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There are two conflicting perceptions that I have regarding the notion of ground state post-SSB. Consider the Higgs mechanism e.g. for the electroweak theory.

On the one hand, we say that the vacuum manifold is just a redundancy: all sectors are related by gauge transformations, and we cannot make a transition between them, even non-perturbatively because tunneling is suppressed by the infinite volume of spacetime posited in QFT. See e.g. the following SE posts (1), (2) and (3). The conclusion is that we must treat all of them as the same state.

On the other hand, we have cosmic strings, which as I understand are theoretically sound even if not observed (and likely won't be any time soon). These, as well as the corresponding domain walls and monopoles require that when the symmetry breaking actually occurs, at different points in spacetime different VEVs will be chosen, in the sense that if we follow some trajectory, we will get varying "phases", i.e. move along the moduli space of vacua. In particular, if $M = G/H$ has nontrivial homotopy groups we will get the mentioned phenomena. But this hinges on the more basic fact that I'm puzzled with, which is that there seems to be physical meaning to the point we are sitting in the moduli space.

  1. How are these two statements compatible? I suppose it has to do with the fact that "gauge-equivalent" isn't the same as "the same", in a similar sense as in other phenomena related to large gauge transformations and the nontriviality of the field configuration space (instantons, sphalerons, anything Chern-Simons).

  2. How, in particular, the statement that you cannot make the transition between different sectors as each defines a different Hilbert space etc. compatible with the fact that one describes interactions between these topological defects, among themselves and with matter, and that particles living in different superselection sectors can interact with one another?

  3. How does one describe interactions between particles living in different sectors? How would you write such a correlation function, for example?

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Both local strings (involving a gauge symmetry) and global strings (involving a global symmetry) go to the vacuum manifold as you get far from the string core. In the case of local strings as in the Abelian-Higgs model, all the points on what would be a non-trivial vacuum manifold are in fact identified with each other, as you point out in the premise of your question. So you are not really winding around a vacuum manifold out at infinity. So the sectors you are asking about in your second and third question are not really there.

So where does topology come in? What keeps the local string stable? The gauge field still has a non-trivial winding number around the core of the string. This is a gauge invariant quantity (i.e. the Wilson loop). Another way to think of this is that there is quantized magnetic flux carried by the string.

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