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

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

1995-06-09

arxiv.org/abs/hep-th/9506060v1

J.Phys. A29 (1996) L17-L22

Physics

High Energy Physics

High Energy Physics - Theory

pp. 9, Latex

Scientific paper

10.1088/0305-4470/29/1/003

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.

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