Strings, Rings, and Other Things

I'm reasonably sure that there are manifolds with nontrivial topology that are still globally hyperbolic. It's just when those topologies involve closed timelike curves that things get messy. The sum over histories should include representatives of each allowed topological sector, regardless. The degree to which any two branches of the wave function will have an effect on each other depends on how similar they are: two branches that differ only in the configuration of the electric field (a "finite" difference, but possibly including topological changes in the field) will tend to have substantial quantum interference. Two branches that have a large number of differences (topological or not) will tend not to have any substantial interference at all. (Ok, technically those statements are mostly true only if you read them as "the neighborhoods of two branches", in the sense that I used the term in my earlier sum over histories post.)