Calabi-Yau manifold

A cut through a Calabi-Yau, the quintic

Calabi-Yau manifolds , Calabi-Yau for short , or Calabi-Yau spaces , are special complex manifolds in mathematics . They 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 for Kaluza-Klein compactification used the theory.

definition

A Calabi-Yau manifold (or a Calabi-Yau space) is a compact Kähler manifold with vanishing first Chern class . According to a conjecture by Eugenio Calabi from 1954, which was proved by Shing-Tung Yau in 1977, the latter condition is equivalent to the existence of a Ricci flat metric for compact manifolds . Equivalent can be a complex with -dimensional Calabi-Yau as a manifold - Holonomy define. This in turn is equivalent to the existence of a globally defined, nowhere vanishing holomorphic ( n , 0) form . ${\displaystyle n}$ ${\displaystyle \operatorname {SU} (n)}$

Examples

• ${\displaystyle n=1}$: The Riemann surfaces that are Calabi-Yau manifolds are the elliptic curves . Since the torus metric is flat, the holonomy group is trivial.
• ${\displaystyle n=2}$: In two complex dimensions there are two different classes of Calabi-Yau manifolds: K3 surfaces (with all of SU (2) as holonomy group) and compact complex tori (with trivial holonomy group).
• ${\displaystyle n=3}$: In three complex dimensions there is no complete classification of Calabi-Yau manifolds. A well-known example is the quintic, i. H. the set of zeros of a 5th degree polynomial, in complex projective space . ${\displaystyle \mathbb {CP} ^{4}}$

Application in string theory

Calabi-Yaus play an important role in the supersymmetric version of string theory , since in its simplest version it is formulated in ten dimensions. In order to obtain the known four spacetime dimensions, it is assumed that the six extra dimensions are compact and sufficiently small and therefore cannot be detected with today's experiments. The theory in the remaining four non-compact directions depends essentially on the chosen geometry of these internal six dimensions.

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

Generalizations

By Nigel Hitchin a generalization of the concept of Calabi-Yau, was a so-called Generalized Calabi-Yau (Generalized Calabi-Yau) proposed in the context of a "generalized complex geometry." This extension is also used in string theory.

literature

• M. Gross, D. Huybrechts, D. Joyce: Calabi-Yau Manifolds and Related Geometries , Springer, Berlin 2003, ISBN 3-540-44059-3 .
• Tristan Hübsch: Calabi-Yau Manifolds: A Bestiary for Physicists World Scientific, Singapore 1992, ISBN 981-02-0662-3 .
• Noriko Yui: Calabi-Yau varieties and mirror symmetry. American Math. Soc., Providence 2003, ISBN 0-8218-3355-3 .

Web links

Commons : Calabi-Yau variety  - collection of images, videos and audio files

• Definition of Calabi-Yau manifolds at mathworld (English)
• Calabi-Yau Home Page at the University of Bonn (English)

swell

1. ^ Eugenio Calabi: The space of Kähler metrics . In: Proceedings of the International Congress of Mathematicians . tape 2 . Amsterdam 1954, p. 206-207 .
2. ↑ Shing-Tung Yau: Calabi's conjecture and some new results in algebraic geometry . In: Proceedings of the National Academy of Sciences . tape 74 , no. 5 , January 5, 1977, p. 1798-1799 , PMID 16592394 .
3. ↑ Philip Candelas , Gary Horowitz , Andrew Strominger , Edward Witten : Vacuum configurations for superstrings . In: Nuclear Physics B . tape 258 , 1985, pp. 46-74 , doi : 10.1016 / 0550-3213 (85) 90602-9 .
4. ↑ Nigel Hitchin: Generalized Calabi – Yau Manifolds . In: The Quarterly Journal of Mathematics . tape 54 , no. 3 , January 9, 2003, p. 281–308 , doi : 10.1093 / qmath / hag025 , arxiv : math.dg / 0209099 .