Trouble with Construction of Sphaleron S

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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 $S^2$ 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:

$$S^1\wedge S^2\sim S^3$$ and the map $$S^3\rightarrow S^3$$ where the target space is the Higgs vacuum manifold three-sphere, now leads to nontrivial topology.

My questions are:

1) What is a "loop" in configuration space in physical terms?

2) Is this a constraint in our theory? If so what is this constraint?

3) Why does considering a loop in configuration space mean considering $$S^1\times S^2$$?

This post imported from StackExchange Physics at 2017-08-02 15:44 (UTC), posted by SE-user Optimus Prime

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