Monopole solutions to the Bogomolny equation as three-dimensional generalizations of the Kronecker series

TL;DR

This paper constructs a three-dimensional generalization of the Kronecker series by analyzing Dirac monopoles on a torus, revealing solutions to the Bogomolny equation with modular invariance.

Contribution

It introduces a novel three-dimensional generalization of the Kronecker series derived from monopole solutions on a torus, extending the mathematical framework of the Bogomolny equation.

Findings

Solution satisfies the functional equation.

Invariance under modular transformations.

Generalizes the Kronecker series to three dimensions.

Abstract

The Dirac monopole on a three-dimensional torus is considered as a solution to the Bogomolny equation with non-trivial boundary conditions. The analytical continuation of the obtained solution is shown to be a three-dimensional generalization of the Kronecker series. It satisfies the corresponding functional equation and is invariant under modular transformations.