Orbital folds

In the topology of a is orbifold (English: orbifold ) is a generalization of a manifold .

definition

Like a manifold, an orbifold is described by local properties. Unlike a manifold, which locally represents an open subset of , an orbifold is described locally by quotients of open subsets of according to finite group operations . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$

A -dimensional orbifold is a topological Hausdorff space , which is called the underlying space, with a cover by open subsets , which is closed under finite sections. For each there is: ${\displaystyle n}$ ${\displaystyle X}$ ${\displaystyle U_{i}}$ ${\displaystyle U_{i}}$

• an open subset des which is invariant under a faithful finite group operation ; ${\displaystyle V_{i}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \Gamma _{i}}$
• a continuous mapping from to , which is invariant under , also called the map of the orbifold fold. ${\displaystyle \phi _{i}}$ ${\displaystyle V_{i}}$ ${\displaystyle U_{i}}$ ${\displaystyle \Gamma _{i}}$

A set of maps is called an orbifold atlas when the following is given

• For each inclusion there is an injective group homomorphism and an - equivariant homeomorphism from to an open subset of (also known as an adhesive map) that is compatible with the cards, i.e. H. ${\displaystyle U_{i}\subset U_{j}}$ ${\displaystyle f_{ij}\colon \Gamma _{i}\rightarrow \Gamma _{j}}$ ${\displaystyle f_{ij}}$ ${\displaystyle \psi _{ij}}$ ${\displaystyle V_{i}}$ ${\displaystyle V_{j}}$

${\displaystyle \phi _{j}\psi _{ij}=\phi _{i}}$.

The adhesive image should be unambiguous except for translation, i.e. H. There should be one with two adhesive images . ${\displaystyle \psi _{ij},\psi _{ij}^{\prime }}$ ${\displaystyle g\in \Gamma _{j}}$ ${\displaystyle \psi _{ij}^{\prime }=g\psi _{ij}}$

Examples

• A simple example is an identification topology for a group effect with fixed points. Let the real number line be parameterized by the coordinate . The identification now creates a fixed point in . The quotient space resulting from identification is the simplest example of an orbiform fold. ${\displaystyle \mathbb {R} }$ ${\displaystyle x}$ ${\displaystyle x\sim -x}$ ${\displaystyle x=0}$
• Orbic folds that arise through the formation of quotients from the action of a finite group on a manifold are called good orbic folds .

Application in string theory

When the 10 + 1 dimensional heterotic string theory is compactified with an underlying manifold, one is mostly interested in when to get for a supersymmetric theory in four dimensions. Given some assumptions, it follows that these underlying manifolds must be Calabi-Yau manifolds . Because the explicit metric for almost all Calabi-Yau manifolds is not known, one tries to construct orbifoldings that are a limit of the respective Calabi-Yau manifolds, whereby the metric is explicitly known here. ${\displaystyle N=1}$

literature

• William Thurston : The Geometry and Topology of Three-Manifolds (Chapter 13) , Princeton University lecture notes 1980. PDF
• Barton Zwiebach : A First Course in String Theory , Cambridge University Press 2004, ISBN 0521831431
• Katrin Becker , Melanie Becker and John H. Schwarz : String Theory and M-Theory, A modern introduction , Cambridge University Press 2006, ISBN 0521860695
• Suhyoung Choi : Geometric structures on orbifolds and holonomy representations. Geom. Dedicata 104: 161-199 (2004).