In string theory, we compactify a 10-dimensional space by a Calabi-Yau 3-fold to reduce the dimension to 4. To get a reasonable theory, a Calabi-Yau 3-fold should satisfy some properties. One is the Euler number must be $\pm6$ so that it is compatible with the generation of the elementary particles.
I heard that the fundamental group of the Calabi-Yau 3-fold should not be trivial. What should it be?
The 4D physics emerging from the Calabi-Yau compactification is at the GUT scale. In the case of a non-simply connected fundamental group, holonomies around the loops become dynamical and can account for the gauge symmetry breaking from the intermediate GUT level to the standard model level. Only the generators commuting with the holonomy element reside in the lower scale. See for example the following article by Andreas and Hoffmann, where this mechanism is used in an $SU(5)$ GUT case.
This mechanism is known as the Hosotani mechanism, or symmetry breaking by Wilson lines.
