Trouble with Construction of Sphaleron S

Question

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

Comments

Pleas see Chapter 11 in Manton and Sutcliffe Topological solitons for a detailed description books.google.co.il/books/about/… I'll try to write an answer in a few days

This post imported from StackExchange Physics at 2017-08-29 09:33 (UTC), posted by SE-user David Bar Moshe

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@DavidBarMoshe: Thanks for the input. The issue I have with Manton's book is that given my current level of knowledge, it usually creates more questions than answers! The toy model he considers there, i.e sphalerons on a circle, is basically what I've been trying to visualize for the past few weeks. I may be wrong, but I think the heart of my problem lies in my inability to form a decent visualization of the configuration space of a field. I'm still eagerly looking forward to your response though.

This post imported from StackExchange Physics at 2017-08-29 09:33 (UTC), posted by SE-user Optimus Prime