In Mathematical, a orbifold is a generalization of the concept of variety container possible singularity S. These spaces were introduced explicitly for the first time by Satake in 1956 under the name of V-manifolds. To pass from the concept of variety (differentiable) to that of orbifold one adds as model buildings all the quotients of open of $R^n$ by the action of finished groups. The interest for these object was revived considerably at the end of the Seventies by William Thurston in bond with her conjecture of geometrization.

In physics, these spaces were considered initially as spaces of compactification in Théorie of the cords because in spite of the presence of singularities the theory is well defined there.
Lorsqu' they are used within the more particular framework of the Théorie of the supercordes the authorized orbifolds must have the additional property to be varieties of Calabi-Yau in order to preserve a minimal quantity of Supersymétrie. But if singularities are present it is an extension of the original definition of spaces of Calabi-Yau because those are in theory spaces without singularity.

Orbifolds in differential topology and geometry

Definition

A orbifold (without edge) $O$ is a metric Espace provided with an atlas orbifold, i.e. of a whole of quadruplets $(U\_i,\; \backslash \; U\_i\; tilde,\; \backslash \; phi\_i,\; \backslash \; Gamma\_i)$ where $U\_i$ is open of $O$, $\backslash \; U\_i$ tilde is open of $R^n$, $\backslash \; Gamma\_i$ is a finished group acting in way smoothes on $\backslash \; U\_i$ tilde and $\backslash \; phi\_i$ is a continuous application which goes down in homeomorphism of $\backslash \; tilde\; U\_i/\backslash \; Gamma\_i$ in $U\_i$. The whole of these quadruplets must check:

1. $\backslash \; displaystyle\; \backslash \; bigcup\_i\; U\_i=O$,

2. if $\backslash \; phi\_i\; (X)\; =\; \backslash \; phi\_j\; (there)\; \backslash ,$ then there exist vicinities $U\_x\; \backslash \; subset\; \backslash \; U\_i$ tilde and $U\_y\; \backslash \; subset\; \backslash \; U\_j$ tilde of $x$ and $y$ and a diffeomorphism $\backslash \; psi:\; U\_x\; \backslash \; to\; U\_y$ such as $\backslash \; phi\_i=\; \backslash \; phi\_j\; \backslash \; circ\; \backslash \; psi$.

Each quadruplet of an atlas is called chart of $O$. Two atlases define the same structure of orbifold if them meeting is still an atlas. When there is a risk of confusion, one will note $|O|$ topological space subjacent with a orbifold $O$.

A point not having a chart for which the acting group is commonplace is known as singular or exceptional.

An application smoothes between two orbifolds $O$ and $O\text{'}$ is a continuous function $f:\; |O|\backslash \; to|O\text{'}|$ such as for any point $x$ of $O$ it exite of the charts $(U,\; \backslash \; tilde\; U,\; \backslash \; phi,\; \backslash \; Gamma)$ and $(U\text{'},\; \backslash \; U\text{'}\; tilde,\; \backslash \; phi\text{'},\; \backslash \; Gamma\text{'})$ where $x\; \backslash \; in\; U$ and $f\; (U)\; \backslash \; subset\; U\text{'}$ and one smooth application $\backslash \; tilde\; F:\; \backslash \; tilde\; U\; \backslash \; to\; \backslash \; U\text{'}$ tilde with the top of $f$ which is équivariante compared to some homomorphism of $\backslash \; Gamma$ in $\backslash \; Gamma\text{'}$. Such an application is called immersion or immersion if all the $\backslash \; tilde\; f$ it are. An immersion which is a homeomorphism on its image is called a plunging. A diffeomorphism is a plunging surjective.

Simple examples

One builds most of the time a orbifold like Espace quotient of a variety (without singularity) by a symmetry discrete of this one. If the operation of symmetry does not have not of point fixes then one knows that the result is still a variety, but so on the other hand there exists one or more fixed points then the quotient has singularities on each one of those and is thus “truly” a orbifold.

