Lagrangian curves on spectral curves of monopoles

TL;DR

This paper explores Lagrangian points on holomorphic curves in a neutral Kähler manifold, revealing their structure as real curves and their relation to ruled surfaces in Euclidean space, with applications to monopole spectral curves.

Contribution

It characterizes Lagrangian curves on holomorphic curves in T${ m P}^1$, links them to ruled surfaces in ${ m E}^3$, and applies these findings to spectral curves of monopoles.

Findings

Lagrangian points form real curves on holomorphic curves

Ruled surfaces generated have zero Gauss curvature

Application to monopole spectral curves

Abstract

We study Lagrangian points on smooth holomorphic curves in T${\mathbb P}^1$ equipped with a natural neutral K\"ahler structure, and prove that they must form real curves. By virtue of the identification of T${\mathbb P}^1$ with the space ${\mathbb L}({\mathbb E}^3)$ of oriented affine lines in Euclidean 3-space ${\mathbb E}^3$, these Lagrangian curves give rise to ruled surfaces in ${\mathbb E}^3$, which we prove have zero Gauss curvature. Each ruled surface is shown to be the tangent lines to a curve in ${\mathbb E}^3$, called the edge of regression of the ruled surface. We give an alternative characterization of these curves as the points in ${\mathbb E}^3$ where the number of oriented lines in the complex curve $\Sigma$ that pass through the point is less than the degree of $\Sigma$. We then apply these results to the spectral curves of certain monopoles and construct the ruled surfaces and edges of regression generated by the Lagrangian curves.