology:

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Scale Definition
Planck scale Length scale of quantum fluctuations of spacetime geometry are important, about 10-33 cm, or 1019 GeV.
String scale Average size of string, once assumed to be Planck scale but now more complicated.
Compactification scale Size of compact extra dimensions, also determines mass of massive oscillations in extra dimensions.
Supersymmetry breaking scale Mass scale where supersymmetry is broken, could be anywhere between the Planck scale and the electroweak scale, depending on the model in question
Grand Unification symmetry breaking scale Mass scale where unified gauge symmetry is broken and remaining symmetries split into three gauge groups of Standard Model, should be about 1016 GeV
Electroweak scale Mass scale below which current particle physics experiments have agreed extremely well with predictions made by Standard Model of particle physics, this is about 100 GeV to 1TeV.
 The attribute of superstring theory that looks the most promising for experimental detection is supersymmetry. Supersymmetry breaking and compactification of higher dimensions have to work together to give the low energy physics we observe in accelerator detectors.   Braneworld models in general are very different from superstring Kaluza-Klein compactification models because they don't require there to be so many steps between the Planck scale and the electroweak scale. The huge difference between the Planck scale and the electroweak scale is called the gauge hierarchy problem.    Supersymmetry was originally interesting to particle physicists because it could address this problem. But some braneworld models need supersymmetry for the brane geometry to be stable.   If supersymmetry is detected at next-generation particle physics experiments, then the details of the supersymmetric physics will have something to say, hopefully, about any underlying superstring model and whether there is Kaluza-Klein compactification of extra space dimensions into some tiny rolled up internal space, or whether we are all living in the four dimensional equivalent of being flies stuck on the wall of a higher dimensional Universe.

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