For example if a circle $S\_1\; \backslash \; is\; considered,$ of ray $R\; \backslash ,$ which is a variety (of Dimension 1) and that one it paramétrise by a Angle $\backslash \; theta\; \backslash \; in\; \backslash ,$ then one can consider the two operations following

• $\backslash \; sigma\_1:\; \backslash \; theta\; \backslash \; rightarrow\; \backslash \; theta\; +\; \backslash \; pi\; \backslash ,$
• $\backslash \; sigma\_2:\; \backslash \; theta\; \backslash \; rightarrow\; -\; \backslash \; theta\; \backslash ,$

Then $\backslash \; sigma\_1\; \backslash ,$, which is a translation of one half-period, does not have a fixed point. The associated quotient, noted $S\_1/\backslash \; sigma\_1$ is thus still a variety and of fact it still acts of a circle but of ray $R/2\; \backslash ,$.

$\backslash \; sigma\_2\; \backslash ,$ on the other hand has two points fixes in $\backslash \; theta=0\; \backslash ,$ and $\backslash \; theta=\; \backslash \; pi\; \backslash ,$. $S\_1/\backslash \; sigma\_2\; \backslash ,$ is thus not a variety but well a orbifold. It is topologically equivalent with one segment $\backslash ,$ and is singular to both extremities.

In dimension 2, the simplest example is that of the quotient of a disc opened by the action of a rotation. The result is topologically a disc but its structure of orbifold includes/understands a singular point whose group is cyclic (the singularity is known as conical).

The first example of surface orbifold which is not the quotient from a smooth variety is obtained by resticking the preceding example and a disc smoothes to obtain a sphere having a point exceptional conical.

Coatings and fundamental group

Orbifolds being introduced in particular to give one structure with the quotient of a variety by the action of a group finished, one wants to be able to say that if $M$ is a variety on which acts a group finished $G$ then the application $M\; \backslash \; to\; M/G=O$ is a coating and that if $M$ is simply related then the orbifold quotient has as a fundamental group $G$.

Coatings

A coating of a orbifold $O$ by a orbifold $\backslash \; bar\; O$ is a continuous application $p:\; |\backslash \; bar\; O|\; \backslash \; to\; |O|$ such as any point $x$ of $O$ admits a vicinity $U$ checking: each related component $V$ of $p^\; \{-\; 1\}\; (U)$ admits a chart $\backslash \; phi:\; \backslash \; tilde\; V\; \backslash \; to\; V$ such as $p\; \backslash \; circ\; \backslash \; phi$ is a chart of $O$.

It is necessary to take guard with the fact that the $p$ application is in general not a coating between space topological.

One calls universal coating of $O$ a coating $p:\; |\backslash \; hat\; O|\; \backslash \; to\; |O|$ such as for any coating $q:\; |\backslash \; bar\; O|\; \backslash \; to\; |O|$ it exist a coating $r:\; |\backslash \; hat\; O|\; \backslash \; to\; |\backslash \; bar\; O|$ such that $p=q\; \backslash \; circ\; r$. According to a theorem of Thurston, all orbifold has a universal coating which is single with diffeomorphism near.

Fundamental group

The fundamental group of a orbifold is the group of automorphisms of its universal coating, i.e. diffeomorphisms of $\backslash \; hat\; O$ such as $p\; \backslash \; circ\; f=p$. It is noted $\backslash \; pi\_1\; (O)$ and one has $O=\; \backslash \; hat\; well\; O\; \backslash \; pi\_1\; (O)$.

Bond with the geometrization of Thurston

The conjecture of geometrization of Thurston affirms, coarsely, that any directional compact variety of dimension three can be cut out in a finished number of pieces carrying a geometrical structure. If the variety admits one not-free action of a finished group, it is enough to provide the orbifold quotient of a geometrical structure then to go up this one on the starting variety. This is why it was important of to include/understand the orbifolds because, paradoxically, it is easier to provide a orbifold (not-smooth) with a geometrical structure that to do it for a general smooth variety.

