In Green-Schwarz-Witten Volume 2, chapter 15, it is argued (roughly) that we need 6-dimensional manifolds of $SU(3)$ holonomy in order to receive 1 covariantly constant spinor field. And it turns out that Calabi Yau manifolds satisfy this property.
Now I understand this reasoning just fine, but I occasionally find in some papers, that Calabi Yau manifolds are "demanded" based on the fact that the sum of all $U(1)$ charges sums to 0? Could someone enlighten me on why this yields the Calabi Yau condition?
