I understand they’re a representation of complex space with flat locality. I just don’t understand how they’re relevant to Physics/how the Calabi-Yau Manifold’s 10 dimensions represent the potential 10 dimensions of the universe
In: 12
A manifold is a space that is locally Euclidean, but globally might be complicated, like a sphere. For instance we usually perceive the earth as flat even though it’s actually round.
General relativity (and string theory etc.) is an applied theory of differential geometry, which loosely speaking is the study of surfaces. The geometric surface used, what we call “spacetime,” is a 4D manifold. In principle you could model spacetime as an object embedded in a higher dimensional Euclidean space but math is hard enough as it is.
String theorists like Calabi-Yau because it maintains [supersymmetry](https://en.m.wikipedia.org/wiki/Supersymmetry) at the string scale. If you want a mathematical explanation for why this is, that’s beyond me. Yau wrote a book called “The Shape of Inner Space” which I understand is directed at general audiences, so maybe check that out if you’re interested.
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