Remarks on the Atiyah-Hitchin metric

TL;DR

This paper explores the Atiyah-Hitchin metric on the moduli space of SU(2) BPS monopoles with charge 2, using algebraic and twistor methods, and discusses connections to quantum moduli spaces and Toda theory.

Contribution

It provides a new twistor-based approach to construct and analyze the Atiyah-Hitchin metric, linking algebraic curves, Toda equations, and Seiberg-Witten theory.

Findings

Constructed the metric via algebraic curves in C^3.

Solved the uniformization problem using twistor methods.

Connected the metric's construction to quantum moduli spaces.

Abstract

We outline the construction of the Atiyah-Hitchin metric on the moduli space of SU(2) BPS monopoles with charge 2, first as an algebraic curve in C^3 following Donaldson and then as a solution of the Toda field equations in the continual large N limit. We adopt twistor methods to solve the underlying uniformization problem, which by the generalized Legendre transformation yield the Kahler coordinates and the Kahler potential of the metric. We also comment on the connection between twistors and the Seiberg-Witten construction of quantum moduli spaces, as they arise in three dimensional supersymmetric gauge theories, and briefly address the uniformization of algebraic curves in C^3 in the context of large N Toda theory. (Based on talks delivered in September 1998 at the 32nd International Symposium Ahrenshoop on the Theory of Elementary Particles, Buckow; the 21st Triangular Meeting on Quantum Field Theory, Crete and the TMR meeting on Quantum Aspects of Gauge Theories, Supersymmetry and Unification, Corfu; to be published in the proceedings in Fortschritte der Physik.)