What are Calabi-Yau Manifolds?

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SUMMARY

Calabi-Yau manifolds are complex manifolds characterized by n-tuples of complex numbers, satisfying specific conditions. They are Kaehler manifolds, meaning they possess a compatible Riemannian metric and Hermitian form. Additionally, they meet the topological requirement of having a vanishing first Chern class, indicating smoothness. The significance of Calabi-Yau manifolds in string theory is underscored by Yau's proof of Calabi's conjecture regarding the vanishing of Ricci curvature, establishing their local flatness.

PREREQUISITES
  • Understanding of complex numbers and n-tuples
  • Familiarity with Kaehler manifolds and Riemannian geometry
  • Knowledge of Chern classes in topology
  • Basic principles of string theory
NEXT STEPS
  • Study the properties of Kaehler manifolds in depth
  • Explore the implications of the vanishing first Chern class
  • Learn about Ricci curvature in Riemannian geometry
  • Read "Geometry, Topology, and Physics" by M. Nakahara for a comprehensive introduction
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This discussion is beneficial for mathematicians, physicists, and students interested in advanced geometry, topology, and their applications in string theory.

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what are they?
 
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They are manifolds, with coordinates of each point being n-tuples. Each coordinate in these n-tuple is a complex number, like x+iy. In addition to this they meet two other conditions:

1) They are Kaehler manifolds. As manifolds they have a Riemannian metric, and as complex spaces they have a Hermitian form, and in Kaehler manifolds these two conditions are compatible. I can't get any more specific than that without giving a course in Kaehler manifolds.

2) They satisfy a topological constraint called the vanishing of the first Chern class. This means that they are pretty smooth.

Calabi conjectured that in manifolds like this the Ricci curvature (from Riemannian geometry) would vanish. They would be "locally flat" in a technical sense.

Yau proved Calabi's conjecture and constructed the family of Calabi-Yau manifolds that string theorists use today.
 
Where to begin to learn such things?
Can you recommend some self contained books about subject.
 
I find Geometry, Topology, and Physics, by M. Nakahara to be an excellent introduction to these topics. It assumes an undergraduate familiarity with set theory, calculus, complex analysis, and linear algebra, but given that it is reasonably self-contained.
 

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