Consider SU(2) YMH theory without fermions. Three-space is compactified by adding the sphere at infinity and configuration space is the space of all static finite-energy 3D gauge and Higgs fields in a particular gauge.
As we are looking for finite-energy solutions to the field equations the SU(2) gauge field must tend to a pure gauge and the Higgs field to its vacuum value. This means we can map the sphere at infinity $S^2_\infty$ into the Higgs vacuum manifold SU(2)~$S^3$.
In order to achieve nontrivial topology we consider a loop in configuration space, which in turn induces a loop in the space of the mappings defined above. By setting appropriate constraints we can go from the Cartesian to smash product and our domain space is now:
$$S^1\wedge S^2_\infty\sim S^3_\infty$$ and the map $$S^3_\infty\rightarrow S^3$$ where the target space is the Higgs vacuum manifold three-sphere, now leads to nontrivial topology.
My questions are:
1) If a "point" in configuration space corresponds to some gauge and Higgs field configuration (in a particular gauge), then a path in this space must be a series of continuously varying configurations with respect to some parameter. What is this parameter? Do two neighboring points along this path somehow correspond to a slightly varied field in physical space?
2) Similarly, a "path" in the space of mappings of $$S^2_\infty\rightarrow S^3\sim SU(2)$$ must mean a set of continuously varying maps from spatial infinity to the Higgs vacuum manifold. What exactly is parametrizing this path?
3) How does a loop in configuration space induce a loop in the space of the above maps?
4) Why does considering a loop in configuration space mean considering $$S^1\times S^2_\infty$$?
This post imported from StackExchange Physics at 2017-08-29 09:33 (UTC), posted by SE-user Optimus Prime