Integrable systems and Special K\"ahler metrics

TL;DR

This paper explores the geometry of Special K"ahler metrics on the base of Hitchin systems, providing explicit formulas and extending to singular spectral curves, with applications to integrable subsystems.

Contribution

It offers a new geometric description of Special K"ahler structures via spectral curves and derives a simple formula for the K"ahler potential, extending to singular cases.

Findings

Derived a simple formula for the K"ahler potential.

Extended the structure to singular spectral curves.

Identified cases with flat metrics.

Abstract

We describe the Special K\"ahler structure on the base of the so-called Hitchin system in terms of the geometry of the space of spectral curves. It yields a simple formula for the K\"ahler potential. This extends to the case of a singular spectral curve and we show that this defines the Special K\"ahler structure on certain natural integrable subsystems. Examples include the extreme case where the metric is flat.