What are Calabi-Yau Manifolds?

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In summary, Calabi-Yau manifolds are complex spaces that meet certain conditions, including being Kaehler manifolds and satisfying a topological constraint. These manifolds were first studied by E. Calabi and later proved by S. T. Yau. They are important in string theory and a good resource for learning about them is the book "Geometry, Topology, and Physics" by M. Nakahara.
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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.
 
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Where to begin to learn such things?
Can you recommend some self contained books about subject.
 
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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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Calabi-Yau manifolds are a type of complex manifold in mathematics that have a significant role in theoretical physics, particularly in string theory. They were first introduced by the mathematician Eugenio Calabi and later studied by Shing-Tung Yau. These manifolds are defined as compact, smooth, and complex manifolds with special geometric properties. They have a unique and intricate structure that allows for the compactification of extra dimensions in string theory. In simpler terms, Calabi-Yau manifolds are mathematical objects that are used to describe the shape and structure of the extra dimensions in string theory. They are an essential tool in understanding the fundamental nature of the universe and are still an active area of research in both mathematics and physics.
 

What are Calabi-Yau Manifolds?

Calabi-Yau manifolds are complex, six-dimensional spaces that are important in string theory and algebraic geometry. They are named after mathematicians Eugenio Calabi and Shing-Tung Yau.

What is the significance of Calabi-Yau Manifolds in string theory?

In string theory, Calabi-Yau manifolds are used to represent the extra dimensions required by the theory. They play a crucial role in compactifying the dimensions and making the theory consistent with observed physical phenomena.

How are Calabi-Yau Manifolds related to algebraic geometry?

Calabi-Yau manifolds have important connections to algebraic geometry, a branch of mathematics that studies geometric objects defined by polynomial equations. In algebraic geometry, these manifolds are known as K3 surfaces and are used to study complex surfaces and their properties.

Are Calabi-Yau Manifolds the only possible extra dimensions in string theory?

No, Calabi-Yau manifolds are just one example of a class of manifolds that can be used to represent the extra dimensions in string theory. Other types of manifolds, such as F-theory or G2-manifolds, have also been studied and proposed as possible extra dimensions.

What are some current areas of research involving Calabi-Yau Manifolds?

Researchers are currently exploring the connections between Calabi-Yau manifolds and other areas of mathematics, such as mirror symmetry and special holonomy manifolds. They are also investigating the role of these manifolds in cosmology and their potential implications for the origins of the universe.

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