Platonic solutions of the discrete Nahm equation

TL;DR

This paper constructs solutions to the discrete Nahm equation with platonic symmetries, linking them to hyperbolic monopoles and calculating their spectral curves directly from these solutions.

Contribution

It introduces a method to obtain solutions of the discrete Nahm equation with platonic symmetries and computes the spectral curves of the associated hyperbolic monopoles.

Findings

Solutions correspond to $SU(2)$ magnetic monopoles of charge $N$ in hyperbolic space.

Spectral curves of hyperbolic monopoles are explicitly calculated from the solutions.

Solutions exhibit platonic symmetries, enriching the understanding of the discrete Nahm equation.

Abstract

The discrete Nahm equation is an integrable nonlinear difference equation for complex $N\times N$ matrices defined on a one-dimensional lattice, with rank and symmetry boundary conditions at the ends of the lattice. Solutions of this system correspond to $SU(2)$ magnetic monopoles of charge $N$ in hyperbolic space, with the curvature related to the number of lattice points. Here some solutions of the discrete Nahm equation are obtained by imposing platonic symmetries, and the spectral curves of the associated hyperbolic monopoles are calculated directly from these solutions.