$\mathcal{N}=2$ supersymmetry and $SU(2)$ holonomy

Question

I was reading this Phys.SE question. I was unable to understand how an $SU(2)$ holonomy would produce $\mathcal{N}=2$ in four dimensions. Could anyone shed some light on this?

Answer 1

As a general rule, compactifaction on a Calabi-Yau $n$-fold results in a preservation of $2^{(1-n)}$ parts of the original supersymmetry. If you start with $\mathcal{N}=4$ and compactify on a 2-fold, you preserve $2^{1-2}=1/2$ of the original supersymmetry, i.e. $\mathcal{N}=2$. One now has to realize what a Calabi-Yau $n$-fold is: it is a Kähler manifold with an $SU(n)$ holonomy. In the case of $n=2$ you get $SU(2)$, leaving us with the conclusion that $\mathcal{N}=2$ supersymmetry is related to $SU(2)$ holonomy.

For comparison, consider compactification on a Calabi-Yau 3-fold: this preserves $2^{1-3}=1/4$ of the original supersymmetry, i.e. $\mathcal{N}=1.$ This leads us to the relation between $\mathcal{N}=1$ supersymmetry and $SU(3)$ holonomy, which was mentioned in the other question.