Generalization of Weierstrassian Elliptic Functions to ${\bf R}^{n}$

TL;DR

This paper extends Weierstrassian elliptic functions to higher dimensions, specifically ${f R}^{n}$, preserving key properties and linking to physical models like spacetime foam and monopole condensates.

Contribution

It introduces higher-dimensional generalizations of Weierstrass functions that maintain periodicity and Legendre relations, connecting to previous quaternionic constructions.

Findings

Functions satisfy higher-dimensional periodicity properties.

Reproduce earlier quaternionic functions for n=4.

Reduction to classical elliptic functions by integrating over lattice points.

Abstract

The Weierstrassian $\wp, \zeta$ and $\sigma $ functions are generalized to ${\bf R}^{n}$. The $n=3$ and $n=4$ cases have already been used in gravitational and Yang-Mills instanton solutions which may be interpreted as explicit realizations of spacetime foam and the monopole condensate, respectively. The new functions satisfy higher dimensional versions of the periodicity properties and Legendre's relations obeyed by their familiar complex counterparts. For $n=4$, the construction reproduces functions found earlier by Fueter using quaternionic methods. Integrating over lattice points along all directions but two, one recovers the original Weierstrassian elliptic functions.