Calabi–Yau manifold

With Calabi -Yau manifold, Calabi - Yau short, or even Calibi -Yau spaces is called in mathematics special complex manifolds, which play a role in algebraic geometry. The theoretical physics, especially string theory, also has a special interest in these objects, as six -dimensional Calabi -Yau manifolds are used for the Kaluza-Klein compactification of the theory.

Definition

A Calabi -Yau manifold (or a Calibi -Yau space ) is a compact Kähler manifold with vanishing first Chern class. The latter condition is for compact manifolds after a conjecture of Eugenio Calabi of 1954, which was proved by Shing-Tung Yau in 1977, equivalent to the existence of a Ricci - flat metric. Equivalent, one can define ( n ) - holonomy of a complex n-dimensional Calabi - Yau as a manifold with SU. This is in turn equivalent to the existence of a globally defined, nowhere vanishing holomorphic (n, 0 )-form.

Examples

• N = 1: The Riemann surfaces, the Calabi -Yau manifolds are the elliptic curves. Since the torus metric is flat, the holonomy is trivial.
• N = 2: In two complex dimensions, there are two different classes of Calabi -Yau manifolds: K3 surfaces ( with all SU (2) as holonomy group ) and compact complex tori ( with trivial holonomy ).
• N = 3: In three complex dimensions, there is no complete classification of Calabi -Yau manifolds. A well-known example is the Quintik, i.e. the zero set of a polynomial of degree 5, in the complex projective space.

Application in string theory

Calabi - Yau play an important role in the supersymmetric version of string theory, as it is formulated in its simplest version in ten dimensions. In order to obtain the known four space-time dimensions, it is assumed that the six extra dimensions are sufficiently small and compact, and therefore can not be detected with the present experiments. The theory in the remaining four non-compact directions depends essentially on the chosen geometry of the internal six dimensions.

The special significance of the Calabi -Yau property is that a compactification of ten-dimensional string theory can lead to a Calabi -Yau geometry to a four-dimensional theory in flat Minkowski space and with unbroken supersymmetry.

Generalizations

From Nigel Hitchin a generalization of the concept of Calabi - Yau, Calabi - Yau a so-called Generalized ( Generalized Calabi - Yau ) has been proposed in the context of a " generalized complex geometry ". Also this extension is applied in string theory.