# Is there any relativistic interpretation on considering Kaehler-Einstein metrics and Calabi-Yau manifolds?

Perhaps I sould ask this question on a physics forum, but I am curious about answers coming from mathematicians.

Calabi-Yau manifolds are examples of Ricci-flat Kaehler manifolds. As we know, in the semi-riemannian case, Ricci flat metrics describes solutions for Einstein field equations on vacuum. So my question is: by considering the additional structure "Kaehler", and consequently, the holomorphic structure on $M$, does there any physical interpretation for such manifolds?

More generally, what is the physical meaning of Einstein-Kahler manifolds? Is this related yet with Einstein field equations? What does the holomorphic sctructure on $M$ offers in addition to real manifolds?

Given a supersymmetric theory on a Minkowski spacetime $\mathbb{R}^{1,D}$, you might want to consider the same theory on $\mathbb{R}^{1,d} \times M$, where $M$ is a compact manifold of dimension $D-d$. To obtain a supersymmetric theory on $\mathbb{R}^{1,d}$, the simplest possbility is to ask for $M$ to admit a covariantly constant spinor. Compact Riemannian manifolds admitting covariantly constant spinors are automatically Ricci-flat and necessarily have reduced holonomy. Calabi-Yau manifolds give a large class of such examples (but there are others, not necessarily obviously related to complex geometry, as 7-manifolds of G2 holonomy for example).