Critical Abelian Higgs fields, winding and divisor data

Let $e>0$ and $\xi >0$ be real parameters, set $\kappa =2e^2\xi$, identify the oriented Euclidean plane ${\mathbb{R}}^2$ with $\mathbb{C}$ using coordinates $(x_1,x_2)$, and use the Hodge star convention $\ast d{x}_1=d{x}_2$, $\ast d{x}_2=-d{x}_1$. A field pair is a smooth real one-form $A=A_{1}d{x}_1+A_{2}d{x}_2$ on the trivial complex line over ${\mathbb{R}}^2$ and a smooth function $q:{\mathbb{R}}^2\to \mathbb{C}$. Its unit-charge covariant derivative is $D_{A}q=dq-iAq$, its curvature is $F_A=dA=Bd{x}_1\wedge d{x}_2$ with $B={\mathrm{\partial }}_1{A}_2-{\mathrm{\partial }}_2{A}_1$, and its transverse energy is $E(A,q)={\int }_{{\mathbb{R}}^2}(\frac{B^2}{2e^2}+\mathrm{\mid }{D}_{A,1}q{\mathrm{\mid }}^2+\mathrm{\mid }{D}_{A,2}q{\mathrm{\mid }}^2+\frac{e^2}{2}(\mathrm{\mid }q{\mathrm{\mid }}^2-\xi {)}^2)d{x}_{1}d{x}_2\mathrm{.}$ The pair has finite energy when this nonnegative Lebesgue integral is finite. An admissible boundary sector consists of an integer $n$, a radius $R_0$, and smooth functions $\rho >0$ and real $2\pi$-periodic $\varphi$ on $r>R_0$ such that $q(r,\theta )=\rho (r,\theta )e^{i(n\theta +\varphi (r,\theta ))}$ and $\rho \to \sqrt{\xi }$, $\varphi \to 0$ uniformly as $r\to \mathrm{\infty }$; $n$ is its winding. A gauge transformation is $(q,A)\mapsto (e^{i\chi }q,A+d\chi )$ for a smooth real-valued function $\chi$ on ${\mathbb{R}}^2$.

For $n\ge 0$, the positive critical first-order equations are $(D_{A,1}+i{D}_{A,2})q=0$ and $B=e^2(\xi -\mathrm{\mid }q{\mathrm{\mid }}^2)$; for $n\le 0$ the negative equations are $(D_{A,1}-i{D}_{A,2})q=0$ and $B=-e^2(\xi -\mathrm{\mid }q{\mathrm{\mid }}^2)$. The adjective critical refers to the displayed energy normalization and these square-saturating equations. For a positive-winding solution, its divisor is an effective finite divisor $D={\sum }_{j=1}^J{m}_j[z_j]$, where the $z_j\in \mathbb{C}$ are distinct, the multiplicities $m_j$ are positive integers, $q$ vanishes exactly at the $z_j$, and near each $z_j$ it has the local form $q(z)=(z-z_j{)}^{m_j}{g}_j(z)$ with $g_j$ smooth and nonzero there. The degree is $N={\sum }_j{m}_j$; the empty divisor has degree zero. Negative winding uses the conjugate field and reversed orientation, with $N=\mathrm{\mid }n\mathrm{\mid }$.

On the complement of the divisor, define $h=\log (\mathrm{\mid }q{\mathrm{\mid }}^2\mathrm{/}\xi )$. The associated scalar Taubes datum is a real function smooth away from the divisor such that $h-2m_j\log \mathrm{\mid }z-z_j\mathrm{\mid }$ extends smoothly across every $z_j$, $h\to 0$ at infinity, and, as a distribution, $\mathrm{\Delta }h+\kappa (1-e^h)=4\pi {\sum }_{j=1}^J{m}_j{\delta }_{z_j},⟨{\delta }_z,\psi ⟩=\psi (z)(\psi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^2))\mathrm{.}$ When the limit exists, the magnetic flux is $\mathrm{\Phi }(A)={\lim }_{R\to \mathrm{\infty }}{\int }_{\mathrm{\mid }x\mathrm{\mid }<R}B(x)d{x}_{1}d{x}_2$. These are mathematical configuration and normalization conventions; they make no assertion that a physical model describes nature.

Critical model, square completion and winding/flux conventions (Tong, §§3.1–3.2)

Independent scalar reduction and full-plane existence treatment (Chen–Yang, §§2–7)