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

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> I just don’t understand how they’re relevant to Physics

Join the rest of the physicists. String theory, even before you get to superstring theory, is an incredible amount of faffing about with equations that ultimately results in something that is unfalsifiable without about 6-7 orders of magnitude more energy or precision than we have access to.

If you want any explanation of 10 dimensional manifolds and such though, I would suggest finding someone specific who works in the field. I know about 30 doctors of physics, and probably only one of them would be able to give a satisfactory answer to your question (and then his answer would be pretty unintelligible without your own doctorate in physics) because it’s far more theoretical than even other areas of theoretical physics.

That is to say, this is the wrong forum to ask.

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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