Why Calabi-Yau manifolds should be complex? I'm aware that mathematically speaking, Calabi-Yau manifolds are complex manifolds with vanishing first Chern number. However from a physics point of view, Calabi-Yau manifolds are related to the solution of Einstein's field equation in vacuum environment (i.e., with vanishing stress–energy tensor). Since Einstein's field equation is on a 4-dimensional real manifold, why Calabi-Yau manifolds are complex? Is there a "real version" of Calabi-Yau manifold?
 A: I think that one possible answer is that a Calabi-Yau manifold is a Riemannian manifold $M$ with $SU(n)$ Riemannian holonomy, where $2n=\dim_\mathbb R M$.
Such a manifold is then necessarily complex, and the Riemannian metric is the real part of a Kähler metric which has zero Ricci curvature. Since the Ricci form in complex geometry is always a representative of the first Chern class of the manifold, what you ask follows.
A: You might consider other real forms of $SU(n)$; a Riemannian manifold with holonomy in such a real form will then arise as extra dimensions in string theories with reduced holonomy and a parallel spinor. The Riemannian manifold need not be complex. But the limit as you make the extra dimensions small will not be Lorentzian, unless the extra dimensions are Riemannian with a parallel spinor. To have a reduced dimension of 4 (for standard general relativity) and a total space dimension of 10 (for type A or type B string theory), you need a 6 dimensional Riemannian manifold of extra dimensions, with a parallel spinor. These conditions give you a Calabi-Yau manifold. If you ask for 7 extra dimensions, you get a G2 holonomy manifold, so extra dimensions are not always complex. There are compact Riemannian Einstein 6-manifolds beside Calabi-Yaus, but not with a parallel spinor.
