Remarks on Nahm's equations

TL;DR

This paper explores Nahm's equations within a broader geometric framework, linking solutions to torsion-free sheaves on spectral curves and revealing new conserved quantities in generalized settings.

Contribution

It generalizes Nahm's equations to a moduli space context and connects solutions to spectral curve sheaves, introducing non-classical conserved quantities.

Findings

Zeros of the vector field correspond to torsion-free sheaves.

Generalizations are necessary for non-reduced spectral curves.

Existence of non-classical conserved quantities.

Abstract

Nahm's equations are viewed in a more general context where they appear as a vector field on a moduli space of co-Higgs bundles on the projective line. Zeros of this vector field correspond to torsion-free sheaves on a singular spectral curve which we translate in terms of a smooth curve in three-dimensional projective space. We also show how generalizations of Nahm's equations are required when the spectral curve is non-reduced and deduce the existence of non-classical conserved quantities in this situation.