This means that we are still examining various types of Calabi-Yau compactifications for the phenomenology they induce in four dimensions. "Realistic" model building is a significant part of on-going string theory research.
So, purely phenomenologically, it is not even absolutely certain we should be compactifying on Calabi-Yau manifolds, but since this is a type of compactification we understand much better than the generic case, we study it nevertheless. Note that it is perfectly valid to compactify with $C$ not Calabi-Yau, but generally, the resulting four-dimensional theory won't have supersymmetry, which is not strictly a problem since we haven't observed supersymmetry so far.
In order for the four-dimensional theory to retain some supersymmetry (which we usually desire), $C$ must be a Calabi-Yau manifold. Formally, one attempts dimensional reduction from the ten-dimensional to the four-dimensional theory, as a higher-dimensional analog of what is done in Kaluza-Klein theory.
The way we extract four-dimensional physics from the ten-dimensional string theories is by supposing that the ten-dimensional "spacetime" is a product $M\times C$ of our spacetime $M$ (usually either supposed to be Minkowski space or deSitter space) and a six-dimensional compact manifold $C$. Yes, it is possible to rule out certain compactification spaces.įirst, it's not "our space" or "our universe" that's a Calabi-Yau manifold.
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