Let's consider the Abelian Higgs model in 1+1 dimensions in Euclidean space-time: $$L_E=\frac{1}{4e^2}F_{\mu\nu}F_{\mu\nu}+D_\mu\phi^\dagger D_\mu\phi+ \frac{e^2}{4}(|\phi|^2-\zeta)^2$$
where $\zeta>0$ and $D_\mu\phi= (\partial_\mu-iA_\mu)\phi$.
We are looking for finite action field configurations, i.e instantons. Because in this case $\phi : S^1_{phys}\rightarrow S^1_{gauge}$ and being $\mathbf{\pi}_1(S^1)=Z$, I expect that for an instanton configuration the action is bounded by its topological number (winding number). More precisely, the finiteness of the action comes from the following behaviour for large $r$: $$ 1) \quad\phi(r,\theta)=\sqrt{\zeta}\,g(\theta)$$ $$ 2) \quad A\mu=g\partial_\mu g^{-1}+O(1/r^2)$$ where $g\equiv g(\theta)$ is an element of U(1).
Therefore I would define the winding number $k$ as the one of pure gauge: $$w=\frac{i}{4\pi}\int d^2x\epsilon_{\mu\nu}F_{\mu\nu}$$
So I would expect, up to numerical factors, something like: $S\ge \zeta w $, where I have inserted $\zeta$ for dimensional reason. I am interested in demonstrating that inequality and in solutions saturating the bound. Any reference on this topic?
