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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?

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It can't be right, because you need the low energy SUSY requirement for CY, but there might be some meaning in this statement that I am missing. Do you remember the source? –  Ron Maimon May 20 '12 at 18:19
    
I more or less found the answer I think. Gauge anomaly cancellation requires $c_2(V) = c_2(T)$ where $V$ is the gauge bundle and $T$ the tangent bundle. This expression can be given in terms of the $U(1)$ charges in certain cases –  Frank Müpe May 22 '12 at 14:35
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Can you post an answer? –  Ron Maimon May 22 '12 at 15:45

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