On Z2 harmonic functions on $\mathbb{R}^2$ and the polynomial Pell's equation

TL;DR

This work analyzes $\mathbb{Z}_2$ harmonic functions on $\mathbb{R}^2$ with point singularities and uncovers a deep link to the polynomial Pell equation. It constructs a two-dimensional correspondence between $ obreak \mathbb{Z}_2$ harmonic functions $f$ on $X=\mathbb{R}^2\setminus K$ and admissible entire functions $u$ via $f(z)=\Re\int_{z_1}^z \frac{u(z)}{\sqrt{D(z)}}\,dz$ with $D(z)=\prod (z-z_i)$, so that $df=\Re\left(\frac{u(z)\,dz}{\sqrt{D(z)}}\right)$ and $f$ is determined by its unbounded part $U(f)$ at infinity. The paper then links this structure to Pell theory by constructing $G_D$ and $U_D$, showing that Pell solutions $(p,q)$ produce $\tilde U(z)=p'(z)/q(z)$ equal to $U_D$ up to a real factor, and that $\int \frac{U_D}{\sqrt{D}}dz= C\ln(p+q\sqrt{D})$, yielding a practical criterion: $D$ is Pellian iff the Abel integrals $\int_{z_1}^{z_j}\frac{U}{\sqrt{D}}dz$ are integer multiples of $\pi i$. The work also analyzes the zero locus of $f$, proving a curved, equiangular structure with order relations to $u$, and illustrating explicit Pellian geometry; together, these results connect branched-cover analysis, complex-analytic methods, and Diophantine Pell theory with potential gauge-theory applications.

Abstract

There has been many studies on Z2 harmonic functions, differential forms or spinors recently. This paper focuses on a very special and relatively simple aspect: Z2 harmonic functions on $\mathbb{R}^2$ with point singularities.

Theorems & Definitions (66)

• Definition 1
• Theorem 2
• Remark 3
• Theorem 4
• Remark 5
• Corollary 6
• Theorem 7
• Theorem 8
• Remark 9
• Lemma 1.1