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In Kaluza's original work, he didn't compactify the fifth dimension, rather imposed the "cylindrical condition" where none of the components in the 4D metric depended on the 5th dimension. It wasn't until Klein that the fifth dimension was compactified to a circle, but as far as I've seen the result is the same, with the same resultant metric, vector and scalar. What does it matter, the geometry of the compactified dimension, in the context of Kaluza Klein theory?

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The significance of the compactified circle as opposed to having a non-compact fifth dimension is that a compact dimension produces the discrete "Kaluza-Klein tower of states" in the effective four-dimensional theory - due to the scalar field then having a discrete Fourier series in the fifth coordinate, which, for small radii of the circle, produces one massless mode and a tower of fields with mass $\propto \frac{n}{R}$ where $n$ is the $n$th mode and $R$ the radius of the circle.

If the circle is very small, this makes it possible that we only see the lowest - i.e. massless - stage of this tower, which is the only way the quantized KK theory can even remotely be hoped to fit to observation. In the non-compact case, there is no such sensible interpretation for the scalar field. Additionally, since the conserved momentum in the fifth direction corresponds to the 4D electric charge, a compact fifth dimension naturally explains charge quantization in this model, since the allowed momenta in a compact dimension (in this case on a closed string, effectively) are discrete.

Furthermore, choosing the dimension compact makes the 5D Kaluza-Klein spacetime a $\mathrm{U}(1)$-principal bundle over the 4D "actual" spacetime, explaining the appearance of the $\mathrm{U}(1)$-gauge field in a naturally geometric way, since the modern geometrical formulation of gauge theory models the gauge field as a connection on such principal bundles.

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The main consideration for the internal space, called a Calabi-Yau manifold, in a Kaluza-Klein (KK) theory is that it be Ricci flat. The reason for this is that if there is a nonzero Ricci tensor then a string that is wound on the internal space will grow in size. This has something to do with the Hamilton equations $$ \frac{dg_{ab}}{dt}~=~-2R_{ab} $$ The string will tend to spread over the space as this first order differential equation is not time reverse invariant. This leads to a divergence of the string densely filling the space.

So the internal space must be Ricci flat. As with Klein's suggestion that the cyclicity condition means there is an internal space of a circle, for six dimensions we can propose that the internal space be a 6-torus. A torus appears to have curvature, but this is really extrinsic curvature for how it is embedded in a high dimension. A video game where an object leaves to top or bottom to emerge at the bottom or top and similarly with the sides is a 2-torus. It is also flat.

The 6-torus turns out not to work out very well. So one makes some additional requirements that the internal space reflect holomorphic conditions on the field equations. This then means the internal space is somewhat more exotic, such as the $K3\times K3$ space for $K3$ standing for Kummer, Kähler and Kodaira, the mathematicians who studied this and related spaces.

The general reason that the geometry and also topology matters is that for a string wound around this space a gauge flux will be incident on the string or the area it encloses in its winding in different ways. It is in a way similar to the equations for magnetic flux and Faraday's equations. Different windings of a string on an internal space, which can be dependent on the geometry and topology of that space, influences the eigenmodes of the string. It is almost a bit like computing EMF for a magnetic field through a conducting loop. Because of that the nature of the particle physics world is then dependent on the shapes of these Calabi-Yau spaces

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  • $\begingroup$ Hmmm... that's not a bad idea... let students study the behavior of different windings on magnetic cores with different genus hands on... that ought to teach something about how topology matters. $\endgroup$ – CuriousOne Jun 14 '16 at 3:00
  • $\begingroup$ 1. The "internal space" is not "called a Calabi-Yau manifold". It is the case that in string theory internal spaces that are CY are of particular interest, but in general dimensional reduction of field theories there is no requirement on the compactified space to be CY. 2. The question is explicitly talking about Kaluza-Klein theory, why are you talking about strings? 3. $K_3$ surfaces are 2D complex manifolds, so $K_3\times K_3$ would be an internal space of 8 real dimensions, hence not a replacement for the 6-torus. $\endgroup$ – ACuriousMind Jun 14 '16 at 13:13

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