Platonic solutions of the discrete Nahm equation

TL;DR

The paper tackles the discrete Nahm equation, an integrable difference system encoding hyperbolic $SU(2)$ monopoles through circle-invariant ADHM data. By enforcing platonic symmetry, the authors construct explicit $m>1$ solutions for $N=3,4,7$ (tetrahedral, octahedral, and icosahedral cases) and derive the associated monopole spectral curves directly from the discrete data. They provide concrete formulas for the spectral-curve coefficients $c_T$, $c_O$, and $c_Y$, together with rank-one boundary conditions that fix the lattice data and reveal how these coefficients behave as the lattice size grows, including comparisons to existing algebraic-geometry results where available. A one-parameter tetrahedral family extends the construction to a broader set of solutions, and the work suggests directions for exploring more symmetry classes and extensions to other gauge groups, with potential connections to the m=1 circle-action description and its possible generalizations for $m>1$.

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.