The Pseudo-Analytic Charge

Abstract

The framed Beltrami--Vekua equation $\hat{I}¦(w_{\bar{z}}-\hat{I}¼w_z)+\hat{I}¨(\overline{w_z}-\hat{I}¼\overline{w_{\bar{z}}})+\mathfrak{a}w+\mathfrak{b}\bar{w}=\mathfrak{f}$, with $|\hat{I}¼|<1$ and $|\hat{I}¦|>|\hat{I}¨|$, carries a numerator field $N=\hat{I}¦\mathfrak{b}-\hat{I}¨\mathfrak{a}-W_L(\hat{I}¦,\hat{I}¨)$ 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 $w=\ddot{I}†w^{\prime }+\ddot{I}ˆ{\bar{w}}^{\prime }$ of the unknown, every scaling of the equation, and every orientation-preserving $C^1$ change of variables: recombinations multiply $N$ by the positive factor $|\ddot{I}†{|}^2-|\ddot{I}ˆ{|}^2$, 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 $\hat{I}¼$-independent wherever $W_{\partial }(\hat{I}¦,\hat{I}¨)\equiv 0$ -- 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.

14 pages. Comments are welcome