In addition the orbifolds of dimension two can play it role of fiber bases in circles called fibers of Seifert and who play a big role like pieces of the decomposition of Thurston (the definition of a fiber to the top of a orbifold is adapted that concerning the varieties by a close step of that used higher for the coatings).

Applications to the theory of the cords

Bond with spaces of Calabi-Yau

See also: Variety of Calabi-Yau

When in theory of the Supercorde S one seeks to build phenomenologic models realistic, it is necessary to carry out a dimensional Réduction because the supercordes are propagated naturally in a space with 10 Dimension S whereas the Espace-temps of the observable Univers has only 4 apparent dimensions. The formal constraints of the theory impose nevertheless restrictions on the space of compactification in which are hidden additional dimensions. In the case or one seeks a realistic model in 4 dimensions which has the Supersymétrie then the space of compactification must be a space of Calabi-Yau thus having 6 dimensions.

There exists of innombrales different possibilities of Calabi-Yau (of tens of thousands, in fact). Their general study is mathematically very complex and for a long time it was difficult to build some much explicitly. Orbifolds are then very useful because when they satisfy the constraints imposed by the supersymmetry which we mentioned then they constitute degenerated examples of Calabi-Yau, because of the presence of singularity S, but nevertheless completely acceptable from the point of view of the physical theory. Such orbifolds is then known as supersymmetric . The advantage of to consider supersymmetric orbifolds rather than Calabi-Yau generals is that their construction is in practical much simpler from a point of view purely technique.

It is then very often possible, as one will see it below, to connect a supersymmetric obifold having singularities with a family Continues spaces of Calabi-Yau without singularity.

Case of $T^4/\backslash \; mathbb\; \{Z\}\; \_2\; \backslash ,$ and K3

The Espace K3 has 16 cycles of dimension 2 which are topologically equivalent to usual spheres. If one makes to tighten the surface of these spheres towards 0 then K3 develops 16 singularities (this limit is at the edge of the Espace of modules of this variety). It is with this singular limit that the orbifold $T^4/\backslash \; mathbb\; \{Z\}\; \_2\; \backslash ,$ corresponds obtained in quotientant the 4-torus by the symmetry of inversion of each one of coordinates (i.e. $x\; \backslash \; rightarrow\; -\; X\; \backslash ,$ in each direction).

Symmetry mirror

See also: Symmetry mirror

Symmetry mirror is a concept whose idea was created in 1988. She says that two Espace of Calabi-Yau lead to same physics if their total number of holes in all dimensions are equal. That wants to say that even if their number of holes is not not equal by dimensions, they lead to the same physics if their total number of holes is identical. I understand by even physical , that of Calabi-Yau with same the numbers of holes with the total leads to a universe with the same number of family.

Operation

This technique was invented in the Eighties by Dixon, Vafa, Witten and Harvey. The operation of orbifold consists in creating a new form of Calabi-Yau by connecting various points of initial Calabi-Yau. It is a method to handle spaces mathematically of Calabi-Yau by connecting some of these points. But these handling is so complicated that the physicists did not plan to reproduce it too on a form complicated that a space of Calabi-Yau in all its splendor.

The operation of orbifold is not a geometrical transition like the Transition from failure or the Transition from conifold which cause monstrous cataclysms like tears of space time.

Consequences

By idantifiant thus certain points, the space of Calabi-Yau of “departure” (that we will call α ) and that of “arrival” (that we will call β ) different by their number of hole in each dimensions: holes of even dimensions (2 E , 4 E , 6 E dimension) of Calabi-Yau β was equal to the number of holes in odd dimensions (1 era , 3 E , 5 E ) of Calabi-Yau α and vice versa! But in other words, the total number of holes does not change. But this inversion par with odd leads to geometrical stuctures extremely disctinctes.

See too

• Theory of Kaluza-Klein
• Space of Calabi-Yau
• Orientifold

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