Calabi-Yau manifolds are an important set of spaces in physics and mathematics. However, some of their geometrical features are yet mysterious or unknown. In particular, beyond the nature of mirror symmetry, we lack an explicit expression for the one of the simplest Calabi-Yau manifolds, the K3.
Context: Kerr black holes are solutions to the Einstein field equations describing rotating black holes. What thing is described by a K3 metric? Due to my background on string theory too, I believe CY spaces describe varieties with singularities, but of course, I can not see the global view (and beyond that of a pure mathematician), and I do know this question matters not only in string theory...But due to some applications, I presume, the K3 metric could help to understand better. Anyway, I am not a pure mathematician...So,
- Why to build a K3 metric is a "hard" problem? In particular, why technically twistor approximations are not good enough to build up the K3 metric, and what is lacking.
- Why is important to solve this problem? I mean, here, what are the kind of problems would be simplified with the aid of an explicit K3 metric.