Revisiting Atiyah-Hitchin manifold in the generalized Legendre transform

TL;DR

This paper analyzes the construction of the Atiyah-Hitchin manifold using the generalized Legendre transform, clarifying differences in K"ahler potentials obtained by different contour choices and deriving explicit forms in holomorphic coordinates.

Contribution

It provides the first derivation of the K"ahler potential and metric for the Atiyah-Hitchin manifold in holomorphic coordinates using Ivanov and Rocek's contour.

Findings

The original approach yields a real K"ahler potential.

The alternative approach produces a complex K"ahler potential.

Explicit formulas for the metric are derived in holomorphic coordinates.

Abstract

We revisit construction of the Atiyah-Hitchin manifold in the generalized Legendre transform approach. This is originally studied by Ivanov and Rocek and is subsequently investigated more by Ionas, in the latter of which the explicit forms of the K\"ahler potential and the K\"ahler metric are calculated. There is a difference between the former and the latter. In the generalized Legendre transform approach, a K\"ahler potential is constructed from the contour integration of one function with holomorphic coordinates. The choice of the contour in the latter is different from the former's one, whose difference may yield a discrepancy in the K\"ahler potential and eventually in the K\"ahler metric. We show that the former only gives the real K\"ahler potential, which is consistent with its definition, while the latter yields the complex one. We derive the K\"ahler potential and the metric for the Atiyah-Hitchin manifold in terms of holomorphic coordinates for the contour considered by Ivanov and Ro\v{c}ek for the first time.