I am a mathematician and reading a physics paper about the holonomy group of Calabi-Yau 3-folds.
In that paper, a Calabi-Yau 3-fold $X$ is defined as a compact 3-dimensional complex manifold with Kahler metric such that the holonomy group $G \subset SU(3)$ but not contained in any $SU(2)$ subgroup of $SU(3)$.
They remark "the condition that $G$ is not contained in $SU(2)$ is a really serious condition for physics since otherwise it would change the supersymmetry".
Could anyone kindly explain this sentence in more detail? I think that if $G\subset SU(2)$ the physics derived from the Calabi-Yau 3-fold has more supersymmetry (because less restriction), but what is wrong about it? One possibility is that too symmetric theory is trivial.
I would appreciate it if someone could kindly explain the physics behind it to a mathematician.