Platonic solutions 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.