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Re: Strings a minumum topology

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Posted by DickT on September 15, 2003 at 07:22:39:

In Reply to: Re: Strings a minumum topology posted by paultrr on September 15, 2003 at 04:15:20:

Paul,

Nonlinear partial differential equations give instanton solutions, along with solitons. Sometimes you can suppress them with boundary conditions, but in general you don't have a choice about having them in your theory or not.

On the orbifolds, we once upon a time had this example of an importnd case.

Start with a circle (a one dimensionl manifold) and think of a diameter through it. Now conceive the group of reflections in that diameter, each point (except two) on the circle has a mirror image point on the other side of the diameter. The two points that don't are the endpoints of the diameter. Yes?

Now consider two points to be in the same "equivalence class" if they are mirror images of each other. And for the end points of the diameter define them to be equivalent to themselves, so you have two equivalence classes with only one point in each. Then you can show that the set of equivalence classes has the topology of a line segment (identified with that diameter). This is the "gluing" operation I spoke of.

But the fact that the equivalence classes are not all the same, that the two endpoints are "fixed points" of the group of reflections makes a lot of difference to the physics. So while the orbifold here might be a topological manifold, it is not one analytically.

Regards,
Dick


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