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 Tstring 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 Ew 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
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 GN is related to the
gravitational coupling G10 of the full ten-dimensional superstring
where VX is the volume of X6.
In terms of the Planck mass MPlanck, which
is the quantum gravity mass scale determined by the gravitational coupling
GN, this relationship becomes
where the mass MS is the fundamental mass scale of the full
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=prc. 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 krc>>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
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.