Kaluza-Klein in string theory

   Superstring theory is a possible unified theory of all fundamental forces, but superstring theory requires a 10 dimensional spacetime, or else bad quantum states called ghosts with unphysical negative probabilities become part of the spectrum and spoil Lorentz invariance. Fermions are very complicated to work with in higher dimensions, so for the sake of simplicity let's consider bosonic string theory, which is Lorentz invariant and ghost-free (albeit tachyonic) in d=26.
    A particle trajectory only has one parameter: the proper time along the path of the particle. Going from particles to strings adds a new parameter: the distance along the string

and that's what makes the outcome of Kaluza-Klein compactification far more interesting in string theory than it is in particle theory.
   If we compactify x²⁵ on a circle of radius R, we get the usual Kaluza-Klein quantized momentum in that direction

We want gravity in the theory, so we need to look at closed strings. A closed string can do something that a particle cannot do: get wrapped around the circle in the compact dimension.
   A closed string can be wrapped around the circle once, twice, or any number of times, and the number of times the string is wrapped around the circle is called the winding number w. The string oscillator sum in the x²⁵ direction changes by a constant piece in a way that is consistent with the periodicity of the closed string and the compact dimension

   The string tension T_string is the energy per unit length of the string. If the string is wound w times around a circular dimension with radius R, then the energy E_w stored in the tension of the wound string is

   The mass of an excited string depends on the number of oscillator modes N and Ñ excited in the two directions of propagation around the closed string, minus the constant vacuum energy. Kaluza-Klein compactification adds the quantized momentum in the compact dimensions, and the tension energy from the string being wrapped w times around the compact dimension, so that the total squared mass becomes

   A very crucial feature of this mass equation is the symmetry under

This is what makes string theory so different from particle theory. The theory doesn't really distinguish between the quantized momentum modes, and the winding modes of the string in the compact dimension. This creates a symmetry between small and large distances that is not present in Kaluza-Klein compactification of a particle theory.
   This symmetry is called T-duality. T-duality is a symmetry that relates different string theories that everyone thought were completely unrelated before T-duality was understood. T-duality preceded the Second Superstring Revolution.
   The theory gains extra massless particles when the radius R of the compact dimension takes the minimum value possible given the above symmetry of T-duality, which is just the string scale itself

This is another purely stringy effect, not occurring with particles.
    The Kaluza-Klein compactification of strings can be done on more than one dimension at once. When n dimensions are compactified into circles, then this is called toroidal compactification, because the product of n copies of a circle is an n-torus, or Tⁿ for short.
    When fermions are added to make superstrings, the mathematics becomes more complicated but the structures and symmetries become more rich. The most studied superstring compactification is heterotic string theory compactified on a Calabi-Yau space in six-dimensions (or three complex dimensions).
    These general models all have in common that the spacetime is a direct product

where M₄ is the four-dimensional noncompact spacetime, and X₆ is some six-dimensional compact internal space. This means that the metric on M₄ doesn't depend at all on the coordinates in the internal space. In this case, the gravitational coupling constant that we measure as Newton's constant G_N is related to the gravitational coupling G₁₀ of the full ten-dimensional superstring theory by

where V_X is the volume of X6.
    In terms of the Planck mass M_Planck, which is the quantum gravity mass scale determined by the gravitational coupling G_N, this relationship becomes

where the mass M_S is the fundamental mass scale of the full ten-dimensional theory_.

Braneworlds

   In the Kaluza-Klein picture, the extra dimensions are envisioned as being rolled up in compact space with a very small volume, with massive excited states called Kaluza-Klein modes whose mass makes them too heavy to be observed in current or future accelerators.
   The braneworld scenario for having extra dimensions while hiding them from easy detection relies on allowing the extra dimensions to be noncompact, but with a warped metric that depends on the extra dimensions and so is not a direct product space. A simple model in five spacetime dimensions is the Randall-Sundrum model, with metric

In this scenario, the three-dimensional space that we experience is a three-dimensional subspace, called a 3-brane, located at f=0, with another 3-brane located at f=p, or y=pr_c. The full four-dimensional space, or five-dimensional spacetime, is referred to as the bulk. The warping or curving of the bulk gives rise to a cosmological constant, which is proportional to the parameter k.
   Since the extra space dimension is noncompact, we would expect the force law of gravity to change. However in this picture, the warping of the brane causes the the graviton to become bound to our brane, so that the graviton wave function falls away very rapidly away in the direction of the extra dimension.
   This spacetime also has oscillations in the extra dimension that are the Kaluza-Klein modes, but in this case there is a continuous spectrum of modes. This would seem to rule the model out, except that the Kaluza-Klein modes here are so weakly coupled that they can't be detected on the brane.
   Why would this model be preferable to having compact extra dimensions? In Kaluza-Klein compactification, the Planck mass in the full ten-dimensional superstring

The parameter M is the fundamental mass scale in the full theory in the bulk, and k is about the same size as M. So for kr_c>>1, the Planck scale measured on our brane would be about the same size as the Planck scale as measured in the full theory. This avoids the situation in the Kaluza-Klein compactification where the Planck mass in four spacetime dimensions depends on the volume of the compactified space, which is hard to control dynamically.

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.
  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 is interesting to particle physicists because it can 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 as the four dimensional equivalent of flies stuck on the wall of a higher dimensional Universe.