In the Yang-Mills-Higgs (also called magnetic Ginzburg-Landau) model in the plane the energy has the form
$$
E(A,\phi)=\int_{\mathbb{R}^2}\left(|(d-iA)\phi|^2+\frac{1}{2}F_{jk}F_{jk}+\frac{1}{4}(1-|\phi|^2)^2\right)d^2x
$$
where $F_{jk}=\partial_jA_k-\partial_kA_j$, $\phi$ is a complex smooth function and $A$ is a real valued 1-form on $\mathbb{R}^2$. The $F_{jk}$'s are the components of $F=dA$ and is known that the number
$$
n=\frac{1}{2\pi}\int_{\mathbb{R}^2}F
$$
is an integer, under suitable hypotheses at infinity for $A$ and $\phi$. The proof of this fact can be found on Jaffe and Taubes' book *Vortices and Monopoles.* Since I couldn't understand deeply the proof, I have several doubts and I want to make things clear.

Taubes and Jaffe's hypotheses are $$ \lim_{R\to\infty}\sup_{|x|=R}|1-|\phi||=0 $$ and that exists $\delta>0$ such that $$ |x|^{1+\delta}|D_A\phi|\leq \text{const}. $$

Now suppose for a second to exaggerate and set both $A$ and $\phi$ compactly supported, for instance in a ball of radius $R$. The hypotheses should be fulfilled and by Stokes' theorem $$ 2\pi n=\int_{B_R}F=\int_{B_R}dA=\int_{\partial{B_R}}A=0. $$ My question is how, relaxing slowly this hypothesis, the integral could "jump" from one integer to another, instead of changing continuously taking also all intermediate values?

In the general case we should compute $$ n=\lim_{R\to\infty}\frac{1}{2\pi}\int_{\partial B_R}A. $$ I would like to understand what is the geometrical concept that hides and integer value in this integral. From Taubes and Jaffe's proof, this integral can indeed split in a part equal to some integer and a decaying part. Moreover, how can I see that this quantity is unchanged by (finite energy) perturbations of the fields?

Any hint or help is extremely appreciated.