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Toroidal Compactification
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Compactification is the process of reconciling 4-dimensional
spacetime with the extra dimensions required by string theory.
For this purpose it is assumed the extra dimensions are "wrapped"
up on themselves, or "curled" up on manifolds. In the limit
where these curled up dimensions become very small, one obtains
a theory in which spacetime has effectively a lower number of
dimensions. To illustrate compactification it is helpful to
consider a cube. Assume that a particle moving in the interior
of the cube reaches a point on a face and is assumed to go
through it and then appears to come forth from the corresponding
point on the opposite face, in the same direction: A -> A and
B -> B. This results in compactifying 2 out of 3 dimensions.
Since a torus is nothing more than a curled up cylinder which
in turn is a curled up rectangle, this is also equivalent to a
torus. Although the torus looks to have a curved geometry, it
can be unfolded, to a Ricci-Flat surface. This concept can be
extended to 3 dimensions by including the C faces of the cube
which corresponds to a 3-torus.
For larger dimensions the manifolds become extremely complex.
A class of manifolds which is of interest to string theorists
are the CALABI-YAU manifolds. A 2D representation of a 6D
Calabi-Yau manifold that exhibits Ricci flatness is shown
below:
Image courtesy of Wikipedia.