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The argument goes like following:

Consider the field configuration such that the action is finite,

therefore we must require the field vanishes at spacetime infinity,

hence, we defined a map from the spacetime manifold to the group manifold of the theory.

I don't understand the last sentence of my previous paragraph.

Let me quote an example from Weinberg's QFT book of volume II.

He considered the Goldstone boson part of action in [tex] d [/tex]-dim Euclidean space(d>2),

[tex] S[\pi] = \int d^dx \left[ \frac{1}{2}\sum_{a,b} g_{ab}(\pi) \partial_i \pi_a \partial_i \pi_b + \cdots \right] [/tex]

To have finite action [tex] S [/tex], we impose the boundary condition that

[tex] \partial_i \pi_a \rightarrow 0[/tex] at spacetime infinity.

He then said something I don't understand:

"

*The Goldstone boson field [tex] \pi_a [/tex] at any point form a homogeneous space, the coset space [tex]G/H[/tex], for which it is possible to transform any one field value to any other by a transformation of [tex]G[/tex], so by a global [tex]G[/tex] transformation it is always possible to arrange that the asymptotic limit [tex]\pi_{a\infty}[/tex] takes any specific value, say 0*"

Then, he made a conclusion which I don't understand how he got this:

"

*The filed [tex] \pi_a [/tex] thus represents a mapping of the whole [tex]d[/tex]-dimensional space, with the sphere [tex]r=\infty[/tex] taken as a single point, into the manifold [tex]G/H[/tex] of all field values.*"

So, after the emergence of this mapping, one could classify it according to homotopy group.

But I don't understand why such a "finite" action consideration leads to such a mapping between two manifolds, i.e. spacetime manifold(with infinity taken as a point) and group manifold.

Any discussion would be appreciated.