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 .
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:
- an open subset des which is invariant under a faithful finite group operation ;
- a continuous mapping from to , which is invariant under , also called the map of the orbifold fold.
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.
- .
- The adhesive image should be unambiguous except for translation, i.e. H. There should be one with two adhesive images .
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.
- 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.
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).