# What is Asymptotic Flatness in Differential Geometry

## encyclopedia

“Calabi-Yau” forwards here. For the piece by Susanna Speier see Calabi-Yau (piece).

Riemannian manifold with SU (n) holonomy A 2D section of a 6D Calabi Yau Quintic manifold.

In algebraic geometry a Calabi-Yau manifold, also known as Calabi-Yau roomis a special type of distributor that has properties such as Ricci flatness, which result in applications in theoretical physics. In superstring theory in particular, it is sometimes assumed that the additional dimensions of spacetime have the shape of a 6-dimensional Calabi-Yau manifold, which led to the idea of ​​mirror symmetry. Her name was taken from Candelas et al. (1985), after Eugenio Calabi (1954, 1957), who first hypothesized that such surfaces could exist, and Shing-Tung Yau (1978), who proved the Calabi conjecture.

Calabi-Yau distributors are complex distributors that are generalizations of K3 surfaces in any number of complex dimensions (i.e., any even number of real dimensions). They were originally defined as compact Kähler distributors with a vanishing first Chern class and a Ricci flat metric, although many other similar but not equivalent definitions are sometimes used.

### Definitions 

The Shing-Tung Yau motivational definition is a compact Kähler manifold with a vanishing first Chern class that is also Ricci-flat.

There are many other definitions of a Calabi-Yau manifold used by different authors, some of which are not equivalent. This section summarizes some of the most common definitions and the relationships between them.

A calabi yau n-fold or Calabi-Yau manifold of (complex) dimension n is sometimes defined as compact n-dimensional Kähler manifold M. meet one of the following equivalent conditions:

These conditions imply that the first integral Chern class of M. disappears. Even so, the opposite is not the case. The simplest examples where this happens are hyperelliptic surfaces, finite quotients of a complex torus of complex dimension 2, which have a vanishing first integral Chern class but no trivial canonical bundle.

For a compact n-dimensional Kähler manifold M. The following conditions are equivalent to each other, but weaker than the above conditions, although they are sometimes used as a definition of a Calabi-Yau manifold:

• M. has disappeared the first real Chern class.
• M. has a Kähler metric with vanishing Ricci curvature.
• M. has a Kähler metric with local holonomy in SU (n).
• A positive force of the canonical bundle of M. is trivial.
• M. has finite coverage that has trivial canonical bundle.
• M. has a finite cover that is a product of a torus and a simply connected distributor with a trivial canonical bundle.

If a compact Kähler manifold is simply connected, the weak definition above corresponds to the stronger definition. Enriques surfaces give examples of complex manifolds with Ricci flat metrics, but their canonical bundles are not trivial, so they are Calabi-Yau manifolds according to the second but not the first definition above. On the other hand, their double covers are Calabi-Yau distributors for both definitions (actually K3 surfaces).

By far the most difficult part of proving the equivalences between the various properties mentioned above is proving the existence of Ricci flat metrics. This follows from Yau's proof of the Calabi conjecture, which implies that a compact Kähler manifold with a vanishing first real Chern class has a Kähler metric in the same class with vanishing Ricci curvature. (The class of a Kähler metric is the cohomology class of the associated 2-form.) Calabi showed that such a metric is unique.

There are many other dissimilar definitions of Calabi-Yau manifolds that are sometimes used and differ in the following ways (among others):

• The first Chern class can disappear as an integral class or as a real class.
• Most definitions claim that Calabi-Yau distributors are compact, but some allow them to be non-compact. When generalizing to non-compact manifolds is the difference must disappear asymptotically. Here, is the Kähler form associated with the Kähler metric. (Gang Tian; Shing-Tung Yau 1990, 1991).
• Some definitions limit the basic group of a Calabi-Yau manifold, e.g. B. the requirement that it is finite or trivial. Each Calabi-Yau manifold has a finite coverage that is the product of a torus and a simply connected Calabi-Yau manifold.
• Some definitions require that the holonomy be exactly equal to SU (n) rather than a subset of what implies that the Hodge numbers disappear for . Abelian surfaces have a flat Ricci metric with a holonomy strictly smaller than SU (2) (trivial, in fact). So they are not Calabi-Yau manifolds according to such definitions.
• Most definitions assume that a Calabi-Yau manifold has a Riemannian metric, but some treat them as complex manifolds with no metric.
• Most definitions assume that the manifold is not singular, but some allow mild singularities. While the Chern class is not well defined for singular Calabi-Yau classes, the canonical bundle and canonical class can still be defined if all singularities are Gorenstein, and therefore can be used to help define a smooth Calabi-Yau - Diversity to expand on a possibly unique Calabi-Yau variety.

### Examples 

The most important basic fact is that every smooth algebraic variety embedded in a projective space is a Kähler manifold, since there is a natural Fubini-Study metric for a projective space that can be constrained to the algebraic variety. By definition, if ω is the Kähler metric for the algebraic variety X and the canonical bundle K.X. is trivial, then X is Calabi-Yau. In addition, there is a unique Kähler metric ω on X such that [ω0] = [ω] ∈ H.2((X.,R.), a fact that was conjectured by Eugenio Calabi and proven by Shing-Tung Yau (see Calabi conjecture).

