Supersymmetric sigma-models, twistors, and the Atiyah-Hitchin metric

TL;DR

This work develops a twistor-space reformulation of the generalized Legendre transform for hyperkähler metrics in four dimensions, demonstrating how a holomorphic data bundle over CP^1 encodes metric information. Central to the approach is a construction where η(ζ) is a polynomial on CP^1 and the Kähler potential arises from a Legendre transform of a contour-integrated generating function F, intimately tied to a holomorphic form ω_h(ζ). The Atiyah-Hitchin metric is presented as a detailed instance: a specific G(η, ζ) yields an elliptic-integral condition F_w = 0 whose solution reproduces AH, while the A_k ALE metrics are revisited to illuminate the role of singularities in χ and the resulting metric data. The authors also propose a broad set of ingredients—flat-space blocks and suitable G-terms—that may generate new complete hyperkähler metrics, including conjectured forms related to D_k ALE metrics and higher-dimensional generalizations, with potential implications for monopole moduli spaces.

Abstract

The Legendre transform and its generalizations, originally found in supersymmetric sigma-models, are techniques that can be used to give constructions of hyperkahler metrics. We give a twistor space interpretation to the generalizations of the Legendre transform construction. The Atiyah-Hitchin metric on the moduli space of two monopoles is used as a detailed example.

Figures (1)

• Figure 1: An identity between integrals.