On the complete integrability of the discrete Nahm equations

TL;DR

This paper proves the complete integrability of the discrete Nahm equations by associating solutions with spectral curves and line-bundles, revealing a geometric structure and implications for hyperbolic monopoles.

Contribution

It establishes the complete integrability of the discrete Nahm equations and connects solutions to spectral curves and line-bundles, advancing understanding of their geometric nature.

Findings

Discrete Nahm equations are completely integrable.

Solutions correspond to spectral curves and line-bundles.

Evolution in solutions relates to straight-line motion in Jacobian.

Abstract

The discrete Nahm equations, a system of matrix valued difference equations, arose in the work of Braam and Austin on half-integral mass hyperbolic monopoles. We show that the discrete Nahm equations are completely integrable in a natural sense: to any solution we can associate a spectral curve and a holomorphic line-bundle over the spectral curve, such that the discrete-time DN evolution corresponds to walking in the Jacobian of the spectral curve in a straight line through the line-bundle with steps of a fixed size. Some of the implications for hyperbolic monopoles are also discussed.