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Superstring theory proposes that our universe includes fundamental strings that vibrate in possibly 10 dimensions. To describe how this occurs in our universe, string theorists (Strominger and Witten among other prominent theorists) propose that space is composed of a particular type of Calabi-Yau (CY) manifold. These manifolds can accommodate the vibrations of strings in the required ten dimensions.

The problem is that there are at least $10^{500}$ possible different CY manifolds and so far there is no theoretical way to determine (if superstring theory is true) which one of these $10^{500}$ (or more) CY manifolds is the CY manifold for the space of our universe. My question is: Is there any way to mathematically completely rule out even one of the possible CY manifolds as the manifold that accommodates string vibrations in our universe?

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Yes, it is possible to rule out certain compactification spaces.

First, it's not "our space" or "our universe" that's a Calabi-Yau manifold. 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 $C$. 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.

In order for the four-dimensional theory to retain some supersymmetry (which we usually desire), $C$ must be a Calabi-Yau manifold. 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. 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.

"Realistic" model building is a significant part of on-going string theory research. This means that we are still examining various types of Calabi-Yau compactifications for the phenomenology they induce in four dimensions. In particular, the particle content of the four-dimensional theory is usually comparatively "easy" to determine from the geometrical properties of the Calabi-Yau manifold, although this is complicated by the possibility of branes and questions about the regimes in which the supergravity approximation to the string theory and therefore the validity of Kaluza-Klein reduction is vaid. There are many examples which do not yield a phenomenology close to the Standard Model and hence should be considered "ruled out" by now. On the other hand, some models do yield Standard Model-type phenomenologies.

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Compactifications of strings theories on $M \times \mathbb R^{1,3}$ yield effective field theories, and if we'd like a theory at least somewhat resembling the Standard Model, then there are some requirements we can deduce to eliminate certain Calabi-Yau manifolds $M$. In particular:

  1. The total number of generations should preferably be three, predicted by $\frac12 \chi_E$, that is, the Euler-Characteristic of the Calabi-Yau manifold.
  2. For phenomenological reasons, $\min(b_{1,1},b_{2,1}) = 2$ (or higher) which are related to the number of generations and mirror generations respectively.
  3. It must be that 'at least' $\pi_1(M) = \mathbb Z_3$ which is related to the ability to break $E_6$ into certain Yang-Mills groups.

Furthermore, $\dim H^1(M, \mathrm{End} \, TM )$ counts the number of light chargeless matter fields of the theory. In addition, the Yukawa couplings are related to the Weil-Petersson metric on the space of complex structures for $TM$.

For more insights into the relation between the choice of Calabi-Yau manifold, $M$, and the effective field theory one obtains, I direct you to Calabi-Yau Manifolds: A Bestiary for Physicists.

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