The Pseudo-Analytic Charge

Abstract

The framed Beltrami--Vekua equation $\Phi(w_{\bar z} - \mu w_z) + \Psi(\overline{w_z} - \mu\overline{w_{\bar z}}) + \mathfrak{a}w + \mathfrak{b}\bar w = \mathfrak{f}$, with $|\mu|<1$ and $|\Phi|>|\Psi|$, carries a numerator field $N = \Phi\mathfrak{b} - \Psi\mathfrak{a} - W_L(\Phi,\Psi)$ whose weighted modulus integrates to the pseudo-analytic mass. This paper extracts the integer carried by the same field. When the zero set of $N$ is compactly contained in a bounded simply connected domain, the winding number of $N$ along any enclosing curve -- the pseudo-analytic charge $n \in \mathbb{Z}$ -- is invariant under every recombination $w = \varphi w' + \psi\bar w'$ of the unknown, every scaling of the equation, and every orientation-preserving $C^1$ change of variables: recombinations multiply $N$ by the positive factor $|\varphi|^2 - |\psi|^2$, so their invariance is exact, while on multiply connected domains the other two actions fix the component charges only in $\mathbb{Z}/2\mathbb{Z}$ and the total charge exactly. The charge is a Brouwer degree: it localizes at the zeros of $N$, vortices which no action of the class creates or destroys; an isolated vortex persists under perturbation of the data precisely when its local charge is non-zero. It involves the Beltrami coefficient only through the $L$-Wronskian of the frame, and is $\mu$-independent wherever $W_\partial(\Phi,\Psi) \equiv 0$ -- in particular at the trivial frame, where $N = \mathcal{B}$ and the charge is the gauge-invariant winding of the coefficient of the Beltrami--Vekua equation. Mass and charge are independent: every pair in $(0,\infty)\times\mathbb{Z}$ is realized.