The Nahm-Schmid equations and Hypersymplectic Geometry

TL;DR

This paper investigates the Nahm-Schmid equations, revealing their integrable structure, explicit solutions, and connection to hypersymplectic geometry, derived from dimensional reduction of Yang-Mills equations in split signature.

Contribution

It provides a comprehensive analysis of the Nahm-Schmid equations, linking their integrability, explicit solutions, and geometric interpretation in hypersymplectic terms.

Findings

Derived explicit solutions of the Nahm-Schmid equations

Established Lax-pair formulation and conservation laws

Connected the equations to hypersymplectic geometry

Abstract

We explore the geometry of the Nahm-Schmid equations, a version of Nahm's equations in split signature. Our discussion ties up different aspects of their integrable nature: dimensional reduction from the Yang--Mills anti-self-duality equations, explicit solutions, Lax-pair formulation, conservation laws and spectral curves, as well as their relation to hypersymplectic geometry.