A Journey Between Two Curves

TL;DR

This paper explores the relationship between two types of algebraic curves arising in integrable systems, specifically comparing the classical scattering problem associated with Nahm equations to the Seiberg-Witten curves in gauge theory.

Contribution

It identifies a framework to relate the curves from Nahm equations and Hitchin systems, bridging classical scattering problems with gauge theory solutions.

Findings

Different natures of the curves in Nahm and Hitchin systems clarified.

A setup to relate classical scattering curves to Seiberg-Witten curves proposed.

Potential implications for understanding integrable systems and gauge theories.

Abstract

A typical solution of an integrable system is described in terms of a holomorphic curve and a line bundle over it. The curve provides the action variables while the time evolution is a linear flow on the curve's Jacobian. Even though the system of Nahm equations is closely related to the Hitchin system, the curves appearing in these two cases have very different nature. The former can be described in terms of some classical scattering problem while the latter provides a solution to some Seiberg-Witten gauge theory. This note identifies the setup in which one can formulate the question of relating the two curves.