Calabi-Yau manifold

Calabi-Yau manifolds are among the 'standard toolkit' for string theorists today.

Together with two-dimensional complex tori, they are the Calabi-Yau manifolds of dimension two.

Calabi-Yau manifold was named after him.

Calabi-Yau manifolds are important in superstring theory.

That leads to the name Calabi-Yau manifolds.

Fano varieties are quite rare, compared to other families, like Calabi-Yau manifolds and general type surfaces.

Similar concepts apply when studying spacetime and Calabi-Yau manifolds.

Instead, one finds that there are two Calabi-Yau manifolds that give rise to the same physics.

The general study of Calabi-Yau manifolds is mathematically complex and for a long time examples have been hard to construct explicitly.

In physics and mathematics, mirror symmetry is a relation that can exist between two Calabi-Yau manifolds.

There are many different inequivalent definitions of a Calabi-Yau manifold used by different authors.

Their double covers are Calabi-Yau manifolds for both definitions (in fact K3 surfaces).

Most definitions assert that Calabi-Yau manifolds are compact, but some allow them to be non-compact.

It is a Calabi-Yau manifold.

A one-dimensional Calabi-Yau manifold is a complex elliptic curve, and in particular, algebraic.

This family of quintic hypersurfaces is the most famous example of Calabi-Yau manifolds.

Such orbifolds are called "supersymmetric": they are technically easier to study than general Calabi-Yau manifolds.

On the other hand, Mirror symmetry allows for the mathematical similarity between two distinct Calabi-Yau manifolds.

Is it interpenetrated by innumerable Calabi-Yau manifolds, which connect our 3-dimensional universe with a higher dimensional space?

It was also realised in 1985 that to obtain supersymmetry, the six small extra dimensions need to be compactified on a Calabi-Yau manifold.

All Calabi-Yau manifolds are spin.

In the late 1980s, it was noticed that given such a physical model, it is not possible to uniquely reconstruct a corresponding Calabi-Yau manifold.

Some definitions put restrictions on the fundamental group of a Calabi-Yau manifold, such as demanding that it be finite or trivial.

In two complex dimensions, the K3 surfaces furnish the only compact simply connected Calabi-Yau manifolds.

Important cases include Calabi-Yau manifolds and hyperkähler manifolds.