What exactly is a Calabi-Yau manifold?

Question

...and where can I get one?

Answer 1

Calabi-Yau manifolds form a special class of surfaces used in some branches of mathematics (such as algebraic geometry) as well as in theoretical physics. For instance, in superstring theory the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi-Yau manifold. The precise definition of a Calabi-Yau manifold, given below, builds on a considerable mathematical background.


Formal definition
A Calabi-Yau manifold is a Kähler manifold with a vanishing first Chern class. A Calabi-Yau manifold of complex dimension n is also called a Calabi-Yau n-fold. The mathematician Eugenio Calabi conjectured in 1957 that all such manifolds admit a Ricci-flat metric (one in each Kähler class), and this conjecture was proved by Shing-Tung Yau (丘成桐) in 1977 and became Yau's theorem. Consequently, a Calabi-Yau manifold can also be defined as a compact Ricci-flat Kähler manifold.

Equivalently one may define a Calabi-Yau n-fold as a manifold with an SU(n) holonomy. Yet another equivalent condition is that the manifold admit a global nowhere vanishing holomorphic (n,0)-form.

The first Chern class vanishes if and only if the canonical bundle is trivial, which in turn is the case if and only if the canonical class is the zero class. While the Chern class fails to be well-defined for singular Calabi-Yau's, the canonical bundle and canonical class may still be defined and so may be used to extend to definition of a smooth Calabi-Yau manifold to a possibly singular Calabi-Yau variety.

Answer 2

You may very well be living in one ☺

There's a pretty good article on them (obviously written by a mathematician who understands manifolds and superstring theory) at
http://en.wikipedia.org/wiki/Calabi-Yau_manifold


Doug

Answer 3

It fits directly into the sub port for the plasma injectors for the hyperdrive on the millenium falcon.For parts try a junk yard on Corellia or Tatooine.I don't think they make them anymore they were in production a long time ago in a ....

Answer 4

Why, you've had them all along! All you have to do is click your heels.