What is an Orbibundle? Explanation

[meteor]

I would appreciate an explanation about what's an orbibundle

 

[Haelfix]

Ok, it depends on how detailed an answer you want. Pick up a book on mathematical topology, and they'll go through a lengthy axiom, defintion, theorem spiel until they get to orbifolds, and then orbibundles. Warning: You'll need to understand category theory. 'I didn't the first time around, and I was immensely confused'

Sloppily speaking, orbifolds are generalizations of manifolds. Basically they are just like manifolds (eg an atlas of charts.. ie the open union of point sets, with every open set homeomorphic to R^N). The main difference, is that the charts (called uniformizers) are like : (I haven't bothered to figure out how to use math yet on these forums)

Psi(a) --> V/group(P).. Where V is a vector space, and group(P) acts on V (often taken to be the complex field). For most trivial group actions, the whole thing resembles a manifold (if we say restrict V to R^N(, but regardless these maps live in neighborhoods of {0} of V/(P). In principle the finite group(P) varies from point to point.

Now an orbibundle, are just what you would naively think they are.. By definition the base space is an orbifold and the fiber (equipped with a local trivilization) over a point p is a vector space modulo the group action of the orbifold.

The whole point of this structure, is to avoid a certain unavoidable redundancy of domain of the exponential map around highly singular points.

I hope this hasn't been too sloppy, and that it helps a bit.

 

[Mike2]

Originally posted by Haelfix
: (I haven't bothered to figure out how to use math yet on these forums)

Get TeXaide from:
http://www.dessci.com/en/products/texaide/

Write an equation and copy and paste the Latex into the forum window.

 

[rick1138]

A quick, intuitive description of an orbifold is that an orbifold is a manifold containing isolated singularities that are isomorphic to a cone.

 

[Mike2]

Originally posted by rick1138
A quick, intuitive description of an orbifold is that an orbifold is a manifold containing isolated singularities that are isomorphic to a cone.

What do you mean "isolated singularities"? Does that mean some function on the manifold goes to infinity at some points on the manifold?

Thanks.

 

[rick1138]

Perhaps singularity is a bad term, no infinities here - the essential point is that the patch in question is isomorphic to a cone, which has a point (at the tip), where there is no derivative. Remember that on a manifold the Jacobian must never vanish, on an orbifold this requirement is relaxed.

 

[rick1138]

In string theory, obifolds are a potential stand-ins for Calabi-Yau manifolds as a candidate for forming the structure of the 6 compacted space dimensions.