 Solution

The Abelian Higgs model has local $G=U(1)$, and a nonzero minimally charged scalar expectation value has trivial stabilizer $H=\{1\}$. Thus $M\simeq U(1)\simeq S^1$ and ${\pi }_1(M)=\mathbb{Z}$, while ${\pi }_0$ and ${\pi }_2$ are trivial. The defects are local strings, or Nielsen-Olesen vortices, whose winding fixes flux $2\pi n/e$.

Put $R=\eta +\varphi /\sqrt{2}$ and $\vartheta =\psi /(\eta \sqrt{2})$. The specified gauge-field shift is $A=A^{\prime }+\partial \vartheta /e$, soExpanding the potential gives $V=\lambda {\eta }^2{\varphi }^2/2+O({\varphi }^3)$. Hence the broken-phase spectrum of the Abelian Higgs model follows fromThe phase is not an additional physical massless scalar: the Higgs mechanism supplies the vector's longitudinal polarization. In the symmetric phase the complex scalar has two real degrees of freedom and the massless vector has two polarizations, a total of four. In the broken phase there are one scalar and three massive-vector polarizations, again four. The symmetric phase is the thermally restored phase; the origin of the given zero-temperature potential is unstable, not a separate stable particle vacuum. The local polar change of variables cannot remove global vortex winding everywhere, so the massive spectrum is consistent with string defects.