In a complex dimension, the only compact examples are tori, which form a one-parameter family. The Ricci flat metric on a torus is actually a flat metric, so holonomy is the trivial group SU (1). A one-dimensional Calabi-Yau manifold is a complex elliptic curve, especially an algebraic one.

In two complex dimensions, the K3 surfaces provide the only compact, easily connected Calabi-Yau distributors. Not simply connected examples are Abelian surfaces. Enriques surfaces and hyperelliptical surfaces have the first Chern class to vanish as an element of the real cohomology group, but not as an element of the integral cohomology group. Hence, Yau's theorem about the existence of a Ricci flat metric still holds true for them, but they are sometimes not considered to be Calabi-Yau manifolds. Abelian surfaces are sometimes excluded from being classified as Calabi-Yau because their holonomy (again, the trivial group) is an appropriate subgroup of SU (2) rather than being isomorphic to SU (2). However, the Enriques surface subset does not fully correspond to the SU (2) subset in the string theory landscape.

In three complex dimensions, the classification of the possible Calabi-Yau manifolds is an open problem, although Yau suspects that there are a finite number of families (albeit a much larger number than his estimate from 20 years ago). In turn, Miles Reid also hypothesized that the number of topological types of Calabi-Yau-3 folds is infinite and that they can all be continuously converted into one another (through certain mild singularizations such as conifolds) - similar to how Riemann surfaces can. An example of a three-dimensional Calabi-Yau manifold is a non-singular quintic that is three-fold CP4This is the algebraic variety that consists of all zeros of a homogeneous quintic polynomial in the homogeneous coordinates of the CP4. Another example is a smooth model of the Barth-Nieto-Quintin. Some discrete quotients of the quintine of different ones Z.5 Actions are also Calabi-Yau and have received a lot of attention in literature. One of them is related to the original Quintic by mirror symmetry.

For any positive integer n, the null set in the homogeneous coordinates of complex projective space CPn+1of a non-singular homogeneous degree n + 2 polynomial in n + 2 variables is a compact Calabi-Yau n-wrinkles. The case n = 1 describes an elliptic curve, while for n = 2 you get a K3 surface.

More generally, Calabi-Yau varieties / orbifolds can be found as weighted complete intersections in a weighted projective space. The main tool for finding such spaces is the additional formula.

All hyper-Kähler manifolds are Calabi-Yau manifolds.

### Applications in superstring theory 

Calabi-Yau manifolds are important in superstring theory. Calabi-Yau manifolds are essentially shapes that fill the space requirements for the six “invisible” spatial dimensions of string theory, which may be smaller than our currently observable lengths because they have not yet been recognized. A popular alternative known as large additional dimensions, and often found in Braneworld models, is that the Calabi-Yau is large, but we limit ourselves to a small subset on which it intersects a D-brane. Further extensions into higher dimensions are currently being investigated with additional consequences for the general theory of relativity.

In the most conventional superstring models, ten putative dimensions are supposed to exist in string theory as four, which we know to carry a kind of vibration with the fiber dimension six. Compaction on Calabi-Yau n- Wrinkles are important because they leave some of the original supersymmetry unbroken. More precisely, without flux, the compression to a Calabi-Yau 3-fold (real dimension 6) leaves a quarter of the original supersymmetry unbroken if the holonomy is the complete SU (3).

More general is a flux-free compaction on one n-Diversity with holonomy SU (n) Leaves 21−n the original supersymmetry unbroken, corresponding to 26−n Charges in the case of a compression of type II supergravity or 25−n Charges in a Type I compaction. When rivers are included, the supersymmetry condition implies instead that the compaction manifold is a generalized Calabi-Yau, a term introduced by Hitchin (2003). These models are known as flux densities.

F-theory compactifications on different Calabi-Yau quadruples offer physicists a method to find a variety of classical solutions in the so-called string theory landscape.

A group of low-energy string vibration patterns is associated with each hole in the Calabi-Yau space. Since string theory says that our known elementary particles correspond to low-energy string vibrations, the presence of multiple holes causes the string patterns to fall into multiple groups or families. Although the following statement has been simplified, it conveys the logic of the argument: if the Calabi-Yau has three holes, then three families of vibrational patterns and thus three families of particles are observed experimentally.

Since strings vibrate through all dimensions, the shape of the coiled strings logically influences their vibrations and thus the properties of the observed elementary particles. For example, Andrew Strominger and Edward Witten have shown that the particle masses depend on the type of intersection of the various holes in a calabi yau. In other words, Strominger and Witten found that the positions of the holes relative to each other and to the substance of the Calabi-Yau space affect the particle masses in some way. This applies to all particle properties.

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